From 70 feet away from the base of a tree on level ground, the angle to the top of a tree is 60⁰. What is the height of the tree? What is the approximate distance from the vantage point to the top of the tree, to one decimal point? (Assume that the tree is perpendicular to the ground.)
14.0
To find the height of the tree, you can use the tangent function, which relates the angle of elevation to the height and the distance from the tree.
Let's denote the height of the tree as 'h' and the distance from the vantage point to the top of the tree as 'd'. We are given that the distance from the base of the tree to the vantage point is 70 feet, and the angle of elevation to the top of the tree is 60 degrees.
Using the tangent function, we have the equation tan(60) = h/70, where h is the height of the tree and 70 is the distance from the base of the tree to the vantage point.
To solve for h, we can rearrange the equation: h = 70 * tan(60).
Now let's calculate the value of h:
h = 70 * tan(60)
h ≈ 70 * 1.7321
h ≈ 121.24 feet
Therefore, the height of the tree is approximately 121.24 feet.
To find the approximate distance from the vantage point to the top of the tree, we already have the height of the tree (h) and the distance from the base of the tree to the vantage point (70 feet).
Using the Pythagorean theorem, we can find the distance (d) as follows:
d = √(h^2 + 70^2)
d = √(121.24^2 + 70^2)
d ≈ √(14681 + 4900)
d ≈ √19581
d ≈ 139.9 feet (approx.)
So, the approximate distance from the vantage point to the top of the tree is approximately 139.9 feet (rounded to one decimal point).
sketch a right-angled triangle with the given information.
height/70 = tan 60°
height = 70tan60 = appr 121.2 ft
for distance to top of tree:
cos 60° = 70/hypotenuse
hypotenuse = 70/cos60 = 140.0 ft