Math / Trigonometry

QuestionA radio tower is located 275 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is $37^{\circ}$ and that the angle of depression to the bottom of the tower is $20^{\circ}$ . How tall is the tower? $\square$ feet
Give your answer rounded to the nearest foot.

Studdy Solution

STEP 1

What is this asking? We need to find the height of a radio tower given the distance from a building and the angles of elevation to the top and depression to the bottom of the tower from a window in the building. Watch out! Don't mix up the angles of elevation and depression!
Also, remember we're looking for the total height of the tower.

STEP 2

1. Split the Tower
2. Top Half Tango
3. Bottom Half Boogie
4. Tower Time!

STEP 3

Imagine slicing the tower horizontally at the level of the window.
This creates two right triangles!
One triangle is formed by the window, the top of the tower, and the horizontal distance to the tower.
The other triangle is formed by the window, the bottom of the tower, and the horizontal distance to the tower.

STEP 4

Let's call the height of the top part of the tower $h_1$ .
We know the angle of elevation to the top of the tower is $37^\circ$ and the horizontal distance to the tower is $\text{275}$ feet.
We can use the tangent function!

STEP 5

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(37^\circ) = \frac{h_1}{\text{275}}$ .

STEP 6

Multiply both sides of the equation by $\text{275}$ to isolate $h_1$ .
This gives us $h_1 = 275 \cdot \tan(37^\circ)$ .

STEP 7

Calculate $h_1$ : $h_1 = 275 \cdot \tan(37^\circ) \approx 275 \cdot \textbf{0.7536} \approx \textbf{207.24}$ feet.
So, the top portion of the tower is approximately $\textbf{207.24}$ feet tall!

STEP 8

Now, let's call the height of the bottom part of the tower $h_2$ .
The angle of depression is $20^\circ$ , which is the same as the angle of elevation from the base of the tower to the window.
The horizontal distance is still $\text{275}$ feet.
We can use the tangent function again!

STEP 9

We have $\tan(20^\circ) = \frac{h_2}{\text{275}}$ .

STEP 10

Multiply both sides by $\text{275}$ to get $h_2 = 275 \cdot \tan(20^\circ)$ .

STEP 11

Calculate $h_2$ : $h_2 = 275 \cdot \tan(20^\circ) \approx 275 \cdot \textbf{0.3640} \approx \textbf{99.10}$ feet.
The bottom portion of the tower is approximately $\textbf{99.10}$ feet tall.

STEP 12

The total height of the tower is the sum of the heights of the top and bottom portions: $h = h_1 + h_2$ .

STEP 13

Substitute the values we found: $h \approx 207.24 + 99.10$ .

STEP 14

Calculate the total height: $h \approx \textbf{306.34}$ feet.
Rounding to the nearest foot, we get $\textbf{306}$ feet.

STEP 15

The radio tower is approximately $\text{306}$ feet tall.

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Studdy Solution

STEP 1

STEP 2

1. Split the Tower
2. Top Half Tango
3. Bottom Half Boogie
4. Tower Time!

STEP 3

STEP 4

Let's call the height of the top part of the tower $h_1$ .
We know the angle of elevation to the top of the tower is $37^\circ$ and the horizontal distance to the tower is $\text{275}$ feet.
We can use the tangent function!

STEP 5

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(37^\circ) = \frac{h_1}{\text{275}}$ .

STEP 6

Multiply both sides of the equation by $\text{275}$ to isolate $h_1$ .
This gives us $h_1 = 275 \cdot \tan(37^\circ)$ .

STEP 7

Calculate $h_1$ : $h_1 = 275 \cdot \tan(37^\circ) \approx 275 \cdot \textbf{0.7536} \approx \textbf{207.24}$ feet.
So, the top portion of the tower is approximately $\textbf{207.24}$ feet tall!

STEP 8

Now, let's call the height of the bottom part of the tower $h_2$ .
The angle of depression is $20^\circ$ , which is the same as the angle of elevation from the base of the tower to the window.
The horizontal distance is still $\text{275}$ feet.
We can use the tangent function again!

STEP 9

We have $\tan(20^\circ) = \frac{h_2}{\text{275}}$ .

STEP 10

Multiply both sides by $\text{275}$ to get $h_2 = 275 \cdot \tan(20^\circ)$ .

STEP 11

Calculate $h_2$ : $h_2 = 275 \cdot \tan(20^\circ) \approx 275 \cdot \textbf{0.3640} \approx \textbf{99.10}$ feet.
The bottom portion of the tower is approximately $\textbf{99.10}$ feet tall.

STEP 12

The total height of the tower is the sum of the heights of the top and bottom portions: $h = h_1 + h_2$ .

STEP 13

Substitute the values we found: $h \approx 207.24 + 99.10$ .

STEP 14

Calculate the total height: $h \approx \textbf{306.34}$ feet.
Rounding to the nearest foot, we get $\textbf{306}$ feet.

STEP 15

The radio tower is approximately $\text{306}$ feet tall.

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Studdy Solution

STEP 1

STEP 2

1. Split the Tower2. Top Half Tango3. Bottom Half Boogie4. Tower Time!

STEP 3

STEP 4

Let's call the height of the top part of the tower h1h_1h1​.We know the angle of elevation to the top of the tower is 37∘37^\circ37∘ and the horizontal distance to the tower is 275\text{275}275 feet.We can use the tangent function!

STEP 5

Remember, tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}tan(θ)=adjacentopposite​.In our case, tan⁡(37∘)=h1275\tan(37^\circ) = \frac{h_1}{\text{275}}tan(37∘)=275h1​​.

STEP 6

**Multiply both sides** of the equation by 275\text{275}275 to **isolate** h1h_1h1​.This gives us h1=275⋅tan⁡(37∘)h_1 = 275 \cdot \tan(37^\circ)h1​=275⋅tan(37∘).

STEP 7

STEP 8

Now, let's call the height of the bottom part of the tower h2h_2h2​.The angle of depression is 20∘20^\circ20∘, which is the same as the angle of elevation from the base of the tower to the window.The horizontal distance is still 275\text{275}275 feet.We can use the tangent function again!

STEP 9

We have tan⁡(20∘)=h2275\tan(20^\circ) = \frac{h_2}{\text{275}}tan(20∘)=275h2​​.

STEP 10

**Multiply both sides** by 275\text{275}275 to get h2=275⋅tan⁡(20∘)h_2 = 275 \cdot \tan(20^\circ)h2​=275⋅tan(20∘).

STEP 11

**Calculate** h2h_2h2​: h2=275⋅tan⁡(20∘)≈275⋅0.3640≈99.10h_2 = 275 \cdot \tan(20^\circ) \approx 275 \cdot \textbf{0.3640} \approx \textbf{99.10}h2​=275⋅tan(20∘)≈275⋅0.3640≈99.10 feet.The bottom portion of the tower is approximately 99.10\textbf{99.10}99.10 feet tall.

STEP 12

The **total height** of the tower is the sum of the heights of the top and bottom portions: h=h1+h2h = h_1 + h_2h=h1​+h2​.

STEP 13

**Substitute** the values we found: h≈207.24+99.10h \approx 207.24 + 99.10h≈207.24+99.10.

STEP 14

**Calculate** the total height: h≈306.34h \approx \textbf{306.34}h≈306.34 feet.Rounding to the nearest foot, we get 306\textbf{306}306 feet.

STEP 15

The radio tower is approximately 306\text{306}306 feet tall.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Split the Tower
2. Top Half Tango
3. Bottom Half Boogie
4. Tower Time!

Let's call the height of the top part of the tower $h_1$ .
We know the angle of elevation to the top of the tower is $37^\circ$ and the horizontal distance to the tower is $\text{275}$ feet.
We can use the tangent function!

Remember, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .
In our case, $\tan(37^\circ) = \frac{h_1}{\text{275}}$ .

Multiply both sides of the equation by $\text{275}$ to isolate $h_1$ .
This gives us $h_1 = 275 \cdot \tan(37^\circ)$ .

Now, let's call the height of the bottom part of the tower $h_2$ .
The angle of depression is $20^\circ$ , which is the same as the angle of elevation from the base of the tower to the window.
The horizontal distance is still $\text{275}$ feet.
We can use the tangent function again!

We have $\tan(20^\circ) = \frac{h_2}{\text{275}}$ .

Multiply both sides by $\text{275}$ to get $h_2 = 275 \cdot \tan(20^\circ)$ .

Calculate $h_2$ : $h_2 = 275 \cdot \tan(20^\circ) \approx 275 \cdot \textbf{0.3640} \approx \textbf{99.10}$ feet.
The bottom portion of the tower is approximately $\textbf{99.10}$ feet tall.

The total height of the tower is the sum of the heights of the top and bottom portions: $h = h_1 + h_2$ .

Substitute the values we found: $h \approx 207.24 + 99.10$ .

Calculate the total height: $h \approx \textbf{306.34}$ feet.
Rounding to the nearest foot, we get $\textbf{306}$ feet.

The radio tower is approximately $\text{306}$ feet tall.