Chapter 9.5, Problem 48PPE

a)

To determine

To evaluate: the expression for the surface area of the prism.

Answer to Problem 48PPE

The expression for surface area of prism is given by $6x^2+28x$ .

Explanation of Solution

Given:

Prism is given as

  

The surface area of prism is equal to the surface area as a cube with edges 8 in. long.

The surface area of cuboid is given as,

  $2\left(lb+bh+hl\right)$

It can be seen that given prism is in the form of cuboids so its surface area will be same as cuboid’s surface area.

The expression for surface area of prism is given by,

  $\begin{array}{c}\text{SA}=2\left(lb+bh+hl\right)\\ =2\left\{\left(x+7\right)x+\left(x\right)\left(x\right)+x\left(x+7\right)\right\}\\ =2\left\{x^2+7x+x^2+x^2+7x\right\}\\ =2\left(3x^2+14x\right)\\ =6x^2+28x\end{array}$

Hence, the expression for surface area of prism is given by $6x^2+28x$ .

b)

To determine

To evaluate: an equation that relates the surface area of the prism to the surface area of the 8 −in- cube.

Answer to Problem 48PPE

The equation that relates the surface area of the prism to the surface area of the 8-in-cube is

  $6x^2+28x=8$ .

Explanation of Solution

Given:

Prism is given as

  

The surface area of prism is equal to the surface area as a cube with edges 8 in. long.

The expression calculated in 1^st part is as $6x^2+28x$ .

Now,

The surface area of cube is given by 8 in. cube.

Equating both the surface areas, then equation becomes,

  $6x^2+28x=8$

So, the equation that relates the surface area of the prism to the surface area of the 8-in-cube is

  $6x^2+28x=8$ .

c)

To determine

To evaluate: the equation in the part b).

Answer to Problem 48PPE

The solution of the above equation is $x=0.27$ .

Explanation of Solution

Equation in the part b) is,

  $6x^2+28x=8$

Solving above equation using quadratic formula,

  $\begin{array}{c}x=\frac{-28\pm \sqrt{{\left(28\right)}^2-4\left(-8\right)\left(6\right)}}{2\left(6\right)}\\ =\frac{-28\pm \sqrt{784+192}}{12}\\ =\frac{-28\pm \sqrt{976}}{12}\\ =\frac{-28\pm 31.24}{12}\\ =\frac{3.24}{12}\mathrm{...............}\left(x\text{ can't be negative}\right)\\ =0.27\end{array}$

The solution of the above equation is $x=0.27$ .