1a).A tower and a building stand on the same horizontal level.
From the point P at the bottom of the building, the angle of elevation of the top T,of the tower is 65°. From the top Q of the building, the angle of elevation of the point T is 25°. If the building is 20m high,Calculate the distance PT.
1b). Hence or otherwise calculatethe height of the tower ( give your answer to 3 significant figures)?
I assume you made your sketch
Look at triangle PQT
angle QPT = 25° , (90-65)
angle PQT = 115°
so you can find angle QTP
then by the sine law:
PT/sin115° = 20/sin(QTP)
all yours from here ....
To solve this problem, we can use trigonometric ratios such as tangent.
Let's solve part 1a) first to find the distance PT.
Step 1: Draw a diagram to visualize the given information.
- Draw a straight horizontal line to represent the ground.
- Draw a vertical line from point P to represent the height of the building (20m).
- Label the top of the building as Q and the top of the tower as T.
- Mark the angles of elevation: angle PQT = 25° and angle PTP = 65°.
Step 2: Identify the right triangle.
In this case, the right triangle is PQT. We want to find the length of PT.
Step 3: Apply the tangent ratio.
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this scenario:
tan(PTP) = opposite side (PT) / adjacent side (PTQ)
Step 4: Substitute the values.
tan(65°) = PT / 20m
Step 5: Solve for PT.
PT = 20m * tan(65°)
PT ≈ 41.90m (approximately)
So, the distance PT is approximately 41.90 meters.
Now, let's move on to part 1b) to calculate the height of the tower.
Step 6: Identify the right triangle.
In this case, the right triangle is QTP. We want to find the height of the tower, which is TP.
Step 7: Apply the tangent ratio.
tan(PQT) = opposite side (TP) / adjacent side (QT)
Step 8: Substitute the values.
tan(25°) = TP / PT
Step 9: Solve for TP.
TP = PT * tan(25°)
TP ≈ 41.90m * tan(25°)
TP ≈ 18.12m (approximately)
So, the height of the tower is approximately 18.12 meters (to 3 significant figures).