Question

pyramid a is a right square pyramid with a height of 131 m and a square base with a perimeter of 344 m. pyramid b has a square base with a perimeter of 244 m and a height of 115 m. which pyramid has the greater volume and by how many times as great? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the volume of pyramid b is times greater than the volume of pyramid a. (round to the nearest tenth as needed) b. the volume of pyramid a is times greater than the volume of pyramid b. (round to the nearest tenth as needed) c. the two pyramids have equal volume.

Explanation:

Step1: Calculate the side length of the base for each pyramid

For Pyramid A:
The perimeter of the square base is \(P = 344\) m. Since \(P=4s\) (where \(s\) is the side - length of the square), then \(s_A=\frac{344}{4}=86\) m.
For Pyramid B:
The perimeter of the square base is \(P = 244\) m. Since \(P = 4s\), then \(s_B=\frac{244}{4}=61\) m.

Step2: Use the volume formula for a pyramid \(V=\frac{1}{3}Bh\) (where \(B\) is the area of the base and \(h\) is the height)

For Pyramid A:
\(B_A=s_A^2=(86)^2 = 7396\) \(m^2\), \(h_A = 131\) m.
\(V_A=\frac{1}{3}\times7396\times131=\frac{7396\times131}{3}=\frac{978876}{3}=326292\) \(m^3\).
For Pyramid B:
\(B_B=s_B^2=(61)^2 = 3721\) \(m^2\), \(h_B = 115\) m.
\(V_B=\frac{1}{3}\times3721\times115=\frac{3721\times115}{3}=\frac{427915}{3}\approx142638.3\) \(m^3\).

Step3: Find the ratio of the volumes

\(\frac{V_A}{V_B}=\frac{326292}{\frac{427915}{3}}=\frac{326292\times3}{427915}=\frac{978876}{427915}\approx2.3\)

Answer:

B. The volume of pyramid A is \(2.3\) times greater than the volume of pyramid B.