Question

two cubes have sides of length 8 cm and 4 cm. what is the ratio of their respective surface areas? their respective surface areas have a ratio of □:□ (type a reduced ratio.)

Explanation:

Step1: Recall the surface area formula for a cube

The surface area formula of a cube is \(S = 6s^{2}\), where \(s\) is the side - length of the cube.

Step2: Calculate the surface area of the first cube

For the cube with side - length \(s_1=8\mathrm{cm}\), its surface area \(S_1 = 6s_1^{2}\). Substitute \(s_1 = 8\) into the formula: \(S_1=6\times8^{2}=6\times64 = 384\mathrm{cm}^{2}\).

Step3: Calculate the surface area of the second cube

For the cube with side - length \(s_2 = 4\mathrm{cm}\), its surface area \(S_2=6s_2^{2}\). Substitute \(s_2 = 4\) into the formula: \(S_2=6\times4^{2}=6\times16 = 96\mathrm{cm}^{2}\).

Step4: Find the ratio of the two surface areas

The ratio of \(S_1\) to \(S_2\) is \(\frac{S_1}{S_2}=\frac{384}{96}\). Simplify the fraction \(\frac{384\div96}{96\div96}=\frac{4}{1}\).

Answer:

\(4:1\)