Question

a. what are the dimensions of each face of the platform?
face a: $3x + 1$ by $x$
face b: $x + 1$ by $x$
face c: $3x + 1$ by $x + 1$
b. what is the area of the top of the platform?
$3x^2 + 4x + 1$
(simplify your answer.)
c. what expression represents the surface area of the entire platform, including the bottom?
(simplify your answer.)

Explanation:

Step1: Identify matching face pairs

A rectangular platform (prism) has 2 of each face type.

• 2 faces of Face A: dimensions $3x+1$ by $x$
• 2 faces of Face B: dimensions $x+1$ by $x$
• 2 faces of Face C: dimensions $3x+1$ by $x+1$

Step2: Calculate area of each pair

Area of 2 Face A: $2\times(3x+1)x = 2(3x^2+x) = 6x^2+2x$
Area of 2 Face B: $2\times(x+1)x = 2(x^2+x) = 2x^2+2x$
Area of 2 Face C: $2\times(3x+1)(x+1) = 2(3x^2+4x+1) = 6x^2+8x+2$

Step3: Sum all pair areas

$$(6x^2+2x)+(2x^2+2x)+(6x^2+8x+2)$$
Combine like terms:
$6x^2+2x^2+6x^2 = 14x^2$
$2x+2x+8x = 12x$
Constant term: $2$

Answer:

$14x^2 + 12x + 2$