A solid has base region in the first quadrant bounded above by the line
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
A solid has base region T in the first quadrant bounded above by the line y=kx, below by the parabola y=x2, and on the left by the y-axis. The parameter k is a positive real number greater than 1. For every x in the interval over which the two curves intersect, the cross section of the solid taken perpendicular to the x-axis is a square whose side lies entirely in T.
A designer wants the solid to have volume exactly 5 cubic units.
Complete the tasks below.
(a) Derive an explicit formula for the volume V(k) of the solid in terms of the parameter k. Show each step in your derivation.
(b) Solve the equation V(k)=5 to determine the value of k (to three decimal places) that yields the desired volume. Provide justification for the numerical method or algebraic process you use.
(c) Discuss how the volume of the solid changes as k increases beyond the value found in part (b). Support your discussion with appropriate mathematical reasoning.