a tree grew at a 3 degree slant from the vertical. at a point 60 feet from the tree, the angle of elevation to the top of the tree is 14 degrees. find the height of the tree to the nearest tenth of a foot.

Depends if the tree is leaning 3° towards the observer or away from the observer.

I will assume "towards"
So the diagram will show a triangle with two base angles of 87° and 14° with the contained side of 60 ft
leaving the vertical angle at the top 79°
by sine law:
x/sin14 = 60/sin79
x = 60sin14/sin79 = appr 14.8

If you had the other option, repeat with the base angle of 93° instead of 87

To find the height of the tree, we need to use the concept of trigonometry and create a right triangle with the given information.

Let's denote the height of the tree as 'h' (in feet).

We can see that the tree grew at a 3-degree slant, meaning the angle between the vertical and the tree trunk is 3 degrees.

From the given information, we know that the angle of elevation to the top of the tree from a distance of 60 feet is 14 degrees.

Now, we can create a right triangle ABC, where:
- Angle A is the angle of elevation (14 degrees)
- Angle B is the angle between the vertical and the tree trunk (3 degrees)
- Side AB is the height of the tree (h)
- Side BC is the distance from the tree to the point where the angle of elevation is measured (60 feet)

We can use the tangent function to relate the angle of elevation with the height and distance:
tan(A) = h / BC

tan(14 degrees) = h / 60

To find the height, we can isolate 'h' in the above equation:
h = tan(14 degrees) * 60

Using a calculator, we find that tan(14 degrees) is approximately 0.246. Plugging this value into the equation, we get:
h ≈ 0.246 * 60 ≈ 14.76

Therefore, the height of the tree is approximately 14.76 feet (rounded to the nearest tenth of a foot).

To find the height of the tree, we can use trigonometry. Let's break down the problem step by step:

Step 1: Draw a diagram.
Draw a diagram illustrating the situation. Label the tree, the point where the angle of elevation is measured, and the angle of elevation itself.

Step 2: Define the given information.
Let's define the given information:
Angle of slant from the vertical = 3 degrees
Distance from the point to the tree = 60 feet
Angle of elevation = 14 degrees

Step 3: Identify the relevant trigonometric relationship.
We can use the tangent function to relate the angle of elevation to the height of the tree. The tangent of an angle is equal to the opposite side divided by the adjacent side.

Step 4: Determine the lengths of the sides.
In this case, the height of the tree is the opposite side, and the distance from the point to the tree is the adjacent side.

Step 5: Set up the equation.
Using the tangent function, we can set up an equation:
tan(angle of elevation) = height of the tree / distance from the point to the tree

Plugging in the values we know:
tan(14 degrees) = height of the tree / 60 feet

Step 6: Solve for the height of the tree.
Rearrange the equation to solve for the height of the tree:
height of the tree = tan(14 degrees) * 60 feet

Using a calculator, evaluate the tangent of 14 degrees and multiply it by 60:
height of the tree ≈ 14.07 feet

Therefore, the height of the tree, to the nearest tenth of a foot, is approximately 14.1 feet.