Question

a prism is created using 2 regular pentagons as bases. the apothem of each pentagon is 2.8 centimeters. (there is an image of a prism with labels (2x + 1), x, and 2.8) which expression represents the volume of the prism, in cubic centimeters? \\(\bigcirc\\) \\(9x^2 + 7x\\) \\(\bigcirc\\) \\(14x^2 + 7x\\) \\(\bigcirc\\) \\(16x^2 + 14x\\) \\(\bigcirc\\) \\(28x^2 + 14x\\)

Explanation:

Step1: Calculate the area of the pentagon base

The formula for the area of a regular polygon is \(A = \frac{1}{2}Pa\), where \(P\) is the perimeter and \(a\) is the apothem. For a regular pentagon with side length \(x\), the perimeter \(P = 5x\), and the apothem \(a=2.8\). So the area of the pentagon \(A=\frac{1}{2}\times5x\times2.8=\frac{1}{2}\times14x = 7x\)

Step2: Calculate the volume of the prism

The formula for the volume of a prism is \(V=Ah\), where \(A\) is the area of the base and \(h\) is the height. The height of the prism \(h=(2x + 1)\). So \(V=7x\times(2x + 1)\)

Step3: Expand the expression

Using the distributive property \(a(b + c)=ab+ac\), we have \(V=7x\times2x+7x\times1=14x^{2}+7x\)

Answer:

\(14x^{2}+7x\)