Optimisation & Rates of Change — practice questions

A rectangular paddock is to be fenced using 240 m of fencing. Find the dimensions that maximise the enclosed area. You may assume your solution gives a maximum.

An open-topped box is made from a 30 cm by 30 cm square of card by cutting a square of side $x$ cm from each corner and folding up the sides. Find the value of $x$ that maximises the volume, and state the maximum volume.

A rectangle is inscribed in a semicircle of radius $R$, with its base on the diameter. Prove that the rectangle of maximum area has area $R^2$, and find its dimensions in terms of $R$.

The radius of a circular oil slick increases at 3 m per hour. Find the rate at which its area is increasing when the radius is 20 m.

A spherical balloon is being deflated so that its volume decreases at 40 cm³ per second. Find the rate at which its surface area is decreasing when the radius is 5 cm.

A ladder 5 m long leans against a vertical wall. The foot of the ladder slides away from the wall at a constant 0.4 m/s. Find the rate at which the top of the ladder is sliding down when the foot is 3 m from the wall, and explain what happens to this rate as the ladder approaches the ground.