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Math

QuestionFind the angle of elevation for a jet that travels 56 miles horizontally and reaches an altitude of 12.6 miles.

Studdy Solution

STEP 1

Assumptions1. The jet travels in a straight line from the point of takeoff to the point where it achieves the altitude. . The distance traveled by the jet is56 miles.
3. The altitude achieved by the jet is12.6 miles.
4. We are asked to find the angle of elevation, which is the angle formed by the horizontal ground and the line of sight to the jet.

STEP 2

We can use the tangent of the angle of elevation to find the angle. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{opposite}{adjacent}

STEP 3

In this case, the opposite side is the altitude of the jet and the adjacent side is the distance traveled by the jet on the ground. So we can write the equation astan⁡(θ)=altitudedistance\tan(\theta) = \frac{altitude}{distance}

STEP 4

Substitute the given values into the equation.
tan⁡(θ)=12.6 miles56 miles\tan(\theta) = \frac{12.6\, miles}{56\, miles}

STEP 5

Calculate the tangent of the angle.
tan⁡(θ)=12.56=0.225\tan(\theta) = \frac{12.}{56} =0.225

STEP 6

To find the angle, we need to use the inverse tangent function (also known as arctan or tan^-1). This function gives the angle whose tangent is a given number.
θ=arctan⁡(tan⁡(θ))\theta = \arctan(\tan(\theta))

STEP 7

Substitute the value of the tangent into the equation.
θ=arctan⁡(0.225)\theta = \arctan(0.225)

STEP 8

Calculate the angle of elevation. Remember that the arctan function gives the result in radians, so we need to convert it to degrees by multiplying by 180π\frac{180}{\pi}.
θ=arctan⁡(0.225)×180π\theta = \arctan(0.225) \times \frac{180}{\pi}

STEP 9

Calculate the angle of elevation.
θ=arctan⁡(.225)×180π=12.94 degrees\theta = \arctan(.225) \times \frac{180}{\pi} =12.94\, degreesThe angle of elevation for the jet's path is approximately12.94 degrees.

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