To calculate the distance the shortstop needs to throw the ball to first base, we can visualize the field as a square with the following bases:
- Second base (2B)
- Third base (3B)
- First base (1B)
Since the bases are typically 90 feet apart in a standard baseball field, we can represent the points on a coordinate plane:
- 2B is at (90, 90)
- 1B is at (90, 0)
- The shortstop is standing between 2B and 3B at a point 30 feet from 2B, which places her at coordinates (90 - 30, 90) = (60, 90).
Now, we can find the distance from this point (60, 90) to first base (90, 0) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where
- \((x_1, y_1) = (60, 90)\) (position of the shortstop)
- \((x_2, y_2) = (90, 0)\) (position of 1B)
Substituting the coordinates into the formula:
\[ d = \sqrt{(90 - 60)^2 + (0 - 90)^2} \] \[ d = \sqrt{(30)^2 + (-90)^2} \] \[ d = \sqrt{900 + 8100} \] \[ d = \sqrt{9000} \] \[ d \approx 94.868 (rounded) \]
Rounding to the nearest tenth, the distance she needs to throw the ball to first base is approximately:
\[ \boxed{94.9} \text{ feet.} \]