Math

Question Find the horizontal distance from a helicopter to a landing pad if the angle of depression is 3737^{\circ} and the helicopter is 1250 feet from the ground.

Studdy Solution

STEP 1

Assumptions
1. The angle of depression from the helicopter to the landing pad is 3737^{\circ}.
2. The helicopter's altitude above the ground is 1250 feet.
3. The situation can be modeled using a right triangle, where the altitude forms the opposite side, the horizontal distance forms the adjacent side, and the angle of depression corresponds to the angle of elevation from the landing pad to the helicopter.

STEP 2

To find the horizontal distance, we will use the trigonometric tangent function, which relates the angle of a right triangle to the ratio of the opposite side over the adjacent side.
tan(θ)=Opposite sideAdjacent side\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}

STEP 3

Since the angle of depression from the helicopter to the landing pad is the same as the angle of elevation from the landing pad to the helicopter, we can use the given angle of depression for our calculations.
θ=37\theta = 37^{\circ}

STEP 4

The opposite side in our right triangle is the altitude of the helicopter, which is given as 1250 feet.
Opposite side=1250 feet\text{Opposite side} = 1250 \text{ feet}

STEP 5

We can rearrange the tangent function formula to solve for the adjacent side, which is the horizontal distance we are looking for.
Adjacent side=Opposite sidetan(θ)\text{Adjacent side} = \frac{\text{Opposite side}}{\tan(\theta)}

STEP 6

Plug in the values for the opposite side and the angle into the formula.
Adjacent side=1250 feettan(37)\text{Adjacent side} = \frac{1250 \text{ feet}}{\tan(37^{\circ})}

STEP 7

Calculate the tangent of 3737^{\circ}.
tan(37)0.75355\tan(37^{\circ}) \approx 0.75355

STEP 8

Now, divide the opposite side by the tangent of the angle to find the adjacent side.
Adjacent side=1250 feet0.75355\text{Adjacent side} = \frac{1250 \text{ feet}}{0.75355}

STEP 9

Perform the division to calculate the horizontal distance.
Adjacent side1250 feet0.753551658.69 feet\text{Adjacent side} \approx \frac{1250 \text{ feet}}{0.75355} \approx 1658.69 \text{ feet}
The horizontal distance from the helicopter to the landing pad is approximately 1658.69 feet.

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