Question

a rectangular pyramid has a height of 5 units and a volume of 50 units³. shannon states that a rectangular prism with the same base area and height has a volume that is three times the size of the given rectangular pyramid. which statement explains whether shannon is correct?
a rectangular prism in which ba = 10 and h = 5 has a volume of 150 units³; therefore, shannon is correct
a rectangular prism in which ba = 30 and h = 5 has a volume of 150 units³; therefore, shannon is correct
a rectangular prism in which ba = 10 and h = 5 has a volume of 50 units³; therefore, shannon is incorrect
a rectangular prism in which ba = 30 and h = 5 has a volume of 50 units³; therefore, shannon is incorrect

Explanation:

Step1: Recall the volume formulas

The volume formula for a rectangular pyramid is \(V_{pyramid}=\frac{1}{3}BA\times h\), and for a rectangular prism is \(V_{prism}=BA\times h\), where \(BA\) is the base - area and \(h\) is the height.
Given \(V_{pyramid} = 50\) units³, \(h = 5\) units. From \(V_{pyramid}=\frac{1}{3}BA\times h\), we can find \(BA\). Substitute the values: \(50=\frac{1}{3}BA\times5\). Then \(BA=\frac{50\times3}{5}=30\) units².

Step2: Calculate the volume of the prism

For a rectangular prism with \(BA = 30\) units² and \(h = 5\) units, using the formula \(V_{prism}=BA\times h\), we get \(V_{prism}=30\times5 = 150\) units³.

Answer:

A rectangular prism in which \(BA = 30\) and \(h = 5\) has a volume of \(150\) units³; therefore, Shannon is correct.