A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 3 mi apart, to be angle x = 31° and angle y = 49°, as shown in the figure. (Round your answers to two decimal places.)
Find the distance of the plane from point A.
Find the elevation of the plane.
A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.7 mi apart, to be 36° and 41°, as shown in the figure
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To find the distance of the plane from point A, we can use trigonometry and the given angles of depression.
Let's assume that point A is located directly below the plane, and point B is the milepost closer to the plane, as shown in the figure. The distance from point B to the plane can be represented as d1, and the distance from point A to the plane can be represented as d2.
Using the tangent function, we can set up the following equation:
tan(x) = d1 / 3
Substituting the given angle x, we have:
tan(31°) = d1 / 3
Using a calculator to find the tangent of 31°, we get:
0.6009 = d1 / 3
Multiplying both sides of the equation by 3, we have:
d1 = 0.6009 * 3
d1 ≈ 1.8027
Therefore, the distance from point B to the plane (d1) is approximately 1.8027 miles.
To find the distance from point A to the plane (d2), we can use the same logic:
tan(y) = d2 / 3
Substituting the given angle y, we have:
tan(49°) = d2 / 3
Using a calculator to find the tangent of 49°, we get:
1.1226 = d2 / 3
Multiplying both sides of the equation by 3, we have:
d2 = 1.1226 * 3
d2 ≈ 3.3678
Therefore, the distance from point A to the plane (d2) is approximately 3.3678 miles.
Now, to find the distance of the plane from point A, we can use the Pythagorean theorem:
distance of plane from point A = √(d1^2 + d2^2)
Substituting the values we found earlier, we have:
distance of plane from point A = √(1.8027^2 + 3.3678^2)
Using a calculator, we get:
distance of plane from point A ≈ √(3.2482 + 11.3093)
distance of plane from point A ≈ √14.5575
distance of plane from point A ≈ 3.8203
Therefore, the distance of the plane from point A is approximately 3.8203 miles.
To find the elevation of the plane, we can use the tangent function:
tan(z) = opposite / adjacent
tan(z) = d2 / d1
tan(z) = 3.3678 / 1.8027
Using a calculator, we get:
tan(z) ≈ 1.8681
Therefore, the tangent of the elevation angle (z) is approximately 1.8681. To find the elevation angle, we take the inverse tangent (arctan) of 1.8681:
z ≈ arctan(1.8681)
Using a calculator, we get:
z ≈ 63.57°
Therefore, the elevation of the plane is approximately 63.57°.
To find the distance of the plane from point A, we can use the tangent function, as the angles of depression are given. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Let's call the distance of the plane from point A as d.
From the given information, we can construct two right triangles: Triangle ABC and Triangle ABD.
In Triangle ABC:
Angle C = 31°
Length of the adjacent side = 3 miles (distance between mileposts)
Length of the opposite side = h (height)
Using the tangent function for angle C, we have:
tan(C) = opposite/adjacent
tan(31°) = h/3
Simplifying the equation, we get:
h = 3 * tan(31°)
Calculating this value, h is approximately 1.63 miles.
In Triangle ABD:
Angle D = 49°
Length of the adjacent side = 3 miles (distance between mileposts)
Length of the opposite side = h + d (height + distance of plane from point A)
Using the tangent function for angle D, we have:
tan(D) = opposite/adjacent
tan(49°) = (h + d)/3
Rearranging the equation to solve for d, we get:
d = 3 * tan(49°) - h
Plugging in the values we calculated previously, we have:
d = 3 * tan(49°) - 1.63
Calculating this value, d is approximately 3.06 miles.
Therefore, the distance of the plane from point A is approximately 3.06 miles.
To find the elevation of the plane, we can substitute the value of d into the equation for the opposite side of Triangle ABD.
Opposite side = h + d = 1.63 + 3.06 = 4.69 miles
Thus, the elevation of the plane is approximately 4.69 miles.