Tree Height 9C6006

1. **State the problem:** We are given a right triangle where the hypotenuse (the slanted side) is $8.63$ m and the angle of inclination at the base is $68^\circ$. We need to find the height of the tree, which is the side opposite the angle, denoted as $x$. 2. **Formula used:** In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ Here, $\theta = 68^\circ$, opposite side = $x$, and hypotenuse = $8.63$ m. 3. **Set up the equation:** $$\sin(68^\circ) = \frac{x}{8.63}$$ 4. **Solve for $x$:** Multiply both sides by $8.63$: $$x = 8.63 \times \sin(68^\circ)$$ 5. **Calculate $\sin(68^\circ)$:** Using a calculator, $\sin(68^\circ) \approx 0.9272$ 6. **Find $x$:** $$x = 8.63 \times 0.9272 = 8.000136$$ 7. **Round the answer:** Rounded to two decimal places, $x = 8.00$ m. **Final answer:** The height of the tree is $8.00$ m.