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?
(b) At what rate is his distance from third base increasing at the same moment?
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To find the rates in this scenario, we can use the concepts of related rates. By using the Pythagorean theorem, we can determine the relationships between the distances involved.
Let's start by labeling the diagram:
```
First Base
*----------* <- batter's current position
/ x \
/ \
/ o \
/ \
Second Base Third Base
*----------* *----------*
```
Let's denote the distance from the batter's current position to second base as y and the distance from the batter's current position to third base as z.
(a) 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 find dy/dt.
At any point in time, the position of the batter can be represented by the coordinates (x, y). Since the baseball diamond is a square, the x-coordinate and y-coordinate are always equal.
Given that the side length of the square is 90 ft, when the batter is halfway to first base, the x-coordinate and y-coordinate will both be equal to half the side length, which is 45 ft.
To find dy/dt, we need to apply the Pythagorean theorem:
(x)^2 + (y)^2 = (90)^2
Since x = y in this case:
(45)^2 + (y)^2 = (90)^2
2025 + (y)^2 = 8100
(y)^2 = 8100 - 2025
(y)^2 = 6075
Differentiating both sides with respect to time (t):
2y(dy/dt) = 0
Simplifying:
2y(dy/dt) = 0
(dy/dt) = 0
The rate at which the batter's distance from second base is decreasing when he is halfway to first base is 0 ft/s. This means that the distance is not changing.
(b) To find the rate at which the batter's distance from third base is increasing at the same moment, we need to find dz/dt.
Using the same method as in question (a), let's find z when x = 45 ft:
(x)^2 + (z)^2 = (90)^2
Since x = z in this case:
(45)^2 + (z)^2 = (90)^2
2025 + (z)^2 = 8100
(z)^2 = 8100 - 2025
(z)^2 = 6075
Differentiating both sides with respect to time (t):
2z(dz/dt) = 0
Simplifying:
2z(dz/dt) = 0
(dz/dt) = 0
The rate at which the batter's distance from third base is increasing when he is halfway to first base is 0 ft/s. This means that the distance is not changing.
Therefore, both the distances from second base (y) and third base (z) are not changing when the batter is halfway to first base.