### Related Rates:

### 1. Suppose \( x^2+4y^2=40\), where \(x\) and \(y\) are functions
of \(t\). If \(\dfrac{dx}{dt}= 1.5\), find \(\dfrac{dy}{dt}\) when \(x=2\) and
\(y=3\). solution

### 2. The radius of a circle is increasing at the rate of 2 inches per sec. At
what rate is the area increasing when the radius is 5 inches?
solution

### 3. Each side of a square is increasing at a rate of 3 inches per min. At
what rate is the area of the square increasing when the side of the square is 7
inches? solution

### 4. In calm waters, the oil spilling from the ruptured hull of a grounded
tanker spreads in all directions. Assuming that the area polluted is circular,
determine how fast is the area increasing when the radius of the circle is 60
feet and is increasing at the rate of 1/2 feet per sec?
solution

### 5. The radius of a sphere is increasing at a rate of 8 mm/s. How fast is the volume increasing when the diameter is 40 mm? solution

### 6. A cylindrical tank with radius 6 m is being filled with water at a rate
of 5 cubic m/min. How fast is the height of the water increasing? solution

### 7. A ladder 10 feet long rests against a vertical wall.
If the bottom of the ladder slides away from the wall at a rate of 1
foot per second, how fast is the top of the ladder sliding down the
wall when the bottom of the ladder is 6 feet from the wall?
solution

### 8. A ladder 17 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.2 ft/sec, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? solution

### 9. Two cars start moving from the same point. One
travels south at 60 mph and the other travels west at 25 mph. At what
rate is the distance between the cars increasing two hours later?
solution

### 10. A street light is mounted at the top of a 15 ft tall
pole. A man 6 ft tall walks away from the pole with a speed of 5 ft per
sec along a straight path. How fast is the tip of his shadow moving
when he is 40 ft from the pole? solution

### 11. A girl flies a kite at a height of 300 ft, the wind carrying
the kite horizontally away from her at a rate of 25 ft/sec. How fast must she
let out the string when the kite is 500 ft away from her? Assume the string is
taut so that it forms a straight line.
solution

### 12. Person A starts walking north at 3 ft/s from a point P. Five minutes later person B starts walking south at 4 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 10 minutes after the person B starts walking? solution

### 13. A balloon is rising at a constant speed of 5 ft per sec. A
boy is cycling along a straight road at a speed of 15 ft per sec. When he passes
under the balloon, it is 45 ft above him. How fast is the distance between the
boy and the balloon increasing 3 sec later?
solution

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