Question

The length, breadth and the height of the cuboid are in ratio 5 : 4 : 2 and the total surface area is 1216 ${cm}^2$, then the volume of cuboid is-
A. 2460 ${cm}^3$
B. 2560 ${cm}^3$
C. 2660 ${cm}^3$
D. 2700 ${cm}^3$

Answer 1

Hint: This is a problem of mensuration in 3D objects. The formula for the total surface area and the volume of the cuboid will be used,
TSA = 2(lb + bh + hl)
V = lbh

Complete step-by-step solution -

The ratio of length, breadth and height is given. We can assume that they are 5x, 4x and 2x respectively. Now, we will apply the formula for the total surface area-
TSA = 2(lb + bh + hl)
1216 = 2[(5x)(4x) + (4x)(2x) + (2x)(5x)]
$20{x}^2 + 8{x}^2 + 10{x}^2$ = 608
38$x^2$ = 608
$x^2$ = 16
x = 4 cm, negative value is rejected

So, the length, breadth and height are respectively 20 cm, 16 cm and 8 cm. Now, we will find the volume of the cuboid by-
V = lbh
V = (20)(16)(8) = 2560 ${cm}^3$

This is the answer is B. 2560 ${cm}^3$

Note: The common mistake is that students forget to mention that they are rejecting the negative root and directly write the positive value. Also, all the formulas should be remembered. Whenever a ratio of quantities is given, we can assume the quantities by multiplying a variable with them.