Two similar pyramids have laterl area 8ft^2 and 18ft^2. The volume of the smaller pyramid is 32ft^3. Find the volume of the larger pyramid.
To find the volume of the larger pyramid, we need to use the given information about the lateral areas of the two pyramids.
Step 1: Understand the problem.
We have two similar pyramids, which means they have the same shape but different sizes. We are given the lateral area (the area of the sides) of both pyramids and the volume of the smaller pyramid. We need to find the volume of the larger pyramid.
Step 2: Determine the relationship between the lateral area and the volume of a pyramid.
The lateral area of a pyramid is related to its volume through a common ratio. In general, if the dimensions of two pyramids are in the ratio of a:b, then the ratio of their volumes is a^3:b^3. In this case, since we are dealing with similar pyramids, we can assume that the ratio of the lateral areas is equal to the ratio of the volumes raised to the power of 2/3.
Step 3: Set up the equation.
Let's denote the lateral area and volume of the smaller pyramid as A1 and V1, respectively, and the lateral area and volume of the larger pyramid as A2 and V2, respectively. We can set up the following equation:
(A2 / A1) = (V2 / V1)^(2/3)
Step 4: Substitute the given information into the equation.
We are given A1 = 8 ft^2, A2 = 18 ft^2, and V1 = 32 ft^3. Let's substitute these values into the equation:
(18 / 8) = (V2 / 32)^(2/3)
Step 5: Solve for V2.
To find V2, we need to isolate it on one side of the equation. Start by raising both sides of the equation to the power of (3/2):
[(18 / 8)^(3/2)] = [(V2 / 32)^(2/3)]^(3/2)
Simplifying further:
[(18 / 8)^(3/2)] = (V2 / 32)
We can rearrange this equation to solve for V2:
V2 = [(18 / 8)^(3/2)] * 32
Step 6: Calculate the volume of the larger pyramid.
Using a calculator, calculate the value of the expression [(18 / 8)^(3/2)] * 32 to find V2. The result will be the volume of the larger pyramid.
I hope this explanation helps you understand how to solve the problem and find the volume of the larger pyramid.