Subjects trigonometry

Kite Height 08941B

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Kite Height 08941B


1. **Problem statement:** Kristoffer's kite string makes an angle of 15° with the ground, and the string length is $\sqrt{6} + \sqrt{2}$ meters. We need to find how high the kite is above the ground. 2. **Formula used:** The height of the kite corresponds to the vertical component of the string length. Using trigonometry, height $h = \text{string length} \times \sin(\theta)$ where $\theta = 15^\circ$. 3. **Calculate the height:** $$ h = (\sqrt{6} + \sqrt{2}) \times \sin(15^\circ) $$ 4. **Evaluate $\sin(15^\circ)$:** $$ \sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ $$ Using known values: $$ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} $$ 5. **Substitute back:** $$ h = (\sqrt{6} + \sqrt{2}) \times \frac{\sqrt{6} - \sqrt{2}}{4} $$ 6. **Simplify the product:** $$ (\sqrt{6} + \sqrt{2})(\sqrt{6} - \sqrt{2}) = (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 $$ 7. **Calculate height:** $$ h = \frac{4}{4} = 1 $$ **Final answer:** The kite is 1 meter above the ground.