Chapter 2.5, Problem 49PPE

(a)

To determine

Write a formula for the surface area A of the prism.

Answer to Problem 49PPE

  $A=2s^2+4sh$

Explanation of Solution

Given :

  

Calculation:

The surface area is the sum of area of all the sides of the solid (including top and bottom).

The top and bottom of the prism are squares with side s .

So, the area of the top and bottom is $s\times s$ each.

There are 4 sides of the prism and each side is a rectangle with dimensions s and h .

So, the area of each side of the prism is $s\times h$

Hence the surface area of the prism:

  $\begin{array}{l}A=(s\times s)+(s\times s)+(s\times h)+(s\times h)+(s\times h)+(s\times h)\\ \Rightarrow A=2(s\times s)+4(s\times h)\\ \Rightarrow A=2s^2+4sh\end{array}$

(b)

To determine

Rewrite the formula to find h in terms of A and s . Find the height of the prism.

Answer to Problem 49PPE

  $h=\frac{A-2s^2}{4s}$ ; 14 cm

Explanation of Solution

Given :

   s is 10 cm and A is 760 $c{m}^2$ ,

Calculation:

From part (a) , the surface area of the prism is given by :

  $A=2s^2+4sh$

Solve it for h in terms of A and s :

  $\begin{array}{l}A=2s^2+4sh\\ \Rightarrow A-2s^2=2s^2+4sh-2s^2\mathrm{..........}[\text{subtract }2s^2\text{ each side ]}\\ \Rightarrow A-2s^2=4sh\\ \Rightarrow \frac{A-2s^2}{4s}=\frac{4sh}{4s}\mathrm{..........................}[\text{divide each side by 4s]}\\ \Rightarrow \frac{A-2s^2}{4s}=h\\ \Rightarrow h=\frac{A-2s^2}{4s}\mathrm{.............................}[\text{flip the equation]}\end{array}$

Given:

s = 10

A = 760

Substitute the values in the equation and find h :

  $\begin{array}{l}h=\frac{A-2s^2}{4s}\\ \Rightarrow h=\frac{760-2{(10)}^2}{4(10)}\\ \Rightarrow h=\frac{760-200}{40}\\ \Rightarrow h=\frac{560}{40}\\ \Rightarrow h=\frac{56}{4}\\ \Rightarrow h=14\text{ cm}\end{array}$

So, the height of the prism is 14 cm.

(c)

To determine

Write the formula of surface area A in terms of s only.

Answer to Problem 49PPE

  $A=6s^2$

Explanation of Solution

Given :

  

Let h = s .

Calculation:

From part (a) , the surface area of the prism is given by :

  $A=2s^2+4sh$

Given:

h = s

Substitute the value in the equation and find surface area in terms of s only:

  $\begin{array}{l}A=2s^2+4sh\\ \Rightarrow A=2s^2+4s(s)\mathrm{..........}[h=s]\\ \Rightarrow A=2s^2+4s^2\\ \Rightarrow A=6s^2\end{array}$