Final answer:
To find the rate at which Kova's distance from third base decreases, use the Pythagorean Theorem and the derivative with respect to time. When halfway between first and second base, the rate is calculated to be 6/sqrt(5) ft/sec based on Kova's speed of 6 ft/sec.
Step-by-step explanation:
The student's question involves using the Pythagorean Theorem and derivatives to solve a related rates problem, which is a classic type of question in calculus. To solve for the rate at which Kova's distance from third base decreases, we need to consider the baseball diamond as two right-angled triangles and use the rate of change of the hypotenuse when Kova is halfway between first and second base.
Let's denote the distance between Kova and third base as d, and the distance between Kova and second base as x which changes with time. Here, when Kova is halfway between the bases, x is equal to 10 ft. Since the sides of the baseball diamond are 20 ft long, we can use the Pythagorean Theorem to establish the relationship between x, d, and the side of the diamond.
When Kova has run 10 ft, the situation forms a right-angled triangle with sides 10 ft, 20 ft (the distance to third base), and d as the hypotenuse. According to the Pythagorean Theorem:
d^2 = 10^2 + 20^2
If we differentiate both sides with respect to time t, we have:
2d*(dd/dt) = 2*10*(dx/dt)
Given that Kova runs at 6 ft/sec, we substitute dx/dt with 6 ft/sec:
2d*(dd/dt) = 2*10*6
To find the value of d, we plug back the values into the Pythagorean Theorem:
d = sqrt(10^2 + 20^2) = √(100 + 400) = √(500) = 10*√(5) ft.
Now we can solve for (dd/dt):
(dd/dt) = (2*10*6) / (2*10√(5)) = 6/√(5) ft/sec
This value is the rate at which Kova's distance from third base decreases when she is halfway between first and second base.