"a. For what values of x is ​f(x)​ > 0?
b. What is the domain of​ f?
c. What is the range of​ f?
d.What are the​ x-intercept(s)?
e.What are the​ y-intercept(s)?
f.How often does the line"

A graphing calculator can be a useful tool for solving systems of linear equations because it is fast, accurate, and can handle more complex equations.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

A graphing calculator can be a useful tool for solving systems of linear equations because it can provide a quick and accurate visual representation of the solution.

Here are some reasons why someone might choose to use a graphing calculator instead of graphing by hand:

Speed:   Graphing by hand can be time-consuming, especially for more complex systems of equations.

Accuracy: When graphing by hand, it's easy to make mistakes in plotting points or drawing lines, which can lead to errors in the final solution

Multiple solutions: When dealing with systems of equations with multiple solutions, graphing by hand can be difficult and time-consuming.

Complex equations: Graphing by hand can become very challenging for systems of equations with more than two variables or for nonlinear equations.

Hence, a graphing calculator can be a useful tool for solving systems of linear equations because it is fast, accurate, and can handle more complex equations.

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Market Price: The point where supply and demand are equal.

Market price is the current price of a good or service that is determined by the interactions of buyers and sellers in a competitive market. It is the result of the forces of supply and demand and is determined by the number of buyers and sellers in the market and their respective levels of demand and supply. Market prices are affected by both external and internal market forces such as government policies, taxation, economic conditions, and availability of resources.

Demand: The quantity of a good or service that consumers are willing and able to buy

Supply: The quantity of a good or service that businesses are willing and able to provide

Producers: Individuals and organizations that determine what products and services will be available for sale

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A. The equation given can be expressed in polar form using the Pythagoras theorem.
Let the polar coordinate of a point be (r, θ).
Let r = √(x^2 + y^2)
=> r^2 = x^2 + y^2
=> r^2 = 2x^2 - 6xy - 2x
Let θ = tan^-1 (y/x)
Now, the equation in polar form is:

r^2 = 2x^2 - 6xy - 2x

tan^-1 (y/x) = θ


To find the exact values, solve the equation for the given values of x and y:


For x = 0 and y = 2:

r^2 = 0 - 6*2 - 0

=> r^2 = -12

=> r = √12

=> r = 2√3

tan^-1 (2/0) = θ

=> θ = 90°

Hence, the point (x, y) = (0, 2) has polar coordinates (2√3, 90°).

B. The given cycle has a period of 12 hours. This means that the time between each high and low tide is 12 hours. The height of high tide is 6 feet and the height of low tide is 2 feet. This means that the amplitude of the cycle is 4 feet (6 feet - 2 feet).

Hence, the equation for the tide cycle can be expressed as:

Height = 4sin (π/6 * t) + 4

Where t is time in hours.

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Answer:

No. It is a rational number.

Step-by-step explanation:

...

Answer: D

Step-by-step explanation: Its D because for every 6 units is 3/2 pi

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The sin of the angle is 0.6 and the cos of the angle is 0.8.

In a right triangle, the sine (sin) of an angle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. The cosine (cos) of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

In this case, the side opposite to the angle is 6 and the side adjacent to the angle is 8. To find the hypotenuse, we can use the Pythagorean theorem:

a^2 + b^2 = c^2

Where a and b are the lengths of the two legs of the right triangle, and c is the length of the hypotenuse.

Plugging in the given values:

6^2 + 8^2 = c^2

36 + 64 = c^2

100 = c^2

c = 10

So the hypotenuse of the right triangle is 10.

Now we can find the sin and cos of the angle:

sin = opposite/hypotenuse = 6/10 = 0.6

cos = adjacent/hypotenuse = 8/10 = 0.8

Therefore, the sin of the angle is 0.6 and the cos of the angle is 0.8.

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Answer: −3x^2+x+2

Step-by-step explanation:

Solution of the expression is,

⇒ - 3x² + x + 2

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.

For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

WE have to given that;

To divide 15x⁴ − 5x³ − 10x² by −5x².

Now, We can solve as;

⇒ (15x⁴ − 5x³ − 10x²) / −5x²

⇒ 5x² (3x² - x - 2) / - 5x²

⇒ - (3x² - x  - 2)

⇒ - 3x² + x + 2

Thus, After divide we get;

⇒ - 3x² + x + 2

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(a) True

(b) True

(c) False

(d) True

(e) False

(a) True. For any n × n invertible matrix A, the equation Ax = 0 admits only the trivial solution. This is because if A is invertible, then there exists an inverse matrix A^-1 such that A^-1A = I, where I is the identity matrix. Therefore, if Ax = 0, then A^-1Ax = A^-10, or x = 0. This means that the only solution to the equation is the trivial solution x = 0.

(b) True. For any two matrices A and B of size n × n, (AB)^T = (B^T) (A^T). This is a property of matrix multiplication and transposition, and can be easily verified by performing the operations on the matrices.

(c) False. Let A be a matrix of size n × n. If for two vectors y and z of R^n we have Ay = Az, it does not necessarily mean that y = z. It is possible for two different vectors to produce the same result when multiplied by the same matrix. For example, consider the matrix A = [[1, 0], [0, 1]] and the vectors y = [1, 0] and z = [0, 1]. Both Ay and Az will result in the same vector [1, 1], but y and z are clearly not equal.

(d) True. If A is a matrix of size n × n such that A^3 = I(n), then A is invertible. This is because A^3 = I(n) implies that A^2 = A^-1, meaning that the inverse of A exists and A is invertible.

(e) False. For any two matrices A and B of size n×n such that AB = 0, it is not necessarily true that A = 0 or B = 0. It is possible for two non-zero matrices to multiply to produce the zero matrix. For example, consider the matrices A = [[1, -1], [1, -1]] and B = [[1, 1], [1, 1]]. Both A and B are non-zero matrices, but AB = 0.

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Answer:

n = 15

Step-by-step explanation:

A

the area (A) of a rectangle is calculated as

A = length × width

here length = n - 3 and width = 2 , then

A = 2(n - 3)

given A = 24 , then

2(n - 3) = 24

B

solving the equation for n

2(n - 3) = 24 ← divide both sides by 2

n - 3 = 12 ( add 3 to both sides )

n = 15

By answering the above question, we may state that We must thus locate function the value of h that makes the equation true in order to estimate the height of the tower platform.

Mathematicians study numbers, as well as their variations, equations, related forms, and conceivable arrangements of these. Function refers to the relationship between a set of inputs, each of which has an associated output. A function is an association of inputs and outputs, where each input leads to a single, identifiable output. Each function is given a domain, codomain, or scope. The letter f is frequently used to represent functions (x). A cross is typed. The four primary types of accessible functions are on functions, one-to-one capabilities, so numerous capabilities, in capabilities, and on functions.

The projectile's starting height above the earth must be determined. Setting the time to zero (t = 0) and finding h will allow us to do this.

The formula for the projectile's height at time t is:

[tex]vt + h + -16t2 = h(t)[/tex]

where h represents the projectile's initial height and v its initial velocity.

t = 0 results in:

[tex]h(0) = -16(0)^2 + v(0) + h h(0) = h[/tex]

Hence, h, the height of the tower platform, is the projectile's beginning height.

We must thus locate the value of h that makes the equation true in order to estimate the height of the tower platform.

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It is false.

The equation for the volume of a prism is V=Bh, where B is the base area and h is the height. A rectangular base supports the prism. The rectangle has a 9 cm length and a 7 cm width. Area A of a rectangle is equal to its length (l) and width (w).

The area of a prism's base is typically represented by the variable B.

A prism's total surface area (which includes the bases) is typically denoted by T or A, while its lateral surface area (which includes the bases) is typically denoted by L.

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The order of the values from least to greatest is -2 1/4, -1 3/4, -1 1/4, 3/4, and 1 1/4.

The values can be compared using order of operations. First, we need to calculate the numerical values of each fraction. -2 1/4 is equal to -2.25, 1 1/4 is equal to 1.25, -1 3/4 is equal to -1.75, 3/4 is equal to 0.75, and -1 1/4 is equal to -1.25.

To order the values from least to greatest, we can use the following formula:

n1 < n2 < n3 < n4 < n5

Where n1 is -2.25, n2 is -1.75, n3 is -1.25, n4 is 0.75, and n5 is 1.25.

Therefore, the order from least to greatest is -2 1/4, -1 3/4, -1 1/4, 3/4, and 1 1/4.

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The percentage of the discount is b) 75%.

Discount is the state of having a bond's price lower than its face value. The difference between the purchase price and the item's par value is the discount.

Discounts are different types of price reductions or deductions from a product's cost. It is frequently employed in consumer transactions when consumers receive discounts on a range of goods. The % discount rate is provided.

Here the original price of picture = 120.00

Sale price after discount = 29.99 ≅ 30

Now using discount percent formula,

=> Discount percentage = (original price- sale price)/(original price)*100

=> Discount percentage = [tex]\frac{120-30}{120}\times 100=\frac{90}{120}\times100 = 75\%[/tex]

Hence the percentage of the discount is b)75%.

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Answer:

6.7+1.2=7.9

Step-by-step explanation:

6+1=7

0.7+0.2=0.9

so 7+0.9=7.9

so u have to break the numbers as tens and units

The cost Analysis for the two proposed plans for building a low ultra-light bicycle concluded on the recommendation that d that Bici Bicycle Company follow the first plan.

The first plan involves a cost of $125,000 to design and build a prototype bicycle. The combined materials and labor costs for each bike made under the first plan will be $225. This means that for every bike Bici Bicycle Company produces under the first plan, it will incur a total cost of $350 ($125,000 + $225).

On the other hand, the second plan involves a cost of $100,000 to design and build the prototype. The combined materials and labor costs for each bike made under the second plan will be $275. This means that for every bike Bici Bicycle Company produces under the second plan, it will incur a total cost of $375 ($100,000 + $275).

Therefore, while the second plan has a lower initial cost of prototype development, it will ultimately result in a higher cost per unit produced. The first plan, although it has a higher initial prototype development cost, will ultimately result in a lower cost per unit produced.

To determine the break-even point, we can set the total cost of producing a bike under each plan equal to each other and solve for the number of bikes produced.

For the first plan:

Total cost = $125,000 + ($225 x n)

For the second plan:

Total cost = $100,000 + ($275 x n)

Setting these equal to each other, we get:

$125,000 + ($225 x n) = $100,000 + ($275 x n)

$25,000 = $50 x n

n = 500

Therefore, Bici Bicycle Company would need to produce at least 500 bicycles for the cost per unit of the first plan to be less than that of the second plan.

In conclusion, based on my analysis, I would recommend that Bici Bicycle Company follow the first plan, as it will ultimately result in a lower cost per unit produced. However, if the company anticipates producing fewer than 500 bicycles, the second plan may be more cost-effective.

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Using Algebraic expression we concluded, In practice, the angle of the pendulum may not be exactly equal to the calculated values due to factors such as air resistance, friction, and variations in the length of the pendulum.

What is Algebraic expression ?

A combination of variables and constants is an algebraic expression.

We can see that the pendulum's height drops with each swing if we make the assumption that each swing is measured from the same starting point.

Using a sequence is one technique to mathematically express this data. Assuming that n = 1, 2, 3, 4, let's define the sequence "a n" as the pendulum's height on the nth swing. Then, we could type:

a_1 = 80

a_2 = 76

a_3 = 72.2

a_4 = 68.59

Another way to represent this information is by using a formula that describes the height of the pendulum on each swing. One common formula for the height of a simple pendulum is:

[tex]h = L - L\times cos(\theta)[/tex]

where h is the height of the pendulum, L is the length of the pendulum, and θ is the angle that the pendulum swings from its resting position. Assuming that the length of the pendulum remains constant, we can use this formula to find the angle θ for each swing:

For the first swing, h = 80 cm, L = constant, so we can solve for θ:

[tex]80 = L - L\times cos(\theta)\\cos(\theta) = 1 - 80/L\\\theta = arccos(1 - 80/L)[/tex]

For the second swing, we have h = 76 cm, so we can use the same formula with h = 76 and solve for θ:

[tex]76 = L - L\times cos(\theta)\\cos(\theta) = 1 - 76/L\\\theta = arccos(1 - 76/L)[/tex]

Similarly, we can use the formula to find θ for the third and fourth swings:

For the third swing, h = 72.2 cm:

θ = arccos(1 - 72.2/L)

For the fourth swing, h = 68.59 cm:

θ = arccos(1 - 68.59/L)

Therefore, in practice, the angle of the pendulum may not be exactly equal to the calculated values due to factors such as air resistance, friction, and variations in the length of the pendulum.

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complete question -

The correct order from least to greatest is: 3.1818181818, 5.55, 5.571, 5.58

To order these numbers from least to greatest, we first need to convert all of them to decimals so that we can compare them easily.

To convert (111)/(20) to a decimal, we can divide 111 by 20 to get 5.55.

To convert 5(7)/(11) to a decimal, we can first multiply 5 by 7 to get 35, and then divide 35 by 11 to get 3.1818181818.

Now we have the following numbers in decimal form:

5.58, 5.55, 5.571, 3.1818181818

Next, we can compare these numbers to determine their order from least to greatest.

3.1818181818 is the smallest number, followed by 5.55, then 5.571, and finally 5.58.

So, the correct order from least to greatest is:

3.1818181818, 5.55, 5.571, 5.58

In the original form of the numbers, the correct order is:

5(7)/(11), (111)/(20), 5.571, 5.58

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Answer: undefined

Step-by-step explanation:

(0+5)/(0+0)

There could be many other combinations of scores that would result in the same mean, median, and mode.

What is mean ?

Mean can be defined as ratio sum of given observations, total number of observations.

Since the median is 7 and the mode is 8, we can assume that the majority of her scores were in the range of 7-8 points. We can also assume that there were a few games in which she scored lower than 7 points and a few games in which she scored higher than 8 points.

Here's one possible set of scores that would result in a mean of 6, a median of 7, and a mode of 8:

Game 1: 3 points

Game 2: 5 points

Game 3: 6 points

Game 4: 7 points

Game 5: 7 points

Game 6: 8 points

Game 7: 8 points

Game 8: 8 points

Game 9: 9 points

Game 10: 10 points

Game 11: 11 points

Note that this is just one possible set of scores that would fit the given statistics.

Therefore, There could be many other combinations of scores that would result in the same mean, median, and mode.

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Hypothesis: The P-value for this test is 0.026251 which is less than the significance level of 0.05.

Hypothesis is a statement or explanation proposed to explain a phenomenon. It is a logical conjecture, based on observations or experiments, made in order to draw out and test its consequences. In scientific research, a hypothesis is used as a starting point for further investigation and is tested through the scientific method. A hypothesis must be testable and falsifiable, meaning it can be tested and disproved using scientific evidence.

Therefore, we can reject the null hypothesis that there is no difference in the proportions of almonds in the new and old recipes. This means that the proportion of almonds in the new recipe is greater than in the previous recipe.

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Area of sector = (60°/360)*pi*(10 in)^2 = 25.13 in^2.

To solve for the Area of the shaded region in each circle, use the formula Area of circle = pi*r^2. For circle 5, Area of circle = pi*(16in)^2 = 201.06 in^2. For circle 6, Area of circle = pi*(22m)^2 = 1520.53 m^2.

To solve for the Arc Length of the following circles, use the formula Arc Length = (angle/360)*(2*pi*r). For circle 2, Arc Length = (150°/360)*(2*pi*3 m) = 11.78 m. For circle 1, Arc Length = (315°/360)*(2*pi*8 cm) = 21.99 cm.

To solve for the Area of the Sector in each circle, use the formula Area of sector = (angle/360)*pi*r^2. For circle 3, Area of sector = (150°/360)*pi*(19)^2 = 133.04. For circle 4, Area of sector = (60°/360)*pi*(10 in)^2 = 25.13 in^2.

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The extraneous solutions of the equation (1)/(x+2)+(3)/(x+7)=(5)/(x²+9x+14) are x = 0, x = -7, and x = -14

To solve this equation, we must first simplify each side of the equation and then use the rule of equality. We can do this by finding the least common denominator (LCD) of the equation. In this case, the LCD is (x+2)(x+7)(x²+9x+14).

To simplify each side, we must multiply each term by the LCD divided by its original denominator. This leaves us with:

(x+2)(x+7)(x^2+9x+14) / (x+2) + (x+2)(x+7)(x²+9x+14) / (x+7) = (x+2)(x+7)(x²+9x+14) / (x²+9x+14)

We can now apply the rule of equality to solve the equation:

x³+11x²+37x+98 = 0

To find the solution of the equation, we must now factor the left side of the equation. We can do this by first factoring out a common factor of x, which gives us:

x(x²+11x+98) = 0

We can now factor the remaining trinomial, which gives us:

x(x+7)(x+14) = 0

Therefore, the extraneous solutions of the equation are: x = 0, x = -7, and x = -14

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The rate of interest per annum will be 3.6%.

The interest earned on savings that are computed using both the original principal and the interest accrued over time is known as compound interest. A = P(1 + r/n)^nt, where P is the principal balance, r is the interest rate, n is the number of times interest is compounded annually, and t is the number of years, which is the formula for compound interest.

Given, Isabelle takes out a loan that gathers compound interest.

At the start, the loan amount is £5500

after the 1-year loan amount is £5698.00

After the 2-year loan amount is £5903.13

thus,

P = 5500

For t = 1 year A = 5698.00

since interest is annually thus, the value of n is 1

So, from the formula

5698 =  5500( 1 + r)

5500r = 5698 -5500

r = 198/5500

r = 0.036

or

rate of interest, r = 3.6%

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To find the probability that at least one apple is rotten, we can first find the probability that none of the apples are rotten and then subtract that from 1 to get the probability that at least one apple is rotten.

First, let's find the number of rotten apples in the box:
2/5 × 10 = 4
So there are 4 rotten apples and 6 good apples in the box.
Now, let's find the probability that none of the apples are rotten:
6/10 × 5/9 × 4/8 = 120/720 = 1/6
So the probability that none of the apples are rotten is 1/6.

Finally, we can find the probability that at least one apple is rotten by subtracting the probability that none of the apples are rotten from 1:
1 - 1/6 = 5/6
So the probability that at least one apple is rotten is 5/6.

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Work Shown:

[tex]45\text{ km}/\text{hr} = \frac{45\text{ km}}{1\text{ hr}}\\\\=\frac{45\text{ km}}{1\text{ hr}}*\frac{1000\text{ m}}{1\text{ km}}*\frac{1\text{ hr}}{60\text{ min}}*\frac{1\text{ min}}{60\text{ sec}}\\\\=\frac{45*1000*1\text{ m}}{1*1*60*60\text{ sec}}\\\\=\frac{45000\text{ m}}{3600\text{ sec}}\\\\=12.5 \text{ m}/\text{s}\\\\[/tex]

Pay close attention to how the conversion fractions are set up on the second line. This placement is done specifically to have the units of "km", "hr", and "min" cancel out. The goal is to be left with "meters" up top and "seconds" down below.

So, the LU-factorization of the given matrix is:

LU = `[1 0 0 0]` `[4 0 1 8]`

`[0.25 1 0 0]` `[0 1 -4.25 6]`

`[0 4.25 1 0]` `[0 0 1.25 -2]`

`[0 0 -1.25 1]` `[0 0 0 1]`

The LU-factorization of a matrix is a method of decomposing a matrix into a lower triangular matrix (L) and an upper triangular matrix (U). The L matrix must have a unit diagonal, meaning that all the entries on the main diagonal are equal to 1.

To find the LU-factorization of the given matrix, we can use the following steps:

1. Begin with the given matrix:

`[4 0 1 8]`

`[1 1 -4 10]`

2. Use Gaussian elimination to transform the matrix into an upper triangular matrix (U):

`[4 0 1 8]` → `[1 1 -4 10]` → `[0 1 -4.25 6]` → U = `[0 0 1.25 -2]`

3. Now, we can use the entries from the Gaussian elimination to create the lower triangular matrix (L):

L = `[1 0 0 0]`

`[0.25 1 0 0]`

`[0 4.25 1 0]`

`[0 0 -1.25 1]`

4. Finally, we can write the LU-factorization as:

LU = `[1 0 0 0]` `[4 0 1 8]`

`[0.25 1 0 0]` `[0 1 -4.25 6]`

`[0 4.25 1 0]` `[0 0 1.25 -2]`

`[0 0 -1.25 1]` `[0 0 0 1]`

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The vertex of a parabola (1.75, -0.25). The vertical intercept of a parabola is (0, 11). The two intercepts of the parabola ((7 + √5)/4, 0) and ((7 - √5)/4, 0).

A) The vertex of a parabola is given by the formula (h, k), where h = -b/2a and k = f(h). In this case, a = 4, b = -14, and c = 11. So, h = -(-14)/2(4) = 14/8 = 1.75. Plugging this value back into the equation, we get k = f(1.75) = 4(1.75)^2 - 14(1.75) + 11 = -0.25. Therefore, the vertex of the parabola is (1.75, -0.25).

B) The vertical intercept is the point where the parabola crosses the y-axis, which occurs when x = 0. Plugging this value into the equation, we get f(0) = 4(0)^2 - 14(0) + 11 = 11. Therefore, the vertical intercept is the point (0, 11).

C) The two intercepts of the parabola are the points where it crosses the x-axis, which occur when f(x) = 0. We can use the quadratic formula to find these values:

x = (-b ± √(b^2 - 4ac))/2a

x = (-(-14) ± √((-14)^2 - 4(4)(11)))/2(4)

x = (14 ± √(196 - 176))/8

x = (14 ± √20)/8

x = (14 ± 2√5)/8

x = (7 ± √5)/4

Therefore, the two intercepts are the points ((7 + √5)/4, 0) and ((7 - √5)/4, 0).

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1. The arithmetic mean of the sample is: d. 3.50

To calculate the arithmetic mean, you need to add up all of the numbers in the sample and divide it by the total number of numbers in the sample. Using the given data, the total number is 20 (3+6+8+2=20). The total of the numbers in the sample is 31 (2+3+4+5=14). Therefore, the arithmetic mean of the sample is 31/20 = 3.50.

2. The average age of the children (x) is: c. 7.81

To calculate the average age of the children, you need to find the sum of the products of age and frequency (xf) and divide it by the total frequency (42). The sum of the products of age and frequency is 2628 (42+84+80+72+50=2628). Therefore, the average age of the children is 2628/42 = 7.81.

3. Standard deviation: a. 0.79

To calculate the standard deviation, you need to find the sum of the squares of the frequency multiplied by the square of the deviation (x²f) and divide it by the total frequency (42). The sum of the squares of the frequency multiplied by the square of the deviation is 6790 (252+588+640+648+500=6790). Therefore, the standard deviation is 6790/42 = 0.79.

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