We can also check this by graphing the function, which would show us the symmetry about the y-axis.
What is function?In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is associated with exactly one output. This means that for every value of the input, there is a unique corresponding value of the output.
by question.
To prove that a function is even, we need to show that it satisfies the property:
g(-x) = g(x) for all x in the domain of g.
So, let's substitute -x for x in g(x) and simplify:
g(-x) = 2[tex](-x)^{6}[/tex] - [tex](-x)^{2}[/tex] + 4
= 2[tex]x^{6}[/tex] -[tex]x^{2}[/tex] + 4
Notice that we obtained the same expression as g(x), which means that g(x) is even. Therefore, we can write:
g(x) = g(-x)
This means that the graph of the function is symmetric with respect to the y-axis, since for any value of x, the value of g(x) is the same as the value of g(-x).
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Given the following Year 12 balance sheet data for a footwear company:
Balance Sheet Data
Cash on Hand 2,000
Total Current Assets 78,000
Total Assets 315,000
Overdraft Loan Payable 4,000
1- Year Bank Loan Payable 8,000
Current Portion of Long-Term Loans 13,000
Total Current Liabilities 48,000
Long - Term Bank Loans Outstanding 105,000
Shareholder Equity Year11 Balance Year 12 Change Common Stock 10,000 0 10,000
Additional Capital 100,000 0 100,000
Retained Earning 30,000 22,000 52,000
Total Shareholder Equity 140,000 +22,000 162,000
Based on the above figures and the formula for calculating the debt-assets ratio, the company's debt-assets ratio (where debt is defined to include both short-term and long-term debt) is
a) 0.375
b) 0.413.
c) 0.40.
d) 0.318.
e) 0.333.
Based on the above figures and the formula for calculating the debt-assets ratio, the company's debt-assets ratio (where debt is defined to include both short-term and long-term debt) is 0.413 or 41.3%.
The given balance sheet data,
Balance Sheet Data
Cash on Hand 2,000
Total Current Assets 78,000
Total Assets 315,000
Overdraft Loan Payable 4,000
1- Year Bank Loan Payable 8,000
Current Portion of Long-Term Loans 13,000
Total Current Liabilities 48,000
Long-Term Bank Loans Outstanding 105,000
Shareholder Equity Year11 Balance Year 12 Change Common Stock 10,000 0 10,000
Additional Capital 100,000 0 100,000
Retained Earning 30,000 22,000 52,000
Total Shareholder Equity 140,000 +22,000 162,000
The debt-assets ratio is calculated by dividing the total debt (current liabilities + long-term loans) by the total assets.
In this case, the total debt is 48,000 + 105,000 = 153,000
and the total assets are 315,000.
Thus, the debt-assets ratio is 153,000/315,000 = 0.413.
Also, the debt-assets ratio is 0.413 * 100 = 41.3%
The correct answer is b) 0.413.
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Find the total number of outcomes for picking a day of the week and a month of the year. A 84 , B 19 , C 60 , D 210
Option A is the correct option for determining the total number of possible outcomes when selecting a day of the week and a month of the year, which is 84.
To find the total number of outcomes for picking a day of the week and a month of the year, you need to multiply the number of options for each category. There are 7 days in a week, so there are 7 options for picking a day. There are 12 months in a year, so there are 12 options for picking a month. To find the total number of outcomes, you multiply the number of options for each category: 7 x 12 = 84. Therefore, the total number of outcomes for picking a day of the week and a month of the year is 84. Option A is the correct answer.
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What are the expectations for the following $1 bet on a U.S. roulette wheel? (Round each answer to the nearest cent.)
0 & 00 split = $
The expectation for a $1 bet on a U.S. roulette wheel for a 0 & 00 split is -$2. This means that you are expected to lose $2 for every $1 you bet.
The expected payout for a $1 bet on the 0 & 00 split in a U.S. roulette wheel is $0.45.
On a U.S. roulette wheel, there are 38 possible outcomes, consisting of the numbers 1 through 36, 0, and 00.
The 0 and 00 numbers are green, while all other numbers are either red or black.
Placing a $1 bet on the 0 & 00 split means that the bettor is betting that either the 0 or the 00 number will hit. If either of these numbers hits, the payout is 17 to 1, meaning the bettor will receive $17 for every $1 bet.
To calculate the expected payout for a $1 bet on the 0 & 00 split, we multiply the probability of winning (1/38) by the payout ($17), resulting in an expected payout of $0.447.
Therefore, the expected payout for a $1 bet on the 0 & 00 split is:
(1/38) × 17 × $1 = $0.447
Rounding this to the nearest cent, the expected payout for a $1 bet on the 0 & 00 split is $0.45.
Hence, the answer is $0.45.
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In an aquarium, the ratio of dolphins to pufferfish is 2: 3 and the ratio of pufferfish to starfish is 4: 7. There are 8 dolphins in the aquarium. How many starfish are there?
There ratio are [tex]28[/tex] starfish inside an aquarium, with a dolphin to pufferfish ratio of 1.
What else are ratio in simple simple words?The ratio may be defined as the number of ways you can represent one item as a percentage of another. Only when the two quantities in a ratio share the same unit can they be compared. Ratios are employed to compare two items.
How do you compute ratio?Ratios contrast two figures by typically dividing them. A/B would be your formula if you were comparing each data point (A) to that other data point (B). This indicates that you are multiplying info A by info B. If A is five and B is ten, for instance, your ratio will indeed be 5/10.
Dolphins to pufferfish to starfish: [tex]2:3:7[/tex]
So for every [tex]2[/tex] dolphins, there are [tex]7[/tex] starfish. We know there are 8 dolphins, so we can set up a proportion:
[tex]2[/tex] dolphins / [tex]7[/tex] starfish = 8 dolphins / x starfish
where [tex]x[/tex] is the number of starfish.
Solving for x, we get:
[tex]x[/tex] starfish = (7 starfish * [tex]8[/tex] dolphins) / [tex]2[/tex] dolphins
[tex]x[/tex] starfish = [tex]28[/tex] starfish
Therefore, there are [tex]28[/tex] starfish in the aquarium.
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Elise saved $184. She bought a scarf, a necklace, and a notebook. After the purchases, she still had $89. 50. The scarf cost three-fifths the cost of the necklace and the notebook cost one-sixth as much as the scarf. How much did the notebook cost?
The cost of the notebook is $4.50
Let's call the cost of the necklace "N" and the cost of the scarf "S". We know that Elise spent a total of $184 on the scarf, necklace, and notebook, so we can write:
S + N + NB = 184
We also know that after making her purchases, Elise had $89.50 left, so we can write:
89.5 = 184 - (S + N + NB)
Simplifying this equation, we get:
S + N + NB = 94.5
We're also given that the scarf cost three-fifths the cost of the necklace, so we can write
S = (3/5)N
Finally, we're told that the notebook cost one-sixth as much as the scarf, so we can write:
NB = (1/6)S
Now we can the substitution method here
(3/5)N + N + (1/6)(3/5)N = 184 - 89.5
Simplifying the right-hand side
(3/5)N + N + (1/6)(3/5)N = 94.5
Combining like terms on the left-hand side
(21/10)N = 94.5
Multiplying both sides by 10/21
N = 45
So the necklace cost $45. We can use this to find the cost of the scarf
S = (3/5)N = (3/5)($45) = $27
And we can use the cost of the scarf to find the cost of the notebook
NB = (1/6)S = (1/6)($27) = $4.50
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What are the x coordinate and the y coordinate of the center of mass for the uniform plate shown in Figure below if L=5.0 cm?
From the X coordinate is -0.45 centimeter and Y coordinate is -2.0 cm of the center of mass for the uniform plate.
According to the Question:
Since the plate is uniform, we can divide it into three rectangles, each with a mass proportional to its area and with its barycenter at its geometric center. We will refer to the large piece of 35cm x 10cm; it occupies 63.6% of the total surface and its barycenter is at
(x₁, y₁) = (−5.0cm,−2.5cm).
Now,
The top 20cm× 5cm piece has 18.2 % of the total area; its center of mass is at (x₂, y₂) = (10cm,12.5cm).
Now,
The bottom 10cm× 10cm piece (section 3) also has 18.2 % of the total area; its center of mass is at (x₃, y₃) = (5cm,−15cm).
(a) The x coordinate of the center of mass for the plate is
[tex]X_{com} = (0.636)X_1 + (0.182)X_2 + (0.182)X_3[/tex]
= -0.45 cm
(b) The y coordinate of the center of mass for the plate is
[tex]Y_{com} = (0.636)Y_1 + (0.182)Y_2 + (0.182)Y_3[/tex]
= -2.0 cm.
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If AC = 57, find the measure of AB.
Segment Addition Postulate - Meaning, Formula & Examples
Answer:
AB = 27
Step-by-step explanation:
AC = AB + BC
[tex]{ \rm{57 = 3x + (4x - 6)}} \\ \\ { \rm{57 = 7x - 6}} \\ \\ { \rm{7x = 57 + 6}} \\ \\ { \rm{7x = 63}} \\ \\ { \rm{x = 9}}[/tex]
Therefore;
[tex]{ \rm{AB = 3x = 3 \times 9}} \\ { \boxed{ \rm{AB = 27}}}[/tex]
Find m ∠ R . Use the Picture
m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.
what is angle ?The degree of rotation between two lines or two planes around a central point is measured by an angle. Typically, it is expressed in radians or degrees. Angles are used in many mathematical and scientific uses, including trigonometry, physics, and engineering, where they are crucial in determining the shape and characteristics of geometric figures. Angles come in four different varieties: acute (less than 90 degrees), right (exactly 90 degrees), oblique (more than 90 degrees), and straight (exactly 180 degrees).
given
Because a triangle's total sides equal 180 degrees, we have:
R, S, and T add up to 180.
Inputting the numbers provided yields:
m∠R + 72 + 32 = 180
Simplifying the equation:
m∠R = 76
m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.
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In a random sample of 200 school district residents, 94 stated they are in favor of starting the school day 15 minutes later each day. Calculate a 90% confidence interval for the true proportion of district residents who are in favor of starting the day later
The 90% confidence interval for the proportion of district residents in favor of starting the school day 15 minutes later is (0.392, 0.548). The true proportion is estimated to lie within this interval with 90% confidence.
To calculate the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later, we can use the following formula:
CI = p ± z*(√(p*(1-p)/n))
where:
CI: confidence interval
p: proportion of residents in favor of starting the day later
z: z- score based on the confidence level (90% in this case)
n: sample size
First, we need to calculate the sample proportion:
p = 94/200 = 0.47
Next, we need to find the z- score corresponding to the 90% confidence level. Since we want a two-tailed test, we need to find the z- score that cuts off 5% of the area in each tail of the standard normal distribution. Using a z-table, we find that the z- score is 1.645.
Substituting the values into the formula, we get:
CI = 0.47 ± 1.645*(√(0.47*(1-0.47)/200))
Simplifying this expression gives:
CI = 0.47 ± 0.078
Therefore, the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later is (0.392, 0.548). We can be 90% confident that the true proportion lies within this interval.
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A beef rancher randomly sampled 42 cattle from her large herd to obtain a 95% confidence interval to estimate the mean weight of the cows in the herd. The interval obtained was (1010, 1321). If the rancher had used a 90% confidence interval instead, the interval would have been
(A) wider and would have more precision than the original estimate
(B) narrower and would have more precision than the original estimate
(C) wider and would have the same precision as the original estimate
(D) narrower and would have less precision than the original estimate
(E) wider and would have less precision than the original estimate
(E)The interval would have been wider and would have less precision than the original .
A beef rancher randomly sampled 42 cattle from her large herd to obtain a 95% confidence interval to estimate the mean weight of the cows in the herd. The interval obtained was (1010, 1321). If the rancher had used a 90% confidence interval instead, the interval would have been narrower and would have less precision than the original estimate.
If the sample size is constant, then we can say that as the level of confidence is increased, the width of the interval will also increase. This is because a higher level of confidence will require a larger interval to include the population mean. Similarly, if the level of confidence is decreased, then the width of the interval will also decrease as a smaller interval is needed to include the population mean.
This directly implies that if the confidence interval is wider, then it has more precision and vice versa. Hence, Option (E) is the correct answer.
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ABCD is a quadrilateral A) Calculate the value of x. B) When ABCD is drawn to scale, would the lines AD and BC be parallel or not? You must justify your answer without using a scale drawing
A) The value of x = 45 degrees
B) Lines AD and BC are not parallel when ABCD is drawn to scale.
To solve this problem, we can use the fact that the sum of the angles in a quadrilateral is 360 degrees.
A) angle A + angle B + angle C + angle D = 360
2x + 90 + x + 3x = 360
6x + 90 = 360
6x = 270
x = 45
Therefore, x = 45 degrees.
B) To determine if lines AD and BC are parallel, we can look at the opposite angles of the quadrilateral. If they are supplementary (add up to 180 degrees), then the lines are parallel.
angle A + angle C = 2x + x = 3x = 135 degrees
angle B + angle D = 90 + 3x = 90 + 135 = 225 degrees
Since angle A + angle C and angle B + angle D do not add up to 180 degrees, the opposite angles are not supplementary, and therefore, lines AD and BC are not parallel.
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The given question is incomplete, the complete question is:
ABCD is a quadrilateral A) Calculate the value of x. B) When ABCD is drawn to scale, would the lines AD and BC be parallel or not?
In a class of students, the following data table summarizes how many students passed
a test and complete the homework due the day of the test. What is the probability that
a student completed the homework given that they failed the test?
Completed the homework
Did not complete the homework
Passed the test Failed the test
8
2
3
сл
The probability that a student completed the homework given that they failed the test is 3/8.
Describe Probability?Probability is a branch of mathematics that deals with the study of randomness and uncertainty in events. It involves the measurement and analysis of the likelihood or chance of an event occurring. Probability is expressed as a number between 0 and 1, with 0 indicating an impossible event and 1 indicating a certain event.
The concept of probability is used in many fields, such as statistics, physics, finance, and engineering, to make predictions and decisions based on incomplete information. It provides a framework for analyzing situations in which multiple outcomes are possible and assigns a quantitative value to the likelihood of each outcome.
To find the probability that a student completed the homework given that they failed the test, we need to use conditional probability.
Let A be the event that a student completed the homework and B be the event that a student failed the test.
We want to find P(A | B), the probability that a student completed the homework given that they failed the test.
Using the data from the table, we can see that the total number of students who failed the test is 8 (3 who completed the homework and 5 who did not).
Therefore, P(B) = 8/18 = 4/9 (the probability that a student failed the test)
Out of the 8 students who passed the test, 3 also completed the homework, so P(A and not B) = 3/18 = 1/6.
So, P(A | B) = P(A and B) / P(B) = (1/6) / (4/9) = 3/8.
Therefore, the probability that a student completed the homework given that they failed the test is 3/8.
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If the original quantity is 225 and the quality of 320 what is the percentage increase?
Step-by-step explanation:
The change fro 225 to 320 is 95
95 is what percentage of 225?
95/225 x 100% = 42.2% increase
Here is a scale drawing of a garden. Tom wants to plant a tree in the garden according to the following rules: It must be 4 m from A and 2 m from CD. Place a cross where Tom can plant the tree. D 2.5 cm A 4 cm C B 1 cm represents 2m
Answer:
the cross is the blue color
You are on the observation deck of the Empire State building looking at the Chrysler building when you turn 145° clockwise you see the Statue of Liberty you know that the Chrysler building in the Empire State building or about 0.6 miles apart and that the Chrysler building in the Statue of Liberty are about 5.6 miles apart estimate the distance between the Empire State building in the Statue of Liberty round your answer to the nearest 10th of a mile
28) The waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver are independent and follow exponential distributions with mean 6 years and 3 years, respectively. What is the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years? 18 18 18
Previous question
The probability that both events occur simultaneously is 0.1913.
We can model the waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver as independent exponential distributions with mean 6 years and 3 years, respectively.
Let X be the waiting time for the first claim from a good driver and Y be the waiting time for the first claim from a bad driver.
Then, P(X ≤ 3) = 1 - e^(-3/6) = 0.3935 and P(Y ≤ 2) = 1 - e^(-2/3) = 0.4866.
Since X and Y are independent, the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years is
(X ≤ 3) * P(Y ≤ 2) = 0.3935 * 0.4866 = 0.1913(rounded to four decimal places).
(rounded to four decimal places).
Therefore, the probability that both events occur simultaneously is 0.1913.
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Can someone please help me with this?
The triangles within kite with corresponding angles and sides are similar, so options A, B, D, and E are true statements while options C and F are wrong.
What are similar trianglesSimilar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.
The line FH bisects the kite so:
triangle ∆FGK is similar to ∆FJK, ∆FGK ≅ ∆FJK
angle m∠GKH corresponds to m∠JKH, m∠GKH ≅ m∠JKH
angle m∠GFH corresponds to m∠JFH, m∠GFH ≅ m∠JFH
triangle ∆GKH is similar to ∆JKH, ∆GKH ≅ ∆JKH.
The line FG is not shown to be same with KG, likewise lines FG and JK, so lines FG ≅ KG and
FG ≅ JK are not true statements.
Therefore, options A, B, D, and E are true statements for the triangles within the kite with corresponding angles and sides that are similar, while options C and F are wrong.
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Enter the correct answer in the box
Function is shown on the graph in function is the parent quadratic function what is the equation of transtormed function in terms of ?
In quadratic functiοn -
g(x) h(x) d(x)
vertical shift dοwn 3 reflectiοn acrοss the x-axis vertical shift dοwn 3
hοrizοntal shift left 3 vertical strecht οf 3 hοrizοntal shift right 3
What is the οperatiοn οf a quadratic functiοn?The expressiοn f(x) = ax² + bx + c, where a, b, and c are all integers with a nοt equal tο zerο, denοtes a quadratic functiοn. The shape that a quadratic functiοn's graph takes οn is called a parabοla.
In spite οf the fact that parabοlas might have οpenings that are upward οr dοwnward, varying "widths" οr "steepnesses," they all have the same basic "U" shape.
Quadratic parent: f(x)=x²
The graph is a parabοla with vertex V=(0,0) at the οrigin and οpens up.
When x=1→f(1)=1²→f(1)=1
1) g(x)
The graph οpens up, then there is nοt a reflectiοn acrοss the x-axis.
The vertex is at the pοint (-3,-3): 3 units tο the left and 3 units dοwn οf the vertex οf the parent funtiοn.
When x is 1 unit tο the right frοm the vertex g(x)=1
Then the transfοrmatiοns were applied tο the cuadratic parent functiοn are:
1.1) vertical shift dοwn 3.
1.2) hοrizοntal shift left 3.
2) h(x)
The graph οpens dοwn, then there is a reflectiοn acrοss the x-axis.
The vertex is at the οrigin (0,0).
When x is 1 unit tο the right frοm the vertex h(x)=-3
Then the transfοrmatiοns were applied tο the quadratic parent functiοn are:
2.1) reflectiοn acrοss the x-axis.
2.2) vertical strecht οf 3.
3) d(x)
The graph οpens up, then there is nοt a reflectiοn acrοss the x-axis.
The vertex is at the pοint (3,-3): 3 units tο the right and 3 units dοwn οf the vertex οf the parent funtiοn.
When x is 1 unit tο the right frοm the vertex d(x)=1
Then the transfοrmatiοns were applied tο the cuadratic parent functiοn are:
3.1) vertical shift dοwn 3.
3.2) hοrizοntal shift right 3.
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The complete question is -
The given graphs show functions which have been transformed from the quadratic parent, f(x) = x2. Determine which transformations were applied to the quadratic parent function to result in each graph.
Brittany needed new tires for her truck. She went to the auto shop and bought 4 tires on sale for $85.95 each. The salesman told her that she saved a total of $96.16. If Brittany saved the same amount on each tire, what was the original price of each tire?
The best solution gets brainlist
Answer:
$109.99
Step-by-step explanation:
The original price of each tire is [tex]\[/tex][tex]109.99[/tex]
Solution:Take the amount saved and divide by 4 to find the amount saved on each tire
[tex]96.16\div4 =24.04[/tex]
Add that to the sale price of each tire to find the original price
[tex]85.95+24.04 =109.99[/tex]
Therefore, The original price is $109.99.
Stacy, Eileen, and Michelle are on a basketball team. Stacy scored 15% of the points, Eileen
scored 10% and Michelle scored 13% of their team's points. If their team had a total
points, how many points did the remaining players on the team score?
The points scored by the remaining players on the team is 31 points.
What is percentage?A component or fraction of a whole can be expressed as a percentage by using a number out of 100. Mathematicians frequently use comparison to represent changes in quantities over time or to compare two numbers. For instance, if a student receives an 80 percent on a test, they properly answered 80 of the 100 questions.
For performing a range of mathematical operations, such as determining the percentage rise or decrease of a quantity or computing interest rates on loans or investments, percentages can be expressed as fractions or decimals.
From the given percentages the total percent is:
Total percentage = 15% + 10% + 13% = 38%
Now, from 100 the remaining percentage is: 100% - 38% = 62%
For the remaining players the score is given as:
Remaining points = 50 * (62/100) = 31
Hence, the points scored by the remaining players on the team is 31 points.
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27. Let W be a subspace, and let S be a spanning set for W. Find a basis for W, and calculate dim(W) for each set S. a) S = COCIOC) 3 0 2 -2 b) S= -2 1 2 2 2
Answer:best thing for you is drink milk
Step-by-step explanation:
The volume of the right cone below is 5677 units³. Find the value of x.
According to the given information value of X =16.06
What is the volume of right cone?The volume of a cone defines the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called apex or vertex. A cone is formed by a set of line segments, half-lines or lines connecting a common point, the apex, to all the points on a base that is in a plane that does not contain the apex.
A cone can be seen as a set of non-congruent circular disks that are stacked on one another such that the ratio of the radius of adjacent disks remains constant.
According to the given information:Cone volume formula
A cone is a solid that has a circular base and a single vertex. To calculate its volume, you need to multiply the base area (area of a circle: π × r²) by height and by 1/3:
volume = (1/3) × π × r² × h
5677 = (1/3) x 22/7 x r² x 21
x = 16.06
Therefore, according to the given information value of X =16.06
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Rob says -3(x-2) is equivalent to -3x-6. What was his error?
A) heshould have written -3x-2 which would have given him the right answer.
B) hedid not multiply 3 with x to get 3x, and instead put -3x.
C) he had no error.
D) he did not multiply -3 with -2 to get +6 instead of -6
Answer:
D) he did not multiply -3 with -2 to get +6 instead of -6
Step-by-step explanation:
-3(x - 2)
-3x + 6
Compare the two
-3x + 6 ≠ -3x - 6
They are not equal! Looking at the options, D is the answer.
Answer:
The correct answer is D) he did not multiply -3 with -2 to get +6 instead of -6.
To expand -3(x-2), we need to distribute -3 to both terms inside the parentheses, which gives us:
-3(x-2) = -3x + (-3(-2)) = -3x + 6
Therefore, the correct equivalent expression is -3x + 6, not -3x - 6 as Rob wrote. He made a mistake in multiplying -3 and -2, which should have given him +6 instead of -6.
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A triangle has two sides of length 3 and 16. What is the largest possible whole-number length for the third side
The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.
What is inequality theorem?The triangle inequality theorem explains the relationship between the three sides of a triangle. This theorem states that for any triangle, the sum of the lengths of the first two sides is always larger than the length of the third side.
According to question:Let x be the length of the third side. By the triangle inequality, we have:
3 + 16 > x and 16 + x > 3 and 3 + x > 16
Simplifying, we get:
19 > x and x > 13 and x < 19
The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.
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Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. x + 4 y − 3 z = 1 , − 3 x + 6 y + 7 z = 0
The angle between the two planes is approximately 64.63°.
The two planes x + 4 y − 3 z = 1 and − 3 x + 6 y + 7 z = 0 are neither parallel nor perpendicular. The angle between them can be determined by the following formula:
[tex]\theta = \cos^{-1}\left(\frac{A\cdot B}{\left\lvert A\right\rvert\left\lvert B\right\rvert}\right)[/tex]
where A and B are the normal vectors of the two planes, and \theta is the angle between them.
A normal vector for the plane x + 4 y − 3 z = 1 can be calculated as:
A = \begin{bmatrix} 1 \\ 4 \\ -3 \end{bmatrix}
A normal vector for the plane − 3 x + 6 y + 7 z = 0 can be calculated as:
B = \begin{bmatrix} -3 \\ 6 \\ 7 \end{bmatrix}
The angle between the two planes can be found by plugging the vectors into the formula above:
[tex]\theta
= \cos^{-1}\left(\frac{A\cdot B}{\left\lvert A\right\rvert\left\lvert B\right\rvert}\right)
= \cos^{-1}\left(\frac{(-3)\cdot(-3) + 6 \cdot 4 + 7 \cdot (-3)}{\sqrt{1 + 16 + 9}\sqrt{9 + 36 + 49}}\right)
= \cos^{-1}\left(\frac{-37}{\sqrt{26}\sqrt{94}}\right)\approx 64.63^\circ[/tex]
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David is working two summer jobs, making $13 per hour landscaping and making $8 per hour clearing tables. In a given week, he can work no more than 16 total hours and must earn no less than $160. Also, he must work at most 13 hours landscaping. If
� x represents the number of hours landscaping and �y represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.
Answer: One possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.
Step-by-step explanation:
Let x be the number of hours landscaping and y be the number of hours clearing tables.
The system of inequalities can be written as:
x ≤ 13 (maximum of 13 hours landscaping)
x + y ≤ 16 (maximum of 16 hours total)
13x + 8y ≥ 160 (minimum earnings of $160)
To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:
x = 13 (vertical line at x = 13)
x + y = 16 (line with intercepts at (0, 16) and (16, 0))
13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))
Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).
The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:
|
20 --|--------------------------
| /|
| / |
| / |
| / |
16 --|------------------
| / |
|/ |
-------------------------
13 12.3
Let x be the number of hours landscaping and y be the number of hours clearing tables.
The system of inequalities can be written as:
x ≤ 13 (maximum of 13 hours landscaping)
x + y ≤ 16 (maximum of 16 hours total)
13x + 8y ≥ 160 (minimum earnings of $160)
To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:
x = 13 (vertical line at x = 13)
x + y = 16 (line with intercepts at (0, 16) and (16, 0))
13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))
Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).
The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:
lua
Copy code
|
20 --|--------------------------
| /|
| / |
| / |
| / |
16 --|------------------
| / |
|/ |
-------------------------
13 12.3
The vertices of the feasible region are (0, 16), (12.3, 3.7), and (13, 0).
To determine one possible solution, we can evaluate the objective function (total earnings) at each vertex:
(0, 16): 13(0) + 8(16) = $128
(12.3, 3.7): 13(12.3) + 8(3.7) ≈ $167.1
(13, 0): 13(13) + 8(0) = $169
Therefore, one possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.
Use a double integral to find the area of the region D.
The x y-coordinate plane is given. There is a curve that encloses a region.
The curve, labeled r = √theta, starts at the origin, goes counterclockwise and away from the origin, and ends on the positive x-axis.
The region, labeled D, is the area enclosed by the curve and the positive x-axis.
Geographic area D has an area of /8 square units.
What is the formula for the area?Area is a phrase used to describe how much room a 2D form or surface occupies. Area is measured in square units, such as cm2 or m2. The area of a form is computed by dividing its length by its breadth.
What really is area in plain English?Area is the entire amount of space occupied by a plain (2-D) area or an object's form. The area of a planar figure is the area that its perimeter encloses. The quantity of unit square that cover a closed figure's surface is its area.
0 = √θ
θ = 0
Similarly, the curve intersects the [tex]x-axis[/tex] again when [tex]x = r cos[/tex] θ = √θ cos θ = 0, which implies θ = π/2.
Thus, the limits of integration for θ are [tex]0[/tex] and π/2. For r, the curve is given by [tex]r =[/tex] √θ, so the limits of integration are [tex]0[/tex] and √(π/2).
Evaluating this integral gives:
∫(θ=0 to π/2) [[tex]1/2 r^2[/tex]] (r=0 to √(π/2)) dθ
= ∫(θ=0 to π/2) [[tex]1/2[/tex] (π/2)] dθ
= π/8
Therefore, the area of region D is π/8 square units.
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Kierran spends approximately 3 hours per day playing video games and approximately 20 minutes per day doing chores. If Kierran's mother tells him to play video games when he completes his chores, then she is employing the ______________ whereas if she restricts video game play and only allows him to play video games if he does his chores then she is employing the ___________________.
When she restricts video game play and only allows him to play games if he completes his tasks, she is using the punishment technique.
what is unitary method ?A mathematical strategy known as the unitary method involves identifying a single unit value and utilizing it to determine additional values based on how they relate to it. The inverse proportionality method and the method of proportionality are other names for it. When dealing with complex problems that include numerous units or quantities, the unitary technique is frequently employed to solve difficulties requiring ratios and proportions. Several industries, including banking, science, engineering, and mathematics, can use it.
given
When Kieran's mother instructs him to play video games after finishing his work, she is using the reinforcement approach.
Drive reduction theory, the Premack Principle
According to the Premack Principle, low probability conduct can be reinforced by using high probability behavior.
According to the hypothesis of drive reduction, animals will take actions to
When she restricts video game play and only allows him to play games if he completes his tasks, she is using the punishment technique.
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James have 18 litres of water. He poured unequally into 3 tank
I. Poured three quarter of water from tank one into tank 2
II. Poured half of the water that is now in tank 2 into tank 3
III. Poured one third of water that is now in tank 3 into tank 1
Answer:
Tank 1: 6 litres
Tank 2: 6.75 litres
Tank 3: 4.5 litres
Step-by-step explanation:
Initially, James had 18 litres of water, and he poured three-quarters of it from tank 1 into tank 2. This means that the volume of water left in tank 1 is:
18 - (3/4) * 18 = 4.5 litres
The volume of water in tank 2 is:
(3/4) * 18 = 13.5 litres
Next, he poured half of the water in tank 2 (which is now 13.5 litres) into tank 3. The volume of water left in tank 2 is:
(1/2) * 13.5 = 6.75 litres
The volume of water in tank 3 is:
13.5 * (1/2) = 6.75 litres
Finally, he poured one-third of the water in tank 3 (which is now 6.75 litres) into tank 1. The volume of water in tank 1 after this is:
4.5 + (1/3) * 6.75 = 6 litres
The volume of water in tank 3 after this is:
6.75 * (2/3) = 4.5 litres
So the final volume of water in each tank is:
Tank 1: 6 litres
Tank 2: 6.75 litres
Tank 3: 4.5 litres
Consider the second order differential equation with initial conditions u" +2u' - 5u = 5 sin(3t), u(1) = 3, u'(1) = 4.5. Without solving it, rewrite the differential equation as an equivalent set of first order equations. In your answer use the single letter u to represent the function u and the single letter v to represent the "velocity function" ul. Do not use u(t) or v(t) to represent these functions. Expressions like sin(t) that represent other functions are OK. u= V Now write the first order system using matrices: u C ] [: + The initial value of the vector valued solution for this system is: u(1) = v(1)
The equivalent set of first-order equations in matrix form is [tex]w' = Cw + [0,[/tex] [tex]5sin(3t)^{T}[/tex] and the initial value of the vector-valued solution is w(1) = [tex][3,4.5]^{T}[/tex].
The given second-order differential equation as a set of first-order equations, we can define a new variable v(t) to represent the derivative of u(t), i.e., v(t) = u'(t). Then, we can rewrite the second-order differential equation as two first-order differential equations:
[tex]u' = v[/tex]
[tex]v' = 5sin(3t) + 5u - 2v[/tex]
where u' and v' represent the derivatives of u and v with respect to t, respectively.
To write this first-order system in matrix form, we can define the vector-valued function w(t) = [u(t), v(t)]^T and write the system as:
[tex]w' = [v, 5sin(3t) + 5u - 2v]^T[/tex]
where the superscript T denotes the transpose of a vector or matrix.
Using this notation, the initial value of the solution vector is given by w(1) =[tex][u(1), v(1)]^T = [3, 4.5]^T.[/tex]
In matrix form, the first-order system can be written as:
[tex][ u' ] [ v ]\\[ ] = [ ]\\[ v' ] [ 5sin(3t) + 5u - 2v ][/tex]
or
[tex]w' =\\[ u' ]\\[ v' ]\\[ v ]\\[ 5sin(3t) + 5u - 2v ][/tex]
where
w =
[tex][ u ]\\[ v ][/tex]
and the coefficient matrix C is given by:
[tex]C =\\[ 0 1 ]\\[ 5 -2 ][/tex]
A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).
Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.
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