To find the distance from the climber to the top of the cliff, we can break the problem into steps.
- First, we will calculate the height of the cliff using the angle of elevation from the rescue team.
Given:
- Distance from the rescue team to the cliff (base) = 700 ft
- Angle of elevation to the top of the cliff = 65°
Using the tangent function: \[ h = d \cdot \tan(\theta) \] where \( h \) is the height of the cliff, \( d = 700 \) ft, and \( \theta = 65° \).
Calculating the height \( h \): \[ h = 700 \cdot \tan(65°) \] Using a calculator: \[ \tan(65°) \approx 2.1445 \] \[ h \approx 700 \cdot 2.1445 \approx 1,501.15 \text{ ft} \]
- Next, we will find the height of the climber using the angle of elevation of 40°.
Using the same formula with the angle of elevation to the climber: \[ h_c = d \cdot \tan(40°) \] Calculating the height \( h_c \): \[ h_c = 700 \cdot \tan(40°) \] Using a calculator: \[ \tan(40°) \approx 0.8391 \] \[ h_c \approx 700 \cdot 0.8391 \approx 587.37 \text{ ft} \]
- Finally, we find the distance from the climber to the top of the cliff by subtracting the height of the climber from the height of the cliff: \[ \text{Distance from climber to top} = h - h_c \] \[ \text{Distance} \approx 1501.15 - 587.37 \approx 913.78 \text{ ft} \]
So, the climber is approximately 913.78 ft from the top of the cliff.
Thus, the correct response is: The climber is 913.78 ft. from the top of the cliff.
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