Question

e) 4 ft 30° y

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the adjacent side to the \(30^{\circ}\) angle is \(y\), and the hypotenuse is \(4\sqrt{3}\).

Step2: Apply the cosine formula

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\). Using the formula \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), we have \(\cos30^{\circ}=\frac{y}{4\sqrt{3}}\).

Step3: Solve for \(y\)

Substitute \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) into \(\frac{\sqrt{3}}{2}=\frac{y}{4\sqrt{3}}\). Cross - multiply: \(y = 4\sqrt{3}\times\frac{\sqrt{3}}{2}\). Since \(\sqrt{3}\times\sqrt{3}=3\), then \(y = 4\times\frac{3}{2}=6\).

Answer:

\(y = 6\)