From the observation deck of a skyscraper, Morgan measures a 67^{\circ}
∘
angle of depression to a ship in the harbor below. If the observation deck is 955 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.

Answer:

$653

Step-by-step explanation:

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Let's use x to represent the cost of one refrigerator and y to represent the cost of one fan.

The equations become:

Equation 1: x + 2y = 1219

Equation 2: 2x + 3y = 2155

Using the same substitution method:

From Equation 1, we have:

x = 1219 - 2y

Substitute this expression for x in Equation 2:

2(1219 - 2y) + 3y = 2155

Simplify the equation:

2438 - 4y + 3y = 2155

-y = 2155 - 2438

-y = -283

===> y = 283

Now substitute the value of y back into Equation 1 to find x:

===> x + 2(283) = 1219

===> x + 566 = 1219

===> x = 1219 - 566

===> x = 653

Therefore, the cost of one refrigerator is $653.

The other options provided, -3x - 1, -x - 12, and -x - 3, do not match the simplified form of the given expression. Only -3x - 4 corresponds to the original expression after simplification. It is important to carefully distribute and combine like terms to simplify expressions correctly.

The expression shown is 2 + 2(x – 3) – 5x. To find an equivalent expression, we need to distribute the 2 to both terms inside the parentheses, resulting in 2x - 6. Now we can simplify the expression further:

2 + 2x - 6 - 5x

Combining like terms, we have:

(2x - 5x) + (2 - 6)

This simplifies to:

-3x - 4

Hence, the expression -3x - 4 is equivalent to 2 + 2(x – 3) – 5x.

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The value of x is equal to 10 degrees.

In Mathematics and Geometry, the Alternate Interior Angles Theorem states that when two (2) parallel lines are cut through by a transversal, the alternate interior angles that are formed are congruent:

By applying the alternate interior angles theorem to parallel lines p and q, we have the following congruent angles:

(5x - 65) = (x - 25)

5x - x = 65 - 25

4x = 40

x = 40/4

x = 10 degrees.

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Answer: Amplitude is 108 degrees

Step-by-step explanation:

To determine the amplitude of the tip of King Arthur's Sword, we need to understand the shape of the sword's blade, which consists of a regular hexagon and a regular pentagon.

A regular hexagon has six equal sides and six equal angles, each measuring 120 degrees. The regular pentagon, on the other hand, has five equal sides and five equal angles, each measuring 108 degrees.

Since the tip of the sword is formed by the meeting point of the hexagon and pentagon, it will be at the apex of the pentagon. The apex of the pentagon will have an angle of 108 degrees.

I hope this helps!!!

The given triangle is a right angled triangle, therefore the rules of basic Trigonometry can be used here to find the solution.

[tex]\qquad\displaystyle \tt \dashrightarrow \: \sin(45 \degree) = \frac{opposite \:\: side}{hypotenuse} [/tex]

[ sin 45° = 1/√2 ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{ \sqrt{2} } = \frac{b}{x} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = b \sqrt{2} \: \: cm[/tex]

So, the side HG is b√2 cm long

To find the measure of each interior and exterior angle of a regular 30-gon, we can use the formula:

In this case, we have a regular 30-gon, so n = 30.

Let's calculate the measures:

Therefore, each interior angle of the regular 30-gon measures 168°, and each exterior angle measures 12°.

Free variables are variables used to represent parameters

Free variables are placeholders or symbols in mathematics and logic that are not constrained by any quantifiers or other specified conditions within an expression, formula, or equation.

When the expression is evaluated, they stand for values that can change or be given different values. In equations or functions, free variables are frequently employed to represent unknowns or parameters.

Free variables enable flexibility and generality in mathematical reasoning since they can be given various values to investigate various situations or solutions. In contrast, bound variables have a specific scope within a certain context or statement and are constrained by quantifiers.

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No, the sides with lengths 7.5, 8.5, and 14.5 cannot form a triangle.

To form a triangle considering three lengths of sides, the sum of the lengths of any two sides must be greater than the length of the third side.

Checking the above condition

1. The sum of 7.5,8.5 is 16, which is greater than 14.5. So, the condition is satisfied.

2. The sum of 7.5, 14.5 is 22, which is greater than 8.5. So, the condition is satisfied.

3.  The sum of 8.5 and 14.5 is 23, which is less than 7.5. The condition is not satisfied.

Since the sum of the two shorter sides 8.5 and 14.5 is not greater than the longest side 7.5 these three sides with given lengths cannot form a triangle.

Therefore, the sides with lengths 7.5, 8.5, and 14.5 cannot form a triangle.

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The length of y in the right triangle is 15 units.

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The sum of angles in a triangle is 180 degrees.

The side y in the triangle XYZ can be found using trigonometric ratios as follows:

Therefore,

sin 45 = opposite / hypotenuse

opposite sides = y

hypotenuse side = 15√2

sin 45 = y / 15√2

cross multiply

y = 15√2 × sin 45

y = 15√2 × 1 / √2

y = 15 units

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No, this shape does not belong in a group of shapes that have more than one pair of perpendicular sides.

In order to determine if the shape belongs to a group of shapes with more than one pair of perpendicular sides, we need to examine the number of right angles in the shape.

A right angle is a 90-degree angle, and shapes with perpendicular sides have right angles at their intersections. By counting the number of right angles in the shape, we can determine if it has more than one pair of perpendicular sides.

If the shape has only one right angle, then it does not have more than one pair of perpendicular sides. But if it has two or more right angles, then it would belong to the group of shapes that have more than one pair of perpendicular sides.

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The perimeter of shaded figure is 10 unit.

We know,

The perimeter of a figure is the total distance around its boundary. To calculate the perimeter, you need to sum the lengths of all the sides of the figure.

From the figure

length of rectangle = 4 unit

width of rectangle =  1 unit

Now, the perimeter of shaded figure

= 2 (l + w)

= 2 (4 +1 )

= 2 x 5

= 10 unit

Thus, the perimeter of figure is 10 unit.

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12) Volume of semi - sphere = 261.67 km³

13) Volume of semi - sphere = 8574.29 feet³

We have to given that;

12) Radius of sphere = 5 km

13) Radius of sphere = 16 feet

Since, We know that;

Volume of sphere = 4/3 πr³

Hence, We get;

Volume of semi - sphere = 2/3πr³

12) Here, Radius of sphere = 5 km

Volume of semi - sphere = 2/3πr³

Volume of semi - sphere = 2/3 × 3.14 × 5³

Volume of semi - sphere = 261.67 km³

13) Here, Radius of sphere = 16 feet

Volume of semi - sphere = 2/3πr³

Volume of semi - sphere = 2/3 × 3.14 × 16³

Volume of semi - sphere = 8574.29 feet³

Thus, We get;

12) Volume of semi - sphere = 261.67 km³

13) Volume of semi - sphere = 8574.29 feet³

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The unit vector for n = (4, -3) is V = (4/5, -3/5)

An unit vector will be a vector that has the same direction than the given one, but a magnitude of 1 unit.

Then we can define the vector V = k*n

Where k > 0 is a real number, then the unit vector is:

V = (4k, -3k)

But notice that this must have a magnitude of 1, then:

1 = √( (4k)² + (-3k)²)

1 = √25k²

1 = 5k

1/5 = k

Then the unit vector is:

V = (4/5, -3/5)

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The distance between Martin's house and the school is 5 kilometers.

To find the distance between Martin's house and the school, we can use the Pythagorean theorem.

Let's represent the distance between Martin's house and the school as x kilometers.

According to the given information:

Distance between Martin's house and Hayley's house (hypotenuse) = 13 kilometers

Distance between the school and Hayley's house (one side of the right triangle) = 12 kilometers

Using the Pythagorean theorem, we have:

x[tex]x^2 + 12^2 = 13^2[/tex]

Simplifying the equation:

[tex]x^2 + 144 = 169x^2 = 169 - 144x^2 = 25[/tex]

Taking the square root of both sides:

x = √25

x = 5

Therefore, the distance between Martin's house and the school is 5 kilometers.

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Answer: i dont know what this is but i  do believe its the second one

Step-by-step explanation:

The  total number of each type of ball:

Football: Given as 8

Basketball: 18

Baseball: 40

Softball: 15

For basketball:

2 + 2(8) = 18

First, we evaluate the expression inside the parentheses: 2(8) = 16

Then, we add 2 to the result: 16 + 2 = 18

So, the total number of basketballs is 18.

For baseball:

5(8) = 40

We simply multiply 5 by 8 to get the total number of baseballs, which is 40.

For softball:

6 + 1/2(18) = 24

First, we evaluate the expression inside the parentheses: 1/2(18) = 9

Then, we add 6 to the result: 9 + 6 = 15

So, the total number of softballs is 15.

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The graph that shows only a translation is (c)

From the question, we have the following parameters that can be used in our computation:

The list of options

The general rule of translation it that

Using the above as a guide, we have the following:

the graph that shows only a translation is (c)

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The length of each side of the square base should be approximately 244.95 meters when rounded to the nearest hundredth.

To calculate the length of the sides of the square base of the pyramid, we can use the formula for the volume of a square pyramid:

V = (1/3) * (side length)^2 * height

Given that the volume of the pyramid is 2,000,000 m³ and the height is 100 m, we can substitute these values into the formula:

2,000,000 = (1/3) * (side length)^2 * 100

To isolate the side length, we can rearrange the equation:

(side length)^2 = (2,000,000 * 3) / 100

(side length)^2 = 60,000

Taking the square root of both sides, we find:

side length ≈ √60,000

side length ≈ 244.95

Therefore, the length of each side of the square base should be approximately 244.95 meters when rounded to the nearest hundredth.

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Answer:

B. 3188.5 cubic inches.

Step-by-step explanation:

The volume of a cone is calculated using the following formula:

Volume = (1/3) * π * r² * h

Where:

In this problem, we are given that r = 17 inches and S.h = 20 inches.

First we need to find height h.

py using Pythagorous theorem,we get

c²=a²+b²

here c= slight height and a is radius

20²=17²+b²

20²-17²=b²

111=b²

b=√(111)

Plugging these values into the formula, we get:

Volume = ⅓*π* 17² *√(111) = 3188.5 cubic inches

Therefore, the volume of the cone is 3188.5 cubic inches.

The probability that a point chosen randomly inside the rectangle and is inside the Square or Trapezoid is 2/15.

We know that, probability of an event

= Number of favourable outcomes/Total number of outcomes.

Here, area of a rectangle = Length × Breadth

= 15×8

= 120 square units

Area of a triangle = 1/2 × Base × Height

= 1/2 ×(√10²-6²)×6

= 0.5×8×6

= 24 square units

Area of a trapezium = 1/2 (Sum of parallel sides)×Height

= 1/2 ×(5+7)×2

= 12 square units

Area of a square = side² = 2²

= 4 square units

(a) Probability of landing in trapezium or Square = 12/120 + 4/120

= 16/120

= 2/15

(b) Probability of landing inside the rectangle but outside the triangle = 16/120

= 2/15

Therefore, the probability that a point chosen randomly inside the rectangle and is inside the Square or Trapezoid is 2/15.

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Based on the given assumptions and calculations, the net present value (NPV) of the investment in the new piece of equipment is -$27,045.76, indicating that the investment is not favorable.

To calculate the after-tax cash flows for each year and evaluate the investment decision, let's use the following information:

Assumptions:

Tax depreciation is straight-line over five years.

Pre-tax salvage value is $10,000 in Year 5 and $15,000 if the asset is scrapped in Year 4.

Tax on salvage value is 30% of the difference between salvage value and book value of the investment.

The cost of capital is 12%.

Given:

Initial investment cost = $50,000

Useful life of the equipment = 5 years

To calculate the depreciation expense each year, we divide the initial investment by the useful life:

Depreciation expense per year = Initial investment / Useful life

Depreciation expense per year = $50,000 / 5 = $10,000

Now, let's calculate the book value at the end of each year:

Year 1:

Book value = Initial investment - Depreciation expense per year

Book value [tex]= $50,000 - $10,000 = $40,000[/tex]

Year 2:

Book value = Initial investment - (2 [tex]\times[/tex] Depreciation expense per year)

Book value [tex]= $50,000 - (2 \times$10,000) = $30,000[/tex]

Year 3:

Book value = Initial investment - (3 [tex]\times[/tex] Depreciation expense per year)

Book value = $50,000 - (3 [tex]\times[/tex] $10,000) = $20,000

Year 4:

Book value = Initial investment - (4 [tex]\times[/tex] Depreciation expense per year)

Book value [tex]= $50,000 - (4 \times $10,000) = $10,000[/tex]

Year 5:

Book value = Initial investment - (5 [tex]\times[/tex] Depreciation expense per year)

Book value [tex]= $50,000 - (5 \times $10,000) = $0[/tex]

Based on the assumptions, the salvage value is $10,000 in Year 5.

If the asset is scrapped in Year 4, the salvage value is $15,000.

To calculate the tax on salvage value, we need to find the difference between the salvage value and the book value and then multiply it by the tax rate:

Tax on salvage value = Tax rate [tex]\times[/tex] (Salvage value - Book value)

For Year 5:

Tax on salvage value[tex]= 0.30 \times ($10,000 - $0) = $3,000[/tex]

For Year 4 (if scrapped):

Tax on salvage value[tex]= 0.30 \times ($15,000 - $10,000) = $1,500[/tex]

Now, let's calculate the after-tax cash flows for each year:

Year 1:

After-tax cash flow = Depreciation expense per year - Tax on salvage value

After-tax cash flow = $10,000 - $0 = $10,000

Year 2:

After-tax cash flow = Salvage value - Tax on salvage value

After-tax cash flow = $0 - $0 = $0

Year 3:

After-tax cash flow = Salvage value - Tax on salvage value

After-tax cash flow = $0 - $0 = $0

Year 4 (if scrapped):

After-tax cash flow = Salvage value - Tax on salvage value

After-tax cash flow = $15,000 - $1,500 = $13,500

Year 5:

After-tax cash flow = Salvage value - Tax on salvage value

After-tax cash flow = $10,000 - $3,000 = $7,000

Now, let's calculate the net present value (NPV) using the cost of capital of 12%.

We will discount each year's after-tax cash flow to its present value using the formula:

[tex]PV = CF / (1 + r)^t[/tex]

Where:

PV = Present value

CF = Cash flow

r = Discount rate (cost of capital)

t = Time period (year)

NPV = PV Year 1 + PV Year 2 + PV Year 3 + PV Year 4 + PV Year 5 - Initial investment

Let's calculate the NPV:

PV Year 1 [tex]= $10,000 / (1 + 0.12)^1 = $8,928.57[/tex]

PV Year 2 [tex]= $0 / (1 + 0.12)^2 = $0[/tex]

PV Year 3 [tex]= $0 / (1 + 0.12)^3 = $0[/tex]

PV Year 4 [tex]= $13,500 / (1 + 0.12)^4 = $9,551.28[/tex]

PV Year 5 [tex]= $7,000 / (1 + 0.12)^5 = $4,474.39[/tex]

NPV = $8,928.57 + $0 + $0 + $9,551.28 + $4,474.39 - $50,000

NPV = $22,954.24 - $50,000

NPV = -$27,045.76

The NPV is negative, which means that based on the given assumptions and cost of capital, the investment in the new piece of equipment would result in a net loss.

Therefore, the investment may not be favorable.

Please note that the calculations above are based on the given assumptions, and additional factors or considerations specific to the business should also be taken into account when making investment decisions.

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The complete question may be like :

Assumptions: Tax depreciation is straight-line over five years. Pre-tax salvage value is $10,000 in Year 5 and $15,000 if the asset is scrapped in Year 4. Tax on salvage value is 30% of the difference between salvage value and book value of the investment. The cost of capital is 12%.

You are evaluating an investment in a new piece of equipment for your business. The initial investment cost is $50,000. The equipment is expected to have a useful life of five years.

Using the given assumptions, calculate the after-tax cash flows for each year and evaluate the investment decision by calculating the net present value (NPV) using the cost of capital of 12%.

To convert from degrees to radians, we use the conversion factor that 180 degrees is equal to π radians (or π/180 radians per degree).

Given that the angle is 45 degrees, we can calculate the radian measure as follows:

Radian measure = (45 degrees) * (π/180 radians per degree)

Radian measure = 45π/180

Simplifying further:

Radian measure = π/4

Therefore, the radian measure of a 45 degree angle is π/4.

The value of x for the second triangle if it is the reduction of the first drawing is 6.3 feet.

Given two triangles.

Also given that, the scale drawings given are the reduction of the figure.

Most probably, the second triangle must be formed after a reduction with a scale factor of k.

Let k be the required scale factor.

The value of k can be found by taking the ratio of the corresponding sides.

k = 5.2 / 10.4

  = 0.5

So all the corresponding sides will be half of the first triangle.

x = 0.5 × 12.6

  = 6.3 feet

Hence the value of x is 6.3 feet.

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f(x) is the quadratic function.  

g(x) is tripling its output for each 1-unit increase in x and g(x) = 3^x.

A quadratic function always grows more slowly than an exponential function.

So the best answer is:
f(x), because it grows slower than g(x)

The answers are given as:

Ai. The price at Store B would be $583 / 1.21 = $481.82

ii. The price at Store A would be $1200 * 1.21 = $1452.

b. The percent increase in membership from 2020 to 2021 is 13.64%

A.

i. If the price at Store A was $583, the price at Store B would be $583 / 1.21 = $481.82 (approximately).

ii. If the price at Store B was $1200, the price at Store A would be $1200 * 1.21 = $1452.

b. The percent increase in membership from 2020 to 2021 is ((12,500 - 11,000) / 11,000) * 100 = 13.64% (approximately).

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a) The volume of the hemisphere is V₁ = 261.80 km³

b) The volume of the hemisphere is V₂ = 1,526.81 feet³

Given data ,

a)

Let the volume of the hemisphere be represented as V₁

where the radius of the hemisphere is r₁ = 5 km

And , volume of hemisphere is V = ( 2/3 )πr³

On simplifying , we get

V₁ = ( 2/3 )π ( 5 )³

V₁ = 261.80 km³

b)

Let the volume of the hemisphere be represented as V₂

The diameter of the hemisphere = 18 feet

where the radius of the hemisphere is r₁ = 9 feet

And , volume of hemisphere is V = ( 2/3 )πr³

On simplifying , we get

V₂ = ( 2/3 )π ( 9 )³

V₂ = 1,526.81 feet³

Hence , the volume of the hemisphere is solved

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In this dataset, the mean is approximately 65.77, the median is 65, and the mode is 5 and 75.

Part A: Finding the mean of the data:

To find the mean, we need to calculate the average of all the data points.

Step 1: Identify the stems and their corresponding leaves:

3 | 5

4 | 4, 5

5 | 3, 6

6 | 2, 5

7 | 5, 5, 6

8 | 2, 5

9 | 2

Step 2: Assign numerical values to each stem-leaf combination:

3 | 5 = 35

4 | 4 = 44, 5 = 45

5 | 3 = 53, 6 = 56

6 | 2 = 62, 5 = 65

7 | 5 = 75, 5 = 75, 6 = 76

8 | 2 = 82, 5 = 85

9 | 2 = 92

Step 3: Calculate the sum of all the numerical values:

35 + 44 + 45 + 53 + 56 + 62 + 65 + 75 + 75 + 76 + 82 + 85 + 92 = 855

Step 4: Determine the count of all the data points:

The count is the total number of data points, which can be determined by adding up the frequencies of each stem-leaf combination:

1 (stem 3) + 2 (stem 4) + 2 (stem 5) + 2 (stem 6) + 3 (stem 7) + 2 (stem 8) + 1 (stem 9) = 13

Step 5: Calculate the mean by dividing the sum of all values by the count:

Mean = Sum of all values / Count = 855 / 13 = 65.77 (rounded to two decimal places)

The mean of the data is approximately 65.77.

Part B: Finding the median of the data:

To determine the median, we need to arrange the data in ascending order and find the middle value.

Arranging the data in ascending order: 35, 44, 45, 53, 56, 62, 65, 75, 75, 76, 82, 85, 92

There are 13 data points, the median will be the value in the middle. In this case, the middle value is the 7th value, which is 65.

The median of the data is 65.

Part C: Finding the mode of the data:

The mode represents the value(s) that occur with the highest frequency.

From the stem-leaf plot, we can see that the leaves with the highest frequency are 5 and 75. Both of these frequencies occur twice.

The mode of the data is 5 and 75.

Part D: Comparing the mean, median, and mode:

In this dataset, the mean is approximately 65.77, the median is 65, and the mode is 5 and 75.

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Answer: The expression can be simplified by combining the fractions with a common denominator:

(x - 9)/(x - 4) + (x^2 - x + 5)/(x - 4)

To add these fractions, we need to find a common denominator, which in this case is (x - 4). Therefore, we can rewrite each fraction with the common denominator:

[(x - 9) + (x^2 - x + 5)] / (x - 4)

Simplifying the numerator:

(x - 9 + x^2 - x + 5) / (x - 4)

Combining like terms:

(x^2 - 2x - 4) / (x - 4)

Hence, the simplified expression is (x^2 - 2x - 4) / (x - 4).

The correct statement that explains the association between being male and favoring the color blue is:

B. There is a negative association because the number of males who chose blue is less than the number of females who chose blue.

In the given table, it can be observed that out of the total 90 students surveyed, 9 students (4 males and 5 females) decided the color blue as their favorite primary color. Since the number of males (4) who selected blue is less than the number of females (5) who chose blue, there is a negative association between being male and favoring the color blue.