Question

The height of a square pyramid shaped gift box is half the length of each side. The volume of the gift box is 4,500in.What is the height of the gift box, in inches?

Answer 1

Answer: the height of the gift box is 9 inches.

Step-by-step explanation:

Let's start by using the formula for the volume of a square pyramid:

V = (1/3) * B * h

where V is the volume, B is the area of the base, and h is the height.

For a square pyramid, the area of the base is:

B = s^2

where s is the length of one side.

We are given that the height of the gift box is half the length of each side, so:

h = (1/2) * s

Substituting this into the formula for the volume, we get:

4500 = (1/3) * s^2 * (1/2) * s

Multiplying both sides by 3/2 to get rid of the fraction, we get:

6750 = s^3

Taking the cube root of both sides, we get:

s = 18

Therefore, the height of the gift box is:

h = (1/2) * s = (1/2) * 18 = 9 inches

So the height of the gift box is 9 inches.

Answer 2

V = (1/3) * base area * height

We know that the volume of the gift box is 4,500 in³, and we can also express the base area in terms of the length of each side:

V = (1/3) * (side length)² * height

Since the height is half the length of each side, we can substitute 2h for the length:

V = (1/3) * (2h)² * h

Simplifying the expression:

4,500 = (4/3) * h³

Multiplying both sides by 3/4:

h³ = 3,375

Taking the cube root of both sides:

h = 15

Therefore, the height of the gift box is 15 inches.