A 15 m flagpole stands on level ground. from point P, due west of the flagpole the angle of elevation of the top of the pole is 38 degrees. From point Q, due north of the flagpole, the flagpole has an angle of elevation of 25 degrees. Find the distance of PQ, correct to one decimal place.
To find the distance PQ, we can use trigonometry.
First, let's draw a diagram to better understand the problem.
Q
|
| .
| .
| .
| . P
|____________________
|---------------------
Flagpole
Let's label some important information:
- Height of the flagpole = 15 m
- Angle of elevation from P = 38 degrees
- Angle of elevation from Q = 25 degrees
To find the distance PQ, we can use the tangent of the angle of elevation from P:
tan(38) = opposite/adjacent
The opposite side is the height of the flagpole (15 m), and the adjacent side is the distance PQ.
tan(38) = 15/PQ
Now, let's solve for PQ:
PQ = 15 / tan(38)
Using a calculator:
PQ ≈ 15 / 0.7813
PQ ≈ 19.2
Therefore, the distance PQ is approximately 19.2 meters.
To find the distance of PQ, we can use trigonometry.
Let's draw a diagram to visualize the situation:
P
|\
| \
h_1 | \ h_2
| \
|_ _ _\_________Flagpole
p q
From the diagram, we can see that we have a right triangle at P and another right triangle at Q.
Let's focus on the triangle at P first, where we know the angle of elevation to the top of the flagpole is 38 degrees. In this triangle, we can identify the opposite side (h_1) and the adjacent side (p).
Now, using trigonometry, we can use the tangent function to find the length of p:
tan(38) = h_1 / p
To find the value of h_1, we need to use the height of the flagpole, given as 15 m. So we have:
tan(38) = 15 / p
Now let's move on to the triangle at Q, where we know the angle of elevation to the top of the flagpole is 25 degrees. In this triangle, we can identify the opposite side (h_2) and the adjacent side (q).
Using the same logic as before, we have:
tan(25) = h_2 / q
Now, let's find the value of h_2. We can relate it to h_1 using the height of the flagpole:
h_2 = h_1 + 15
Substituting this into the equation above, we have:
tan(25) = (h_1 + 15) / q
Now, we have two equations:
(1) tan(38) = 15 / p
(2) tan(25) = (h_1 + 15) / q
We can solve these two equations simultaneously to find the values of p and q.
Once we have the values of p and q, we can use the Pythagorean theorem to find the distance PQ:
PQ = √(p^2 + q^2)
Now, let's solve these equations and calculate the distance PQ.
Label the top of the pole as T and its base as O
in triangle OTP,
tan38 = 15/OP
OP = 15/tan 38
in triangle TOP,
cos 25 + 15/OQ
OQ = 15/tan25
in the right-angled triangle POQ
PQ^2 = OP^2 + OQ^2
I will let you finish it.