An object with a weight of 100 N is suspended by two lengths of rope from the


ceiling. The angles that both lengths make with the ceiling are the same. The


tension in each length is 50 N. Determine the angle that the lengths of ropes make


with the ceiling.

The exact value of sin 45° is [tex]\dfrac{\sqrt{2} }{2}[/tex]

Since we have given that

[tex]\text{sin} \ 45^\circ[/tex]

We need to find the exact value of sin 45°.

From the trigonometric table,

[tex]\text{sin} \ 45^\circ=\dfrac{1}{\sqrt{2}}[/tex]

We need to write it as a simplified fraction,

So, for this, we will rationalize the denominator:

[tex]\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{2} }{\sqrt{2}}[/tex]

[tex]=\dfrac{\sqrt{2} }{2}[/tex]

Hence, the exact value of sin 45° is [tex]\dfrac{\sqrt{2} }{2}[/tex]

Answer: 1 divided by the square root of 2

Step-by-step explanation:

Let's set up an example, if the angle is forty five degrees, and the opposite length is 1, we can solve this as sin to get to the hypotenuse,

1. sin(45) = 1/hyp

2. sin(45) times hyp = 1

3. hyp = sin(45)/1

If we take any answer and put it over the hypotenuse as sin, we can see that it is going to end up as 1/√2, or 0.707

I did 1 because you are just asking for sin(45).

Sin(z) = 4/5, Cos(z) = 3/5, tan(z) = 4/3

We know that

sin(z) = perpendicular/hypotenuse

cos(z) = base/hypotenuse

tan(z) = perpendicular/base

Now putting we get,

Sin(z) = 4/5

Cos(z) = 3/5

tan(z) = 4/3

A) The total commission she earned is $475

B) Total sales for commission of $1375 is $20000

A) Total Commission = Commission 1+ Commission 2

Where:

Commission 1 = 5% of first $5000

Commission 2 = 7.5% of the amount left after $5000 is subtracted

thus

Commission 1 = $5000 * 0.05 = $250

Commission 2= $3000 * 0.075 = $225

Commission total = $250 + $225 = $475

The total commission she earned is $475

B) Total sales = Sales with 5% commission + Sales with 7.5% commission

Sales with 5% commission = $5000

Commission At 7.5% = Total commission -Commission with 5% =    $1375 - $250

Sales * 0.075 =  $1125

Sales with 7.5% commission = $15000

Total sales = $5000+$15000

Total sales = $20000

Total sales for commission of $1375 is $20000

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Complete question is:

A salesperson earns commission on the sales that she makes each month. The salesperson earns a 5% commission on the first $5,000 she has in sales.

The salesperson earns a 7.5% commission on the amount on her sales that are greater than $5,000.

Part A:

This month the salesperson had $8,000 in sales. What amount of commission, in dollars, did she earn?

Part B:

The salesperson earned $1,375 in commission, last month. How much money, in dollars, did she have in sales last month?

when t=1, y is approximately equal to 5.

To solve for y when t=1 in the equation y=1.5 In(et.t+5), we first need to plug in t=1:

y=1.5 In(e(1)(1)+5)

We simplify the exponent e(1)(1) to just e:

y=1.5 In(e+5)

Using the properties of natural logarithms, we can simplify this further:

y=1.5(1+ln(5+e))

We can use a calculator to evaluate ln(5+e) to be approximately 2.063, so we can plug that in and simplify:

y=1.5(1+2.063)

y=1.5(3.063)

y=4.5945

Rounding this answer to the nearest whole number, we get:

y=5

Therefore, when t=1, y is approximately equal to 5.

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Hunter needs 3 ounces of seeds and 7 ounces of dried fruit, which will cost him $22 in total. The system of equations is 1.5x + 2.5y = 22 and x + y = 10. The solution is (x,y) = (3,7).

The total amount of snack mix required is 10 ounces. So, the sum of the amount of seeds and dried fruit should be 10.

x + y = 10 ---(Equation 1)

The cost of seeds is $1.50 per ounce and the cost of dried fruit is $2.50 per ounce. The total cost of snack mix should be $22.

1.50x + 2.50y = 22 ---(Equation 2)

To solve the system, we can use substitution method. Solving Equation 1 for y, we get

y = 10 - x

Substituting this value of y in Equation 2, we get

1.5x + 2.5(10 - x) = 22

Simplifying and solving for x, we get

1.5x + 25 - 2.5x = 22

-x = -3

x = 3

So, Hunter needs 3 ounces of seeds and 7 ounces of dried fruit to make 10 ounces of snack mix with a total cost of $22.

The ordered pair is (3, 7).

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We know that the diameter of the wheel is 1215 inches

Since CD is a perpendicular bisector of AB, it means that CD passes through the center of the circle. Let O be the center of the circle. Then OD is the radius of the circle.

Since chord CE passes through the center O, it is a diameter of the circle. Therefore, CE = 2OD.

Let's use the intersecting chords theorem to find OD.

According to the intersecting chords theorem,

AC * CB = EC * CD

We know that AC = CB (since they are radii of the same circle) and CD = 4 inches. We also know that AB = 12 inches. Let's call the length of segment AE x. Then the length of segment EB is 12 - x.

So we have:

x * (12 - x) = EC * 4

Simplifying:

12x - x^2 = 4EC

Rearranging:

EC = 3x - x^2/4

Now let's use the intersecting chords theorem again, but this time for chords AB and CD:

AC * CB = AD * DB

We know that AC = CB and AB = 12 inches. Let's call the length of segment AD y. Then the length of segment DB is 12 - y.

So we have:

x^2 = y * (12 - y)

Simplifying:

y^2 - 12y + x^2 = 0

Using the quadratic formula:

y = (12 ± sqrt(144 - 4x^2))/2

We can discard the negative solution (since y is the length of a segment, it cannot be negative), so:

y = 6 + sqrt(36 - x^2)

Now let's use the fact that CD is a perpendicular bisector of AB to find x.

Since CD is a perpendicular bisector of AB, it divides AB into two segments of equal length. Therefore,

AD = DB = 6

Using the Pythagorean theorem in triangle ACD:

AC^2 + CD^2 = AD^2

Substituting the values we know:

x^2 + 4^2 = 6^2

Solving for x:

x = sqrt(20)

Now we can find EC:

EC = 3x - x^2/4

Substituting x:

EC = 3sqrt(20) - 5

Finally, we can find OD:

AC * CB = EC * CD

Substituting the values we know:

(2OD)^2 = (3sqrt(20) - 5) * 4

Simplifying:

OD^2 = 12sqrt(20) - 20

OD = sqrt(12sqrt(20) - 20)

We are asked to find the diameter of the circle, which is twice the radius:

Diameter = 2OD = 2sqrt(12sqrt(20) - 20)

This is approximately equal to 1215 inches.

So the answer is:

The diameter of the wheel is 1215 inches.

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The top industries are agriculture, mining, tourism, and manufacturing.

The quadratic equations are as given.A manufacturing company in my fiefdom produces and sells ceramic pots.

The company has fixed costs of$ 10,000 per month and variable costs of$ 5 per pot. The company's profit is given by the quadratic function R( x) = -0.2 x2 50x, where x is the number of pots produced and vended in a month.

What's the maximum profit that the company can induce in a month: To break this problem, we can use the formula for chancing the maximum value of a quadratic function, which is given by x = - b/ 2a. In this case, the measure of the x2 term is-0.2, and the measure of the x term is 50. Plugging these values into the formula, we get x = -50/( 2 *(-0.2)) = 125 Hence we obtain the quadratic equation.

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Solving the equation, Boris needs to read 34 novels in 17 months.

Given that Boris needs to read 2 novels each month.

To relate the number of novels Boris needs to read (n) to the number of months he has to read them (m), we can use the equation:
n = 2m

This equation states that the number of novels (n) is equal to two times the number of months (m) since Boris needs to read 2 novels each month.

Now, to find the number of novels Boris needs to read in 17 months, we can substitute m = 17 into the equation:
n = 2m
n = 2(17)
n = 34

Therefore, Boris needs to read 34 novels in 17 months to meet his goal of reading 2 novels each month.

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To solve the inequality -1/4a > 3a, we need to first multiply both sides by -4 to get rid of the fraction:
-1a > 12a
Next, we can subtract 12a from both sides to get:
-13a > 0
Dividing both sides by -13 gives us:
a < 0

To solve the inequality b – 12 > -3, we can add 12 to both sides:
b > 9

Now we need to find values of a and b that satisfy both inequalities. Since a < 0, we can try any negative value of a. Let's try a = -1:

-1/4(-1) > 3(-1)
1/4 > -3

This inequality is true, so we can move on to the next inequality. Let's plug in a = -1 and see if it satisfies b > 9:

b – 12 > -3
b > 9

Since -1 satisfies both inequalities, the values that make both inequalities true are: a = -1 and any value of b greater than 9.

Answer:12.4

Step-by-step explanation:

18^2-13^2=155
Square root of 155 to the nearest tenth is 12.4

The biologist predicts that the population would be fewer than 50 in approximately 30 years from the initial count.

The biologist likely used the exponential decay function to model the coyote population over time. This function takes the form:

P(t) = P0 * (1 - r)^t

Where:
P(t) is the population at time t,
P0 is the initial population (100 coyotes),
r is the rate of decrease (0.045 or 4.5%),
t is the time in years.

To predict when the population would be fewer than 50, the biologist would solve the equation:

50 = 100 * (1 - 0.045)^t

t = 30

This equation can be used to find the value of t, which represents the number of years it takes for the population to decrease to fewer than 50 coyotes in the tri-county area, which is 30 years.

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Therefore, Lance's average velocity was 15.43 miles per hour.

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, left-hand side (LHS) and right-hand side (RHS), connected by an equal sign (=). The LHS and RHS can contain numbers, variables, operators, and functions, and the equal sign indicates that the value of the expression on the LHS is equal to the value of the expression on the RHS.

Here,

We can compute Lance's average velocity by dividing the total distance he cycled by the time it took him, and then converting the units to miles per hour.

Total distance: 22 miles

Time: 70 minutes = 70/60 hours

= 7/6 hours

Average velocity = Total distance / Time

= 22 / (7/6)

= 15.43 miles per hour (rounded to two decimal places)

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Answer:

--------------------------

Since CD is diameter, therefore the angle CAD opposite to it is a right angle.

We are given the lengths of two legs, AD = 16 and AC = 30.

Use Pythagorean theorem to find the length of the hypotenuse CD:

The linear approximation of f(x) = ln(x) at x = 1 is L(x) = x - 1. Using this approximation, we can estimate ln(1.16) to be approximately 0.16.

The formula for the linear approximation of a function f(x) at a point x = a is given by L(x) = f(a) + f'(a)(x - a), where f'(a) is the derivative of f(x) evaluated at x = a.

In this case, f(x) = ln(x), so f'(x) = 1/x by the derivative of natural logarithm.

We are asked to find the linear approximation of f(x) = ln(x) at x = 1, so a = 1 in the formula.

Plugging in the values, we get L(x) = ln(1) + 1( x - 1) = x - 1.

Now, we can use this linear approximation L(x) = x - 1 to estimate ln(1.16) by plugging in x = 1.16, as given in the question.

L(1.16) = 1.16 - 1 = 0.16, which is our estimated value for ln(1.16) using the linear approximation.

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The measure of angle D is 25°

Exterior angle theorem states that the measure of an exterior angle of a triangle is greater than either of the measures of the opposite interior angles.

Angle D and angle C are the two opposite angles.

Therefore;

40+9x-2 = 20x +5

38+9x = 20x +5

38-5 = 20-9x

11x = 33

divide both sides by 11

x = 33/11

x = 3

Therefore since angle D = 9x-2

substitute 3 for x

D = 9(3) - 2

D = 27 -2

D = 25°

Therefore the measure of angle D is 25°

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The radius of an eyebrow window with a width of 62.8 inches and a height of 18.5 inches is approximately 36.7 inches.

To find the radius of an eyebrow window, we first need to understand its shape. An eyebrow window is a type of arched window that has a curved shape similar to that of an eyebrow. The shape of an eyebrow window is created by a combination of a circular arc and a straight line.

To find the radius of an eyebrow window with a width of 62.8 inches and a height of 18.5 inches, we need to use some geometry formulas. The height of the eyebrow window represents the height of the circular arc, and the width represents the diameter of the circle.

The formula for the radius of a circle is r = d/2, where r is the radius and d is the diameter. To find the diameter, we divide the width by pi (3.14). So, the diameter is 62.8/3.14 = 20 inches.

The height of the circular arc is half of the width, which is 18.5/2 = 9.25 inches. To find the radius, we use the formula for the height of a circular arc, h = r(1-cos(a/2)), where h is the height, r is the radius, and a is the angle of the arc.

The angle of the arc can be found using trigonometry. The sine of half the angle is equal to the height divided by the radius. So, sin(a/2) = h/r. Solving for a, we get a = 2arcsin(h/r).

Plugging in the values, we get a = 2arcsin(9.25/r). To find the radius, we solve for r using a calculator or algebra. The radius is approximately 36.7 inches.

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If angle of arc EA is 112 degrees then value of arc IV is 36 degrees by  outside angles theorem

Given that Arc EA measure is One hundred twelve degrees

By Outside Angles Theorem states that the measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs

(112-x)/2=38

112-x=38×2

112-x=76

112-76=x

36 degrees = angle IV or x

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ANSWER:

(x+7)^2=16

Step-by-step explanation:

x^2-14x+49=16

x^2-14x+49-16=0

x^2-14x+33=0

subtract -33 on both sides

x^2-14x+33-33=-33

x^2-14x=-33

Add 49 on both sides

x^2-14x+49=-33+49

x^2-14x+49=16

x^2-7x-7x+49=16

x(x-7)-7(x-7)=16

(x-7)(x-7)=16

(x-7)^2=16

5 or more Xs

To complete the activities based on the given data:

Compare the sizes: By looking at the shooting star lengths, we can determine the number of Xs that would appear on the line plot. The shooting star lengths "10" and "8" appear more than 5 times, so they would be placed in the "More than 5 Xs" box. The shooting star lengths "6" and "4" appear fewer than 5 times, so they would be placed in the "Fewer than 5 Xs" box.

Complete the line plot: Using the given set of data, we can create a line plot to represent the lengths of shooting stars. We mark each measurement on the number line and place an X above the corresponding value.

The line plot would have an X above the number 10, 8, 6, and 4, each representing the occurrence of shooting stars with those lengths.

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The equations to find the amount of red fabric Moses has are X = x + 2 and X + 222 = 1/3x + 223. These equations are obtained by using x as the amount of blue fabric and setting up equations based on the given information. So, the correct answer is A) and E).

There are two equations that can be used to find the amount of red fabric Moses has

X = x + 2, This equation represents the fact that Moses bought 2 more yards of red fabric than he originally had of blue fabric. So, the amount of red fabric (X) is equal to the amount of blue fabric (x) plus 2.

X + 222 = 1/3x + 223, This equation represents the fact that after buying 2 more yards of red fabric, Moses has equal amounts of red and blue fabric.

So, the amount of red fabric (X) plus 222 (the additional 2 yards he bought) is equal to one-third of the amount of blue fabric (1/3x) plus 223 (the original 2 yards of red fabric he had).

Therefore, the equations that could be used to find the amount of red fabric Moses has are A) and E).

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--The given question is incomplete, the complete question is given

" 6 Moses makes a school spirit flag. He has as many yards of red fabric as blue fabric. He buys 2 yards more red fabric. Now he has equal amounts of red and blue fabric. Use x to represent the amount of blue fabric. Which equations could you use to find the amount of red fabric Moses has? Select all that apply.

A X = x + 2

B x = 2

C 1/x + 273

D x= 1/3x - 223

E x + 2 2 2 = 1/3 x + 2 2 3

F x- 27 28 = 1/3 x

G = x+ {{22}*2}^3 "--

Answer:

38 degrees

Step-by-step explanation:

Vertical angles are congruent(equal measures), so mLA = mLB

STEP 1:

Let's use some simple substitution.

mLA = mLB

mLA = x+21, mLB = 4x-30

You plug these two in and get:

x+21 = 4x-30

This is your equation.

STEP 2:

Let's solve our equation!

x+21 = 4x-30

(add 30 to both sides)

x+51 = 4x

(subtract x from both sides)

51 = 3x

(switch order for comprehension)

3x = 51

(divide both sides by 3)

x = 17

Ta-da! You get the measure of x = 17 degrees.

STEP 3:

Let's plug in our value of x to get the value of LB.

mLB = 4x - 30

mLB = 4(17) - 30

mLB = 68 - 30

mLB = 38

This is your answer.

The volume of this cone is approximately 127.2 cubic inches

Hi! To calculate the volume of the Styrofoam cone used by Dylan to make a floral arrangement, we can use the formula for the volume of a cone: V = (1/3)πr²h. The cone had a radius of 4.5 inches and a height of 6 inches.

Substituting these values into the formula, we have:
V = (1/3)π(4.5)²(6)

V ≈ 127.2 cubic inches (rounded to the nearest tenth).

So, the volume of this cone is approximately 127.2 cubic inches.

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The following are the updates for the given ordinary differential equation using Euler's method, the implicit Euler method, and the midpoint method.

To compute the update Ax for the given ordinary differential equation using Euler's method, we first need to discretize the time domain. Let t0 be the initial time and tn = t0 + nh be the time after n steps of size h. Then, using Euler's method, we have:

xn+1 = xn + hf(xn, tn)

where f(xn, tn) = 1 - ƛxn³/6. Therefore,

Ax = xn+1 - xn = h(1 - ƛxn³/6)

Using the implicit Euler method, we have:

xn+1 = xn + hf(xn+1, tn+1)

where f(xn+1, tn+1) = 1 - ƛxn+1³/6. Solving for xn+1, we get:

xn+1 = (xn + h)/[1 + ƛh/6(xn+1)²]

which is a nonlinear equation that needs to be solved iteratively at each step. Therefore, the update Ax becomes:

Ax = xn+1 - xn

Using the midpoint method, we have:

xn+1 = xn + hf(xn+½h, tn+½h)

where f(xn+½h, tn+½h) = 1 - ƛ(xn+½h)³/6. Therefore,

xn+1 = xn + h(1 - ƛxn³/6 + 3ƛx²n h/4)

and the update Ax becomes:

Ax = xn+1 - xn = h(1 - ƛxn³/6 + 3ƛx²n h/4)

These are the updates for the given ordinary differential equation using Euler's method, the implicit Euler method, and the midpoint method.

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The probability that a senior drives to school is 0.3 or 30%.

To find the indicated probability using the two-way table, we need to locate the row for "senior" and the column for "drive to school" and then find the corresponding cell.

Let's assume that the two-way table shows the number of students who either drive or take the bus to school based on their grade level. We are interested in finding the probability that a student drives to school given that they are a senior.

So, we locate the row for "senior" and the column for "drive to school". Let's say that the cell in the intersection of these two is labeled "30". This means that there are 30 seniors who drive to school.

Next, we need to find the total number of seniors in the sample. Let's say that the total number of seniors in the sample is 100.

To find the probability that a senior drives to school, we divide the number of seniors who drive to school by the total number of seniors in the sample:

P(drive to school | senior) = 30/100 = 0.3 or 30%

Therefore, the probability that a senior drives to school is 0.3 or 30%.

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The estimated radius of the object is about 1.42 mm.

The given information is that the circumference( C) of the object is8.9 mm.

We know that the formula for the circumference of a circle is given by  

C =  2πr  

where r is the compass of the circle.  

To estimate the compass, we can rearrange the formula as

  r =  C/ 2π  

Substituting the given value of C, we get

  r = 8.9/ 2π

we can  estimate this expression to get  

r ≈1.42 mm( rounded to two decimal places)

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The constant of proportionality represents the rate at which Issac earns money while babysitting and can be found by dividing the amount of money he earns by the time spent babysitting.

Let's say that Issac earns $10 per hour of babysitting. Then, the constant of proportionality would be:

$10 per hour = $10/1 hour = 10

Therefore, the constant of proportionality is 10, which means that Issac earns $10 for every hour of babysitting. This relationship is an example of proportionality because the amount of money earned is directly proportional to the time spent babysitting. As Issac spends more time babysitting, he will earn more money in a proportional relationship.

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After the discount and the tax, the amount that Roger pays is $45.41

We know that Roger purchased a pair of pants for 34.50 and a new a new for 12.00 he had a 10% discount on his total purchased and paid 8.5% sales tax, then the total cost before the discount and tax is:

C = 12.00 + 34.50 = 46.50

Now we apply the discount and the tax (as factors in a product) to get:

C' = 46.50*(1 - 0.1)*(1 + 0.085) = 45.41

That is the amouint that Roger pays for the two items.

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Answer:

Step-by-step explanation:

Congruent triangles are triangles that are the same shape and same size.

So if they look the same and have the same dimension like area and perimeter then they are congruent.

Rylee did approximately 58.6 sit-ups.

We are given that the mean number of sit-ups is 46 and the standard deviation is 7. We are also given that Rylee's Z-score was 1.8, we can use the formula for Z-score to find how many sit-ups she did.

The formula for Z-score is [tex]Z = \frac{X-\mu}{\sigma}[/tex]

Z = Z-score

μ = mean

σ = standard deviation

X = ?

Substituting these values into the formula

1.8 = (X - 46)/7

1.8 × 7 = X - 46

X - 46 = 12.6

X = 12.6 + 46

X = 58.6

Therefore, Rylee did approximately 58.6 sit-ups.

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The variance of the given data set is 18603.39.

To find the variance of the given data set {4, 44, 404, 244, 4, 74, 84, 64}, follow these steps:

Step 1: First, we need to find the mean of the data set:

Mean = (4 + 44 + 404 + 244 + 4 + 74 + 84 + 64) / 8 = 120.5

Step 2: Next, we calculate the deviation of each data point from the mean:

(4 - 120.5) = -116.5

(44 - 120.5) = -76.5

(404 - 120.5) = 283.5

(244 - 120.5) = 123.5

(4 - 120.5) = -116.5

(74 - 120.5) = -46.5

(84 - 120.5) = -36.5

(64 - 120.5) = -56.5

Step 3: Now, we square each deviation:

[tex](-116.5)^2 = 13556.25\\(-76.5)^2 = 5852.25\\(283.5)^2 = 80322.25\\(123.5)^2 = 15252.25\\(-116.5)^2 = 13556.25\\(-46.5)^2 = 2162.25 \\(-36.5)^2 = 1332.25\\(-56.5)^2 = 3192.25[/tex](-116.5)^2 = 13556.25

Step 4: We add up all the squared deviations:

13556.25 + 5852.25 + 80322.25 + 15252.25 + 13556.25 + 2162.25 + 1332.25 + 3192.25 = 130223.75

Step 5: We divide the sum of the squared deviations by the number of data points minus 1 to get the variance:

Variance = 130223.75 / 7 = 18603.39 (rounded to two decimal places)

Therefore, the variance of the data set is 18603.39.

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