Question

What is the width of a rectangular prism when the surface area is $208$ square centimeters and it has a height of $8$ and a depth of $6$?

Answer 1

Hint: In this problem we need to calculate the width of the given rectangular prism. So, we will assume the width of the rectangular prism as $w$. Now we have the height of the rectangular prism as $8$ and a depth as $6$. We will represent the height of the rectangular prism as $h$, depth as $d$. We will calculate the surface area of the rectangular prism of having width $w$, height $h$ and depth $d$ by using the formula $S.A=2(wh+wd+hd)$. We will substitute all the values we have and simplify the equation. In the problem they have mentioned that the surface area of the rectangular prism is $208$. So, we will equate the calculated surface area with the given surface area. Now simplify the obtained equation by using basic mathematical operation to get the value of width.

Complete step by step answer:
Given that height of the rectangular prism is $h=8$.
Depth of the rectangular prism is $d=6$.
Let the width of the rectangular prism would be $w$.

Now the surface area of the rectangular prism of height $h=8$, depth $d=6$ and width $w$ is calculated by using the formula $S.A=2(wh+wd+hd)$, then we will get
$S.A=2(w\left(8\right)+w\left(6\right)+8\left(6\right))$
Simplifying the above equation by performing the multiplication operation in right hand side, then we will get
$\begin{array}{rl}& S.A=2(8w+6w+48)\\ & \Rightarrow S.A=2(14w+48)\end{array}$
But in the problem, we have given the surface area of the rectangular prism as $208$ square centimeters. So, equating the both the values, then we will have
$2(14w+48)=208$
Dividing the above equation with $2$ on both sides, then we will get
${\displaystyle \frac{2(14w+48)}{2}}={\displaystyle \frac{208}{2}}$
We have the value of ${\displaystyle \frac{208}{2}}$ as $104$. Substituting this value in the above equation, then we will have
$14w+48=104$
Simplifying the above equation by using basic mathematical operations, then we will get
$\begin{array}{rl}& 14w=104-48\\ & \Rightarrow w={\displaystyle \frac{56}{14}}\\ & \therefore w=4\end{array}$

Hence the width of the rectangular prism is $4$ centimeters.

Note: In this problem we have given the surface area of the rectangular prism so we have used the formula $S.A=2(wh+wd+hd)$ and solved the problem. If they have mentioned the volume of the rectangular prism, then we need to use the formula $V=whd$ and solve the problem as mentioned above.