Sue Pemberton Solutions for Chapter: Further Differentiation, Exercise 8: EXERCISE 8E

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Show that the volume of water in the cone, $V {\mathrm{cm}}^3$, when the height of the water is $h$ cm is given by the formula $V=\frac{\pi h^3}{27}$.

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of change of $h$, when $h=20$ cm.

Oil is poured onto a flat surface and a circular patch is formed. The radius of the patch increases at a rate of $2\sqrt{r} \mathrm{cm}/\mathrm{s}$. Find the rate at which the area is increasing when the circumference is $8\pi \mathrm{cm}$.

Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the radius of the patch after $8$ seconds.

Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the rate of increase of the radius of the patch after $8$ seconds.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. If $V$ represents the volume of water, in ${\mathrm{cm}}^3$, in the cone at time $t$ seconds, find $\frac{dV}{dt}$ in terms of $\mathrm{\pi }$.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, find the rate of change of the height of the water in the cone.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, Find the rate of change of the horizontal surface area of the water in the cone.