To solve this problem, we can use trigonometry, particularly the tangent function, since we are dealing with angles of elevation and distances. The steps can be broken down as follows:

Step 1: Label the elements of the problem

Let:

• $h$ be the height of the cliff from the ground to its top.
• $d_1$ be the horizontal distance Sarah is from the base of the cliff when she first sights it (at 60° angle).
• $d_2 = d_1 + 50$ be the horizontal distance when she steps back 50 meters (at 30° angle).
• $1.8$ meters is Sarah's height, so the angle of elevation will be to the top of the cliff from her eye level, not her entire height.

Step 2: Use the tangent function

For both angles, we can use the tangent formula: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ where the opposite side is the height of the cliff minus Sarah's height (since the angle of elevation is measured from her eye level), and the adjacent side is the horizontal distance from her position to the base of the cliff.

First position (60° angle):

From Sarah's first position: $\tan(60^\circ) = \frac{h - 1.8}{d_1}$ Since $\tan(60^\circ) = \sqrt{3} \approx 1.732$, we have: $1.732 = \frac{h - 1.8}{d_1}$ Thus, $d_1 = \frac{h - 1.8}{1.732} \quad \text{(Equation 1)}$

Second position (30° angle):

From her second position, stepping back 50 meters: $\tan(30^\circ) = \frac{h - 1.8}{d_2}$ Since $\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577$, we have: $0.577 = \frac{h - 1.8}{d_2}$ Since $d_2 = d_1 + 50$, we substitute $d_2$ into the equation: $0.577 = \frac{h - 1.8}{d_1 + 50}$ Thus, $d_1 + 50 = \frac{h - 1.8}{0.577} \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

Now, substitute $d_1$ from Equation 1 into Equation 2.

From Equation 1: $d_1 = \frac{h - 1.8}{1.732}$

Substitute this into Equation 2: $\frac{h - 1.8}{1.732} + 50 = \frac{h - 1.8}{0.577}$

Now, solve for $h$.

Step 4: Simplify and solve for $h$

First, subtract 50 from both sides: $\frac{h - 1.8}{1.732} = \frac{h - 1.8}{0.577} - 50$

Now multiply through by 1.732 and 0.577 to eliminate the fractions. This will give you the exact height of the cliff.

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