Question

find the lengths of w, x, y, and z shown in the figure below if wy=62. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Find \( w \) using cosine

In the right triangle with hypotenuse 8 and angle \( 52^\circ \), \( \cos(52^\circ)=\frac{w}{8} \), so \( w = 8\cos(52^\circ) \approx 8\times0.6157 = 4.9256\approx4.9 \)

Step2: Find \( x \) using sine

\( \sin(52^\circ)=\frac{x}{8} \), so \( x = 8\sin(52^\circ) \approx 8\times0.7880 = 6.304\approx6.3 \)

Step3: Find \( y \)

Given \( w + y = 62 \), so \( y = 62 - w \approx 62 - 4.9256 = 57.0744\approx57.1 \)

Step4: Find \( z \) using Pythagorean theorem

\( z=\sqrt{x^{2}+y^{2}}=\sqrt{6.304^{2}+57.0744^{2}}\approx\sqrt{39.74 + 3257.58}\approx\sqrt{3297.32}\approx57.4 \)

Answer:

\( w\approx4.9 \), \( x\approx6.3 \), \( y\approx57.1 \), \( z\approx57.4 \)