To find the rate at which the distance between the runner and the pitcher is changing, we need to find the derivative of the distance function with respect to time (t). Let's call the distance between the runner and the pitcher D.
Since the baseball diamond is a 90-foot square, the distance D is the hypotenuse of a right triangle formed by the runner, the pitcher, and the baseline. The length of each side of the square is 90 feet, and the runner is crossing the baseline.
To calculate this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, for the triangle formed by the pitcher, runner, and baseline, we have:
D^2 = B^2 + R^2
Where B is the distance from the pitcher's mound to the baseline (45 feet), and R is the distance from the runner to the baseline. We want to find dD/dt, the rate at which D is changing with respect to time.
Differentiating both sides of the equation with respect to time, we get:
2D(dD/dt) = 0 + 2R(dR/dt)
Since the baseline distance B is constant, its derivative is 0. So, the equation becomes:
2D(dD/dt) = 2R(dR/dt)
Dividing both sides by 2D, we get:
dD/dt = (R * dR/dt) / D
Now, we need to substitute the given values into the equation:
R = 90 feet (since the runner is crossing the baseline, which is one side of the square)
dR/dt = 25 ft/sec (given)
To find D, we can use the Pythagorean theorem:
D = sqrt(B^2 + R^2)
D = sqrt((45 feet)^2 + (90 feet)^2)
D = sqrt(2025 + 8100)
D = sqrt(10125)
D ≈ 100.62 feet
Substituting these values into the equation, we have:
dD/dt = (90 feet * 25 ft/sec) / 100.62 feet
dD/dt ≈ 22.36 ft/sec
Therefore, the rate at which the distance between the runner and the pitcher is changing as he crosses the plate is approximately 22.36 ft/sec.