Calculating the distance of a baseball throw using the distance formula.

Let's solve the problem step by step and verify the solution.

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Problem Statement:

A player throws the ball from first base to a point located between third base and home plate, and 10 feet from third base. We are to find the distance of the throw, to the nearest tenth of a foot.

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Step 1: Understand the Baseball Field Layout

In a standard baseball diamond:
- The bases form a square with sides of 90 feet.
- The coordinates of the bases are typically set as:
- Home plate: (0, 0)
- First base: (90, 0)
- Second base: (90, 90)
- Third base: (0, 90)

So:
- Home plate = (0, 0)
- First base = (90, 0)
- Third base = (0, 90)

The point the ball is thrown to is 10 feet from third base, toward home plate.

Since third base is at (0, 90) and home plate is at (0, 0), the line between them is vertical along the y-axis.

So moving 10 feet from third base toward home plate means going down the y-axis by 10 feet:

> $$
> (0, 90) \to (0, 90 - 10) = (0, 80)
> $$

✔ So the target point is (0, 80).

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Step 2: Use Distance Formula

We want the distance from first base (90, 0) to the point (0, 80).

Use the distance formula:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Plug in:
- $ x_1 = 90, y_1 = 0 $
- $ x_2 = 0, y_2 = 80 $

$$
d = \sqrt{(0 - 90)^2 + (80 - 0)^2}
= \sqrt{(-90)^2 + (80)^2}
= \sqrt{8100 + 6400}
= \sqrt{14500}
$$

Now compute $ \sqrt{14500} $:

$$
\sqrt{14500} \approx 120.415...
$$

Rounded to the nearest tenth:

$$
\boxed{120.4} \text{ feet}
$$

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✔ Final Answer:

$$
\boxed{120.4} \text{ feet}
$$

This matches the handwritten solution.

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📌 Summary:

- The throw is from first base (90, 0) to a point 10 ft from third base toward home plate, which is (0, 80).
- Using the distance formula:
$$
d = \sqrt{(90)^2 + (80)^2} = \sqrt{14500} \approx 120.4 \text{ ft}
$$
- ✔ Correct answer: 120.4 feet.