Mickey determines that the angle of elevation from his position to the top of the tower is 52°. He measures the angle of elevation again from a point 47 meters farther from the tower and finds to be 31°. Both positions are due East of the tower. Find the height of the tower.
Your previous 3 posts are all single trig ratios in a right-angled triangle. You MUST know the basic 3 trig ratios and how they are applied in such cases.
This one is the only one that requires more than one step.
make a sketch, mark the top of the tower P and its base Q
Mark his first position A, and his 2nd position B
Fill in the information,
In triangle ABP,
angle A = 31°
angle B = 180-52 = 128°
then angle P = 180-128-31 = 21°
by the Sine Law:
BP/sin31 = 47/sin21
BP = 47sin31/sin21 = .....
Now back to the right-angled triangle PBQ
PQ/PB = sin52
PQ = BPsin52
= (you know BP from above)
or in one string of calculations
= (47sin31)(sin52)/sin21 = appr 53.23 m
or 53 m to the nearest metre
To find the height of the tower, we can use the trigonometric relationships between the angle of elevation, the distance from the tower, and the height of the tower.
Let's start by labeling the given information.
Angle of elevation from Mickey's position: 52°
Angle of elevation from a point 47 meters farther from the tower: 31°
Let's denote the distance from Mickey's position to the tower as "x" meters. So, the distance from the second position to the tower would be "x + 47" meters.
Now, we can use the tangent function to set up an equation for each position:
For Mickey's position:
tan(52°) = height / x
For the second position:
tan(31°) = height / (x + 47)
We have two equations with two variables. We can solve this system of equations to find the height of the tower.
Using the first equation, we can rewrite it as:
height = x * tan(52°)
Substitute this expression for height in the second equation:
tan(31°) = (x * tan(52°)) / (x + 47)
Now we can solve for x. Multiply both sides of the equation by (x + 47):
(x + 47) * tan(31°) = x * tan(52°)
Distribute and isolate x:
x * tan(31°) + 47 * tan(31°) = x * tan(52°)
Now, subtract x * tan(31°) from both sides:
47 * tan(31°) = x * tan(52°) - x * tan(31°)
Factor out x on the right side:
47 * tan(31°) = x * (tan(52°) - tan(31°))
Divide both sides by (tan(52°) - tan(31°)):
x = (47 * tan(31°)) / (tan(52°) - tan(31°))
Now, we can substitute the value of x into the expression for height:
height = x * tan(52°)
Solve for height:
height = [(47 * tan(31°)) / (tan(52°) - tan(31°))] * tan(52°)
Evaluate this expression to find the height of the tower.