https://file.aiquickdraw.com/mathbot/file/1729473733954wi8zi06u.jpg

Solve for X to the nearest 10th Pl. and include the units

The problem shows a right triangle with angles and sides given. Based on the image, we have the following details:

- One angle is \( 40^\circ \) at the top of the triangle.
- Another angle is \( 25^\circ \) at the bottom right.
- The horizontal side (adjacent to the \( 25^\circ \) angle) is 87 feet long.
- We need to solve for the vertical height \( x \) (opposite the \( 40^\circ \) angle).

This is a typical trigonometry problem, and we can use the sine function because we know one angle and need to relate the opposite side (vertical height \( x \)) to the hypotenuse. To solve:

### Step 1: Find the hypotenuse using the sine function

From the triangle:
\[
\sin(25^\circ) = \frac{\text{opposite (height)}}{\text{hypotenuse}}
\]
The opposite side to the \( 25^\circ \) angle is 87 feet. Solving for the hypotenuse:
\[
\text{Hypotenuse} = \frac{87}{\sin(25^\circ)} = \frac{87}{0.4226} \approx 205.8 \, \text{feet}
\]

### Step 2: Use the hypotenuse to find \( x \) (the vertical height)

Now, use the sine of the \( 40^\circ \) angle to find \( x \):
\[
\sin(40^\circ) = \frac{x}{205.8}
\]
Solving for \( x \):
\[
x = 205.8 \times \sin(40^\circ) = 205.8 \times 0.6428 \approx 132.3 \, \text{feet}
\]

### Final Answer:
The height \( x \) is approximately **132.3 feet**.

Would you like further details or clarifications on the solution?

Here are some follow-up questions for deeper understanding:
1. What is the difference between using sine and cosine in triangle problems?
2. How would the calculation change if we were given a different side of the triangle?
3. Can you explain the significance of the right angle in this problem?
4. What if you had been asked to find the length of the hypotenuse directly?
5. How can the law of sines be applied in more complex triangle problems?

**Tip:** Always check which trigonometric ratio (sine, cosine, or tangent) is appropriate based on the sides and angles you're given.

Solve for x to the nearest tenth place and include the units.

Learn how to solve a trigonometry problem involving a right triangle, where you're given angles and one side length. This solution covers the use of the sine function to find the height (x) of a right triangle. Suitable for high school students.

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