Understanding the Ratio of Surface Areas of Cubes

The question asks for the ratio of the surface areas of two cubes with different edge lengths. We are given the edge lengths of the two cubes.

Let the edge length of the first cube be \(a_1\) and the edge length of the second cube be \(a_2\).

• Edge of the first cube, \(a_1 = 7\) cm
• Edge of the second cube, \(a_2 = 14\) cm

The formula for the total surface area of a cube with edge length 'a' is given by \( A = 6a^2 \). This is because a cube has 6 equal square faces, and the area of each face is \(a \times a = a^2\).

Calculating Surface Areas

Let's calculate the surface area of each cube:

• Surface area of the first cube, \( A_1 = 6 \times a_1^2 \)
• Surface area of the second cube, \( A_2 = 6 \times a_2^2 \)

Now, we can substitute the given edge lengths:

• \( A_1 = 6 \times (7 \text{ cm})^2 = 6 \times 49 \text{ cm}^2 = 294 \text{ cm}^2 \)
• \( A_2 = 6 \times (14 \text{ cm})^2 = 6 \times 196 \text{ cm}^2 = 1176 \text{ cm}^2 \)

Finding the Ratio of Surface Areas

The ratio of their surface areas is \( \frac{A_1}{A_2} \). Let's find this ratio:

\( \frac{A_1}{A_2} = \frac{294 \text{ cm}^2}{1176 \text{ cm}^2} \)

We can simplify this fraction. Both numbers are divisible by 6:

\( 294 \div 6 = 49 \)

\( 1176 \div 6 = 196 \)

So the ratio is \( \frac{49}{196} \). Both 49 and 196 are divisible by 49:

\( 49 \div 49 = 1 \)

\( 196 \div 49 = 4 \)

Thus, the simplified ratio is \( \frac{1}{4} \).

Alternatively, we can find the ratio of the edge lengths first and then square it, since the ratio of the surface areas of two similar solids is the square of the ratio of their corresponding linear dimensions.

Ratio of edges \( = \frac{a_1}{a_2} = \frac{7 \text{ cm}}{14 \text{ cm}} = \frac{1}{2} \)

Ratio of surface areas \( = \left( \text{Ratio of edges} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \)

The ratio of their surface areas is \( 1:4 \).