Question

A solid consists of a square-based prism of height 6.5 cm, exactly fitting of a right pyramid of slant height 5 cm on the top. If the volume of the lower part is 416 cm³, find (i) the height of the pyramid. (ii) the volume of the pyramid.

A) (i) 8.1 cm (ii) 108.33 cm³.
B) (i) 5.7 cm (ii) 66.67 cm³.
C) (i) 6.5 cm (ii) 54.16 cm³.
D) (i) 7.2 cm (ii) 81 cm³.

Answer 1

To find the height of the pyramid, we need to find the value of x that makes the sum of the volumes of the lower and upper parts equal to 416 cm³. Once we find x, we can substitute it into the formula for the height of the pyramid to find its value.

Step-by-step explanation:

The volume of the lower part, which is a square-based prism, is given as 416 cm³. Let's assume that the side length of the square base of the prism is x cm. Therefore, the volume of the prism can be expressed as V = x² * 6.5 cm³.

The volume of the upper part, which is a pyramid, can be calculated using the formula V = 1/3 * A * h, where A is the area of the base of the pyramid and h is its height. Since the slant height of the pyramid is given as 5 cm, we can use Pythagoras' theorem to find the height of the pyramid: h = √(5² - x²) cm.

To find the height of the pyramid, we need to find the value of x that makes the sum of the volumes of the lower and upper parts equal to 416 cm³. Once we find x, we can substitute it into the formula for the height of the pyramid to find its value.