Answer:
See below
Step-by-step explanation:
Solve the equation using the quadratic formula
[tex]\displaystyle x^2-2x+2=0\\\\x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(2)}}{2(1)}\\ \\x=\frac{2\pm\sqrt{4-8}}{2}\\ \\x=\frac{2\pm\sqrt{-4}}{2}\\ \\x=\frac{2\pm2i}{2}\\ \\x=1\pm i[/tex]
Discriminant and Solution Analysis
As we determined in the quadratic formula, our discriminant is -4 because [tex]b^2-4ac=(-2)^2-4(1)(2)=-4[/tex], under the radical. Because it is negative, our solutions must be imaginary since the square root of a negative number is not real.
A set of eight cards were labeled as M, U, L, T, I, P, L, Y. What is the sample space for choosing one card? S = {I, U, Y} S = {M, L, T, P} S = {I, L, M, P, T, U, Y} S = {I, L, L, M, P, T, U, Y}
The sample space for choosing one card is, S={I, L, L, M, P, T, U, Y}
What is a sample space?A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.
Given that,
A set of eight cards were labeled with M, U, L, T, I, P, L, Y.
Here, the sample space is;
{ S, U, B, T, R, A, C, T}
Now, Elements in order is,
⇒ S = {I, L, L, M, P, T, U, Y}
Therefore, the sample space for the given cards is,
S = {I, L, L, M, P, T, U, Y}
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Answer: I think the answer would be C
Calculate the length and width of the original figure with a scale factor of 50%.
Answer:To calculate the length and width of the original figure with a scale factor of 50%, you need to reverse the scaling by multiplying the new dimensions by 100% / 50% = 2.
So if the new length is L, the original length would be L * 2 = 2L. Similarly, if the new width is W, the original width would be W * 2 = 2W.
Step-by-step explanation:here's a step-by-step explanation:
Identify the new length (L) and new width (W) of the figure.
Multiply the new length by a factor of 2 to find the original length: 2L = L * 2
Multiply the new width by a factor of 2 to find the original width: 2W = W * 2
The original length is 2L and the original width is 2W.
Note: This process works for any scale factor, not just 50%. To find the original dimensions, simply multiply the new dimensions by 100% / the scale factor.
(x+y)²+8(x+y)+12 factorize
The expression is factorized to give x(x+ 12) + 8( y + 12)
How to factorize the expressionFrom the information given, we have that the expression as;
(x+y)²+8(x+y)+12
Now, expand the bracket, we get;
(x + 2) (x+2) + 8x + 8y + 12
expand the squared bracket, we get;
x² + 2x + 2x + 4 + 8x + 8y + 12
Now, collect like terms
x² 2x + 2x + 8x + 8y + 4 + 12
Add the collected like terms, we get;
x² + 12x + 8y + 16
Now, let's factorize in airs by grouping, we get;
(x² + 12x) + (8y + 16)
Factor the common terms
x(x+ 12) + 8( y + 12)
Hence, the expression is x(x+ 12) + 8( y + 12)
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Which multiplication equation can be used to explain the solution to 15/ by 1/3 of a fraction?
Answer:
Multiply by the inverse.
Answer: 45
Step-by-step explanation:
(15)/(1/3) is the same thing as multiplying the top (numerator), by the inverse of the bottom (denominator).
So we have the following: [tex]\frac{15/1}{1/3}[/tex]
Taking the inverse means flipping the fraction. The inverse of [tex]\frac{1}{3}[/tex] is [tex]\frac{3}{1}[/tex]. So we can keep the top, and multiply by the inverse.
[tex]\frac{15}{1}[/tex] X [tex]\frac{3}{1}[/tex] = 15 x 3 = 45.
Does this make sense?
Solve each of the following inequalitie and illutrate the olution on number line a) -4x < 28
The inequality and number line that shows the solution is x>-7.
What is inequality?A relationship between two expressions or values that is not equal to each other is known as inequality in mathematics. So inequality emerges from an imbalance. Less than symbol (), greater than symbol (>), more than or equal to symbol (), less than symbol (), and not equal to symbol () are the five inequality symbols used in mathematics.A connection in mathematics that compares two numbers or other mathematical expressions in an unequal way is known as an inequality. It is most frequently used to compare the sizes of two numbers on the number line.Simply put, a compound inequality is more than one inequality that we want to address simultaneously. If we want to refer to the answer to both of the inequalities, we can use the word "and," or if we want to refer to the solution to just one of the inequalities, we can use the word "or" (or).Given,
-4x <28
(-4x) (-1) > 28 (-1)
Simplify
4 x>-28
Divide both sides by 4
[tex]\frac{4 x}{4} > \frac{-28}{4}[/tex]
Simplify
x>-7
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Luis jogged 9 3/13 laps the morning and 3 5/26 in the evening how many did he do in all?
Answer: 12 11/26 laps
Step-by-step explanation:
9 3/13= 9 6/26
9 6/26+ 3 5/26=12 11/26.
help this is urgent hurry pls
Answer:
Step-by-step explanation:
You're right; it's 97 degrees. The angles of a triangle add up to 180, so to solve for each triangle's unknown angle value, subtract the known angles from 180. For the left triangle, that's 33 and for the right triangle it's 50. Those missing angles plus the one the question is asking for should add up to 180 degrees as well. Hence, 180 - 33 - 50 = 97.
You decide to invest in government bonds to save money for your eventual wedding. You have an
initial investment of $7,000 and when you choose to invest for 14 years you get a 10% rate
of return.
What will be the value of your investment at maturity?
Round off to nearest whole number
The value of your investment at maturity is $16,800. The interest is $9,800.
What is the meaning of initial investment?
The initial sum of money required to open an account or start a buy-in relationship is known as an initial investment. The two distinct but related industries of banking and long-term investment brokering are the main uses of the term "initial investment."
Given that the initial investment is $7,000. The rate of interest is 10% = 0.10. The number of years of investment is 14 years.
The formula of simple interest is
I = Prt
I = interest, P = principal, r = rate of interest, t = time.
Putting P = 7000, r = 0.10, t = 14 in I = Prt:
I = 7000 × 0.10 × 14
I = 9,800
The amount will be interest + principal = 9800+ 7000 = 16,800.
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Work out the hortet ditance between planet Q and plant R. Give your anwer in tandard form
The shortest distance between planet Q and planet R is 1.64 × 10⁷ .
What is planet distance?RP = 1.6 * 10⁷
most compact distance
The Pythagoras theorem x₂ = R * P² + Q * P² is then used to determine the shortest distance (x).
This results in x = (3.6 * 10⁶) + (1.6 * 10⁷) = 2.
x² = 12.96 * 10¹² + 2.56 * 10¹⁴ should be evaluated.
Write the equation as: x²= 12.96 * 10¹² + 256 * 10¹²
Sums x² = 268.96*10¹² should be evaluated.
multiply both sides' square roots.
x = 16.4 * 10⁶
Replace with: x = 1.64 * 10⁷
In light of this, the distance between planet Q and planet R is 1.64 * 10⁷ km.
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The complete question is
The distances between some planets are given below.
Planet Q is 3.6 x 10⁷ to the power of 6 km from planet P.
Planet R is 1.6 x 10⁷ to the power of 7 km from planet P.
Work out the shortest distance between planet Q and planet R. Give your answer in standard form.
Find the area of the shaded region. Round your answer to the nearest tenth.
The area of the shaded region is 17.4 rounded up to the nearest tenth.
How to evaluate for the area of the shaded regionThe unshaded region forms a circle with a radius of 9/2, so to get the area of the shaded region, we subtract the area of the circle from the area of the square as follows:
area of square = 9 m × 9 m = 81 m²
area of the circle = 22/7 × 9/2 × 9/2
area of the circle = 891/14 m²
area of the shaded region = 81 m² - 891/14 m²
area of the shaded region = (1134 - 891)/14
area of the shaded region = 243/14
area of the shaded region = 17.3571
Therefore, the area of the shaded region is 17.4 rounded up to the nearest tenth.
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As the marketing designer for Burger King, you are given a 4 inch by 6 inch logo and have been told to create an advertisement with sides six times larger. What is the area of your advertisement (in square feet)?
The area of the logo is 6 square feet.
What is the area of your advertisement?We start wit a logo of 4 inches by 6 inches, and we want to make it 6 times larger, so the new dimensions are:
6*(4 inches) = 24 inches.
6*(6 inches) = 36 inches.
Remember that:
1ft = 12 in
Writting that in feet, we will get:
24 inches = (24/12) ft = 2ft
36 inches = (36/12) ft = 3ft
Then the area of the logo is:
A = 2ft*3ft = 6 ft²
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standard deviation and varianc'e remember that the standard deviation and variance can only be used when the mean is used as the measure of center. True/False ?
True. The variance and standard deviation both quantify how far spaced apart a set of data is from the mean. They cannot be used to calculate the dispersion around the median or mode, or any other center-of-mass metric.
The variance and standard deviation both quantify how far spaced apart a set of data is from the mean. They can therefore be used to calculate the difference between the highest and lowest values in a data set. The standard deviation, which is the square root of the variance, is used to calculate how far away from the mean each result is on average. The average of the squared deviations between each data point and the mean is the variance. The variance and standard deviation cannot be used to quantify the dispersion of data around other measures of center, such as the median or mode, because they only measure the dispersion of data around the mean.
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determine the equation of each radical function which has been tranformed
The equation of a radical function can be determined by using the transformation rules for x- and y- shifts, stretches/compressions, reflections, and rotations.
The equation of a radical function can be determined by applying various transformation rules. The most common transformations are shifts, stretches/ compressions, reflections, and rotations. For example, a radical function that is shifted left two units would have an equation of y = sqrt(x + 2). This is because the function is shifted two units to the left of the origin, so the equation must include an x + 2 term. Similarly, if the function is reflected in the x-axis, the equation would be y = -sqrt(x). This is because the reflection inverts the sign of the y-coordinate, so the equation must include a negative sign in the radical. For stretches and compressions, the equation must include a coefficient in front of the radical. For example, if the function is stretched horizontally by a factor of 3, then the equation would be y = 3 sqrt(x). This is because the function is stretched by a factor of three, so the equation must include a coefficient of 3 in front of the radical. Finally, for rotations, the equation must include a term for the angle of rotation. For example, if the function is rotated counterclockwise by 45 degrees, then the equation would be y = sqrt(x + y/tan45). This is because the angle of rotation is 45 degrees, so the equation must include a tan45 term.
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(3) pilot altitude 1m angle of depression 3°
d=?
The length d for the angle of depression 3° and altitude 1m is equal to 19 m to the nearest metre using the trigonometric ratio of cosine for the angle 87°
What is an angle of depressionThe angle of depression is an angle formed between the horizontal line and the line of sight when an observer looks downwards.
Considering the right triangle formed by the length d and the altitude of 1 m, we shall derive a complementary angle as:
90° - 3° = 87°
such that;
cos 87° = 1/d {adjacent/hypotenuse}
d = 1/cos 87°
d = 19.1073.
Therefore, the length d for the angle of depression 3° and altitude 1m is equal to 19 m to the nearest metre using the trigonometric ratio of cosine for the angle 87°.
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I am a multiple of 7 le then 50. When my digit are revered,I am a multiple of 8. What number am I
The number which satisfies the given condition is 28.
According to the question
The number 28 is less than 50 and has a multiple of 7. It becomes 82, which is also a multiple of 8 when its digits are inverted. As a result, the criteria can only be satisfied by the number 28. This may be verified by quickly determining whether any other number less than 50 that is a multiple of 7 has its digits switched around to become a multiple of 8. However, it is clear that none of the other figures satisfy both requirements.
In light of this, the answer to this issue is 28.
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The number which satisfies the given condition is 28.
According to the question
The number 28 is less than 50 and has a multiple of 7. It becomes 82, which is also a multiple of 8 when its digits are inverted. As a result, the criteria can only be satisfied by the number 28. This may be verified by quickly determining whether any other number less than 50 that is a multiple of 7 has its digits switched around to become a multiple of 8. However, it is clear that none of the other figures satisfy both requirements.
In light of this, the answer to this issue is 28.
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compute the number of ways you can select n elements from n elements. n = 2, n = 10
The number of ways you can select n elements from N elements by using factorial (!) when n=2 and N=10 is 45.
Describe a factorial.Factorial is a crucial mathematical operation that is used to determine the number of possible arrangements or the ordered set of numbers. Daniel Bernoulli is credited with discovering the factorial function's well-known interpolating feature. Numerous mathematical ideas, including probability, permutations and combinations, sequences and series, etc., use the factorial concept. A factorial function, in essence, multiplies a number by each number below it until it equals 1. For instance, the factorial of 3 is equal to 6 and reflects the multiplication of the integers 3, 2, and 1.
Now,
The number of ways you can select n elements from N elements when n=2 and N=10 is [tex]\frac{N!}{n!(N-n)!}[/tex]
=[tex]\frac{10!}{2!(8)!}[/tex]
=[tex]\frac{10 \times 9 \times \times 8!}{8! \times 2 !}[/tex]
=45
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rosie sold 19 catendars.Suzy sold 26 jaun sold 4 times as many calenars as rosie and sulzy combined how many calendars did juan sell
First, we can find the total number of calendars sold by Rosie and Suzy combined:
19 (calendars sold by Rosie) + 26 (calendars sold by Suzy) = 45 calendars
Next, we can find the number of calendars sold by Juan:
Juan sold 4 times as many calendars as Rosie and Suzy combined, so Juan sold 4 * 45 = 180 calendars.
For which function is the average rate of change over the interval 1 < x < 5 greater than the average rate of change over the same interval for the function g(x) = 1. 8x2?
A. F(x) = x2 B. G(x) = 1. 2x2 C. H(x) = 1. 5x2 D. K(x) = 2x2
The average rate of change over for the interval 1 < x < 5 k(x) greater than the average rate of change over of g(x)
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
We have,
g(x) = 1.8 [tex]x^{2}[/tex]
The average rate of change of g(x) over the interval 1 < x < 5:
= [tex]$\frac{g(5)-g(1)}{5-1}[/tex]
= [tex]$\frac{1.8 \times (5)^{2} -1.8 \times(1)^{2} }{5-1}[/tex]
= [tex]$\frac{43.2}{4}[/tex]
= 10.8..............(1)
The average rate of change of f(x) = [tex]x^2[/tex] in the interval 1 < x < 5:
= [tex]$\frac{25-1}{4}[/tex]
= [tex]$\frac{24}{4}[/tex]
= 6..............(2)
The average rate of change of g(x) = 1.2 [tex]x^2[/tex] in the interval 1 < x < 5:
= [tex]$\frac{28.8}{4}[/tex]
= 7.2.................(3)
The average rate of change of h(x) =1.5 [tex]x^2[/tex] in the interval 1 < x < 5:
= [tex]$\frac{36}{4}[/tex]
= 9...........(4)
The average rate of change of k(x) = 2 [tex]x^{2}[/tex] in the interval 1 < x < 5:
= [tex]$\frac{48}{4}[/tex]
= 12.........(5)
From (1), (2), (3), (4), and (5) we see that,
The average rate of change of [tex]$\mathbf{k}(x)=2 x^2$[/tex] is larger than the average rate of change of [tex]$g(x)=1.8 x^2$[/tex].
12 > 10.8
Thus,
Option D is the correct answer.
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If the two spinners below are spun, what is the probability that the numbers will add to more than 4?
Answer: The answer is 1/4.
Step-by-step explanation:
A class used cars(x) and vans(y) to go on a field trip. They used 11 cars and vans. Each car holds 5 students and each van holds 8. If 73 students went on the trip then how many of each vehicle did the class use.
Answer:
X=5 Cars
Y=6 Vans
Step-by-step explanation:
Let's assume the number of cars used is x and the number of vans used is y.
We can set up two equations based on the information given:
x + y = 11 (the total number of vehicles used)
5x + 8y = 73 (the total number of students that went on the trip)
We can use substitution to solve for one of the variables in terms of the other:
y = 11 - x
5x + 8(11 - x) = 73
5x + 88 - 8x = 73
-3x = -15
x = 5
So the class used 5 cars and 11 - 5 = 6 vans.
(1 point) solve the system using elimination ⎧⎩⎨⎪⎪6x5x−4x−5y−4y 3y 3z 6z 6z===−70−755 x= y= z=
The solution to the system of linear equations is: x = 5, y = (6x + 7 - 10/4)/(-5) = -4/5, z = -5/4.
To solve the system of linear equations using elimination, we can manipulate the coefficients of the variables to make one of the variables zero in one of the equations. Then, we can use this equation to eliminate that variable from the other equations.
Starting with the first two equations:
6x - 5y - 4z = -7
5x - 4y - 4z = -5
We can multiply the first equation by 5 and the second equation by 6 to get rid of the coefficients of x in both equations:
30x - 25y - 20z = -35
30x - 24y - 24z = -30
Now, we can subtract the first equation from the second equation to eliminate x:
0y - 4z = 5
So, z = -5/4
Next, we can use this value of z to solve for y in either of the first two equations:
6x - 5y - 4(-5/4) = -7
Expanding the right side:
6x - 5y + 10/4 = -7
And solving for y:
y = (6x + 7 - 10/4)/(-5)
Finally, we can use the value of y and z to solve for x in any of the first two equations:
6x - 5((6x + 7 - 10/4)/(-5)) - 4(-5/4) = -7
Expanding and solving for x:
6x - (6x + 7 - 10/4)/5 - 10/4 = -7
Multiplying both sides by 5:
30x - 6x - 7 + 2 = -35
And solving for x:
x = 5
Finally, we can use the values of x, y, and z to check the third equation:
3y + 6z = 6
Substituting the values we found:
3((6x + 7 - 10/4)/(-5)) + 6(-5/4) = 6
Expanding and solving:
The left side is equal to 6 and the right side is equal to 6, so the solution is consistent.
Therefore, the solution to the system of linear equations is:
x = 5, y = (6x + 7 - 10/4)/(-5) = -4/5, z = -5/4
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Which theorem or postulate proves that △ABC and △DEF are similar? SSS Similarity Theorem AA Similarity Postulate SAS Similarity Theorem Two triangles with the same shape. In the first triangle, the vertices are labeled as A, B, and C. Base is B C and the top vertex is A. Side A B is labeled 3. Side B C is labeled 7. Side A C is labeled 6. In the second triangle, the vertices are labeled as D, E, and F. Base is E F and the top vertex is D. Side D E is labeled 18. Side E F is labeled 42. Side D F is labeled 36
The theorem that proves that the two triangles are similar is the SAS Similarity Theorem. The theorem states that if two sides of one triangle are proportional to two sides of another triangle, and the included angle between these sides is the same in both triangles, then the two triangles are similar.
In this case, the ratio of the corresponding sides of the two triangles are:
AB/DE = 3/18 = 1/6
BC/EF = 7/42 = 1/6
Since the ratio of corresponding sides is equal, and the included angle between them (angle B and angle E) is the same, the two triangles are similar by the SAS Similarity Theorem.
You pick a card at random.
5
6
7
8
What is P(odd or less than 6)?
Write your answer as a percentage
The probability of odd or less than 6 is 0.75 which is 75%
What is probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
From the question, there are 4 numbers
These include: 5,6,7,8
Pr(odd) 2/4
Pr(x<6) 1/4
Adding the fractions we have
1/2 + 1/4 = 3/4 = 0.75
Converted to percentage = 75%
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September 20, 2019: Sanford pays a $15,000 balance due to the supplier upon delivery of the helmets to its warehouse. how should the transaction be recorded in sanford’s financial accounting system?
The transaction should be recorded as an Accounts Payable (A/P) liability in Sanford’s financial accounting system.
Accounts Payable is a liability account that records the amount of money a company owes to its suppliers for goods and services that have been received, but not yet paid for. In this case, Sanford is responsible for paying the supplier $15,000 for the helmets that have been delivered to its warehouse.
This payment should be recorded in the Accounts Payable account as a debit of $15,000 and a credit to the Cash account of $15,000. By doing this, Sanford’s Accounts Payable balance will be reduced to reflect that the invoice has been paid, and the Cash account will be increased to reflect the payment made. This will keep the company’s financial records up to date and accurate.
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Can someone solve this step by step and with a graph?
earth is approximately a sphere of radius 6.37 x 10 6 m. what are (a) its circumference in kilometers, (b) its surface area in square kilometers, and (c) its volume in cubic kilometers?
(a) Circumference of Earth in kilometers: 40,075.017 km
(b) Surface area of Earth in square kilometers: 510.1 million km^2
(c) Volume of Earth in cubic kilometers: 1.08 trillion km^3
To calculate these values, we use mathematical formulas that relate to the sphere shape of the Earth.
For (a), we use the formula for the circumference of a circle, C = 2 * pi * r, where r is the radius of the Earth in meters (6.37 x 10^6 m). We then convert this result to kilometers by dividing by 1000.
For (b), we use the formula for the surface area of a sphere, A = 4 * pi * r^2, where r is the radius of the Earth. Again, we convert this result to square kilometers.
For (c), we use the formula for the volume of a sphere, V = (4/3) * pi * r^3, where r is the radius of the Earth. We then convert this result to cubic kilometers by dividing by 10^9.
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‼️‼️PLEASEE HELP‼️‼️
When rotated about it’s center, a regular heptagon has __ rotational symmetries. In addition to rotational symmetry, a regular heptagon has __ lines of reflectional symmetry
When rotated about its center, a regular heptagon has 7 rotational symmetries. In addition to rotational symmetry, a regular heptagon has 1 line of reflectional symmetry.
What do you mean by rotation symmetry?Rotation symmetry, also known as radial symmetry or circular symmetry, is a fundamental concept in geometry that refers to the property of a shape to remain unchanged after rotating it about a fixed point. This point is typically the center of the shape, although it can be any point within the shape.
The amount of rotation required for the shape to become congruent to its original form is called the "angle of rotation." If a shape can be rotated multiple times and still remain unchanged, it is said to have multiple rotational symmetries.
For example, a circle has an infinite order of rotational symmetry, as it can be rotated 360 degrees and still look exactly the same. On the other hand, a square has 4 rotational symmetries, as it can be rotated 90 degrees and still look the same 4 times.
Rotational symmetry is a useful concept in mathematics and science as it provides a way to describe the symmetry of objects and shapes in the real world. It also helps us understand how objects and shapes change when they are rotated, and it provides a way to classify objects based on their symmetrical properties.
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from your descriptive statistics, describe your standard deviations of salary, hryly rate, yrs worked, education, and age. what does this tell you about the variables
It is not possible to describe the standard deviations of salary, hourly rate, years worked, education, and age without having the data.
Standard deviation is a measure of the spread or dispersion of a set of data. It is calculated as the square root of the variance and is used to give an idea of how far the individual values in a data set are from the mean or average.
In order to describe the standard deviations of salary, hourly rate, years worked, education, and age, we would need to have the actual data for those variables. Without the data, it is not possible to calculate the standard deviations and describe what they tell us about the variables.
It is important to note that the standard deviation can give us information about the distribution of the data and the variability of the values. For example, if the standard deviation is small, this means that the majority of the values are close to the mean, while a larger standard deviation indicates that the values are more spread out. This information can be useful in making inferences about the variables and in making predictions about future values.
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If tan x° = 3 divided by y and co x° = y divided by z ,what i the value of in x°?
The value of in x° is sin x = 3/z
The ratio of an angle's opposing side to its adjacent side, or its tan
tan x = 3/y
cos x = y/z
We choose 3 and y to represent the opposite and adjacent sides, respectively, because this ratio is 3:y.
Technically, we ought to write 3t and yt, but we'll take a chance since y appears in the second ratio as well!
Since y is already used to represent the adjacent side, z is used to represent the length of the hypotenuse.
The right triangle's sides are now finished.
Because tan x = sin(x)/cos(x), therefore
sin x = tan(x)*cos(x)
= (3/y)*(y/z)
= 3/z
Answer: a) sin x = 3/z
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Use RAM to estimate the area of the region enclosed between the graph of Æ and the x-axis for a ⤠x ⤠b. Æ(x) = 1/x, a = 1, b = 3
The area enclosed between the graph of Æ and the x-axis for 1 ≤ x ≤ 3 is estimated to be 4/3.
Ram's Rule is a numerical integration method used to estimate the area under a curve by dividing it into trapezoids. The formula for calculating the area under the curve is given by A = (1/2) x (b - a) x (f(a) + f(b)), where A is the area, b and a are the limits of the integration and f(a) and f(b) are the values of the function at the limits a and b respectively.
In this case, the area enclosed between the graph of Æ and the x-axis for 1 ≤ x ≤ 3 can be estimated using Ram's Rule as follows:
A = (1/2) x (3 - 1) x (1/1 + 1/3)
= (1/2) x (2) x (4/3)
= 4/3.
Therefore, the area enclosed between the graph of Æ and the x-axis for 1 ≤ x ≤ 3 is estimated to be 4/3.
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