Question

a carpenter designs two cabinets: one in the shape of an oblique rectangular prism and one in the shape of a right rectangular prism. the volume of each cabinet is 4,608 cubic inches. the oblique rectangular prism is 48 inches tall and has an edge length of 64 inches. the right rectangular prism has a height of 48 inches. which statements about the cabinets are true? check all that apply.
the cabinets have the same base area.
the base area of the oblique prism is smaller than the base area of the right prism.
the cabinets may have the same base dimensions.
the cabinets may have different base dimensions.
the base area of the right prism is smaller than the base area of the oblique prism.

Explanation:

Step1: Recall the volume formula for prisms

The volume formula for a prism is \(V = B\times h\), where \(V\) is the volume, \(B\) is the base area, and \(h\) is the height (or length of the perpendicular segment from the base for an oblique prism).
For the oblique prism, \(V = 4608\) cubic inches and \(h = 48\) inches. Using the formula \(B=\frac{V}{h}\), we have \(B=\frac{4608}{48}=96\) square inches.
For the right - rectangular prism, \(V = 4608\) cubic inches and \(h = 48\) inches. Using the formula \(B=\frac{V}{h}\), we also have \(B=\frac{4608}{48}=96\) square inches.

Step2: Analyze the base - area relationship

Since the base area \(B\) of both prisms (calculated as \(96\) square inches) is the same, the statement “The cabinets have the same base area” is True.

Step3: Analyze the base - dimension relationship

The base area \(B = l\times w\) (for a rectangular base). If \(B = 96\), we can have different pairs of \(l\) and \(w\) values (e.g., if \(l = 16\) and \(w = 6\) or \(l= 32\) and \(w = 3\)). So, the cabinets may have the same base dimensions (if \(l_1=l_2\) and \(w_1 = w_2\)) or different base dimensions (if \(l_1
eq l_2\) and/or \(w_1
eq w_2\)) as long as \(l\times w=96\).

Answer:

• The cabinets have the same base area.
• The cabinets may have the same base dimensions.
• The cabinets may have different base dimensions.