Question

1. find the distance from the boat to the base of the lighthouse, to the nearest metre. 46 m 42° x

Explanation:

Step1: Identify the trig - ratio

We have a right - triangle where the height of the lighthouse is the opposite side to the given angle and the distance from the boat to the base of the lighthouse is the adjacent side. We use the tangent function, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Set up the equation

Given $\theta = 42^{\circ}$ and the opposite side $y = 46$ m. Let the adjacent side be $x$. So, $\tan42^{\circ}=\frac{46}{x}$.

Step3: Solve for $x$

We know that $\tan42^{\circ}\approx0.9004$. Then $x=\frac{46}{\tan42^{\circ}}$. Substituting the value of $\tan42^{\circ}$, we get $x=\frac{46}{0.9004}\approx51$ m.

Answer:

51 m