Practice section 6 of 8

A cone of radius $r$ and height $h$ has volume $\tfrac13\pi r^2h$. A sphere of fixed radius $a$ and centre $O$ is inscribed in a right cone, touching its base and curved surface, as shown.

(i) Show that $r^2=\dfrac{a^2h}{h-2a}$. [2]

(ii) Use differentiation to find the minimum volume of the cone in terms of $a$, proving it is a minimum. [6]

(a) A factory produces disposable cups in the shape of inverted right cones of radius $4$ cm and height $8$ cm. For a quality check, the supervisor fills one cup completely with water. Water leaks from its vertex at a constant $1.5\text{ cm}^3\text{ s}^{-1}$. Find the exact rate at which the water level is decreasing when its depth is $2$ cm. [4]

The volume of a cone of base radius $r$ and height $h$ is $\tfrac13\pi r^2h$.

(b) Trapezium $ABCD$ has height $5$ cm and $DA+AB+BC=20$ cm. Points $M,N$ are the feet of the perpendiculars from $A,B$ to $DC$. Given $\angle DAM=\angle CBN=\theta$, where $0<\theta<\pi/2$:

(i) Show that its area is $A=100-50\sec\theta+25\tan\theta\text{ cm}^2$. [3]

(ii) Hence use differentiation to find its maximum area in exact form. You do not need to verify that it is a maximum. [4]

The diagram, not to scale, shows the graph of $y=f(x)$. The vertical asymptotes are $x=0$ and $x=2$. Sketch the graph of $y=f^{\prime}(x)$, showing all corresponding coordinates and asymptotes where possible. [3]

The graph of $y=f(x)$ has asymptotes $x=2$, $y=-3$ and minimum $(-2,-5)$, as shown. Sketch $y=f^{\prime}(x)$, clearly indicating any asymptotes, turning points and axial intercepts. [3]

The supplied graph also labels x-intercepts $1,4$ and y-intercept $-3$.

The diagram shows $y=f(x)$ with a maximum at $x=a$, minimum at $x=b$, and stationary point of inflection at the origin. On a separate diagram draw $y=f^{\prime}(x)$. [3]

The diagram shows $y=f(x)$ with asymptotes $x=3$ and $y=-x+2$, y-intercept $7/3$, minimum $(2,1)$ and maximum $(4,-3)$.

Sketch the curve $y=f^{\prime}(x)$. [3]