Related rates: geometric models Problem set A

The sides of a square sheet of metal are increasing at 3 cm s\(^{-1}\). Find the rate of increase of the area at the instant when the area is 64 cm².

An oil slick is a circle. When its radius is 5 m, the radius is increasing at 0.4 m per minute. Find the rate of increase of the area at this instant.

A spherical balloon is inflated so that its volume increases at 60 cm³ s\(^{-1}\). Find the rate of increase of the radius at the instant when the volume is \(36\pi\) cm³.

A rectangular photo frame is changing size. Its width x cm increases at 3 cm s\(^{-1}\) and its height y cm decreases at 1 cm s\(^{-1}\). When \(x=6\) and \(y=8\), find

1. (a) the rate of increase of the area;
2. (b) the rate of increase of the diagonal.

A circular patch of damp on a wall has radius r cm and area A cm². At one moment the radius is increasing at 0.5 cm s\(^{-1}\) and the area is increasing at 20 cm² s\(^{-1}\). Find the radius at this moment.

A 13 m ladder rests against a vertical wall on horizontal ground. The foot of the ladder slides away from the wall. When the foot is 5 m from the wall it is moving at 0.6 m s\(^{-1}\). Find the rate at which the top of the ladder is moving down the wall at that instant.

A kite flies at a constant height of 30 m above a student's eye level. It moves horizontally at 4 m s\(^{-1}\), directly away from the student. Find the rate at which the distance D between the student and the kite is increasing when the kite is 40 m horizontally from the student.

A conical filter funnel (point down) has top diameter 8 cm and height 10 cm. Liquid runs in at 8 cm³ s\(^{-1}\).

1. (a) Show that \(V=\frac{4\pi}{75}h^3\), where V cm³ is the volume of water when its depth is h cm.
2. (b) Find the rate at which the depth is increasing when \(h=5\).

A tracker on the ground is 500 m from the base of a weather balloon launch. The balloon rises vertically. When the balloon is 650 m up it is rising at 60 m s\(^{-1}\). Find the rate at which the angle of elevation \(\theta\) is increasing at that moment. Give your answer in radians per second to 3 significant figures.

The density of a gas leaking from a sealed syringe is \(\rho=\frac{m}{V}\), where m g is the mass and V cm³ is the volume. When \(\rho=4\) g cm\(^{-3}\), the mass is decreasing at 3 g s\(^{-1}\) and the volume is decreasing at 1 cm³ s\(^{-1}\). Determine, with justification, whether the density is increasing or decreasing.

A rigid 5 m metal rod slides with one end on the floor and the other against a wall. When the top end is 4 m above the floor it is moving down at 0.2 m s\(^{-1}\). Find the speed at which the lower end is moving away from the wall.