Related rates: geometric models Problem set B

The sides of a square patch of ice on a pond are increasing at 1.5 cm s\(^{-1}\). Find the rate of increase of the area at the instant when the area is 49 cm².

A ripple on a pond is a circle. When its radius is 8 cm, the radius is increasing at 2.5 cm s\(^{-1}\). Find the rate of increase of the area at this instant.

A spherical soap bubble is blown so that its volume increases at 45 cm³ s\(^{-1}\). Find the rate of increase of the radius at the instant when the volume is \(288\pi\) cm³.

A rectangular screen is being resized. Its width x cm decreases at 2 cm s\(^{-1}\) and its height y cm increases at 4 cm s\(^{-1}\). When \(x=5\) and \(y=12\), find

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

A circular stain on a carpet has radius r cm and area A cm². At one moment the radius is increasing at 0.8 cm s\(^{-1}\) and the area is increasing at 30 cm² s\(^{-1}\). Find the radius at this moment.

A 10 m plank leans against a vertical fence on horizontal ground. The foot of the plank slides away from the fence. When the foot is 6 m from the fence it is moving at 0.5 m s\(^{-1}\). Find the rate at which the top of the plank is moving down the fence at that instant.

A drone flies at a constant height of 24 m above a camera on the ground. It moves horizontally at 5 m s\(^{-1}\), directly away from the camera. Find the rate at which the distance D between the camera and the drone is increasing when the drone is 7 m horizontally from the camera.

A conical paper cup (point down) has top diameter 6 cm and height 12 cm. Water runs in at 9 cm³ s\(^{-1}\).

1. (a) Show that \(V=\frac{\pi}{48}h^3\), 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 \(h=6\).

An observer stands 120 m from the foot of a tall crane. The crane's hook rises vertically. When the hook is 90 m up it is rising at 4 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 sample of salt solution evaporating in a dish is \(\rho=\frac{m}{V}\), where m g is the mass and V cm³ is the volume. When \(\rho=6\) g cm\(^{-3}\), the mass is decreasing at 7 g s\(^{-1}\) and the volume is decreasing at 1 cm³ s\(^{-1}\). Determine, with justification, whether the density is increasing or decreasing.

A 13 m pole is leaned against a wall and slides, with its foot on horizontal ground. When the top of the pole is 12 m above the ground it is moving down at 0.15 m s\(^{-1}\). Find the speed at which the foot of the pole is moving away from the wall.