Question

Pyramid A is a square pyramid with a base side length of 18 inches and a height of 9 inches. Pyramid B has a volume of 3,136 cubic inches. How many times bigger is the volume of pyramid B than pyramid A? Give your answer as a percentage. Provide an explanation and proof for your answer to receive full credit.

Answer 1

To solve the problem, we first need to calculate the volume of Pyramid A and then compare it to the volume of Pyramid B to find how many times bigger Pyramid B's volume is than Pyramid A's volume.

Step 1: Calculate the volume of Pyramid A.

The formula for the volume $V$ of a pyramid is: $$V=\frac{1}{3}\times \text{Base Area}\times \text{Height}$$

Base Area of Pyramid A:

Since Pyramid A has a square base, we can calculate the base area as: $$\text{Base Area}={\text{side length}}^2={18}^2=324\text{ square inches}$$

Height of Pyramid A:

The height of Pyramid A is given as 9 inches.

Volume of Pyramid A:

Now we can substitute these values into the volume formula: $$V_A=\frac{1}{3}\times 324\times 9$$ Calculating this: $$V_A=\frac{1}{3}\times 2916=972\text{ cubic inches}$$

Step 2: Compare volumes of Pyramid A and Pyramid B.

The volume of Pyramid B is given as 3,136 cubic inches.

Step 3: Determine how many times bigger the volume of Pyramid B is than that of Pyramid A.

To compare the volumes, we can calculate the ratio: $$\text{Ratio}=\frac{V_B}{V_A}=\frac{3136}{972}$$

Calculating this ratio: $$\text{Ratio}\approx 3.225(roundedtothreedecimalplaces)$$

Step 4: Convert the ratio to a percentage.

To find out how many times bigger Pyramid B is than Pyramid A in percentage, we take the ratio calculated, subtract 1 (the original volume of Pyramid A), and multiply by 100: $$\text{Percentage Increase}=(\text{Ratio}-1)\times 100$$ Calculating this: $$\text{Percentage Increase}=(3.225-1)\times 100\approx 222.5$$

Conclusion

Pyramid B is approximately 222.5% bigger than Pyramid A's volume.

This indicates that Pyramid B's volume is more than three times the volume of Pyramid A, making it significantly larger in comparison.