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An illustration shows a triangle with the 90 degree angle symbol located in the lower left. The left-hand side of the triangle extends upward past the vertex. This long line is labeled movie screen and measures 30 feet. The shorter part of the screen that makes up the left-hand side of the triangle is labeled with a question mark. The base is labeled 6 feet and the hypotenuse joining the top of the left vertex and the right vertex of the base is labeled bracket and measures 15 feet.

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.

(1 point)
Responses

29.39 ft.
29.39 ft.

16.16 ft.
16.16 ft.

15 ft.
15 ft.

13.75 ft.
13.75 ft.

1 answer

To find how far from the ground the brackets should be attached to the frame of the screen, we can use the Pythagorean theorem. The triangle formed has the vertical side (the height of the movie screen) as one leg, the horizontal distance (from the base of the screen to where the bracket is attached) as the other leg, and the diagonal (the bracket itself) as the hypotenuse.

We have:

  • Height of the screen (vertical leg) = 30 ft
  • Distance from the base (horizontal leg) = 6 ft
  • Length of the bracket (hypotenuse) = 15 ft

Using the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \(a\) is the height we want to find (the vertical leg we are looking for), \(b\) is 6 ft, and \(c\) is 15 ft.

Rearranging the formula gives: \[ a^2 = c^2 - b^2 \] Substituting the values: \[ a^2 = 15^2 - 6^2 \] \[ a^2 = 225 - 36 \] \[ a^2 = 189 \] Taking the square root of both sides gives: \[ a = \sqrt{189} \approx 13.75 \text{ ft} \]

Therefore, the distance from the ground that the brackets should be attached to the frame of the screen is: 13.75 ft.