(a) If a 8-ft distance produces a 64 ft^2 picture, what is the area of the picture when the projection unit is moved to a distance of 20 ft from the wall?
(b) If the projected image is 256 ft, how far is the projector from the wall?
(a) To find the area of the picture when the projection unit is moved to a distance of 20 ft from the wall, we can use the concept of similar triangles.
Let x be the area of the picture when the projection unit is moved to a distance of 20 ft from the wall.
Using the concept of similar triangles, we have:
\( \frac{8}{20} = \frac{64}{x} \)
Solving for x, we get:
\( x = \frac{20 \times 64}{8} = 160 \, ft^2 \)
Therefore, the area of the picture when the projection unit is moved to a distance of 20 ft from the wall is 160 ft^2.
(b) To find how far the projector is from the wall when the projected image is 256 ft, we can again use the concept of similar triangles.
Let y be the distance of the projector from the wall when the projected image is 256 ft.
Using the concept of similar triangles, we have:
\( \frac{8}{y} = \frac{64}{256} \)
Solving for y, we get:
\( y = \frac{8 \times 256}{64} = 32 \, ft \)
Therefore, the projector is 32 ft from the wall when the projected image is 256 ft.