Question

The volume of a rectangular pyramid is 60 units³. If the width of the base is scaled by a factor of 3, the length by a factor of 2, and the height by a factor 'k' to achieve a volume 12 times that of the original, what is the scale factor 'k' for the height?

a. 2
b. 3
c. 4
d. 6

Answer 1

The scale factor 'k' for the height is 12.

Step-by-step explanation:

To find the scale factor 'k' for the height, we need to set up an equation using the given information. Let the original height of the rectangular pyramid be h. The volume of the original pyramid is given as 60 units³, so the volume of the new pyramid is 12 times that, which is 720 units³.

Since we scaled the width by a factor of 3 and the length by a factor of 2, the new base area is (3w)(2l) = 6wl times the original base area.

Using the formula for the volume of a rectangular pyramid, V = (1/3) base area x height, we can set up the equation:

(1/3)(6wl)(k*h) = 720

Simplifying, we find 2wlkh = 720.

Since we know the volume is proportional to the original volume, we can set up a ratio:

2wlk/60 = 12/1.

Cross multiplying and simplifying, we get 2k = 24, so k = 12.