Question

a baseball diamond is a square with side 90 feet. a batter hits the ball and runs toward first base with a speed of 28 ft/s. (a) at what rate (in ft/s) is his distance from second base decreasing when he is halfway to first base? (round your answer to one decimal place.)

Answer 1

12.5 ft/s

Explanation:

You want to know the rate of decrease of distance to 2nd base when a runner is halfway to first base, running at 28 ft/s on a baseball diamond with baselines that are 90 ft.

Speed made good

The speed in the direction of 2nd base is the speed in the direction of first base multiplied by the cosine of the angle between those directions.

Angle

The angle between the first base line and second base can be found from the inverse tangent function:

Tan = Opposite/Adjacent

tan(α) = (90 ft)/(1/2·90 ft) = 2

α = arctan(2)

Application

Then the speed of interest is ...

(-28 ft/s) · cos(arctan(2)) ≈ -12.5 ft/s

The distance to 2nd base is decreasing at 12.5 ft/s.

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Additional comment

The rate of change can also be found using the distance formula to write an expression for the distance to 2nd base as a function of time. Then the derivative of that distance is the desired rate of change. The second attachment shows this approach.

The third attachment is a diagram showing the geometry of the problem.