Question

find d.

(there is a right - angled triangle in the image, with one angle of 30°, one angle of 60°, the side adjacent to the 30° angle is (5sqrt{6}) mm, and the side we need to find is (d))

write your answer in simplest radical form.

millimeters

Explanation:

Step1: Identify cosine relationship

For the 30° angle, $\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5\sqrt{6}}{d}$

Step2: Substitute $\cos(30^\circ)$ value

$\cos(30^\circ) = \frac{\sqrt{3}}{2}$, so $\frac{\sqrt{3}}{2} = \frac{5\sqrt{6}}{d}$

Step3: Solve for $d$ via cross-multiplication

$d = \frac{5\sqrt{6} \times 2}{\sqrt{3}}$

Step4: Simplify the radical expression

$d = \frac{10\sqrt{6}}{\sqrt{3}} = 10\sqrt{\frac{6}{3}} = 10\sqrt{2}$

Answer:

$10\sqrt{2}$ millimeters