Question Bank

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.*

The graph below shows the function $y = f(x)$.

For the function above:

(i) Find the value(s) of $x$ that meet the following conditions:

1. $f(x)$ is not defined:

2. $f(x)$ is not differentiable:

3. $f''(x) > 0$:

(ii) What is the value of $f(-1)$?

State clearly if the value does not exist.

(iii) What is the value of $\lim_{x\to2} f(x)$?

State clearly if the value does not exist.

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 $= \dfrac{1}{3} \times$ base area $\times$ height

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