The Cube
Cube _A_ has a volume of 8 in3. Cube _B_ has a surface area that is 25 times the surface area of Cube _A_. What is the volume of Cube _B_?
A simpler way to approach the problem is to realize that by increasing the surface area of the cube by a factor of 25 is to increase the length of each side, _s_, by a factor of $$\sqrt{25} = 5$$. Therefore, Cube _B_ would have a side of length 5_s_.
Find _s_ by taking the cube root of 8. Therefore, _s_ = 2. So the length of the side of Cube _B_ is 5_s_ = 5(2) = 10. The volume becomes $$10^3=1000$$.
Incorrect.
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If Cube _B_ had a volume of $$100 \mbox{ in}^3$$, it would have a side of $$\sqrt[3]{100} \approx 4.64$$ and a surface area of
$$\displaystyle SA_{Cube B} = 6s^2=6 \cdot (4.64)^2 = 6 \cdot (21.53) = 129.18$$.
That is not 25 times the surface area of Cube _A_.
Incorrect.
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If Cube _B_ had a volume of $$250 \mbox{ in}^3$$, it would have a side of $$\sqrt[3]{250} \approx 6.30$$ and a surface area of
$$\displaystyle SA_{Cube B} = 6s^2=6 \cdot (6.30)^2 = 6 \cdot (39.69) = 238.14$$.
That is not 25 times the surface area of Cube _A_.
Incorrect.
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If Cube _B_ had a volume of $$2000 \mbox{ in}^3$$, it would have a side of $$\sqrt[3]{2000}=10\sqrt[3]{2}$$ and a surface area of
$$\displaystyle SA_{Cube B} = 6 \cdot (10\sqrt[3]{2})^2 = 6 \cdot (100\sqrt[3]{4}) = 600 \sqrt[3]{4}$$.
That is not 25 times the surface area of Cube _A_.
Incorrect.
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If Cube _B_ had a volume of $$200 \mbox{ in}^3$$, it would have a side of $$\sqrt[3]{200} \approx 5.85$$ and a surface area of
$$\displaystyle SA_{Cube B} = 6s^2=6 \cdot (5.85)^2 = 6 \cdot (34.22) = 205.32$$.
That is not 25 times the surface area of Cube _A_.
Correct!
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The volume of Cube _A_ is given as $$8 \mbox{ in}^3$$. Set this value equal to the formula and solve for _s_.
$$\displaystyle 8 = s^3$$
$$\displaystyle s = 2$$
Find the surface area of Cube _A_:
$$\displaystyle SA_{Cube A} = 6s^2 = 6(2)^2 = 6(4) = 24$$.
Multiply this value by 25 to get a surface area 25 times larger than the original surface area.
$$\displaystyle 24 \cdot 25 = 600$$
Set this value equal to the formula for surface area and solve for _s_ to find the length of a side of Cube _B_.
$$\displaystyle 600 = 6s^2$$
$$\displaystyle 100 = s^2$$
$$\displaystyle 10 = s$$
Use this value in the formula to find the volume of Cube _B_.
$$\displaystyle V_{Cube B} = s^3 = 10^3 = 1000$$
Therefore, the volume of Cube _B_ is $$1000 \mbox{ in}^3$$.
$$100 \mbox{ in}^3$$
$$200 \mbox{ in}^3$$
$$250 \mbox{ in}^3$$
$$1,000 \mbox{ in}^3$$
$$2,000 \mbox{ in}^3$$
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