To find the distance the shortstop needs to throw the ball to first base, we can visualize the baseball diamond as a square where each side represents 90 feet (the distance between each base).
Since the shortstop is 35 feet from second base, we can calculate her position relative to first and second base. She is positioned on the line that connects second and third base.
- The distance from second base to first base is 90 feet.
- The shortstop is positioned 35 feet away from second base, meaning she is \((90 - 35) = 55\) feet away from third base, along the line connecting the bases.
Now, we can use the Pythagorean theorem to determine the distance from the shortstop's position to first base.
The coordinates of the bases can be imagined as follows:
- Second base: \((45, 90)\) assuming home plate is at \((0, 0)\)
- First base: \((90, 0)\)
- The position of the shortstop is \((45 - 35, 90)\) or \((10, 90)\).
Now we can find the distance from point \((10, 90)\) (the shortstop's position) to point \((90, 0)\) (first base).
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1) = (10, 90)\) and \((x_2, y_2) = (90, 0)\):
\[ d = \sqrt{(90 - 10)^2 + (0 - 90)^2} \] \[ d = \sqrt{(80)^2 + (-90)^2} \] \[ d = \sqrt{6400 + 8100} \] \[ d = \sqrt{14500} \] \[ d \approx 120.4 \text{ feet} \]
Thus, the shortstop needs to throw the ball approximately 120.4 feet to first base.
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