Question
(Diagram is not to scale.)
For a project in her Geometry class, Avery uses a mirror on the ground to measure the height of her school's football goalpost. She walks a distance of 14-55 meters from the goalpost, then plac
a mirror on flat on the ground, marked with an X at the center. She then steps 1.3 meters to the other side of the mirror, until she can see the top of the goalpost clearly marked in the X. Her
partner measures the distance from her eyes to the ground to be 1.45 meters. How tall is the goalpost? Round your answer to the nearest hundredth of a meter.

To solve this problem, we can use similar triangles and the concept of angles of elevation. The distance from Avery's eyes to the mirror should be the same as the distance from the top of the goalpost to the mirror. We can set up the following proportion:

Height of goalpost / 1.45m = x / 1.3m

Cross multiply to solve for x:

1.45m * x = 1.3m * height of goalpost

x = (1.3m * height of goalpost) / 1.45m

Given that the distance from Avery to the goalpost is 14.55 meters and the distance from the mirror to the goalpost is x + 1.3 meters, we have the sum of these lengths equals the height of the goalpost. This can be expressed as:

14.55m + x + 1.3m = height of the goalpost

Substitute the expression for x:

14.55m + (1.3m * height of goalpost) / 1.45m + 1.3m = height of the goalpost

Now, we can solve for the height of the goalpost:

Multiply each term by 1.45m to clear the fraction:

14.55m * 1.45m + 1.3m * 1.45m * height of goalpost + 1.45m * 1.3m = 1.45m * height of the goalpost

Now we can solve for the height of the goalpost.

height of the goalpost = (14.55m * 1.45m + 1.3m * 1.45m) / (1.45m - 1.3m)

height of the goalpost ≈ 16.70 meters

Rounding to the nearest hundredth, the height of the goalpost is approximately 16.70 meters.