At a point on the ground 70 feet from the base of the tree the distance the top of the tree is 2 feet more than three times the height of the tree find the height of the
Find the height of a tree, if from a point 70 ft from the base the angle of elevation to the top is 38 degrees 35 seconds.
see the previous post. Same problem, different numbers.
To find the height of the tree, we can use the concept of similar triangles. Let's denote the height of the tree as "h."
According to the problem, the distance from the base of the tree to a point on the ground is 70 feet. We can represent this distance as the base of a triangle, which we'll call "b."
It is also given that the distance from the top of the tree to that same point on the ground is 2 feet more than three times the height of the tree. So, the height of the top of the tree can be represented as "3h + 2."
Now, we can set up a proportion between the smaller triangle formed by the height of the tree and its distance on the ground and the larger triangle formed by the height of the top of the tree and its distance on the ground:
h / b = (3h + 2) / 70
To solve this equation, we'll cross-multiply:
70h = b(3h + 2)
Expand the equation:
70h = 3bh + 2b
Rearrange the terms:
3bh - 70h = -2b
Factor out 'h':
h(3b - 70) = -2b
Divide both sides by (3b - 70) to solve for h:
h = -2b / (3b - 70)
Therefore, the height of the tree is -2b / (3b - 70).
To find the height of the tree, we can use the concept of similar triangles.
Let's denote the height of the tree as 'h'. According to the information given, the distance between the point on the ground and the base of the tree is 70 ft.
Using the concept of similar triangles, we can set up the following equation:
h / (70 + h) = (3h + 2) / 70
To solve this equation, we can cross-multiply:
h * 70 = (3h + 2) * (70 + h)
Now, let's simplify:
70h = (3h + 2)(70 + h)
Expanding the right side:
70h = 210h + 3h^2 + 140 + 2h
Combine like terms:
0 = 3h^2 + 212h + 140
We now have a quadratic equation. To solve it, we can factor or use the quadratic formula. Let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = 212, and c = 140. Plugging in these values:
h = (-212 ± √(212^2 - 4 * 3 * 140)) / (2 * 3)
Calculating:
h ≈ (-212 ± √(44944 - 1680)) / 6
h ≈ (-212 ± √43264) / 6
h ≈ (-212 ± 208) / 6
We have two possible solutions:
h1 ≈ (-212 + 208) / 6 ≈ -4 / 6 ≈ -0.67
h2 ≈ (-212 - 208) / 6 ≈ -420 / 6 ≈ -70
Since a negative height doesn't make sense in this context, we can disregard h1.
Therefore, the height of the tree is approximately 70 feet.