At takeoff, a plane flies at an angle of 10° with the runway. After it has traveled a ground distance of 2,800 feet, find the vertical distance the plane has gained from takeoff. Round your answer to the nearest foot.

(vertical distance) / (ground distance) = tan(10º)

To find the vertical distance the plane has gained from takeoff, we need to use trigonometry.

First, we need to understand the given information. The angle of 10° with the runway is the angle of elevation. It means the plane is flying upwards at an angle of 10° with respect to the horizontal (runway).

The ground distance traveled by the plane is 2,800 feet. This is the horizontal distance traveled by the plane along the ground.

To find the vertical distance gained, we can use the trigonometric function tangent (tan), which is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle (in this case, the angle of 10°).

In our scenario, the horizontal distance traveled (adjacent side) is 2,800 feet. We want to find the vertical distance gained (opposite side).

Using the tangent function:
tan(10°) = vertical distance gained / 2,800

We can rearrange the equation to solve for the vertical distance gained:
vertical distance gained = tan(10°) × 2,800

Using a calculator:
tan(10°) ≈ 0.1763269807

Substituting the value into the equation:
vertical distance gained ≈ 0.1763269807 × 2,800

Calculating the result:
vertical distance gained ≈ 493.08 feet

Therefore, the vertical distance the plane has gained from takeoff is approximately 493 feet (rounded to the nearest foot).

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