A baseball diamond has the shape of a square with sides 90ft long. A player 60ft from second base is running towards third base at a speed of 28ft/min. At what rate is the player’s distance from the home plate changing?
To find the rate at which the player's distance from the home plate is changing, we can use the concept of related rates. This involves using the Pythagorean theorem and differentiating with respect to time.
Let's label the baseball diamond as follows:
- Home plate is labeled H.
- First base is labeled A.
- Second base is labeled B.
- Third base is labeled C.
Given:
- The baseball diamond is a square with sides 90ft long.
- The player is 60ft from second base (B).
- The player is running towards third base (C) at a speed of 28ft/min.
To find the rate at which the player's distance from the home plate is changing, we need to determine the speed at which the distance HC is changing.
We can use the Pythagorean theorem to relate the distances:
AC^2 = AH^2 + HC^2
Differentiating both sides of the equation with respect to time (t), we get:
2AC * dAC/dt = 2AH * dAH/dt + 2HC * dHC/dt
Since we are interested in finding dHC/dt (the rate at which HC is changing), we can solve for it:
dHC/dt = (AC * dAC/dt - AH * dAH/dt) / HC
Now let's calculate the values:
- AC: The diagonal of a square can be found using the formula AC = side * sqrt(2). In this case, AC = 90ft * sqrt(2).
- AH: Since the distances from home plate to first base (AH) and from home plate to second base (AB) are equal, AH = AB = 90ft.
- dAC/dt: The rate at which AC is changing is zero since the baseball diamond does not change size.
- dAH/dt: The rate at which AH is changing is also zero since the distance to first base does not change.
Substituting the values, we get:
dHC/dt = [(90ft * sqrt(2)) * 0 - (90ft * 0)] / HC
Simplifying further, we have:
dHC/dt = 0
Therefore, the rate at which the player's distance from the home plate is changing is 0ft/min. This means the player's position relative to the home plate remains constant.