Pythagorean theorem and its converse to prove that triangle ABC is a right triangle.
The converse of Pythagorean theorem states that if the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.
Let's begin by visualizing the given triangle ABC and the line CN that is perpendicular to side AB. Since CN is perpendicular to AB, we can say that triangle ACN and triangle BCN are right triangles (as the angle at C is 90 degrees).
Now, let's focus on the point H that is the geometric mean of points D and E. This means that DH multiplied by HE is equal to CH squared (as per the definition of geometric mean).
We can represent this relationship using the Pythagorean theorem as well. Since triangle CDH and triangle CEH are right triangles (as angle C is 90 degrees), we can say that:
CD² + DH² = CH² (using Pythagorean theorem for triangle CDH)
CE² + EH² = CH² (using Pythagorean theorem for triangle CEH)
From the given relationship of geometric mean, we can substitute CH² in the above equations and get:
CD² + DH² = HE*DH
CE² + EH² = HE*EH
Adding these two equations, we get:
CD² + CE² + DH² + EH² = HE*(DH + EH)
AB² = 4HE² (using the fact that DH + EH = DE = 2HE)
This equation gives us an important insight about the triangle ABC. It tells us that the sum of the squares of the sides (AB²) is equal to four times the square of the length HE (4HE²).
Now, we can use the converse of Pythagorean theorem to prove that triangle ABC is a right triangle.
In our case, we have AB² = 4HE², which satisfies the converse of Pythagorean theorem. Therefore, we can conclude that triangle ABC is a right triangle.
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Suppose that a population of fish P(t) grows according to the differential equation P' = 4P - 4P^2 - h representing logistic growth with a constant rate of harvesting h. What is the maximum sustainable harvest rate: the largest value of h for which there is still an equilibrium.
The maximum sustainable rate of harvest which is possible for the growth of fish P(t) according to the differential equation " P' = 4P - 4P^2 - h " will be at the harvest rate of h=3.
In order to calculate the maximum sustainable harvest rate possible, we are required to find the equilibrium solutions of the differential equation firstly. Equilibrium solutions occur when P' = 0, so we can set the right-hand side of the differential equation at 0.
4P - 4P² - h = 0
Now a quadratic equation is formed in terms of P, which can be used to obtain P.
P = (1 ± √(1 + h))/2
There are two equilibrium solutions, obtained by these two values of P.
Now, in order to obtain a maximum sustainable harvest rate, we are required to have both equilibrium solutions to as positive ( as negative populations don't make any sense in this context).
Therefore, we have to find the largest value of h for which both of these equations have positive solutions:
(1 + √(1 + h))/2 > 0
(1 - √(1 + h))/2 > 0
On camparing both the inequalities, it can be concluded that the first inequality is always true, since the square root of a positive number is always positive. The second inequality is true when h < 3.
Hence, the maximum sustainable harvest rate is is found to be at h = 3. If the harvesting rate is greater than 3, then there will be no positive equilibrium solutions and the population will eventually be perished.
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10. Suppose that you are checking your work on a test, and see that you have computed the cross product of
v=i+2j−3k
and
w=2i−j+2k
. You got
v×w=i+8j−5k
. Without actually redoing
v×w
, how can you spot a mistake in your work?
We clearly seen that a little sign mistake here in 'j'.
In the given question you have computed the cross product of v=i+2j−3k and w=2i−j+2k. You got v×w=i+8j−5k Without actually redoing v×w and we have to tell how can we spot a mistake in my work.
cross product of v and w i.e v×w is orthogonal to v and w
i.e (v×w)×v = 0 = (v×w)×w
Now v×w = i+8j₋5k
v = i+2j−3k
w = 2i−j+2k
then,
(v×w)×v = ( i+8j₋5k)×(i+2j−3k)
(v×w)×v = 1+16+15
(v×w)×v = 32≠0
and
(v×w)×w = ( i+8j₋5k)×(2i−j+2k)
(v×w)×w = 2₋8₋10
(v×w)×w = ₋16≠0
hence given cross product is wrong
Also we clearly seen that a little sign mistake here in 'j'
i.e if (v×w) = ( i₋8j₋5k)
(v×w)×v = ( i₋8j₋5k)×(i+2j−3k)
(v×w)×v = 1₋16+15
(v×w)×v = 0
and
(v×w)×w = ( i₋8j₋5k)×(2i−j+2k)
(v×w)×w = 2+8₋10
Hence, (v×w) = ( i₋8j₋5k)
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You buy these for 50.00 and then sell them for 67.00
5. (9 points) A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of on another. For a certain 4 out of 6 system, assume that on a rainy day each component has probability 0.7 of functioning, and that on a non-rainy day each component has probability 0.9 of functioning. (a). (3 points) What is the probability that the system functions on a rainy day? (b). (3 points) What is the probability that the system functions on a non-rainy day? (c). (3 points) Assume that the probability of rain tomorrow is 0.2. What is the prob- ability that the system will function tomorrow?
(a) The probability that the system functions on a rainy day is the probability that at least 4 out of the 6 components function, with each component having a probability of 0.7 of functioning.
We can calculate this probability using the binomial distribution:
P(system functions on a rainy day) = P(X ≥ 4), where X ~ Binomial(n=6, p=0.7)
Using a calculator or statistical software, we find:
P(system functions on a rainy day) = 0.8482
(b) The probability that the system functions on a non-rainy day is the probability that at least 4 out of the 6 components function, with each component having a probability of 0.9 of functioning. Again, we can use the binomial distribution:
P(system functions on a non-rainy day) = P(X ≥ 4), where X ~ Binomial(n=6, p=0.9)
Using a calculator or statistical software, we find:
P(system functions on a non-rainy day) = 0.9970
(c) To find the probability that the system will function tomorrow, we need to use the law of total probability. Let R be the event that it rains tomorrow and NR be the event that it doesn't rain tomorrow. Then:
P(system functions tomorrow) = P(system functions on a rainy day)P(R) + P(system functions on a non-rainy day)P(NR)
Using the probabilities from parts (a) and (b), and the fact that P(R) = 0.2 and P(NR) = 0.8, we can calculate:
P(system functions tomorrow) = (0.8482)(0.2) + (0.9970)(0.8) = 0.9746
Therefore, the probability that the 4 out of 6 systems will function tomorrow is 0.9746, assuming a 0.2 probability of rain.
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In calculus, the _______ form of angles is used most often.
radian
gradian
degree
revolution
Which equation has y= 8.1 as the solution
y+2.1=11
12-y=8.1
9+7.2-y=8.2
28.4-y=17.2
Answer:c
Step-by-step explanation:
because
the x-intercept of the graph of f(x)=3log(x+5)+2 is:
10^-2/3 - 5 is the x-intercept of the given equation.
Determine the x-intercept of a functionThe x-intercept of a function is the point where the function f(x) is equivalent to zero.
Given the function below;
f(x)=3log(x+5)+2
3log(x+5)+2 = 0
Make x the subject of the formula:
3log(x+5) = -2
log(x+5) = -2/3
x + 5 = 10^-2/3
x = 10^-2/3 - 5
Hence the x-intercept of the graph of f(x)=3log(x+5)+2 is 10^-2/3 - 5
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The following is the average daily temperature for Frederick, Maryland for the month of June: 94 82 83 86 91 72 88 92 82 75 82 72 | 77 93 71 | 78 | 74 | 81 91 85 90 79 82 72 83 86 85 89 90 94 (a) Complete the frequency distribution for the data. Age Frequency Relative Frequency 70-74 75-79 80-84 85-89 90-94 (b) Which of the following is the correct histogram for this data? Frequency 7080 Temperature Frequency 70 75 80 85 Temperature 90 o Frequency 70 90 80 Temperature 0 Frequency 70 90 75 80 85 Temperature
The data is analyzed and frequency table is created. The correct option for the histogram is the second option.
Here 30 individual samples are given. We are first categorizing the values in a frequency distribution table. It is shown as an image below.
Temperature frequency Relative frequency
70-74 5 5/30 = .166 = 16.6%
75-79 4 4/30 = .133 = 13.3%
80-84 7 7/30 = 0.233 =23.3%
85-89 6 6/30 = 0.2 = 20%
90-94 8 8/30 = 0.266 = 2.66%
From the given histogram the second one represents the data given in the frequency table. It is marked in the image.
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The complete question is included as image
find the selling for 52.75 karaoke machine with a 49.5markup
Answer: To find the selling price of the karaoke machine with a 49.5% markup, you need to first calculate the amount of the markup. You can do this by multiplying the cost of the machine (52.75) by the markup rate (49.5%) expressed as a decimal:
52.75 * 0.495 = 26.067375
Next, add this markup amount to the cost of the machine:
52.75 + 26.067375 = 78.817375
So, the selling price of the karaoke machine with a 49.5% markup would be $78.82.
Step-by-step explanation:
A variable needs to be eliminated to solve the system of equations below. Choose the correct first step.
4x + 5y = -6
-4x + 9y = -22
Answers:
Subtract to eliminate y.
Subtract to eliminate x.
Add to eliminate y.
Add to eliminate x.
Answer:
Step-by-step explanation:
Right Answer: Add to eliminate X
[tex]4x+ (-4x)+5y+9y=-6+(-22)\\14y=-28\\y=\frac{-28}{14} \\y=2[/tex]
Can you please help with this problem
Answer:
x = 30, -40
Step-by-step explanation:
This a right triangle, so you can use the Pythagorean Theorem.
[tex]a^{2}+ b^{2}=c^{2}[/tex], now in place of the a, b and c use x, x +10 and 50.
so a = x, b =x+10 and c = 50
[tex]x^{2} + (x+10)^{2} =50^{2}[/tex]
[tex]x^{2} +x^{2} +20x +100 = 2500[/tex]
An ancient egyptian stone tablet is found .it is correctly detucted that it is a sum with each symbol representing a different number .suprisingly there are 2 ways of doing it .what are they?
one possible explanation for why there are two different ways of adding up the symbols on the tablet is given below.
What is expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
The ancient Egyptian number system was based on hieroglyphs, and it used a decimal system with hieroglyphs representing the powers of 10. However, without seeing the specific symbols on the tablet, it is difficult to provide an accurate answer.
Assuming that the tablet is using a basic arithmetic operation such as addition, and each symbol represents a different number,
it is possible that there are two different ways to add up the numbers represented by the symbols on the tablet.
One way to approach this problem is to try to find two different sets of numbers that when added, produce the same result.
For example, if the symbols represent the numbers 2, 3, and 5, then one way to add them up would be:
2 + 3 + 5 = 10
Another way to add them up would be:
3 + 7 = 10
In this case, the symbol for 2 would represent the number 7, and the symbol for 5 would represent the number 3. This would be one possible explanation for why there are two different ways of adding up the symbols on the tablet.
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sorry but without seeing the actual tablet then we cant work it out to get a correct answer
i need help with this correct answer only please
Answer:
Step-by-step explanation:
1.
Find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. (If an answer does not exist, enter DNE.) h(t) = t t ? 6 , [7, 13] absolute maximum (t, y) = absolute minimum (t, y) =
A graphing utility such as Desmos or Wolfram Alpha can be used to graph the function. The graph shows that the maximum value occurs at t = 13 and the minimum value occurs at t = 7, which confirms our calculations.
To find the absolute extrema of the function h(t) = t^2 - 6 on the closed interval [7, 13], we can start by taking the derivative of the function and setting it equal to zero to find any critical points:
h'(t) = 2t = 0
t = 0
However, t = 0 is not in the interval [7, 13], so we only need to check the endpoints of the interval and any other critical points that may lie within the interval.
Since there are no critical points in the interval, we only need to evaluate the function at the endpoints:
h(7) = 7^2 - 6 = 43
h(13) = 13^2 - 6 = 163
Thus, the absolute maximum of h(t) on the interval [7, 13] is 163, which occurs at t = 13, and the absolute minimum is 43, which occurs at t = 7.
To verify our results using a graphing utility, we can graph the function h(t) = t^2 - 6 on the interval [7, 13] and visually observe the maximum and minimum values.
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Sampling frame Ideally, the sampling frame in a sample survey should list every individual in the population, but in practice, this is often difficult.
Suppose that a sample of households in a community is selected at random from the telephone directory. Explain how this sampling method results in under coverage that could lead to bias.
Using a telephone directory as the sampling frame can lead to under-coverage and bias, and it is generally not considered a representative sampling method.
Other more representative sampling methods, such as random digit dialing, address-based sampling, or a combination of methods, may be used to obtain a more accurate representation of the population.
Selecting a sample of households from the telephone directory can result in under-coverage and bias for several reasons.
Concept: When random sampling is not used, bias or regular flaws in the way the sample mean the population can occur.
Voluntary reply to samples, in which respondents choose their own respondents, and benefit samples, in which people who live nearby are included in the sample, are particularly biased.
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The graph represents the volume of a cylinder with a height equal to its radius. When the diameter is 2 cm, what is the radius of the cylinder? Express the volume of a cube of side length as an equation. Make a table for volume of the cube at 0 cm, 1 cm, 2 cm, and 3 cm. Which volume is greater: the volume of the cube when 3 cm, or the volume of the cylinder when its diameter is 3 cm?
Answer:
Comparing this to the volume of the cube when the side length is 3 cm, which is 27 cm^3, we can see that the volume of the cube is greater than the volume of the cylinder.
Step-by-step explanation:
I'm sorry, but I cannot see the graph you are referring to. However, I can still answer some of your questions based on the information provided.
When the diameter of the cylinder is 2 cm, the radius is equal to half the diameter, which is 1 cm.
To express the volume of a cube of side length s as an equation, we use the formula for the volume of a cube:
Volume of cube = s^3
Making a table for the volume of the cube at different side lengths, we get:
Side Length (cm) Volume (cm^3)
0 0
1 1
2 8
3 27
To compare the volume of the cube when the side length is 3 cm and the volume of the cylinder when the diameter is 3 cm, we need to find the radius of the cylinder first.
When the diameter is 3 cm, the radius is half the diameter, which is 1.5 cm. The height of the cylinder is also equal to the radius, so the volume of the cylinder can be found using the formula:
Volume of cylinder = πr^2h
Substituting r = 1.5 cm and h = 1.5 cm, we get:
Volume of cylinder = π(1.5)^2(1.5) ≈ 10.602 cm^3
Comparing this to the volume of the cube when the side length is 3 cm, which is 27 cm^3, we can see that the volume of the cube is greater than the volume of the cylinder.
Geometry- scale factor and similar triangles
can someone please explain these questions to me (see picture)
The values missing sides of the figures are calculated below.
How to solve for the missing sides of the figures?
The scale factor is the size by which the shape is enlarged or reduced. It is used to increase the size of shapes like circles, triangles, squares, rectangles, etc.
NUMBER 7
Scale factor is the ratio of two corresponding sides of similar figures. Since the scale factor from A to B = 3:5. We have:
A : B = 3:5
(x+11) : 30 = 3 : 5
(x+11) /30 = 3 / 5
x + 11 = (30*3)/5
x + 11 = 90/5
x + 11 = 18
x = 18 - 11
x = 7
NUMBER 8
(2x-12) : 12 = 1:3
(2x-12) /12 = 1/3
2x-12 = 12/3
2x-12 = 4
2x = 4 + 12
2x = 14
x = 14/2
x = 7
NUMBER 9
AB : FG = BC:GH
104/39 = 112/x
x = 42
NUMBER 9
TK:KL = KU:KM
14/91 = 12/x
x = 78
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Braden's salary used to be $148,520. After changing the amount of time he works, Braden has begun earning 45% less. What is Braden's salary now?
Answer: $81,686
Step-by-step explanation:
$148,520 - 45% = $81,686
Alternative
$148,520 x 0.55 = $81,686
Select the true statement(s): a.Any statistic is a random variable. b.An exact sampling distribution can never be obtained. c.A Statistics are used to estimate parameters.
A: The correct statement is c. Statistics are used to estimate parameters.
Statistics is the science of collecting and analyzing data to gain insight into a population of interest. It involves the collection, organization, analysis, interpretation, and presentation of data. The goal of using statistics is to draw conclusions about a population of interest and to estimate parameters of the population.
A parameter is a numerical value that is used to describe a population. For example, the mean of a population is a parameter. Statistics are used to estimate parameters of a population from a sample of the population. This process is called estimation. A statistic is a numerical value that is used to describe a sample.
For example, the sample mean is a statistic that is used to estimate the population mean. Estimation is done using the formula for a sample statistic, which is given by:
Estimate= (Statistic) / (Sample Size)
Here, the statistic is the sample mean, and the sample size is the number of observations in the sample. For example, if the sample mean is 10 and the sample size is 5, then the estimate of the population mean is 2 (10/5).
Statistics can also be used to construct confidence intervals to describe population parameters, such as means and proportions. A confidence interval is an interval of values around a sample statistic, such as a mean or a proportion, that is expected to contain the population parameter with a certain level of confidence.
In conclusion, statistics are used to estimate parameters of a population from a sample of the population. Estimation is done using the formula for a sample statistic, and confidence intervals can be used to describe population parameters.
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9687 rounded to the nearest thousand
Answer:
Step-by-step explanation:
10000
Answer:10000
Step-by-step explanation:
dimosure
I need help with these questions. Ignore my failed attempts
The values of all 9 angles shown were found using the property of a kite they are:
m∠1=55°
m∠2=35°
m∠3=35°
m∠4=90°
m∠5=55°
m∠6=67°
m∠7=67°
m∠8=23°
m∠9=23°
What is an angle ?
In Kite figure, it is given that:
∠2+∠3 = 70°
Since they are equal so ∠2=∠3=35
∠8+∠9 = 46°
Since they are equal so ∠8=∠9=23°
∠5 = 180-∠3-∠4 = 180-35-90=55°
Similarly, ∠1 = 55°
∠6+∠7 = 180-∠8-∠9 = 180-46 = 134°
Since they are equal
∠6=∠7 = 67°
Therefore, all 9 angles were calculated using property of a kite.
m∠1=55°
m∠2=35°
m∠3=35°
m∠4=90°
m∠5=55°
m∠6=67°
m∠7=67°
m∠8=23°
m∠9=23°
In mathematics, an angle is a geometric figure formed by two rays or line segments that share a common endpoint, called the vertex. The rays or line segments are called the sides or legs of the angle, and the distance between the sides at the vertex is called the angle's measure.
Angles are typically measured in degrees or radians, and they can be classified by their measures as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees and less than 180 degrees), straight (exactly 180 degrees), reflex (greater than 180 degrees and less than 360 degrees), or full (exactly 360 degrees).
Kites have several properties related to their angles, including:
Two pairs of opposite angles in a kite are congruent. That is, the angles formed between the pairs of congruent sides are equal.
One diagonal of a kite bisects the other diagonal. This means that the diagonal that connects the non-congruent vertices of the kite divides the other diagonal into two equal segments.
The sum of the measures of the two non-congruent angles in a kite is 180 degrees. This is because the kite can be divided into two congruent triangles by drawing the diagonal that bisects the other diagonal, and the sum of the angles in a triangle is always 180 degrees.
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HELP!!!! Please #10
Thank you so muchhh
The lengths of the missing sides associated with geometric systems formed by similar triangles are listed below:
Case 10: 49
Case 12: 36
How to determine missing lengths in geometric systems formed by similar triangles
In this problem we have the case of two geometric systems formed by two similar right triangles. These figures are similar if they have congruent internal angles but their sides are not congruent though proportional. Hence, missing sides can be found by proportion formulas. Now we proceed to determine the missing side for each case:
Case 10
x / 7√33 = 7√33 / 33
x = 7²
x = 49
Case 12
x / 6√13 = 6√13 / 13
x = 6²
x = 36
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by half time, curry scored 22 of the teams 56 points. What percent of his teams points did curry score?
The percentage scored by Curry is 39.3% oby halftime.
How to determine the percentage scoredTo find the percentage of his team's points that Curry scored, we can use the following formula:
percentage = (part / whole) x 100
Where
Part = 22 points
Whole = 56 points
Substitute the known values in the above equation, so, we have the following representation
Percentage = (22 / 56) x 100%
Evaluate
Percentage = 39.3%
Hence, Curry scored 39.3% of his team's points by halftime.
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What are the measurements of the missing angels?
Angel 1, Angel 2
The missing angles in triangle RST and LMN are 121° and 15° respectively.
What are angles?When two straight lines or rays intersect at a single endpoint, an angle is created.
The vertex of an angle is the location where two points come together.
The Latin word "angulus," which means "corner," is where the term "angle" originates.
Angle position describes how a line interacts with another line or plane. The amount of rotation of the body relative to the reference position is used to calculate the angle of position.
Theta (), the sign used to represent the angular position, can represent the angle in degrees (°), radians (rads), or revolutions.
So, in the triangle RST:
Missing angle:
44 + 15 + x = 180
x = 180 - 59
x = 121°
In triangle LMN:
121 + 44 + x = 180
x = 180 - 165
x = 15°
Therefore, the missing angles in triangle RST and LMN are 121° and 15° respectively.
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FILL IN THE BLANK Quantitative data that measure how many are ________; quantitative data that measure how much are ________.
Answer:
Caca
Step-by-step explanation:
Caca
Jane’s class gave examples of absolute value in distances traveled. Which example demonstrates the greatest distance traveled? Responses Jim jumped on a trampoline to a height of 8.67 feet. Jim jumped on a trampoline to a height of 8.67 feet. Teresa rode an elevator to a depth of −10 feet. Teresa rode an elevator to a depth of - 10 feet. Brenan dove to a depth of −12.76 feet. Brenan dove to a depth of − 12.76 feet. Kailey hopped a length of 7.612 feet. Kailey hopped a length of 7.612 feet.
Answer:
The example that demonstrates the greatest distance traveled is Brenan diving to a depth of -12.76 feet, as it has the highest absolute value.
Which of the following mathematical relationships could be found in a linear programming model? (Select all that apply.) (a) -1A + 2B ≤ 20 (b) 24 - 2B = 30 (c) 14 - 6B2 ≤ 10 (d) 3√A + 2B ≥15 (e) 1A + 1B = 9 (f) 24 + 68 + 1AB ≤ 36 For the relationships that are unacceptable for linear programs, state why. ___ could not be found in a linear programming model because __
Mathematical relationships could be found in a linear programming model are (b) 2A - 2B = 30 (e) 1A + 1B = 9 (f) 2A + 6B + 1AB ≤ 36
Linear programming, mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints.
(a) -1A + 2B ≤ 20 : Cannot be found in linear programming model as linear programming model can only consists of positive linear numbers and this equation contain negative number.
(b) 2A - 2B = 30 : Can be found in linear programming model
(c) 1A - 6B2 ≤ 10 : Cannot be found in linear programming model as equation includes square variable.
(d) 3√A + 2B ≥15 : Cannot be found in linear programming model as equation includes square root variable.
(e) 1A + 1B = 9 : Can be found in linear programming model
(f) 2A + 6B + 1AB ≤ 36 : Can be found in linear programming model
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Ms Núñez sells fruit from a fruit stand. She starts the day with 80 pieces of fruit. In the morning, 5 customers each buy (p) pieces of fruit. In the afternoon, Pascal feeds each bird (b) for bird. She sees 2 pieces of fruit.
Write an algebraic expression that represents the amount of pieces of fruit Ms Núñez has after her customers and feeding the birds.
How many pieces of fruit would Ms Núñez have left of each customer bought 6 pieces of fruit and she gave 5 birds the 2 pieces of fruit?
Answer:
Step-by-step explanation:
please help me.
Evaluate f (-2)where is the piecewise function graphed below.
f(-2)=
After evaluating f (-2), where f(x )is the piecewise function graphed given, the value of f(-2) = -3.
What is the piecewise function?
A function that is defined piecemeal is one that has numerous subfunctions, each of which applies to a distinct interval in the domain. Instead of being a property of the function itself, piecewise definition is a means to express the function.
Here, we have
The problem asks for f(-2), which is the value of y, that is, the vertical axis, when x, that is, the horizontal axis is -2.
We can see that at x = -2, the function changes the definition, thus it is called a piece-wise function.
To find the numeric value at x = -2, we have to look at the definition with the closed circle, which is at y = -3.
Hence, the value of f(-2) = -3.
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A person invested $3, 700 in an account growing at a rate allowing the money to
double every 6 years. How much money would be in the account after 14 years, to the
nearest dollar?
Answer:
Step-by-step explanation:10299