the angle of elevation of a top of a tree from point P on horizontal ground is 30 degrees.Fromanother point Q 8 METRES FROM THE BASEof the tree,the angle of elevation of the top of the tree is 48 degrees. a)calculate to one decimal place the height of the tree b)calculate the distance between P and Q
Did you make your sketch?
let the height of the tree be h metres.
I see a right-angled triangle where
tan 48° = h/8 ----> h = 8tan48°
let PQ = x, then
tan 30° = h/(x+8)
(x+8)tan30° = h, but you know h, so you can find x
To calculate the height of the tree, we can use trigonometric ratios. Let's break down the steps:
a) Calculate the height of the tree:
1. Draw a diagram to visualize the problem. Label the tree as T, point P as the point on the ground, and point Q as the point 8 meters from the base of the tree.
2. Use basic trigonometry. We have a right-angled triangle with the height of the tree as the vertical leg. The angle of elevation from point P is 30 degrees, and the opposite side is the height of the tree (h).
3. We can use the tangent (tan) function to find the height:
tan(30 degrees) = h / distance from point P to the base of the tree.
tan(30) = h / (distance from point Q to the base of the tree)
Since point Q is 8 meters from the base of the tree, we need to subtract the distance from point P to point Q (which will give us the distance from point P to the base of the tree):
h = tan(30) * (distance from Q to the base of the tree - distance from P to Q)
h = tan(30) * (8 - 0) (we subtract 0 since point P is the origin on the ground)
Solve the equation to find the height of the tree:
h ≈ tan(30) * 8
Using a calculator, the height of the tree (h) ≈ 4.62 meters.
Therefore, the height of the tree is approximately 4.62 meters.
b) Calculate the distance between P and Q:
Using the given angles, we can find the distance between P and Q by creating another right-angled triangle.
1. Again, refer to the diagram you drew in the previous step.
2. In this right-angled triangle, the base will be the distance between P and Q (x), and the angle of elevation from point Q is 48 degrees.
3. Use the tangent (tan) function to find the distance between P and Q:
tan(48 degrees) = height of the tree / distance between P and Q.
tan(48) = 4.62 / x (height of the tree is the value we calculated in part a)
Solve the equation to find the distance between P and Q:
x ≈ 4.62 / tan(48)
Using a calculator, the distance between P and Q (x) ≈ 4.89 meters.
Therefore, the distance between points P and Q is approximately 4.89 meters.