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

13.75 ft.
13.75 ft.

29.39 ft.
29.39 ft.

16.16 ft.
16.16 ft.

15 ft.

Answer 1

To solve this problem, we can visualize a right triangle formed by the screen, the metal bracket, and the ground.

1. The height of the screen (vertical leg of the triangle) is 30 ft.
2. The length of the metal bracket (hypotenuse of the triangle) is 15 ft.
3. The distance from the base of the screen to where the bracket is anchored on the ground (horizontal leg of the triangle) is 6 ft.

We need to determine where to attach the bracket on the screen, which will be the height (h) at which the top of the bracket meets the screen.

Using the Pythagorean theorem:

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

Where:

• \(a\) is the height from the ground to the attachment point,
• \(b\) is the distance from the base of the screen to where the bracket is anchored (6 ft),
• \(c\) is the length of the bracket (15 ft).

This translates to:

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

Calculating the squares:

\[ h^2 + 36 = 225 \]

Subtract 36 from both sides to isolate \(h^2\):

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

Now, take the square root of both sides:

\[ h = \sqrt{189} ≈ 13.75 \]

Therefore, the height from the ground where the brackets should be attached to the frame of the screen is approximately 13.75 ft.

The correct response is: 13.75 ft.