A baseball diamond is a square with side 90 ft. a batter hits the ball and runs toward first base with a speed of 24 ft/s.
(a) At what rate is his distance from second base decreasing when he is halfway to first base?
Is the answer 0.0932ft/s?
I don't see that it is logical because I did not get a negative answer..
To find the rate at which the batter's distance from second base is decreasing when he is halfway to first base, we need to use the concept of related rates. We can start by defining some variables:
Let x be the distance between the batter and first base.
Let y be the distance between the batter and second base.
Given that the baseball diamond is a square with a side length of 90 ft, we can determine the relationship between x and y:
x^2 + y^2 = 90^2
To find the rate at which y is changing with respect to time, we can differentiate both sides of the equation with respect to time (t):
2x(dx/dt) + 2y(dy/dt) = 0
Simplifying the equation gives:
x(dx/dt) + y(dy/dt) = 0
We are given that dx/dt, the rate at which the batter is running towards first base, is 24 ft/s. When the batter is halfway to first base, x = 45 ft.
Substituting the known values into the equation:
45(24) + y(dy/dt) = 0
Solving for dy/dt, we get:
y(dy/dt) = -45(24)
dy/dt = -45(24)/y
To find the rate at which the batter's distance from second base is decreasing, we need to substitute y = half of the distance between the bases. Since the distance between bases is 90 ft, y = 90/2 = 45 ft.
Substituting this value into the equation gives:
dy/dt = -45(24)/45 = -24 ft/s
Therefore, the batter's distance from second base is decreasing at a rate of 24 ft/s when he is halfway to first base.
So, the correct answer is -24 ft/s, not 0.0932 ft/s. The negative sign indicates that the distance from second base is decreasing.