As a baseball player runs from second to third base, he is going 20 feet per second. What is the rate of change from the runner to home when he is halfway to third base? (The sides of a baseball field are 90 feet)
x^2 = (45-20*t)^2 + 90^2
2 x dx/dt = 2 *(45-20t) (-20)
at x=sqrt(45^2+90^2) when t = 0
2 (100.6) dx/dt = 90*-20
dx/dt = -8.94 ft/s
To find the rate of change from the runner to home when he is halfway to third base, we need to determine the distance from the runner to home at that point.
Given that the sides of a baseball field are 90 feet, the distance from second base to third base would be 90 feet. Since the runner is halfway to third base, the distance from the runner to third base would be 90 feet divided by 2, which is 45 feet.
Now, to find the rate of change from the runner to home, we need to calculate how fast the distance from the runner to home is changing. We already know the speed of the runner, which is 20 feet per second.
Since the runner is halfway to third base, the distance from the runner to home is also halfway between second base and home plate, which is 90 feet divided by 2, or 45 feet.
To determine the rate of change, we can divide the distance from the runner to home (45 feet) by the speed of the runner (20 feet per second):
Rate of change = Distance / Time
Rate of change = 45 feet / 20 feet per second
Simplifying this expression, we find:
Rate of change = 2.25 seconds
Therefore, the rate of change from the runner to home when he is halfway to third base is 2.25 seconds.