Chapter 12.8, Problem 26HP

To determine

To find: the length of the side of the other cube.

Answer to Problem 26HP

The length of the side of the new cube is $24.22cm$ .

Explanation of Solution

Given:

The ratio of the surface areas of two cubes is $1:12$ .

The length of one side of the smaller cube is $7$ centimeters.

Calculation:

The objective is to find the length of the side of the other cube.

Let the length of the side of the other cube be $x$ .

Two solid figures are similar if they have the same shape and their corresponding linear measurements are proportional.

Since both are cubes and their corresponding linear measurements are proportional, so the cubes are similar.

The ratio of the side of smaller cube to larger cube is $\frac{7}{x}$ .

The scale factor is $\frac{7}{x}$ .

Now, use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the surface area is in the ratio ${\left(\frac{a}{b}\right)}^2$

Thus,

  $\frac{\text{Surface area of small cube}}{\text{Surface area of larger cube}}\text{=}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}$ ,

where $\frac{a}{b}$ is the scale factor.

  $\begin{array}{l}\frac{1}{12}={\left(\frac{7}{x}\right)}^2\\ \frac{1}{12}=\frac{49}{x^2}\\ x^2=12\left(49\right)\\ =3\left(4\right)\left(49\right)\\ x=\sqrt{3\left(4\right)\left(49\right)}=\sqrt{3\left(2^2\right)\left(7^2\right)}\\ =14\sqrt{3}\approx 14(1.73)\\ =24.22\end{array}$

Therefore, the length of the side of the new cube is approximately $24.22cm$ .

Conclusion:

Therefore, the length of the side of the new cube is approximately $24.22cm$ .