To solve problem 12, we'll also use trigonometric ratios. The diagram shows a right triangle where:

• The angle at the base is $29^\circ$,
• The opposite side (height of the kite from the ground) is 28 units,
• The hypotenuse is $x$, which we need to find.

Step-by-Step Solution:

1. Identify the Trigonometric Function: Since we know the angle and the opposite side, and we're looking for the hypotenuse, we use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

2. Set Up the Equation: $\sin(29^\circ) = \frac{28}{x}$

3. Solve for $x$: $x = \frac{28}{\sin(29^\circ)}$

4. Calculate the Value: $x \approx \frac{28}{0.4848} = 57.74$

5. Round to the Nearest Hundredth: $x \approx 57.74$

Final Answer:

The value of $x$ is approximately 57.74.

Would you like any further details on this solution, or do you have any other questions?

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Relative Questions:

1. How would you solve for $x$ if the angle were given as $45^\circ$?
2. What if the height of the kite (opposite side) was changed to 35 units, how would that affect the value of $x$?
3. Can you explain why the sine function is appropriate for this problem, instead of cosine or tangent?
4. How would you find the length of the side adjacent to the angle given the same problem?
5. What is the cosine of a $29^\circ$ angle, and how could you use it to find the adjacent side?

Tip: When dealing with right triangles, always make sure you're using the correct trigonometric ratio based on the sides you know and what you're solving for.