## Section 3.9: Related Rates (additional questions)

### 1. If \(V\) is the volume of a cube with edge length \(x\) and the cube expands as time passes, find \(dV / dt\) in terms of \(d x / d t\).

### 2. (a) If \(A\) is the area of a circle with radius \(r\) and the circle expands as time passes, find \(d A / d t\) in terms of \(d r / d t\).

(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of \(1 \, \mathrm{m} / \mathrm{s},\) how fast is the area of the spill increasing when the radius is \(30 \, \mathrm{m} \) ?

### 3. Each side of a square is increasing at a rate of \( 6 \, \mathrm{cm} / \mathrm{s}\). At what rate is the area of the square increasing when the area of the square is \(16 \, \mathrm{cm}^{2}\) ?

### 4. The length of a rectangle is increasing at a rate of \( 8 \, \mathrm{cm} / \mathrm{s} \) and its width is increasing at a rate of \( 3 \,\mathrm{cm} / \mathrm{s}\). When the length is \(20 \, \mathrm{cm} \) and the width is \( 10 \, \mathrm{cm},\) how fast is the area of the rectangle increasing?

### 5. A cylindrical tank with radius \( 5 \, \mathrm{m}\) is being filled with water at a rate of \( 3 \, \mathrm{m}^{3} / \mathrm{min}\). How fast is the height of the water increasing?

### 6. The radius of a sphere is increasing at a rate of \( 4 \, \mathrm{mm} / \mathrm{s}\). How fast is the volume increasing when the diameter is \(80 \, \mathrm{mm}\) ?

### 7. The radius of a spherical ball is increasing at a rate of \( 2 \, \mathrm{cm} / \mathrm{min} \). At what rate is the surface area of the ball increasing when the radius is \( 8 \, \mathrm{cm} \).

### 8. A plane flying horizontally at an altitude of 1 mi and a speed of \( 500 \, \mathrm{mi} / \mathrm{h}\) passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

### 9. If a snowball melts so that its surface area decreases at a rate of \( 1 \mathrm{cm}^{2} / \mathrm{min}, \) find the rate at which the diameter decreases when the diameter is \(10 \, \mathrm{cm}\).

### 10. The height of a triangle is increasing at a rate of 3 cm per min while the base of the triangle is increasing at a rate of 5 cm per min. At what rate is the area of the triangle changing when the height is 10 cm and the area is 180 square cm?

### 11. The altitude (height) of a triangle is increasing at a rate of 1 cm per min while the area of the triangle is increasing at a rate of 2 square cm per min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 square cm?

### 12. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock ?

### 13. The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

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