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**Homework:9.3 HW - The Law of Cosines**

**Question 28,** **9.3.57**

**HW Score:** 92.24%, 26.75 of 29 points

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Part 1

The distance from the home plate to the fence in dead center at a certain baseball field is

408408

feet. How far is it from the fence in the dead center to third​ base?

408 ft408 ft

90 ft90 ft

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Part 1

The fence in the dead center is about

enter your response here

feet from third base.

​(Round to two decimal places as​ needed.)

The fence in the dead center is about(Round to two decimal places as needed.)

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To solve this problem, we can use the **Law of Cosines**. Here’s a breakdown:

### Problem Setup:
1. We have a triangle where:
   - The distance from home plate to the fence (in dead center) is **408 feet**.
   - The distance from home plate to third base is **90 feet**.
   - The angle between the line from home plate to third base and the line from home plate to the center fence is **90 degrees** (since it forms a right angle on a standard baseball field).

### Applying the Law of Cosines:
The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \( c \) is the side opposite the angle \( C \),
- \( a \) and \( b \) are the other two sides,
- \( C \) is the angle between sides \( a \) and \( b \).

In this case:
- Let \( a = 408 \) feet (from home plate to the fence).
- Let \( b = 90 \) feet (from home plate to third base).
- Let \( C = 90^\circ \).

Since \( \cos(90^\circ) = 0 \), the formula simplifies to:
\[
c^2 = a^2 + b^2
\]
So:
\[
c^2 = 408^2 + 90^2
\]
\[
c^2 = 166464 + 8100
\]
\[
c^2 = 174564
\]
Taking the square root:
\[
c = \sqrt{174564} \approx 417.33 \text{ feet}
\]

### Final Answer:
The distance from the fence in the dead center to third base is approximately **417.33 feet**.

Would you like a deeper explanation or have any other questions?

---

### Related Questions:
1. How would the calculation change if the angle was not 90 degrees?
2. What is the Law of Cosines used for in real-world applications?
3. How is the Law of Cosines derived from the Pythagorean Theorem?
4. Can we use the Pythagorean Theorem directly for this problem?
5. How would we approach this problem if it involved a three-dimensional space?

#### Tip:
The Law of Cosines is very helpful in non-right triangles, especially when you know two sides and the included angle.

The distance from the home plate to the fence in dead center at a certain baseball field is 408 feet. How far is it from the fence in the dead center to third base? (Round to two decimal places as needed.)

Find the distance from the fence in dead center to third base on a baseball field using the Law of Cosines. Given a right-angle triangle with distances of 408 feet and 90 feet, this problem provides a step-by-step calculation with rounding to two decimal places. Ideal for students in Grades 10-12 studying trigonometry.

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