|7. A surveyor needs to determine the distance across the pond shown below. She determines that the distance from her position to point P on the south shore of the pond is 15 feet and the angle from her position to point X on the north shore is 32° on the map. What will be the actual length of the pond from north to south, to the nearest foot?
|7. A surveyor needs to determine the distance across the pond shown below. She determines that the distance from her position to point P on the south shore of the pond is 15 feet and the angle from her position to point X on the north shore is 32° on the map. What will be the actual length of the pond from north to south, to the nearest foot?
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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![**Geometry Problems: Distance and Circles**
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**Problem 7: Determining Distance Across a Pond**
*A surveyor needs to determine the distance across the pond shown in the diagram. She determines that the distance from her position to point \( P \) on the south shore of the pond is 15 feet, and the angle from her position to point \( X \) on the north shore is \( 32^\circ \) on the map. What will be the actual length of the pond from north to south, to the nearest foot?*
*Explanation:*
This problem involves right triangle trigonometry. Given:
- The distance from the surveyor’s position to point \( P \) (adjacent side) = 15 feet.
- The angle (\( \theta \)) at the surveyor’s position = \( 32^\circ \).
To find the opposite side (north to south distance):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substitute the known values:
\[
\tan(32^\circ) = \frac{X}{15}
\]
Solving for \( X \):
\[
X = 15 \times \tan(32^\circ)
\]
Using a calculator to evaluate:
\[
X \approx 15 \times 0.6249 \approx 9.37
\]
Thus, the actual length of the pond from north to south is approximately **9 feet**.
*Diagram Description:*
The diagram depicts a triangle with the surveyor at the bottom-left corner, point \( P \) on the south shore to the right, and point \( X \) on the north shore vertically aligned above point \( P \). The angle at the surveyor's position is marked as \( 32^\circ \).
---
**Problem 8: Length of OP in a Circle**
*In circle \( O \), radius \( OC \) is 5 cm. Chord \( AB \) is 8 cm and is perpendicular to \( OC \) at point \( P \). What is the length of \( OP \)?*
*Explanation:*
This problem involves properties of circles and the Pythagorean Theorem. Given:
- Radius (\( OC \)) = 5 cm.
- Length of chord \( AB \) = 8 cm.
- \( AB \) is perpendicular to \( OC \) at point](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccc6c07f-901f-4b9b-9990-bf05e0ed87d5%2F1b9b293c-21ab-4103-8109-69461a32554d%2Ftn98knq_processed.png&w=3840&q=75)
Transcribed Image Text:**Geometry Problems: Distance and Circles**
---
**Problem 7: Determining Distance Across a Pond**
*A surveyor needs to determine the distance across the pond shown in the diagram. She determines that the distance from her position to point \( P \) on the south shore of the pond is 15 feet, and the angle from her position to point \( X \) on the north shore is \( 32^\circ \) on the map. What will be the actual length of the pond from north to south, to the nearest foot?*
*Explanation:*
This problem involves right triangle trigonometry. Given:
- The distance from the surveyor’s position to point \( P \) (adjacent side) = 15 feet.
- The angle (\( \theta \)) at the surveyor’s position = \( 32^\circ \).
To find the opposite side (north to south distance):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substitute the known values:
\[
\tan(32^\circ) = \frac{X}{15}
\]
Solving for \( X \):
\[
X = 15 \times \tan(32^\circ)
\]
Using a calculator to evaluate:
\[
X \approx 15 \times 0.6249 \approx 9.37
\]
Thus, the actual length of the pond from north to south is approximately **9 feet**.
*Diagram Description:*
The diagram depicts a triangle with the surveyor at the bottom-left corner, point \( P \) on the south shore to the right, and point \( X \) on the north shore vertically aligned above point \( P \). The angle at the surveyor's position is marked as \( 32^\circ \).
---
**Problem 8: Length of OP in a Circle**
*In circle \( O \), radius \( OC \) is 5 cm. Chord \( AB \) is 8 cm and is perpendicular to \( OC \) at point \( P \). What is the length of \( OP \)?*
*Explanation:*
This problem involves properties of circles and the Pythagorean Theorem. Given:
- Radius (\( OC \)) = 5 cm.
- Length of chord \( AB \) = 8 cm.
- \( AB \) is perpendicular to \( OC \) at point
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