Use calculus to find the volume generated when the line $y=kx$, for $0\leq x\leq h$, is rot

Question

Use calculus to find the volume generated when the line $y=kx$, for $0\leq x\leq h$, is rotated through $2\pi$ radians about the $x$-axis.[3] Hence show that the volume of a cone of height $h$ and base radius $r$ is $\dfrac{1}{3}\pi r^2h$.[1] The straight line $y=kx$ cuts the curve $(x-k)^2+y^2=k^2$ at the point where $x=0.6$. Find exactly the possible values of $y$ at this point.[3]

Accepted Answer

(a)(i) $\dfrac{1}{3}\pi k^2h^3$ (ii) Shown, using $r=kh$ (b) $y=\dfrac{1}{5}$ or $y=\dfrac{9}{5}$

N2026 P2 Q2

N2026 P2 Q2