To solve this problem, we can use trigonometry, specifically the tangent function, because we have a right triangle formed by the ship, the base of the lighthouse, and its top.

Given:

1. Height of the lighthouse ($h$) = 100 ft
2. Angle of elevation to the base of the lighthouse ($\theta_1$) = $17^\circ 24'$
3. Angle of elevation to the top of the lighthouse ($\theta_2$) = $21^\circ 6'$

Step 1: Convert angles to decimal form

To calculate trigonometric functions accurately, we convert degrees and minutes to decimal degrees:

• $\theta_1 = 17^\circ 24' = 17 + \frac{24}{60} = 17.4^\circ$
• $\theta_2 = 21^\circ 6' = 21 + \frac{6}{60} = 21.1^\circ$

Step 2: Distance relationships using tangent

Let the distance from the ship to the cliff be $d$.

1. For the angle to the base of the lighthouse: $\tan(\theta_1) = \frac{\text{height of the base of the lighthouse (ground level)}}{d}$ Since the base is at ground level, we write: $d = \frac{0}{\tan(17.4^\circ)} = d$

To clarify the problem and continue the explanation properly, in this case, we're given the cliff high-to-point-between-lighthouses.The tan**) to proceedLet us proceed methodically with the correct calculation to determine the distance from the ship to the cliff using trigonometry.

Step-by-step Solution

We are tasked to find the horizontal distance $d$ between the ship and the base of the cliff.

Geometry of the problem:

1. From the ship, the angles of elevation to the base of the lighthouse and its top are given.
2. The lighthouse's height $h$ = 100 ft.
3. The two triangles involved are as follows:
   • One formed with the base of the lighthouse and the horizontal distance $d$.
   • The other formed with the top of the lighthouse and the same horizontal distance $d$.

We now apply trigonometric relationships to calculate $d$.

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1. Calculate the distance to the base of the lighthouse using the smaller angle $\theta_1 = 17.4^\circ$:

Using the tangent function: $\tan(\theta_1) = \frac{\text{height of the base of the lighthouse}}{d}$ Here, the height of the base is 0 ft (cliff ground level): [ d =