Question

The total surface area of a cuboid is 1332 square cm, and its sides are in the ratio 4:5:6. What will be the length of the sides?

• A. 12, 15, 18
• B. 20, 25, 30
• C. 8, 10, 12
• D. None of the above

Hint:

When the sides of a cuboid are in a given ratio, use the surface area formula to solve for the value of \( x \), then calculate the actual lengths of the sides.

Answer 1

The Correct Option is A

Let the sides of the cuboid be \( 4x \), \( 5x \), and \( 6x \), where \( x \) is the constant of proportionality. The surface area of a cuboid is given by the formula: \[ A = 2(lw + lh + wh) \] where \( l \), \( w \), and \( h \) are the length, width, and height of the cuboid, respectively. Substitute the values of \( l = 4x \), \( w = 5x \), and \( h = 6x \) into the formula: \[ A = 2 \left( (4x)(5x) + (4x)(6x) + (5x)(6x) \right) \] \[ A = 2 \left( 20x^2 + 24x^2 + 30x^2 \right) \] \[ A = 2 \times 74x^2 = 148x^2 \] We are given that the total surface area is 1332 square cm: \[ 148x^2 = 1332 \] \[ x^2 = \frac{1332}{148} = 9 \] \[ x = 3 \] Now, substitute \( x = 3 \) into the sides: - \( l = 4x = 12 \) - \( w = 5x = 15 \) - \( h = 6x = 18 \) Thus, the sides of the cuboid are \( 12 \), \( 15 \), and \( 18 \).