A radio tower is located 325 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 39° and that the angle of depression to the bottom of the tower is 29° . How tall is the tower? feet Give your answer rounded to the nearest foot.

Intermediate Algebra
19th Edition
ISBN:9780998625720
Author:Lynn Marecek
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Chapter7: Rational Expressions And Functions
Section7.5: Solve Applications With Rational Equations
Problem 7.92TI: A pine tree casts a shadow of 80 feet next to a 30 foot tall building which casts a 40 feet shadow....
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How would I figure out which trig function to use based on the given values, to find the value in foot?

**Problem Statement:**

A radio tower is located 325 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 39° and that the angle of depression to the bottom of the tower is 29°. How tall is the tower?

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**Answer Submission:**

[__] feet

*Give your answer rounded to the nearest foot.*

---

**Explanation:**

To solve this problem, we can use trigonometry. Here are the steps:

1. **Identify Variables:**
   - Distance between the building and the tower \( d = 325 \, \text{feet} \)
   - Angle of elevation to the top of the tower \( \theta_1 = 39^\circ \)
   - Angle of depression to the bottom of the tower \( \theta_2 = 29^\circ \)

2. **Set Up Right Triangles:**
   - From the angle of elevation and the distance \( d \), we can find the height from the window to the top of the tower \( h_1 \).
   - From the angle of depression and the distance \( d \), we can find the height from the window to the bottom of the tower \( h_2 \).

3. **Apply Trigonometric Functions:**
   - Use the tangent function (since tangent relates the angle of a right triangle to the ratio of the opposite side and adjacent side).

4. **Calculations:**
   - \( \tan(39^\circ) = \frac{h_1}{325} \)
   - \( h_1 = 325 \times \tan(39^\circ) \)
   
   - \( \tan(29^\circ) = \frac{h_2}{325} \)
   - \( h_2 = 325 \times \tan(29^\circ) \)
   
5. **Sum Heights to Find Total Tower Height:**
   - Total height of the tower \( H = h_1 + h_2 \)

Lastly, round the final result to the nearest foot. Use a calculator to perform the trigonometric calculations and to obtain the numerical result.

---

Make sure to verify your calculations carefully to ensure accuracy. Understanding these steps helps in solving similar problems involving angles of elevation and depression.
Transcribed Image Text:**Problem Statement:** A radio tower is located 325 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 39° and that the angle of depression to the bottom of the tower is 29°. How tall is the tower? --- **Answer Submission:** [__] feet *Give your answer rounded to the nearest foot.* --- **Explanation:** To solve this problem, we can use trigonometry. Here are the steps: 1. **Identify Variables:** - Distance between the building and the tower \( d = 325 \, \text{feet} \) - Angle of elevation to the top of the tower \( \theta_1 = 39^\circ \) - Angle of depression to the bottom of the tower \( \theta_2 = 29^\circ \) 2. **Set Up Right Triangles:** - From the angle of elevation and the distance \( d \), we can find the height from the window to the top of the tower \( h_1 \). - From the angle of depression and the distance \( d \), we can find the height from the window to the bottom of the tower \( h_2 \). 3. **Apply Trigonometric Functions:** - Use the tangent function (since tangent relates the angle of a right triangle to the ratio of the opposite side and adjacent side). 4. **Calculations:** - \( \tan(39^\circ) = \frac{h_1}{325} \) - \( h_1 = 325 \times \tan(39^\circ) \) - \( \tan(29^\circ) = \frac{h_2}{325} \) - \( h_2 = 325 \times \tan(29^\circ) \) 5. **Sum Heights to Find Total Tower Height:** - Total height of the tower \( H = h_1 + h_2 \) Lastly, round the final result to the nearest foot. Use a calculator to perform the trigonometric calculations and to obtain the numerical result. --- Make sure to verify your calculations carefully to ensure accuracy. Understanding these steps helps in solving similar problems involving angles of elevation and depression.
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