1. The area of a square is increasing at a rate of \( 8\; cm/s\). At what rate the sides of the square is changing when the area is \( 25 \;cm^2 \)? Are the side lengths increasing or decreasing?

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2. The length of a rectangle is decreasing at a rate of \( 6 \;cm/s\) and its width is increasing at a rate of \( 4 \;cm/s\). How fast is the area of the rectangle changing, when the length is \( 12 \;cm\) and the width is \(5 \;cm\)? Is the area increasing or decreasing?

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3. A cylindrical tank of radius \( 3 \;ft\) is being filled with water at a rate of \( 4 \;ft^3/s \). How fast is the height of the water increasing?

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4. The volume of a sphere is decreasing at a rate of \( 12\;cm^3/s \). How fast is the radius of the sphere is changing when the radius is \( 6\;cm\). How fast is the surface area changing at this time?

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5. Adam and Bob start biking from the same point. Adam travels north at \(24\;km/h \) and Bob travels east at \( 10 \;km/h \). At what rate is the distance between the bikers changing two hours later?

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6. Adam starts biking north at \( 24 \; km/h \). An hour later Bob starts biking east from the same point at \( 10\; km/h \). At what rate is the distance between the bikers changing two hours after Adam starts biking?

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7. At noon, Adam is \(10\; km \) west of Bob. Adam is biking north at \( 24 \; km/h \), and Bob is biking south at \( 10\; km/h \). How fast is the distance between the two bikers changing at 3:00 pm.

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8. At 4:00 pm Adam is \(100 \;km \) north of Bob. Adam is biking south at \( 24 \; km/h \), and Bob is biking west at \( 10\; km/h \). How fast is the distance between these people changing at 7:00 pm.

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9. Suppose Oak Avenue is a north-south road, and Maple Boulevard is an east-west road so that they intersect at a right angle. At 1:00pm, Adam is 200 km north of the intersection and is biking south at 24 km/h, and Bob is 150 km east of the intersection and is biking west at 10 km/s. How fast is the distance between the two bikers changing at 5:00pm?

Solution