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A good description of the cross section shown that is parallel to the edge of the pyramid that measures 5​ millimeters is a triangle with base of 5 millimeters and height of 16 millimeters.

In Mathematics and Geometry, a square pyramid can be defined as a type of pyramid that has a square base, four (4) triangular sides, five (5) vertices, and eight (8) edges.

In Mathematics and Geometry, a triangle can be defined as a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

In this context, we can reasonably infer and logically deduce that the edge of the prism that measures 5 millimeters represents a triangle with base of 5 millimeters and height of 16 millimeters.

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Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.

If Ben filled his 20-gallon tank of gas with regular fuel for $59.80, then the cost per gallon can be found by dividing the total cost by the number of gallons:

cost per gallon = total cost / number of gallons

cost per gallon = $59.80 / 20 gallons

cost per gallon = $2.99/gallon

If the gas station across the street sold regular fuel for $2.84 a gallon, then the amount Ben could have saved per gallon is:

savings per gallon = cost per gallon at initial station - cost per gallon at other station

savings per gallon = $2.99/gallon - $2.84/gallon

savings per gallon = $0.15/gallon

Therefore, Ben could have saved $0.15 per gallon if he had gone to the gas station across the street.

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The mass of copper in the platter is 45 grams, which corresponds to option (B).

What is ratio ?

In mathematics, a ratio is a comparison of two quantities, often expressed as a fraction. Ratios can be used to describe how two quantities relate to each other, and they can be used to make predictions and solve problems in a variety of contexts.

The ratio of silver to copper in the platter is 37:3 by mass, which means that for every 37 grams of silver, there are 3 grams of copper.

Let's call the mass of silver in the platter "s" and the mass of copper "c". We know that the total mass of the platter is 600 grams, so:

s + c = 600

We also know that the ratio of silver to copper is 37:3, which means that:

s÷c = 37÷3

We can use this second equation to solve for s in terms of c:

s:c = 37:3

s = (37÷3)c

Now we can substitute this expression for s into the first equation:

s + c = 600

(37÷3)c + c = 600

(40÷3)c = 600

c = (3÷40) * 600

c = 45

Therefore, the mass of copper in the platter is 45 grams, which corresponds to option (B).

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The area of the shaded region is 422.8 m², rounded to the nearest tenth.

To find the area of the shaded region, we first need to determine the areas of the two shapes that make up the region. The first shape is a rectangle with dimensions of 18.6 m by 30 m, which has an area of:

Area of rectangle = length x width = 18.6 m x 30 m = 558 m²

The second shape is a semi-circle with a diameter of 18.6 m, which has a radius of 9.3 m. The area of a semi-circle is half the area of a full circle, so we can use the formula for the area of a circle to find the area of the semi-circle:

Area of semi-circle = (1/2) x π x r² = (1/2) x π x 9.3² = 135.2 m²

To find the area of the shaded region, we need to subtract the area of the semi-circle from the area of the rectangle:

Area of shaded region = Area of rectangle - Area of semi-circle
Area of shaded region = 558 m² - 135.2 m²
Area of shaded region = 422.8 m²

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Ms. Brock can now fill up to 88 spray bottles with her preparation.

It is calculated that:

hence we have that

1 gallon = 128 fluid ounces

1 pint = 16 fluid ounces  

cross multiplying

16 fluid ounces x 1 gallon = 128 fluid ounces x 1 pint

1 gallon = (128 fluid ounces x 1 pint) / 16 fluid ounces

1 gallon = 8 pints

Consequently, there are 8 pints in one full gallon.

Ms. Brock blended 1 gallon of cleaner with 10 gallons of water, thus resulting in 11 gallons.

To figure out the total number of pints Ms. Brock has in her solution, we must multiply 11 gallons by 8 pints/gallon:

11 gallons x 8 pints/gallon = 88 pints

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If t is less than -1.699, we reject the null hypothesis.

To test the claim by the owner of the large dealership, we will use a one-sample t-test with the following hypotheses:

Null Hypothesis: H0: µ >= µ0 (The population mean time of ownership is greater than or equal to µ0)

Alternative Hypothesis: Ha: µ < µ0 (The population mean time of ownership is less than µ0)

where µ is the population mean time of ownership, µ0 is the claimed mean time of ownership by the owner of the dealership.

The significance level is α = 0.05.

We can calculate the t-value as:

t = ([tex]\bar{x}[/tex] - µ0) / (s / √n)

where [tex]\bar{x}[/tex] is the sample mean time of ownership, s is the sample standard deviation, n is the sample size.

Plugging in the values given in the problem, we get:

t = ([tex]\bar{x}[/tex] - µ0) / (s / √n) = (5.7 - µ0) / (1.8 / √n)

Since the alternative hypothesis is one-tailed (less than), we need to find the critical t-value from the t-distribution table with n-1 degrees of freedom and a significance level of 0.05. For a sample size of n = 30 (assuming it is large enough), the critical t-value is -1.699.

If the calculated t-value is less than the critical t-value, we reject the null hypothesis and conclude that there is evidence to support the claim that the mean time of ownership for all cars is less than the claimed mean time of ownership by the owner of the dealership.

If the calculated t-value is greater than the critical t-value, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim that the mean time of ownership for all cars is less than the claimed mean time of ownership by the owner of the dealership.

So, if we assume that the sample is representative of the population and meets the assumptions of the t-test, we can calculate the t-value as:

t = (5.7 - µ0) / (1.8 / √30)

If t is less than -1.699, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Note that we don't have any information about the claimed mean time of ownership by the owner of the dealership, so we cannot calculate the t-value or make any conclusions.

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The value of x is approximately 61.7 degrees using the Law of Sines.

To find the value of x, we can use the Law of

Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides. Mathematically, we can write:

a/sin A = b/sin B = c/sin C

where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite those sides.

Using the given information, we can set up the

equation as follows:

12/sin 42° = 22/sin x

Multiplying both sides by sin x°, we get:

sin x = (22/12) x sin 42°

sin x = 1.6977

Taking the inverse sine of both sides, we get:

x* = sin" (1.6977)

x = 61.7°

Rounding to the nearest tenth, we get:

x = 61.7°

Therefore, the value of x is approximately 61.7

degrees.

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The complete question is:

In the given figure , find the X (round to the nearest tenth) , where the sides are marked as 12cm and 22 cm with the angle of 42°.

Therefore , the solution of the given problem of unitary method comes out to be  length (12 ft) that is less than its width (20 ft), which is the case with the tent.

The well-known simple approach, real variables, and any crucial elements from the very initial And specialised inquiry can all be used to finish the work. Customers may then be given another chance to try the product in response. If not, significant impacts on our understanding of algorithms will vanish.

Here,

The hypotenuses of two right triangles created by halving the triangle can be viewed as the two sides of the triangle. Next, we have

=> h² = (12/2)² - (20/2)²

=> h² = 36 - 100

=> h² = -64

The fact that the outcome is bad indicates that we made a mistake. In this instance, it is because a legitimate triangular prism cannot be formed using the specified dimensions.

A triangular prism cannot have a length (12 ft) that is less than its width (20 ft), which is the case with the tent.

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After the duration of 2 hours the distance between the first pair and the second pair is 3.07 km, under the condition that the second pair decide to head 60° west of north travelling at the same pace.

In order to evaluate the distance between two points with given coordinates, we can apply the distance formula. The distance formula is

d = √ [ (x₂ − x₁)² + (y₂ − y₁)² ]

Here,

(x₁, y₁) and (x₂, y₂) = coordinates of the two points.

For the given case, we can consider that the first pair of team members start at the origin (0, 0) and cover the distance towards  north for 2 hours at a pace of 4 km/h.

Hence, their final position is (0, 8).

The second pair of team members take  the origin (0, 0) and travel 60° west of north for 2 hours at a pace of 4 km/h.

Now to evaluate their final position, we have  to find their coordinates. Let us consider their final position (x, y).

We can apply trigonometry to find x and y.

The angle between their direction of travel and the y-axis is 60°.

sin(60°) = y / d

cos(60°) = x / d

Here,

d = distance travelled by the second pair of team members.

It is given that they travelled for 2 hours at a pace of 4 km/h.

d = 2 hours × 4 km/h

= 8 km

Staging this value into the above equations

y = d × sin(60°) = 8 km × sin(60°)

≈ 6.93 km

x = d × cos(60°) = 8 km × cos(60°)

≈ 4 km

Hence, the final position regarding the second pair of team members is approximately (4 km, 6.93 km).

Now we can apply the distance formula to evaluate the distance between the two pairs of team members

d = √ [ (x₂ − x₁)² + (y₂ − y₁)² ]

d = √ [ (4 − 0)² + (6.93 − 8)² ]

d ≈ 3.07 km

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Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.

First, let's determine the assessed value of the property. To do this, we'll multiply the market value by the rate of assessment:

Assessed Value = Market Value × Rate of Assessment
Assessed Value = $119,340 × 0.42
Assessed Value = $50,122.80

Now, we need to find the tax rate in mills. One mill is equal to $1 per $1,000 of assessed value. To find the tax rate, we'll divide the yearly real estate taxes by the assessed value and multiply by 1,000:

Tax Rate (in mills) = (Yearly Real Estate Taxes / Assessed Value) × 1,000
Tax Rate = ($4,388.65 / $50,122.80) × 1,000
Tax Rate ≈ 87.6 mills

Therefore, Ted Pappas' tax rate to the nearest tenth of a mill is approximately 87.6 mills.

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The tax on a bicycle costing $700, with the tax remaining the same as a bicycle costing $400 with a tax of $32, will be $56.

To calculate the tax on a bicycle costing $700, we need to know the percentage of tax charged on the $400 bicycle. The tax on the $400 bicycle is $32. To find the tax rate, we divide the tax by the cost of the bicycle and multiply by 100 to get a percentage.

tax rate = (tax / cost of bicycle) x 100%

tax rate = (32 / 400) x 100%

tax rate = 8%

Therefore, the tax rate is 8%. We can use this tax rate to calculate the tax on a bicycle costing $700.

tax on $700 bicycle = (tax rate / 100) x cost of bicycle

tax on $700 bicycle = (8 / 100) x $700

tax on $700 bicycle = $56

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a) When the radius is 4 cm, it is changing at a rate of 1/(4π) cm/s.

b) When the circumference is 3 cm, the radius is changing at a rate of 2/3 cm/s.

a. Given that the area of a circle increases at a rate of 2 cm²/s, let's denote this rate as dA/dt.

The formula for the area of a circle is A = πr²,

where A is the area and r is the radius.

We want to find the rate at which the radius is changing, or dr/dt, when the radius is 4 cm.

Using implicit differentiation with respect to time t, we get:

dA/dt = d(πr²)/dt 2 = 2πr(dr/dt)

Now, we'll plug in the radius value of 4 cm:

2 = 2π(4)(dr/dt)

Solving for dr/dt, we get:

dr/dt = 1/(4π) cm/s

b. We are given the circumference, which is 3 cm.

The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.

First, we need to find the radius when the circumference is 3 cm: 3 = 2πr r = 3/(2π)

Now, we'll plug this value for the radius back into the formula from part a:

2 = 2π(3/(2π))(dr/dt)

Solving for dr/dt, we get:

dr/dt = 2/3 cm/s

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Answer:

$64

Step-by-step explanation:

This would be the answer because if we figure out what needs to be added to -21 to reach 43 we get 64.

(-21+64 - $43.)

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The problem involves analyzing the cost of production for a company that produces a certain quantity of units. Specifically, we are given the marginal cost function C'(q) = q^2 - 17q + 70, where q is the quantity produced, and we need to find the total cost of producing 20 units, as well as the fixed cost and variable cost for this quantity. To find the total cost, we need to integrate the marginal cost function from 0 to 20, which will give us the total variable cost. We can then find the fixed cost by subtracting the total variable cost from the initial cost C(0), which is given in the problem. Cost analysis is an important concept in economics and business, and is used to optimize production and pricing decisions for companies. Understanding the relationship between marginal cost, fixed cost, and variable cost is crucial for making informed decisions about production and pricing strategies.

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The marginal cost function for a company is given by C'(q) = [tex]q^2[/tex] - 17q + 70 dollars/unit; where q is the quantity produced. If C(0) = 650, then total cost of producing 20 units = 2,600 dollars. Fixed cost for the quantity = 650 dollars. Variable Cost of producing 20 units = 1,950 dollars

To find the total cost function C(q), integrate the marginal cost function C'(q):

C(q) = ∫[tex](q^2 - 17q + 70)[/tex] dq = [tex](1/3)q^3 - (17/2)q^2 + 70q + K[/tex]

To find the constant K, use the given information: C(0) = 650

650 = [tex](1/3)(0)^3 - (17/2)(0)^2 + 70(0) + K[/tex]
K = 650

So the total cost function is:

C(q) = [tex](1/3)q^3 - (17/2)q^2 + 70q + 650[/tex]

Now, we find the total cost of producing 20 units:

C(20) = [tex](1/3)(20)^3 - (17/2)(20)^2 + 70(20) + 650[/tex]
C(20) = 2,600

Total cost of producing 20 units = 2,600 dollars.

Fixed cost is the cost incurred when producing 0 units, which is given as C(0) = 650 dollars.

To find the total variable cost for producing 20 units, subtract the fixed cost from the total cost:

Variable Cost = Total Cost - Fixed Cost
Variable Cost = 2,600 - 650
Variable Cost = 1,950 dollars

To summarize:
Fixed cost = 650 dollars
Variable cost of producing 20 units = 1,950 dollars
Total cost of producing 20 units = 2,600 dollars

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The best prediction for the bacteria population after 8 days is approximately 14,301.67 bacteria.

To find the predicted population of bacteria after 8 days, we need to apply the given growth rate of 220% per day to the initial population of 50 bacteria for each day, starting from day 1 and continuing to day 8.

For each day, the population of bacteria is expected to be 220% or 2.2 times the population of the previous day. So, we can use the formula:

P = P0 [tex]x (1 + r)^n[/tex]

where P is the predicted population after n days, P0 is the initial population, r is the growth rate per day (as a decimal), and n is the number of days.

Substituting the given values, we get:

P = 50[tex]x (1 + 2.2)^8[/tex]

P ≈ 14,301.67

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The line crosses the y-axis at -9 and has a slope of 4, meaning it increases 4 units in the y-direction for every 1 unit increase in the x-direction.

A graph is a visual representation of data, typically involving the use of points, lines, and curves to show how different values are related to each other. Graphs can be used to display a wide range of information, from mathematical functions and scientific data to business trends and social networks.

Here,

To graph y = 4x - 9, we can start by plotting the y-intercept, which is -9 on the y-axis. Then, we can use the slope of 4 to find other points on the line.

Slope = 4 can be written as rise/run = 4/1. This means that for every increase of 1 in the x-direction, the y-value increases by 4.

Starting from the y-intercept (-9,0), we can go up 4 units and to the right 1 unit to get the point (1,-5). We can continue this pattern and plot more points on the line, or we can simply draw a straight line through the y-intercept and (1,-5) to represent the equation y = 4x - 9.

Here's what the graph looks like:

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The debt increased between 1996 and 2003. Then the national debt increased by approximately $4,903.73 billion between 1996 and 2003.

To find how much the debt increased between 1996 and 2003, we need to find the value of the function D'(t) for t=7 (since 2003 is 7 years after 1996).

D'(t)=858.29+819.48t-184.32t^2+12.12t^3

D'(7)=858.29+819.48(7)-184.32(7^2)+12.12(7^3)
D'(7)=858.29+5,736.36-8,132.32+3,458.68
D'(7)=1,921.01

Therefore, the annual rate of change in the national debt in 2003 was $1,921.01 billion per year.

To find how much the debt increased between 1996 and 2003, we need to integrate the function D'(t) from t=1 to t=7:

∫(D'(t))dt = ∫(858.29+819.48t-184.32t^2+12.12t^3)dt
= 858.29t + 409.74t^2 - 61.44t^3 + 3.03t^4 + C

where C is the constant of integration.

Evaluating this expression at t=7 and t=1 and taking the difference, we get:

(858.29(7) + 409.74(7)^2 - 61.44(7)^3 + 3.03(7)^4 + C) - (858.29(1) + 409.74(1)^2 - 61.44(1)^3 + 3.03(1)^4 + C)
= 6,111.09 - 1,207.36 = 4,903.73

Therefore, the national debt increased by approximately $4,903.73 billion between 1996 and 2003.

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Projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).

We'll use the following formula:

projection of u onto v = (u•v / ||v||²) * v

First, we need to calculate the dot product (u•v) and the magnitude squared (||v||²) of vector v.

1. Dot product (u•v):
u•v = (5 * -7) + (-9 * -4) + (-5 * 3) = -35 + 36 - 15 = -14

2. Magnitude squared (||v||²):
||v||^2 = (-7)² + (-4)² + (3)² = 49 + 16 + 9 = 74

Now, we'll plug these values into the projection formula:

projection of u onto v = (-14 / 74) * v

We'll multiply each component of vector v by the scalar (-14/74):

projection of u onto v = (-14/74) * (-7, -4, 3) = (1.33, 0.76, -0.57)

So, the projection of vector u onto vector v is approximately (1.33, 0.76, -0.57).

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(a) The critical numbers are x = 0 and x = 6.

(b) The function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).

(a) To find the critical numbers of f(x), we need to find the values of x where f'(x) = 0 or f'(x) does not exist.

f'(x) = 12x² - 72x

Setting f'(x) = 0, we get:

12x² - 72x = 0

12x(x - 6) = 0

So, the critical numbers are x = 0 and x = 6.

(b) To determine where the function is increasing or decreasing, we need to examine the sign of f'(x) on different intervals.

For x < 0, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (-∞, 0).

For 0 < x < 6, f'(x) = 12x² - 72x > 0, which means the function is increasing on (0, 6).

For x > 6, f'(x) = 12x² - 72x < 0, which means the function is decreasing on (6, ∞).

So, the function is increasing on the interval (0, 6) and decreasing on the intervals (-∞, 0) and (6, ∞).

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The measurement closest to the number of meters Chelsea sprinted is F) 109.1 yd.

To determine the number of meters Chelsea sprinted, we need to convert the distance she sprinted from yards to meters.

Given:

Chelsea sprinted 120 yards.

Conversion:

1 meter is approximately equal to 1.1 yards.

To convert yards to meters, we divide the distance in yards by the conversion factor:

120 yards / 1.1 yards/meter ≈ 109.1 meters.

Therefore, the measurement closest to the number of meters Chelsea sprinted is F) 109.1 yd.

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Two possible values of x are 10 and 40.

To find two possible values of x, we need to first determine the highest and lowest numbers in the set.

Highest number: 46
Lowest number: 4

To get a range of 55, we need the highest number minus the lowest number to equal 55.

46 - 4 = 42

But we know that there are 8 numbers in the set, so the range must be spread out over those 8 numbers. To get an idea of how much each number should increase, we can divide 42 by 7 (the number of gaps between the numbers) to get an average increase of 6.

Now we can use this average increase to find two possible values of x.

1) If x is 6 more than the lowest number (4 + 6 = 10), then the set becomes:

4, 9, 10, 32, 35, 38, 46, 46

And the range is:

46 - 4 = 42

2) If x is 6 less than the highest number (46 - 6 = 40), then the set becomes:

4, 9, 35, 38, 40, 46, 46, 32

And the range is:

46 - 4 = 42

Therefore, two possible values of x are 10 and 40.

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Answer:

C) [tex]\sf \sqrt{45}[/tex]

Step-by-step explanation:

Pythagorean theorem:

               AB = 6 units

              BC = 3 units

AC is hypotenuse and AB is the base and BC is the altitude.

Hypotenuse² = base² + altitude²

               AC² = AB² + BC²

                      [tex]\sf = 6^2 + 3^2\\\\ = 36 + 9\\\\ = 45[/tex]

               [tex]\sf AC= \sqrt{45}[/tex]

The given radian measure -1.7 pi is equivalent to -306 degrees.

The correct answer is option (b), Negative 306 degrees. This conversion takes into account the negative sign of the radian measure, resulting in a negative degree measure to convert a radian measure to a degree measure, we use the conversion factor that 180 degrees is equal to π radians.

Given the radian measure -1.7π, we can calculate the corresponding degree measure by multiplying -1.7π by the conversion factor:

Degree measure = (-1.7π) * (180 degrees / π)

The π in the numerator and denominator cancels out, resulting in:

Degree measure = -1.7 * 180 degrees

Calculating the value, we have:

Degree measure = -306 degrees

Therefore, the correct answer is option b) Negative 306 degrees.

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Answer:

y=60

Step-by-step explanation:

In a school, a teacher surveys 50 randomly selected 6th grade students and 50 randomly selected 7th grade students. c) More seventh-graders than sixth-graders enjoy horror films.

The first statement is false as the total 6th graders which like horror and comedy movie is 19 + 9 = 28 students which is more than 6th graders who like action movies which is 22, hence the first statement is false. this is interpreted from given data set.

The second statement is also false as it says that 6th graders prefer comedy films to action films, whereas 7th graders prefer action films but from the data given, it can be seen that the number of 6th graders who like comedy films is same as the number of 7th graders who like action movies which is 19, hence statement is false.

The third statement is true as 6th graders who like horror movies is 9 while 7th graders who like horror movies is 14 and hence, the statement is true.

Fourth statement is also false as 17, 7th graders like comedy movies in contrast to 14, 7th graders who like horror movies.

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Correct question:

A teacher conducts a survey of 50 randomly selected 6th grade and 50 randomly selected 7th grade students in a school. The survey asks the students about the type of movies they like to watch. The table shows the number of students who selected each type of movie. Select the correct statement.

a) 6th graders like action movies more than horror and comedy movies.

b) 6th graders like comedy movies more than 7th graders like action movies.

c) More 7th graders than 6th graders like horror movies.

d) 7th graders like horror movies more than comedy movies.

The distance traveled by the surveyor is √73 miles. The height of the building that the ladder reaches is √15 meters. The longest line that can be drawn is 5√34 cm. The length of the guy wire that is needed is 5√5 meters. The length of the hypotenuse of the given right triangle is 6√3 inches.

In a right-angled triangle that is a triangle with one of the angles with magnitude 90° following is true according to Pythogaras' Theorem:

[tex]A^2=B^2+C^2[/tex]

where A is the hypotenuse

B is the base

C is the height

1. According to the question,

the distance between the starting and the ending point is the hypotenuse of a right-angled triangle

B = 8 miles

C = 3 miles

A = √(64 + 9)

= √73 miles

2. Hypotenuse in the given question is the length of the ladder, thus,

A = 4 m

B = 1 m

16 = 1 + [tex]C^2[/tex]

C = √15 meters

3. The longest line that can be drawn on the paper is described as the hypotenuse of the triangle

C = √225 + 625

= 5√34 cm

4. The length of the guy wire is the hypotenuse of the triangle.

C = √100 + 25

= 5√5 meters

5. Let the base of the triangle be x

the hypotenuse be 2x

height = 9 inches

[tex]4x^2=x^2[/tex] + 81

[tex]3x^2[/tex] = 81

x = 3√3 inches

Hypotenuse = 2x = 6√3 inches

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We can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4, missing monomial is -49a4.

To find the missing monomial, we need to use the distributive property to expand the left side of the equation:

(... - b 4)(64 + ....) = 121a 10 – 68

Expanding the left side gives:

64(... - b 4) + ....(... - b 4) = 121a 10 – 68

Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.

Looking at the constants in the equation, we can see that 121 and 68 have a common factor of 17. Therefore, we can divide both sides of the equation by 17 to simplify the coefficients:

7a 10 - 4 = 4(... - b 4) + 4

Simplifying further, we get:

7a 10 - 8 = 4(... - b 4)

Dividing both sides by 4, we get:

(7/4)a 10 - 2 = ... - b 4

Now we need to find a monomial to replace the ellipsis that will make the left side of the equation equal to the right side of the equation, no matter what value is substituted for a and b.

Since we want the left side of the equation to have a degree of 4 (because the right side has a term of -b4), we need to choose a monomial of degree 4 that will cancel out the terms of degree 10 on the left side. One possible monomial is:

-49a4

Therefore, we can replace the ellipsis with -49a4 to get: (7/4)a 10 - 2 = -49a4 - b 4

So the missing monomial is -49a4.

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(a) The triangle is a right triangle when x is equal to 24.

(b) The side lengths do not form a Pythagorean triple.

(a) To determine when the triangle is a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Using the given side lengths, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

To find the value of x, we take the square root of both sides:

x = √74

Approximating the square root of 74, we get:

x ≈ 8.60

Therefore, when x is approximately 8.60, the triangle is a right triangle.

(b) For the side lengths to form a Pythagorean triple, they must satisfy the condition of the Pythagorean theorem. In this case, we have:

5^2 + 7^2 = x^2

Simplifying the equation:

25 + 49 = x^2

74 = x^2

Since the sum of the squares of the two smaller sides (25 + 49 = 74) is not equal to the square of the longest side (x^2 = 74), the side lengths of 5, 7, and x do not form a Pythagorean triple.

In conclusion, the triangle is a right triangle when x is equal to approximately 8.60, and the side lengths do not form a Pythagorean triple.

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