THERE IS A SMALL ISLAND IN THE MIDDLE OF A 100M WIDE RIVER AND A TALL TREE STANDS ON THE ISLAND.pAND q RE POINTS DIRECTLY OPPOSITE EACH OTHER ON THE TWO BANKS AND IN LINE WITH THE TREE.IF THE ANGLE OF ELEVATION OF THE TOP OF THE TREE FROM p AND q ARE 30 DEGREES AND 45 DEGREES.FIND THE HEIGHT OF THE TREE.
Tan30= h/p
Tan45=h/q
p + q= 100
Use substitution
(h/tan30) + (h/ tan45 )= 100
You should be able to take over
To find the height of the tree, we can use trigonometry and the concept of similar triangles. Let's break down the problem step by step.
1. Draw a diagram: Draw a diagram representing the situation. Label the points as follows:
- Let O be the base of the tree on the island.
- Let T be the top of the tree.
- Let P and Q be the points on the banks of the river, directly opposite each other and in line with the tree.
- Label the 100m distance across the river as AB, with A being on the same bank as point P and B being on the same bank as point Q.
2. Identify the triangles: In this diagram, we have two right-angled triangles. One triangle is OAP and the other is OTQ.
3. Determine the relationships: We are given the angles of elevation of the top of the tree from points P and Q, which are 30 degrees and 45 degrees, respectively.
4. Use trigonometry: The tangent function relates the angle of elevation to the height and distance. Therefore, we can set up the following equations:
- In triangle OAP:
- tan(30°) = height of the tree / distance AB.
- In triangle OTQ:
- tan(45°) = height of the tree / distance AB.
5. Solve for height: To find the height of the tree, solve the equations for the height. Since both equations are equal to the same height, we can set them equal to each other:
- tan(30°) = tan(45°) = height of the tree / distance AB.
- Substitute the actual values of tangent:
- 1/√3 = 1 = height of the tree / 100.
- Solve for the height: Cross-multiply and solve for the height of the tree:
- height of the tree = 100 * 1 = 100 meters.
Therefore, the height of the tree is 100 meters.