Question

the figure shows a right prism with 2 congruent square bases and 4 congruent lateral faces. the length, in units, of the prisms base edge is different from its height, in units. the sum of the areas of the 4 lateral faces is 4,220 square units. if all the dimensions, in units, of this prism are whole numbers greater than 1, which of the following could be the volume of this prism?
note: figure not drawn to scale.
a 211 cubic units
b 3,376 cubic units
c 21,100 cubic units
d 222,605 cubic units

Explanation:

Step1: Recall the formula for the volume of a prism

The volume \(V\) of a prism is given by \(V = B\times h\), where \(B\) is the area of the base and \(h\) is the height of the prism. Since the bases are squares, if the side - length of the square base is \(s\), then \(B=s^{2}\). The sum of the areas of the 4 lateral faces of a prism with base perimeter \(P\) and height \(h\) is \(S = P\times h\). For a square base with side - length \(s\), \(P = 4s\), so \(S=4s\times h\). We know that \(S = 4220\) square units, so \(4s\times h=4220\), and \(s\times h=\frac{4220}{4}=1055\).

Step2: Analyze the volume formula in terms of the given information

The volume \(V=s^{2}\times h=s\times(s\times h)\). Since \(s\times h = 1055\) and \(s\) is a whole number greater than 1.
Let's check the options:

• Option A: If \(V = 211\), then \(s\times(s\times h)=211\). Since \(211\) is a prime number and \(s\times h = 1055\), \(211\) is not a factor of \(1055\) (\(1055\div211 = 5\) but \(s\times(s\times h)

eq211\) as \(s>1\)).

• Option B: If \(V = 3376\), then \(s\times(s\times h)=3376\). Since \(s\times h = 1055\), \(3376\div1055=\frac{3376}{1055}\approx3.2\) (not a whole number).
• Option C: If \(V = 21100\), then \(s\times(s\times h)=21100\). Since \(s\times h = 1055\), \(s=\frac{21100}{1055}=20\) (a whole number greater than 1).
• Option D: If \(V = 222805\), then \(s\times(s\times h)=222805\). \(222805\div1055 = 211.2\) (not a whole number).

Answer:

C. 21,100 cubic units