From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of the tower is 78°, as shown in the accompanying diagram. Find the height of the tower, to the nearest tenth of a foot. 78° 25 ft Tower
From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of the tower is 78°, as shown in the accompanying diagram. Find the height of the tower, to the nearest tenth of a foot. 78° 25 ft Tower
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![**Problem Statement:**
From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of the tower is 78°, as shown in the accompanying diagram. Find the height of the tower, to the nearest tenth of a foot.
**Accompanying Diagram:**
The diagram is a right triangle where:
- One leg is labeled "25 ft" and represents the distance from the observer to the base of the tower.
- The angle between this leg and the hypotenuse is 78°, representing the angle of elevation from the observer to the top of the tower.
- The other leg, which we need to find, represents the height of the tower and is labeled "Tower".
**Solution Approach:**
1. Recognize the need to use a trigonometric function to solve for the height of the tower.
2. Use the tangent function, which is the ratio of the opposite side (height of the tower) to the adjacent side (distance from the observer to the base of the tower).
\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
3. Insert the known values into the equation:
\[
\tan(78^\circ) = \frac{\text{height}}{25 \, \text{ft}}
\]
4. Solve for the height:
\[
\text{height} = 25 \, \text{ft} \times \tan(78^\circ)
\]
Use a calculator to find \(\tan(78^\circ)\):
\[
\tan(78^\circ) \approx 4.7046
\]
5. Multiply these values together to solve for the height:
\[
\text{height} \approx 25 \, \text{ft} \times 4.7046 \approx 117.6 \, \text{ft}
\]
**Final Answer:**
The height of the tower is approximately 117.6 feet.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccc6c07f-901f-4b9b-9990-bf05e0ed87d5%2F90d82f24-1349-4b69-9b28-5f8a63f7c63a%2F9ixrxhd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of the tower is 78°, as shown in the accompanying diagram. Find the height of the tower, to the nearest tenth of a foot.
**Accompanying Diagram:**
The diagram is a right triangle where:
- One leg is labeled "25 ft" and represents the distance from the observer to the base of the tower.
- The angle between this leg and the hypotenuse is 78°, representing the angle of elevation from the observer to the top of the tower.
- The other leg, which we need to find, represents the height of the tower and is labeled "Tower".
**Solution Approach:**
1. Recognize the need to use a trigonometric function to solve for the height of the tower.
2. Use the tangent function, which is the ratio of the opposite side (height of the tower) to the adjacent side (distance from the observer to the base of the tower).
\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
3. Insert the known values into the equation:
\[
\tan(78^\circ) = \frac{\text{height}}{25 \, \text{ft}}
\]
4. Solve for the height:
\[
\text{height} = 25 \, \text{ft} \times \tan(78^\circ)
\]
Use a calculator to find \(\tan(78^\circ)\):
\[
\tan(78^\circ) \approx 4.7046
\]
5. Multiply these values together to solve for the height:
\[
\text{height} \approx 25 \, \text{ft} \times 4.7046 \approx 117.6 \, \text{ft}
\]
**Final Answer:**
The height of the tower is approximately 117.6 feet.
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