the angle elevation of the top of a cliff from point p is 45 from a point Q which is 10m from P towards the root of the cliff the angle of elevation is 48 calculate the height of the cliff.
Locate point R at foot of cliff.
Tan45 = h/(PQ+QR) = h/(10+QR).
h = (10+QR)Tan45.
Tan48 = h/QR.
h = QR*Tan48.
(10+QR)Tan45 = QR*Tan48.
10+QR = 1.11QR
0.11QR = 10
QR = 90.91m.
h = QR*Tan48 = 90.91*Tan48 = 100.91 m.
Having seen this type of problem hundreds of times, here is a re-phrasing:
The angle elevation of the top of a cliff from point P is 45°.
From a point Q, which is 10m from P towards the root of the cliff, the angle of elevation is 48°.
Calculate the height of the cliff.
Make your diagram, label the "root" of the cliff as R, and the top of the cliff as T
Since angleTPR = 48°, angle TPQ = 132° , angle PTQ = 3°, and we know PQ =10 metres
Using the sine law:
TP/sin45 = 10/sin3
TP = 10sin45°/sin3° = ....
now in triangle TPR, which is right-angled, you know angle TPR and you know TP
so TR/TP = sin48°
your height of the cliff TR = TPsin48° = .... metres
If you draw the diagram and review your basic trig functions, it should be clear that if the height is h, then
h cot45° - h cot48° = 10
why wrong answer when using cos
Thank s
To calculate the height of the cliff, we can use basic trigonometry principles. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram representing the cliff, point P, point Q, and the angles of elevation.
Cliff
-----
/ | \
/ | H \
P / θ1| \ Q
/_____|_____\
Note: H represents the height of the cliff.
Step 2: Identify the given information
From the problem statement, we know the following:
- The angle of elevation at point P is 45 degrees.
- The angle of elevation at point Q is 48 degrees.
- The distance between point P and point Q is 10 meters.
Step 3: Identify the trigonometric function to use
Based on the given information and the angles involved, we can use the tangent function (tan), as it relates the height (opposite side) to the distance (adjacent side) from the angle of elevation.
Step 4: Calculate the height of the cliff
Using the tangent function, we can set up the following equation:
tan(θ1) = H / x
where θ1 is the angle of elevation at point P, and x is the distance between point P and the root of the cliff.
By substituting the given values in the equation, we get:
tan(45) = H / x
Since x is not given, we need to find it. To do so, we can use the information that the distance between point P and point Q is 10 meters.
x = 10 meters
Now we can solve for H:
tan(45) = H / 10
H = 10 * tan(45)
Use a calculator to find tan(45) ≈ 1. Hence:
H ≈ 10 * 1
H ≈ 10 meters
Therefore, the height of the cliff is approximately 10 meters.