b The surface area is quadrupled. Explanation: See solution.
Practice makes perfect
a In this exercise, we are give the surface area of a cube and asked to find the length of the side. Let's start by looking at the formula for the surface area.
S = 6g^2
We can substitute S= 96 for the surface area, and then solve for the length of the edge, g.
Since g represents a measurement of length, we can ignore the negative solution and say the length of the edge of the cube is 4 ft.
b A common mistake is to think that when the edge of something doubles, so will everything else. Let's consider a second cube whose sides are twice as long as the sides from Part A. We can multiply our edge length, g=4, from Part A by two, 2g=8. For clarity lets call this new edge length m, and substitute it into the formula.
Now let's compare our new surface area to our original surface area in Part A.
384/96 = 4
The new surface area is 4 times that of the original. From this we can say that the surface area quadruples when the edge length is doubled.
Alternative Solution
New Surface Area
There are ratios of proportionality that relate to surface area and volume. Let's have a look at the formulas and then how we can use that for our solution.
Type of Measurement
Ratio
Example
Length
a/b
1/2
Area
(a/b)^2 = a^2/b^2
1^2/2^2=1/4
Volume
(a/b)^3 = a^3/b^3
1^3/2^3=1/8
In our case, surface area is a type of area. We can see that if the sides increase by a factor of 2, then the area will increase by a factor of 4. We can determine the surface area of the new cube by multiplying the original surface area by 4. 96 ft^2 * 4= 384 ft^2
Therefore, we can say that the surface area is quadrupled when the edge length is doubled.
