Comprehensive

Jamie is doing some baking and pouring
the flour to form a conical pile.

The height of the pile is always the same
as the diameter of the base of the cone.

If the flour is being added at a constant
rate of $3{\mathrm{cm}}^3$ per second, at what rate is
the height increasing when the pile is
$4\mathrm{cm}$ in height?

*You must use calculus and show any*
*derivatives that you need to find when*
*solving this problem.*

Note that volume of a cone $=\frac{1}{3}\pi r^{2}h$.

A police helicopter is flying above a straight horizontal section of motorway chasing a speeding
car.

The helicopter is flying at a constant speed of $72\mathrm{m}{\mathrm{s}}^{-1}$ and at a constant height of $400$ metres
above the ground. The helicopter is attempting to catch up with the car.

When the direct distance from the helicopter to the car is $2500$ metres, the angle of depression, $\theta$,
between the horizontal and the line of sight from the helicopter to the car is increasing at a rate of
$0.002\mathrm{rad}{\mathrm{s}}^{-1}$.

Calculate the speed of the car at this instant.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

The diagram below shows the cross-section of a bowl containing water.

When the height of the water level in the bowl is $h$ cm, the volume, $V$ cm${}^3$, of water in the bowl is
given by $V=\pi \left(\frac{3}{2}{h}^2+3h\right)$.

Water is poured into the bowl at a constant rate of $20$ cm${}^3$ s${}^{-1}$.

Find the rate, in cm s${}^{-1}$, at which the height of the water level is increasing when the height of the
water level is $3$ cm.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Megan cycles from her home, H, to school, S, each day.

She rides along a path from her home to point P at a constant speed of 10 kilometres per hour.
At point P, Megan cuts across a park, heading directly to school. When cycling across the park,
Megan can only cycle at 6 kilometres per hour.

At what distance from her home should she choose to cut across the park in order to make her
travelling time a minimum?

You must use calculus and show any derivatives that you need to find when solving this problem.

A rectangle has one vertex at $(0,0)$ and the opposite vertex on the curve $y=6e^{1-0.5x}$, where $x>0$, as
shown on the graph below.

Find the maximum possible area of the rectangle.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

*You do not have to prove that the area you have found is a maximum.*

A cylinder of height $h$ and radius $r$ is inscribed, as shown to the
right, inside a sphere of radius 20 cm.

Find the maximum possible volume of the cylinder.

*You must use calculus and show any derivatives that you need to
find when solving this problem.*

*You do not need to prove that the volume you have found is a
maximum.*

A rocket is fired vertically upwards. Its height above the launch point is given by the formula
$h(t)=4.8t^2$, where $h$ is the height in metres, and $t$ is the time in seconds from firing.

An observer at point A is watching the rocket. She is at the same level as the launch point of
the rocket, and 500 m from the launch point.

Find the rate at which the angle of elevation at A of the rocket is increasing when the rocket is
480 m above the launch point.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

The Wynyard Crossing bridge in Auckland can be raised and lowered to allow tall boats to
sail through when open, and pedestrians to walk across when closed. The bridge consists of
two arms, each of length 22 metres.

When the bridge is rising, the angle of the bridge arm above the horizontal increases at the
rate of $0.01\mathrm{rad}{\mathrm{s}}^{-1}$.

Find the rate at which the height, BH, is increasing when H is 15 metres above the horizontal,
FB.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A car is being pulled along by a rope attached to the tow-bar at the back of the car.

The rope passes through a pulley, the top of which is 3 m further from the ground than the tow-bar.

The pulley is $x$ m horizontally from the tow-bar, as shown in the diagram above.

The rope is being winched in at a speed of $0.6$ m s${}^{-1}$.

The wheels of the car remain in contact with the ground.

At what speed is the car moving when the length of the rope, $L$, between the tow-bar and the pulley is $5.4$ m?

*You must use calculus and show any derivatives that you need to find when solving this problem.*

The diagram below shows the graph of the function $y=15-x^2$, inside which an isosceles
triangle OAB has been drawn.

Find the maximum possible area, $A$, of the triangle.
You may assume that your answer is a maximum.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A water tank is in the shape of an inverted right-circular cone.

The height of the cone is 200 cm and the radius of the cone is 80 cm.

The tank is being filled with water at a rate of $150{\mathrm{cm}}^3$ per second.

At what rate will the surface area of the water in the tank be increasing when the depth of
water in the tank is 125 cm?

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A building has an external elevator. The elevator is rising at a constant rate of $2\mathrm{m}{\mathrm{s}}^{-1}$.

Sarah is stationary, watching the elevator from a point $30\mathrm{m}$ away from the base of the
elevator shaft.

Let the angle of elevation of the elevator floor from Sarah's eye level be $\theta$.

Find the rate at which the angle of elevation is increasing when the elevator floor is $20\mathrm{m}$
above Sarah’s eye level.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Salt harvested at the Grassmere Saltworks forms a cone as it falls from a conveyor belt.
The slant of the cone forms an angle of $30^{\circ }$ with the horizontal.
The conveyor belt delivers the salt at a rate of $2{\mathrm{m}}^3$ of salt per minute.

Find the rate at which the slant height is increasing when the radius of the cone is $10\mathrm{m}$.
*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A corridor is 2 m wide.

At the end it turns 90° into another corridor.

What is the minimum width, $w$, of the second corridor if a ladder of length 5 m can be carried
horizontally around the corner?

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A street light is 5 m above the ground, which is flat.

A boy, who is 1.5 m tall, is walking away from the point directly below the streetlight at
2 metres per second.

At what rate is the length of his shadow changing when the boy is 8 m away from the point
directly under the light?

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

A water container is constructed in the shape of a square-based pyramid. The height of the
pyramid is the same as the length of each side of its base.

A vertical height of 20 cm is then cut off the top of the pyramid, and a new flat top added.
The pyramid is then inverted and water is poured in at a rate of $3000{\text{ cm}}^3$ per minute.

Find the rate at which the surface area of the water is increasing when the depth of the water
is $15\text{ cm}$.

Volume of pyramid $=\frac{1}{3}\times$ base area $\times$ height

*You must use calculus and show any derivatives that you need to find when solving this problem.*

What is the maximum volume of a cone if the slant length of the cone is 20 cm?

You do not need to prove that the volume you have found is a maximum.

*Show any derivatives that you need to find when solving this problem.*

A container is winched up vertically from a point P at a constant rate of $1.5\mathrm{m}{\mathrm{s}}^{-1}$.
It is being observed from point Q, which is $20\mathrm{m}$ horizontally from point P.
$\theta$ is the angle of elevation of the container from point Q.

At what rate is the angle of elevation increasing when the object is $20\mathrm{m}$ above point P?

*Show any derivatives that you need to find when solving this problem.*

A cone has a radius of 20 cm and a height of 40 cm.

A cylinder fits inside the cone, as shown below.

What must the radius of the cylinder be to give the cylinder the maximum volume?
You do not need to prove that the volume you have found is a maximum.
*Show any derivatives that you need to find when solving this problem.*

A rectangle is drawn inside a right angled triangle, as shown in the diagram below.

Point B moves along the base of the triangle AC, beginning at point A, at a constant speed of 3 cm s${}^{-1}$.

At what rate is the area of the rectangle changing when point B is 20 cm from point A?

*Show any derivatives that you need to find when solving this problem.*

A copper sheet of width 24 cm is folded, as shown, to make spouting.

Cross-section:

Find angle $\theta$ which gives the maximum cross-sectional area.

*You do not need to prove that you have found a maximum.*

*Show any derivatives that you need to find when solving this problem.*