7.4.3 The Baseball Problem

Question 1

The Baseball Problem

Answer 1

• Related rate problems involve using a known rate of change to find an associated rate of change.
• Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.
• A negative value for the derivative means the original function is decreasing.

Question 2

note

Answer 2

• As a runner is moving toward first base, the distance between the runner and second base changes.
• Suppose a runner is moving at 24 feet per second down the first baseline. How fast is the distance between the runner and second base changing at the instant the runner is halfway between home and first?
• Since the distance between the runner and first base is shrinking, it makes sense to define the velocity at which the runner is moving as a negative number instead of a positive one.
• The Pythagorean theorem gives you the distance between the runner and second base. Notice that the Pythagorean theorem relates three lengths together, one of which is a constant in this situation.
• Use implicit differentiation to find the derivative of s with respect to time.
• Substitute the known values into the equation and solve for ds/dt.
• Notice that the rate is negative. This shows that the distance is decreasing as the runner moves.

Question 3

Suppose you are told that z = x ^2y^ 3, where x and y are changing with time. Suppose also that x is increasing at a constant rate of 3 units / second and that y is decreasing at a constant rate of 2 units / second. What is the rate of change of z with respect to time when x = 5 and y = 7?

Answer 3

2,940 units / second

Question 4

Sand is flowing out of a hopper at a constant rate of 2/3 cubic feet per minute into a conical pile whose height is always twice its radius. What is the rate of change of the radius of the cone when the cone is 4 feet high?

Answer 4

0.027 feet per minute

Question 5

The distance between home plate and first base on a baseball diamond is 90 ft. A runner is moving towards first base at 24 ft / sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 60 ft away from first base?

Answer 5

dh/dt=−48/√13 ft/sec

Question 6

Peter is running on a circular track with a radius of 400 meters. If he is running at a constant rate of 6 meters per second, what is the rate of change of his distance from the center of the circular track?

Answer 6

0 meters / second

Question 7

As the second hand of a clock moves around the dial, it sweeps out a sector with a constantly increasing area.

Pablo’s clock has a second hand that is 5 centimeters long. How fast is the area of the sector that is swept out by the second hand increasing?

Answer 7

1.3 square centimeters per second