Question

The formula for the surface area of a rectangular prism is $S = 2wl + 2wh + 2lh$ . How do you solve for $w$?

Answer 1

Hint: For solving this particular question for $w$ where formula for the surface area of a rectangular prism is given as $S = 2wl + 2wh + 2lh$ , you have to isolate $w$terms while keeping the equation balanced.

Complete step-by-step solution:
The formula for the surface area of a rectangular prism is $S = 2wl + 2wh + 2lh$ .
See below for a process to resolve this formula for $w$ ,
First, divide all sides of the equation by two while keeping the equation balanced:
\[
   \Rightarrow \dfrac{S}{2} = \dfrac{{2wl + 2wh + 2lh}}{2} \\
   \Rightarrow \dfrac{S}{2} = \dfrac{{2(wl + wh + lh)}}{2} \\
   \Rightarrow \dfrac{S}{2} = wl + wh + lh \\
 \]
Next, subtract $lh$ from either side of the equation to isolate the $w$ terms while keeping the equation balanced:
$
   \Rightarrow \dfrac{S}{2} - lh = wl + wh + lh - lh \\
   \Rightarrow \dfrac{S}{2} - lh = wl + wh \\
 $
Then, take common $w$ from each term on the right side of the equation giving:
$ \Rightarrow \dfrac{S}{2} - lh = w(l + h)$
Now, divide all sides of the equation by $(l + h)$ to resolve for w while keeping the equation balanced:
$
   \Rightarrow \dfrac{1}{{(l + h)}}\left( {\dfrac{S}{2} - lh} \right) = \dfrac{1}{{(l + h)}}\left( {w(l + h)} \right) \\
   \Rightarrow \dfrac{1}{{(l + h)}}\left( {\dfrac{S}{2} - lh} \right) = w \\
 $
We can also rewrite this as:
$ \Rightarrow w = \dfrac{1}{{(l + h)}}\left( {\dfrac{S}{2} - lh} \right)$

Additional Information:
A rectangular prism could be a six-faced, three-dimensional solid during which all the faces are rectangles. All six faces meet at right angles to at least one another. Opposite faces are congruent.
A special sort of rectangular prism may be a cube, within which all six faces are congruent.
The extent of an oblong prism is that the total area of all six faces.

Note: Finding extent for all rectangular prisms (including cubes) involves both addition and multiplication. you need to know the width, length and height of the prism before you'll be able to apply this formula:
\[A{\text{ }} = {\text{ }}2\left( {width{\text{ }} \times {\text{ }}length} \right){\text{ }} + {\text{ }}2\left( {length{\text{ }} \times {\text{ }}height} \right){\text{ }} + {\text{ }}2\left( {height{\text{ }} \times {\text{ }}width} \right)\] .
We can also rewrite this as:
$A = 2wl + 2wh + 2lh$ .