John wishes to determine the height of a flagpole. He takes a sighting of the top of the flagpole from point P. He then moves further away from the flagpole by 20 metres to point Q and takes a second sighting. Given the angle Q is 28 degrees and angle P is 53 degrees.
Find how high is the flagpole?
Well, determining the height of a flagpole can be a real "pole" in the neck! But, fear not, my friend. We can use a little bit of trigonometry to solve this conundrum.
Let's call the height of the flagpole "h". Now, since John moved from point P to point Q, we've got ourselves a right-angled triangle. The angle at point P is 53 degrees, and the angle at point Q is 28 degrees. The side opposite the angle at P is our desired height "h", while the adjacent side is the distance John moved, which is 20 metres.
Now, we can use the tangent function to solve for "h". The tangent of an angle is equal to the opposite side divided by the adjacent side. Therefore, we have:
tan(53 degrees) = h / 20
To solve for "h", we can multiply both sides of the equation by 20:
h = 20 * tan(53 degrees)
And if we plug this into a calculator, we find that h is approximately equal to 25.25 metres.
So, the height of the flagpole is roughly 25.25 metres high. I hope that puts a "flag" on your excitement!
To find the height of the flagpole, we can use trigonometry and the concept of similar triangles.
Let's represent the height of the flagpole as h, the distance between point P and the base of the flagpole as x, and the distance between point Q and the base of the flagpole as y.
From point P, we have an angle of 53 degrees, and from point Q, we have an angle of 28 degrees. We can set up the following proportions using the tangent function:
tan(53 degrees) = h / x equation (1)
tan(28 degrees) = h / y equation (2)
Now we need to relate x and y. From the problem statement, we know that John moved 20 meters from point P to point Q. Therefore, we have:
y = x + 20 equation (3)
To find the height of the flagpole, we need to solve the system of equations (1), (2), and (3). Let's simplify equation (3) by substituting y with x + 20:
tan(28 degrees) = h / (x + 20) equation (4)
Now, we can solve the system of equations (1), (2), and (4) for h.
First, let's rearrange equation (1) to solve for x:
x = h / tan(53 degrees) equation (5)
Next, substitute equation (5) into equation (4):
tan(28 degrees) = h / (h / tan(53 degrees) + 20)
Simplifying further:
tan(28 degrees) = tan(53 degrees) / (h / tan(53 degrees) + 20)
Cross-multiply:
tan(28 degrees) * (h / tan(53 degrees) + 20) = tan(53 degrees)
Distribute:
h + 20 * tan(28 degrees) = tan(53 degrees) * tan(28 degrees)
Now, solve for h:
h = (tan(53 degrees) * tan(28 degrees) - 20 * tan(28 degrees)) / (1 - tan(53 degrees))
To find the height of the flagpole, we can use trigonometry. Let's draw a diagram to visualize the problem.
1. Draw a right-angled triangle ABC, where A represents the top of the flagpole, C is point P, and angle P is 53 degrees.
. A
/ |
/ |
/ |
C-----
2. Extend side AC further away from the flagpole by 20 meters to point Q.
. A
/ |
/ |
/ |
C-----Q
3. Now, we have a second right-angled triangle ACQ, where AC represents the height of the flagpole, angle P is still 53 degrees, and angle Q is 28 degrees.
AC
. A |
/ | 28°
/ |
/ |
C-----Q
To find the height of the flagpole (AC), we can use the tangent function.
Tangent (angle) = Opposite / Adjacent
In this case, the opposite side is CQ, and the adjacent side is AQ.
Tangent (angle Q) = CQ / AQ
Since we know angle Q is 28°, we can write the equation as:
Tangent (28°) = CQ / 20
To find CQ, rearrange the equation:
CQ = 20 * Tangent (28°)
CQ ≈ 10.209 meters
Now, we need to find AQ. We know that the angle P is 53 degrees. Using the same tangent function:
Tangent (angle P) = AC / AQ
Tangent (53°) = AC / AQ
To find AQ, rearrange the equation:
AQ = AC / Tangent (53°)
Substituting the known values:
AQ = (10.209) / Tangent (53°)
AQ ≈ 7.246 meters
Finally, to find the height of the flagpole (AC), we can subtract AQ from AC:
AC ≈ CQ - AQ
AC ≈ 10.209 - 7.246
AC ≈ 2.963 meters
Therefore, the height of the flagpole is approximately 2.963 meters.
If P is x meters from the base of the pole of height h, then
h/x = tan 53°
h/(x+20) = tan 28°
eliminate x and you have
h cot 53° = h cot 28° - 20
h = 20/(cot28°-cot53°)