An outdoor movie screen is 30 ft. tall. It needs additional support to keep it upright and prevent it from leaning. The town has decided to attach 15 ft. metal brackets to the back of the screen. The best spot to anchor the brackets to the ground is 6 ft. from the base of the screen. How far from the ground should the brackets be attached to the frame of the screen to ensure that the screen is at a right angle with the ground? Round to the nearest hundredth.

Question

(1 point)
Responses

15 ft.
15 ft.

29.39 ft.
29.39 ft.

16.16 ft.
16.16 ft.

13.75 ft.

Answer 1

To find the height at which to attach the brackets to the frame of the screen, we can use the Pythagorean theorem. The screen, the ground, and the bracket form a right triangle.

1. Let \( a \) be the height from the ground to the point where the bracket attaches to the screen.
2. The distance from the base of the screen to where the bracket is anchored is given as \( b = 6 \) ft.
3. The length of the bracket itself is \( c = 15 \) ft.

According to the Pythagorean theorem:

\[ c^2 = a^2 + b^2 \]

Substituting in the values:

\[ 15^2 = a^2 + 6^2 \]

This simplifies to:

\[ 225 = a^2 + 36 \]

Now, isolate \( a^2 \):

\[ a^2 = 225 - 36 \] \[ a^2 = 189 \]

Taking the square root of both sides to find \( a \):

\[ a = \sqrt{189} \approx 13.75 \text{ ft} \]

Thus, the height from the ground at which the brackets should be attached to the frame of the screen is approximately 13.75 ft.

Final Answer: 13.75 ft.