For a project in his Geometry class, Mamadou uses a mirror on the ground to measure the height of his school’s football goalpost. He walks a distance of 13.75 meters from his school, then places a mirror on flat on the ground, marked with an X at the center. He then steps 2.6 meters to the other side of the mirror, until he can see the top of the goalpost clearly marked in the X. His partner measures the distance from his eyes to the ground to be 1.75 meters. How tall is the goalpost? Round your answer to the nearest hundredth of a meter. ### Problem StatementA person is standing on the ground and is looking at the top of a goalpost. The height from the person's eyes to the ground is 1.75 meters. The horizontal distance from the person to the base of the goalpost is 13.75 meters. The distance from the person's eyes to the top of the goalpost (along the line of sight) is 14 meters.**Question:** How tall is the goalpost? Round your answer to the nearest hundredth of a meter.**Diagram:**- The diagram is not to scale.- There is a stick figure representing the person, with a line from their eyes to the top of the goalpost.- The figure's height from eyes to ground: 1.75 meters.- The horizontal distance from the person to the goalpost: 13.75 meters.- A dotted line represents the line of sight with a length of 14 meters.**Steps to Solve:**To solve for the height of the goalpost, we can use the Pythagorean Theorem in the triangle formed by the person's eyes, the base of the goalpost, and the top of the goalpost.Using the Pythagorean theorem:\[c^2 = a^2 + b^2\]Where:- $c$ = line of sight (14 meters)- $a$ = horizontal distance (13.75 meters)- The difference in height ($b$) from the eyes to the top of the goalpost is what we need to find. Once found, add the person’s eye height to find the total height of the goalpost.Rearrange to solve for $b$:\[b = \sqrt{c^2 - a^2}\]\[b = \sqrt{14^2 - 13.75^2}\]Calculate:\[14^2 = 196\]\[13.75^2 = 189.0625\]\[b = \sqrt{196 - 189.0625} = \sqrt{6.9375} \approx 2.63 \text{ meters}\]Finally, add the eye-level height of the person:\[\text{Height of the goalpost} = 2.63 \text{ meters} + 1.75 \text{ meters} = 4.

To find height of the school's football goalpost.

Answered: project in his Geometry class, Mamadou…

PREALGEBRA

15th Edition

ISBN:9781938168994

Author:OpenStax

Publisher:OpenStax

Chapter9: Math Models And Geometry

Section9.3: Use Properties Of Angles, Triangles, And The Pythagorean Theorem

Problem 108E: On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides the shown in the...

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Question

For a project in his Geometry class, Mamadou uses a mirror on the ground to measure the height of his school’s football goalpost. He walks a distance of 13.75 meters from his school, then places a mirror on flat on the ground, marked with an X at the center. He then steps 2.6 meters to the other side of the mirror, until he can see the top of the goalpost clearly marked in the X. His partner measures the distance from his eyes to the ground to be 1.75 meters. How tall is the goalpost? Round your answer to the nearest hundredth of a meter.

### Problem Statement

A person is standing on the ground and is looking at the top of a goalpost. The height from the person's eyes to the ground is 1.75 meters. The horizontal distance from the person to the base of the goalpost is 13.75 meters. The distance from the person's eyes to the top of the goalpost (along the line of sight) is 14 meters.

**Question:** How tall is the goalpost? Round your answer to the nearest hundredth of a meter.

**Diagram:**
- The diagram is not to scale.
- There is a stick figure representing the person, with a line from their eyes to the top of the goalpost.
- The figure's height from eyes to ground: 1.75 meters.
- The horizontal distance from the person to the goalpost: 13.75 meters.
- A dotted line represents the line of sight with a length of 14 meters.

**Steps to Solve:**
To solve for the height of the goalpost, we can use the Pythagorean Theorem in the triangle formed by the person's eyes, the base of the goalpost, and the top of the goalpost.

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]

Where:
- $c$ = line of sight (14 meters)
- $a$ = horizontal distance (13.75 meters)
- The difference in height ($b$) from the eyes to the top of the goalpost is what we need to find. Once found, add the person’s eye height to find the total height of the goalpost.

Rearrange to solve for $b$:
\[
b = \sqrt{c^2 - a^2}
\]
\[
b = \sqrt{14^2 - 13.75^2}
\]
Calculate:
\[
14^2 = 196
\]
\[
13.75^2 = 189.0625
\]
\[
b = \sqrt{196 - 189.0625} = \sqrt{6.9375} \approx 2.63 \text{ meters}
\]

Finally, add the eye-level height of the person:
\[
\text{Height of the goalpost} = 2.63 \text{ meters} + 1.75 \text{ meters} = 4.

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