A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4 mi apart, to be x = 27° and y = 53°, as shown in the figure. (Round your answers to two decimal places.)
(a) Find the distance of the plane from point A.
mi
(b) Find the elevation of the plane.
mi
I THINK I GOT IT
okay so draw a triangle (not a right triangle)
the bottom number is 4
draw an imaginary line straight down from the tip of the triangle to the bottom
half a circle is 180. so the total middle area of the triangle is 180-27-53 which is 100.
keep in mind that 27 and 53 are the ANGLES OF DEPRESSION. so draw a straight line across the top of your triangle and the area between this straight top line and the side of your triangle is the angle of depression.
now look at your imaginary line.
we know that 1/4 of a triangle is 90 deg
90-27=63.
so your top angle in THIS HALF of the triangle is 63degrees
basically treat each half like 2 different triangles.
we can determine the 3rd angle (i have this marked as A) by 180-90-63 (the 90 is where our straight line down is)
so we know that A is 27
now we have all 3 angles to one side of our triangle
now repeat this process to the other half of the triangle
(keep in mind we can NOT divide the 4 miles)
so we should have 90, 37 and 53 for triangle half #2.
NOW look at the triangle as one big triangle
the top is 180, the sides are 27 and 53
the bottom length is 4
look at this as a ASA triangle
27deg, 4, 53deg
now use law of sines
a/sinA= c/sinC
plug in numbers
a/sin27= 4/sin180
now a= (4x sin27)/ sin180
plug that into your calculator and you have side a
repeat this to find side b
the answer to your first question will be the length of side b
for the second question you need to take these same numbers
divide your triangle in half again
solve for the new "side a" aka the line you drew down the middle
tada you have your answers!
yay math
HOLD UP HOLD UP
okay sorry y'all one minor error.... ignore my last post here is the new and improved and correct explanation
I THINK I GOT IT
okay so draw a triangle (not a right triangle)
the bottom number is 4
draw an imaginary line straight down from the tip of the triangle to the bottom
half a circle is 180. so the total middle area of the triangle is 180-27-53 which is 100.
keep in mind that 27 and 53 are the ANGLES OF DEPRESSION. so draw a straight line across the top of your triangle and the area between this straight top line and the side of your triangle is the angle of depression.
now look at your imaginary line.
we know that 1/4 of a triangle is 90 deg
90-27=63.
so your top angle in THIS HALF of the triangle is 63degrees
basically treat each half like 2 different triangles.
we can determine the 3rd angle (i have this marked as A) by 180-90-63 (the 90 is where our straight line down is)
so we know that A is 27
now we have all 3 angles to one side of our triangle
now repeat this process to the other half of the triangle
(keep in mind we can NOT divide the 4 miles)
so we should have 90, 37 and 53 for triangle half #2.
NOW look at the triangle as one big triangle
the top is 100, the sides are 27 and 53
the bottom length is 4
look at this as a ASA triangle
27deg, 4, 53deg
now use law of sines
a/sinA= c/sinC
plug in numbers
a/sin27= 4/sin100
now a= (4x sin27)/ sin100
plug that into your calculator and you have side a
repeat this to find side b
the answer to your first question will be the length of side b
for the second question you need to take these same numbers
divide your triangle in half again
solve for the new "side a" aka the line you drew down the middle
tada you have your answers!
yay math
To solve this problem, we can use trigonometry, specifically the tangent function.
(a) Find the distance of the plane from point A:
Let's assume point A is below the plane and point B is above the plane. We can form a right triangle with the plane as the hypotenuse and the sides representing the distances from the plane to points A and B. The angle of depression from point B will be y = 53°, and the distance between points A and B is 4 miles.
Using the tangent function, we can write the following equation:
tan(y) = (distance from plane to B) / (distance between A and B)
tan(53°) = AB / 4
Rearranging the equation, we get:
AB = 4 * tan(53°)
AB ≈ 4 * 1.327
Therefore, the distance of the plane from point A is approximately 5.306 miles.
(b) Find the elevation of the plane:
To find the elevation of the plane, we need to calculate the vertical distance from point A to the plane. We can use the tangent function again, this time with the angle of depression x = 27°.
tan(x) = (elevation of the plane) / AB
tan(27°) = elevation / 5.306
Rearranging the equation and solving for elevation, we have:
elevation = 5.306 * tan(27°)
elevation ≈ 5.306 * 0.5095
The elevation of the plane is approximately 2.703 miles.
i don't get it
I'm strugglin so hard on this
let me start you off
draw a straight line representing 4miles apart..... now at the ends of the line make angles of 27 and 53
now figure it out