A survey asked 3, 800 students how they are likely
to spend their money. The original results of the
survey are shown.
Category
Food
Drinks
Clothing
Shoes
Accessories
Video
Games
Electronics
Personal
Care
Other
Number of
Students
967
895
816
455
59
143
237
105
123
Complete the statement.
Based on the results of the survey, a circle graph
would have [DROP DOWN 1] sectors that are labeled
as 10% or less and [DROP DOWN 21 sectors that are are labeled as
or less and [DROP DOWN 2] sectors that are labeled as
or more.

The exponential function in the form y = [tex]ab^x[/tex] that goes through the points (0, 13) and (5, 416) is y = 13 * [tex]2^x[/tex].

To write an exponential function in the form y = [tex]ab^x[/tex] that goes through the points (0, 13) and (5, 416), follow these steps:

1. Use the point (0, 13) to determine the value of a. Since the x-coordinate is 0, we have y = a * b⁰, which simplifies to y = a. Therefore, a = 13.

2. Use the point (5, 416) to determine the value of b. Plug in the values for x and y, as well as the value of a we just found: 416 = 13 * b⁵.

3. Solve for b: b⁵ = 416 / 13 = 32. Taking the fifth root of both sides, we get b = 2.

So, the exponential function in the form y = [tex]ab^x[/tex] that goes through the points (0, 13) and (5, 416) is y = 13 * [tex]2^x[/tex].

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There is 20% decrease in scholars number who went to the morning homework help.

To get the scholar's percent decrease, we need to calculate the difference between them on Thursday and Friday and theb divide that by the number of scholars on Thursday.

Data:

Number of scholars on Thursday = 30

Number of scholars on Friday = 24

Difference = 30 - 24 = 6

The percent decrease = (6/30) x 100%

The percent decrease = 0.2 * 100%

The percent decrease = 20%

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Horseshoe Falls has a flow rate of approximately 2.52 x 10⁷ cubic meters of water per hour, but the given options do not accurately represent this value.

Horseshoe Falls is one of the three waterfalls that make up Niagara Falls near Buffalo, New York. It has an average flow of 7 x 10³ cubic meters per second. To determine the amount of water that goes over Horseshoe Falls hourly, we need to convert the flow rate from cubic meters per second to cubic meters per hour.

There are 3600 seconds in an hour. To convert the flow rate to cubic meters per hour, we simply multiply the flow rate by the number of seconds in an hour:

(7 x 10³ cubic meters/second) x (3600 seconds/hour) = 25.2 x 10⁶ cubic meters/hour

The closest value to this among the given options is:

b. 4.20 x 10⁵ cubic meters per hour

However, there seems to be a typo or error in the provided options. The correct answer should be:

25.2 x 10⁶ cubic meters per hour (approximately 2.52 x 10⁷ cubic meters per hour)

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The critical numbers of f(x) = -2x^3 + 39x^2 - 240x + 2 are A = 5 and B = 8.

To find the critical numbers of a function, we need to find the values of x where the derivative of the function is zero or undefined.

For the function f(x) = 5x^2 + 7x - 10, the derivative is:

f'(x) = 10x + 7

To find the critical numbers, we need to set f'(x) = 0 and solve for x:

10x + 7 = 0

10x = -7

x = -7/10

So the critical number of f(x) = 5x^2 + 7x - 10 is x = -7/10.

For the function f(x) = -2x^3 + 39x^2 - 240x + 2, the derivative is:

f'(x) = -6x^2 + 78x - 240

To find the critical numbers, we need to set f'(x) = 0 and solve for x:

-6x^2 + 78x - 240 = 0

We can simplify this equation by dividing both sides by -6:

x^2 - 13x + 40 = 0

Now we can factor the quadratic:

(x - 5)(x - 8) = 0

So the solutions are x = 5 and x = 8.

To determine which critical point is A and which is B, we need to check the sign of the second derivative of f(x) at each critical point.

The second derivative of f(x) is:

f''(x) = -12x + 78

Plugging in x = 5, we get:

f''(5) = -12(5) + 78 = 18

Since f''(5) is positive, we know that f(x) has a local minimum at x = 5. Therefore, x = 5 is the critical point A.

Plugging in x = 8, we get:

f''(8) = -12(8) + 78 = -6

Since f''(8) is negative, we know that f(x) has a local maximum at x = 8. Therefore, x = 8 is the critical point B.

Therefore, the critical numbers of f(x) = -2x^3 + 39x^2 - 240x + 2 are A = 5 and B = 8.

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a) ∫ (1/x + 3/x^2 - 4/x^3) dx
To solve this indefinite integral, we need to use the power rule and the fact that the derivative of ln(x) is 1/x.

∫ (1/x + 3/x^2 - 4/x^3) dx = ln|x| - 3/x + 2/x^2 + C

b) ∫ (x^2 + 2x - 5) / √x dx
To solve this indefinite integral, we can simplify the integrand by multiplying the numerator and denominator by √x. Then, we can use the power rule and u-substitution.

∫ (x^2 + 2x - 5) / √x dx = ∫ (x^(5/2) + 2x^(3/2) - 5x^(1/2)) dx

= (2/7)x^(7/2) + (4/5)x^(5/2) - (10/3)x^(3/2) + C

c) ∫ x e^x dx
To solve this indefinite integral, we need to use integration by parts.

Let u = x and dv/dx = e^x. Then, we can find v by integrating dv/dx.

v = e^x

Using integration by parts, we get:

∫ x e^x dx = xe^x - ∫ e^x dx

= xe^x - e^x + C

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The dimensions of the pasture that require the least amount of fence are approximately 600 meters by 300 meters.

To minimize the amount of fence needed, we want to maximize the length of the side next to the canal. Let's call this side x and the other two sides y.

We know that the area of the rectangle must be 180,000 m², so we have x*y = 180,000. We want to minimize the amount of fence, which is the perimeter of the rectangle: P = x + 2y

To solve for the dimensions that require the least amount of fence, we need to eliminate one variable. We can do this by using the area equation to solve for one variable in terms of the other:
y = 180,000/x

Substituting this into the perimeter equation, we have:
[tex]P = x + 2(180,000/x)[/tex]

To find the minimum value of P, we take the derivative with respect to x and set it equal to zero:
[tex]P' = 1 - 360,000/x^2 = 0x = sqrt(360,000) ≈ 600[/tex]

Substituting this back into the area equation, we find:
[tex]y = 180,000/x ≈ 180,000/600 ≈ 300[/tex]

So, the dimensions of the pasture which require the least amount of fence are approximately 600 meters by 300 meters.

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If the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches

A residual is the difference between the predicted value of a variable (in this case, height) and the actual value of that variable. Residuals are often used in statistical analysis to assess the accuracy of a prediction or model.

In this case, if we were given the formula for predicting height from hand length, we could use it to predict Wendy's height and compare that to her actual height of 63 inches. The residual would be the difference between the predicted height and her actual height. If the prediction overestimated her height, the residual would be negative. If it underestimated her height, the residual would be positive.

For example, if the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches (predicted height minus actual height). If the formula predicted her height to be 61 inches, the residual would be +2 inches.

Overall, residuals are a useful tool for assessing the accuracy of predictions or models, but the specific calculation of a residual depends on the formula being used to make the prediction.

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A. 52 = 8n + 4, B. 4(2n + 1) = 52 and C. 4 = 52 – 8n are the equivalent equations.

Simplifying the equations, we have

A. 52 = 8n + 4

8n = 48

n = 6

To see this, first simplify equation B:

4(2n + 1) = 52

8n + 4 = 52

8n = 48

n = 6

C. 4 = 52 – 8n

8n = 48

n = 6

Then simplify equation D:

4n = 48

n = 12

As you can see, equations A, B and C are equivalent and both simplify to n = 6.

Therefore, the correct answers are A, B and C.

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Answer:

The mean would decrease in value to about 17.1 inches.

Step-by-step explanation:

If we add a smaller data to the mean of a numerical data set, the mean will decrease.

Hope this helps.

Answer:

h = 15 cm

Step-by-step explanation:

Given:

C (base's circumstance) = 48π cm

V (volume) = 8640π cm^3

Find: h (height) - ?

[tex]c = 2\pi \times r[/tex]

[tex]2\pi \times r = 48π[/tex]

[tex]r = 24[/tex]

We found the radius

[tex]v = \frac{1}{3} \times \pi {r}^{2} \times h[/tex]

[tex] \frac{1}{3} \times \pi \times {24}^{2} \times h = 8640π[/tex]

Multiply the whole equation by 3 to eliminate the fraction:

[tex]1728\pi \times h = 25920\pi[/tex]

[tex]h = 15[/tex]

The condition that will result in more than one triangle is C. Student III.

The conditions listed by the third student will result in more than one triangle because we are given all three angles. As a rule in math, some conditions will determine if there is more than one triangle. One of them is this:

Rule 1:

If all three angles of the triangle are given and they all add up to exactly 180°, it is possible to get more than one triangle.  In the third option, we are given angles 62°, 36°, and 82°, so different triangles can be constructed. Also, they all add up to give 180°. The condition is satisfied.

Rule 2:

Also, as a rule, if we have two angles that do not add up to 180° and one side, then only one unique triangle can be obtained. This is the case for student A who is given angles A and B and a side length of 5cm. (ASA)

Rule 3:

Student 2 will also produce a unique triangle because there are three sides that meet the triangle inequality theorem. (SSS)

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The area of S is:

[tex]A(S) = 16\pi \sqrt{(500/11)}= 128.8[/tex]

The surface S can be described in terms of cylindrical coordinates by setting:

x = r cos(θ)

y = r sin(θ)

z = z

Using these coordinates, we can rewrite the equation for C as:

r² = 22/100(x² + y²) = 22/100r²

Simplifying this equation, we get:

[tex]r = \sqrt{(500/11)}[/tex]

Thus, the surface S is the portion of the cylinder of radius [tex]\sqrt{(500/11)}[/tex] between z = 1 and z = 5.

To calculate the area of S, we can use the formula:

A(S) = ∫∫∂S ||n|| [tex]dA[/tex]

where ||n|| is the magnitude of the normal vector to the surface, and [tex]dA[/tex] is the area element on the surface.

For the cylinder, the normal vector is simply the radial unit vector pointing outward from the origin:

n = (cos(θ), sin(θ), 0)

The magnitude of the normal vector is ||n|| = 1, so we can simplify the formula for the area to:

A(S) = ∫∫∂S [tex]dA[/tex]

To evaluate this integral, we need to parameterize the surface S. We can use the cylindrical coordinates we defined earlier:

x = r cos(θ)

y = r sin(θ)

z = z

with 0 ≤ θ ≤ 2π and 1 ≤ z ≤ 5.

The area element in cylindrical coordinates is given by:

[tex]dA = r \ dz\ d\theta[/tex]

Substituting in our parameterization of S, we get:

A(S) = ∫∫∂S r [tex]dz[/tex] dθ

[tex]= \int\limits^{2\pi }_0 \int\limits^5_1 {\sqrt{(500/11)} dz d\theta}\\= \sqrt{(500/11)} \int\limits^{2\pi }_0 {(5 - 1) d\theta}\\= 16\pi \sqrt{(500/11)[/tex]

Therefore, the area of S is:

[tex]A(S) = 16\pi \sqrt{(500/11)}= 128.8[/tex]

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To draw a line segment with endpoint at 1.6 and length 1.2, draw a number line and mark 1.6. Measure 1.2 units to left of 1.6 and mark the starting point. Connect the starting and endpoint.

To draw a line segment with an endpoint at 1.6 and a length of 1.2, we can follow these steps

Draw a number line and mark the point 1.6.

From the point 1.6, measure a distance of 1.2 units in the direction of the negative numbers.

Mark the endpoint of the line segment at the point where the distance of 1.2 units ends.

Draw the line segment connecting the endpoint at 1.6 to the starting point.

In the diagram, the starting point is marked with at 0.4, and the endpoint is marked with at 1.6, which is 1.2 units away from the starting point. The line segment connecting the starting point to the endpoint is shown as a horizontal line.

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Answer:

y = (x + 5)/3

Step-by-step explanation:

To make y the subject, you need to isolate y on one side of the equation.

x = 3y - 5

Add 5 to both sides:

x + 5 = 3y

Divide both sides by 3:

y = (x + 5)/3

Therefore, y is the subject of the formula when it is expressed as:

y = (x + 5)/3

The volume of the prism is 2.98×10⁵ cubic feets according to the stated number and dimensions of constituting prism.

The volume of any shape is it's capacity to contain the item in it. It is the product of all its sides.

Volume of cube = side × side × side

Since there are multiple prisms of specific sides completely contained in the prism, their number will also be multiplied.

Volume of cube = 136 × 13 × 13 × 13

Performing multiplication on Right Hand Side of the equation

Volume of cube = 298,792 cubic feets

Hence, the volume of cube is 2.98×10⁵ cubic feet.

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(a) The three different trigonometric ratios that could have been used to find the distance x between Emma and the base of the tree are sin(60°) = 43.3/50, cos(60°) = x / 50, and tan(60°) = 43.3 / x. The distance is 25 ft.

(b) Using the Pythagorean theorem, the distance x between Emma and the base of the tree is 25 ft.

(a) We can use sine, cosine, and tangent trigonometric ratios to find the distance x between Emma and the base of the tree.

1. Sine:
sin(60°) = opposite/hypotenuse = (tree height) / 50
tree height = 50 * sin(60°) = 43.3 ft (since it's given that the tree is 43.3 ft tall)

2. Cosine:
cos(60°) = adjacent/hypotenuse = x / 50
x = 50 * cos(60°) = 25 ft

3. Tangent:
tan(60°) = opposite/adjacent = (tree height) / x
x = (tree height) / tan(60°) = 43.3 / tan(60°) ≈ 25 ft

(b) To find the distance x between Emma and the base of the tree using the Pythagorean theorem, we can consider the triangle formed by Emma, the base of the tree, and the top of the tree.

Let's call the distance from the base of the tree to the top of the tree (tree height) y.

Emma's distance to the cat (50 ft) is the hypotenuse, the distance x is one leg, and the tree height y (43.3 ft) is the other leg of the right triangle.

Using the Pythagorean theorem: x² + y² = hypotenuse²
x² + 43.3² = 50²
x² + 1874.89 = 2500
x² = 625.11
x ≈ 25 ft

So, the distance between Emma and the base of the tree is approximately 25 ft.

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A circular plate fragment is found by an archaeologist. The plate is 13.9 inches in diameter.

A chord in mathematics is a piece of a straight line that joins two points on a curve. A chord is a line segment with its endpoints on the curve, to be more precise.

The word "chord" is most frequently used in relation to the geometry of circles, where a chord is a line segment that joins two points on a circle's circumference. Given the lengths of the circle's radii and the separation between the chord's ends, the Pythagorean theorem can be used to determine the chord's length in this situation.

The relationship between two notes in music can also be represented by chords. A chord is, in this context, a grouping of three or more notes performed simultaneously to produce a harmonic sound. The structure of musical compositions can be analysed and understood using the mathematical concepts of chord progressions.

The equidistant chords theorem states that two chords are congruent in the same circle or a congruent circle if and only if they are equidistant from the centre.

Additionally, as depicted in the illustration, the supplied chords are equally spaced apart; as a result, they must meet in the circle's centre.

LHE is formed by connecting the points L and E to make a right-angled triangle ΔLHE.

The perpendicular chord bisector theorem states that if a circle's diameter is perpendicular to a chord, the diameter will also bisect the chord's arc HE≅ HD.

and  IF≅IG

hence , HE=7/2

HD=7/2

IF=7/2

And IG=7/2

Applying the Pythagorean theorem, simplifying by addition, and substituting 6 for the perpendicular P, LE for the hypotenuse H, and 7/2 for the base B in the equation P²+B²=H².

P²+B²=H²

6²+7/2²=LE²

LE²= 36+ 12.25

LE²= 48.25

Since LE represents the radius of a circle, it is impossible for it to be negative, hence the negative value LE=6.94 is disregarded.

Since LE is the circle's radius, the circle's radius is 6.94. As a result, the circle has a 13.88 diameter.

The diameter should be rounded out to the closest tenth.

d≈13.9.

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The slope of the graph is the same positive value that is 0.75.

Hence the correct option is (1).

We know that the equation of a straight line with slope 'm' and y intercept 'c' is given by,

y = mx + c

Here the model equation

c = 0.75h, where c is the total cost to buy hotdogs

h is the number of hotdogs bought

And 0.75 is the price of one hotdog

Now we can clearly say that c = 0.75h will make a straight line coordinate plane.

Now comparing the equation with slope intercept equation of straight line we get,

m = 0.75 and c = 0

So the slope of the line represented by model equation = 0.75 which is a positive number.

y intercept  = 0.

We know that the slope of one particular straight line on cartesian plane is unique.

So, the slope of the graph is the same positive value.

Hence the correct option is (1).

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Answer: 6

Step-by-step explanation:42/7=6

The real-world problem is modeled by the proportion 66/100 = x/2,500, where 66 is the sample proportion and 2,500 represents the population size.

To find the value of x, which represents the number of individuals with a specific characteristic in the population, follow these steps:

1. Cross-multiply the terms in the proportion:
  66 * 2,500 = 100 * x
2. Simplify the equation:
  165,000 = 100x
3. Divide both sides by 100 to isolate x:
  x = 1,650

Thus, 1,650 individuals in the population share the specific characteristic represented by the sample proportion. This proportion helps us understand and predict the prevalence of a certain characteristic or behavior within a larger population, based on the information gathered from a smaller sample.

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The distance that shows how much farther (in km) from the sun is Saturn than Venus is: [tex]1.2 * 10^9[/tex] km.

According to the table, the distance of Saturn from the Sun is [tex]1.4 * 10^{9}[/tex] and the distance of Venus from the Sun is [tex]1.1 * 10^{8}[/tex] .

Now to determine how much further from the Sun is Saturn than Venus, we will subtract the distance of the planet with the higher distance span from the one with the lower distance.

So our calculation will go thus:

[tex]1.4 * 10^9 - 1.1 * 10^8 = \\140000000 - 11000000 = 1290000000\\= 1.29 * 10^9[/tex]

From the calculation above, we can see how much further from the sun, is Saturn than Venus.

Complete Question:

The table shown below gives the approximate distance from the sun for a few different planets. How much farther (in km) from the sun is Saturn than Venus? Express your answer in scientific notation.

_______km

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During the oil crisis, Americans took various measures to reduce their dependency on oil from the Middle East. One of the key steps they took was to increase fuel efficiency standards for cars and trucks.

This helped to reduce the amount of oil needed to power vehicles. Additionally, they started to produce more of their own oil by opening up new oil fields and investing in alternative energy sources such as wind and solar power.

They also implemented policies to encourage conservation and reduce wasteful energy consumption. However, they did not decrease the price of oil and gas by four times, nor did they allow people to buy as much gas as they wanted.

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This is the derivative of the given function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule.

To find the derivative of the function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule, we will need to apply both the Product Rule and the Chain Rule.

The Product Rule states that if you have a function f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).

First, let's identify g(x) and h(x) in your function:
g(x) = (3x + 1)^5
h(x) = (3x - 1)^6

Next, we'll find the derivatives g'(x) and h'(x) using the Chain Rule, which states that if you have a function y = [u(x)]^n, then y' = n[u(x)]^(n-1) * u'(x).

For g'(x):
u(x) = 3x + 1
n = 5
u'(x) = 3
g'(x) = 5(3x + 1)^(5-1) * 3 = 15(3x + 1)^4

For h'(x):
u(x) = 3x - 1
n = 6
u'(x) = 3
h'(x) = 6(3x - 1)^(6-1) * 3 = 18(3x - 1)^5

Now, we apply the Product Rule:
f'(x) = g'(x)h(x) + g(x)h'(x) = 15(3x + 1)^4(3x - 1)^6 + (3x + 1)^5 * 18(3x - 1)^5

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The area of the object is 40 square centimeters

The correct answer is an option (B)

We know that the formula for the area of trapezoid is,

A = ((a + b)/2) × h

where a and b are the two parallel bases

h is the height of the trapezoid

From the attached figure, first we determine the length of the two parallel bases.

Let us assume that 'a' represents the length of the upper base and 'b' represents the length of the bottom side of the trapezoid

We can observe that a = 2x

so the length of a = 6 centimeters

And b = 2z

So, the length of b = 2(5)

                              = 10 cm

Here, the height h of the trapezoid is given by y = 5 cm

Using above formula for the area of trapezoid, the area of the object would be,

A = ((a + b)/2) × h

A = ((6 + 10)/2) × 5

A = (16/2)  × 5

A = 40 cm²

Therefore, the correct answer is an option (B)

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Find the complete question below.

Calculated test statistic of 2.08 is greater than the critical value of 1.645, we reject the null hypothesis and conclude that there is convincing evidence at the 0.05 level that more than 55% of California residents use Netflix.

We can use a hypothesis testing approach to answer this question. The null hypothesis is that the true proportion of California residents who use Netflix is the same as the national proportion, or p = 0.55. The alternative hypothesis is that the true proportion of California residents who use Netflix is greater than 0.55, or p > 0.55.

We can use the sample proportion of Netflix users in California, which is 360/600 = 0.6, as an estimate of the true proportion p. The standard error of the sample proportion is:

SE = √[(p*(1-p))/n] = √[(0.55*(1-0.55))/600] = 0.024

The test statistic is:

z = (p - 0.55)/SE = (0.6 - 0.55)/0.024 = 2.08

Assuming a significance level of 0.05 and a one-tailed test (since the alternative hypothesis is one-sided), the critical z-value is 1.645.

Since our calculated test statistic of 2.08 is greater than the critical value of 1.645, we reject the null hypothesis and conclude that there is convincing evidence at the 0.05 level that more than 55% of California residents use Netflix. However, we should keep in mind that this conclusion is based on a sample of 600 adults from California, and there is always some degree of uncertainty involved in statistical inference based on samples.

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Answer:

a b e

Step-by-step explanation:

A) The amount of interest he will earn in a year is $3,000.

B) The amount he spent on furniture is $600.

Part A: To calculate the interest Alex will earn in one year, use the formula for simple interest:

Interest = Principal × Rate × Time.

In this case, Principal = $30,000, Rate = 10% (0.10), and Time = 1 year. So,

Interest = $30,000 × 0.10 × 1 = $3,000.

Part B: Alex spends 20% of the interest on furniture. To calculate this amount, multiply the interest by 20% (0.20): $3,000 × 0.20 = $600.

Therefore, in one year, Alex will earn $3,000 in interest. He will spend $600 on furniture for his new house.

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The triangle above has the following measures. The length of sider is 13.3 inches.

q = 8 inches

m ∠Q = 37°

sin (Q) = q/r

r = q / sin(Q)

= 8 / sin (37°)

= 13.3 inches

In Math, a triangle is a three-sided polygon that comprises of three edges and three vertices. The main property of a triangle is that the amount of the inward points of a triangle is equivalent to 180 degrees. This property is called point total property of triangle.

There are three points in a triangle. These points are framed by different sides of the triangle, which meets at a typical point, known as the vertex. The amount of every one of the three inside points is equivalent to 180 degrees.

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