MAT Exam 2021 Paper

Concept:The largest square base that can fit inside the circular cross-section of the trunk is used after trimming.The diagonal of this square equals the diameter of the cylinder.Explanation:The trunk is a right cylinder with radius $1.5 \mathrm{~m}$.So, its diameter is $2 \times 1.5 = 3 \mathrm{~m}$.When trimmed to a square base, the square must fit inside the circular cross-section.Hence, the diagonal of the square base equals the diameter of the circle.Let each side of the square base be $x$ m.Using Pythagoras theorem for the square:$$x^2 + x^2 = (3)^2$$$$2x^2 = 9$$$$x = \frac{3}{\sqrt{2}} \mathrm{~m}$$The height of the parallelopiped remains the same as the cylinder, $10 \mathrm{~m}$.Therefore, volume of timber remaining is:$$V = x^2 \times 10 = \left(\frac{3}{\sqrt{2}}\right)^2 \times 10$$$$V = \frac{9}{2} \times 10 = 45 \mathrm{~m}^3$$Answer:$45 \mathrm{~m}^3$Correct option: C.