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Problem 427

# At a point on the ground 40 feet from the foot of a tree, the angle of elevation to the top of the tree is $$42^{\circ}$$. Find the height of the tree to the nearest foot.

Expert verified
The height of the tree is approximately 36 feet.
See the step by step solution

## Step 1: Write down the tangent function formula

The tangent function formula is given by: $$tan(\theta) = \frac{opposite}{adjacent}$$ In this problem, the opposite side is the height of the tree (h), the adjacent side is the distance between the point on the ground and the foot of the tree (40 feet), and the angle θ is given as $$42^{\circ}$$.

## Step 2: Plug in the given values

Now, let's plug the given values into the tangent function formula: $$tan(42^{\circ}) = \frac{h}{40}$$

## Step 3: Solve for h

To find the height of the tree (h), we will multiply both sides of the equation by 40: $$h = 40 \times tan(42^{\circ})$$ Now, use a calculator to find the value of $$tan(42^{\circ})$$: $$h = 40 \times 0.9004$$

## Step 4: Calculate the height of the tree

Multiply 40 by 0.9004 to find the approximate height of the tree: $$h \approx 36.016$$ Since we are asked to find the height of the tree to the nearest foot, we can round the value of h to the nearest whole number: $$h \approx 36$$ Thus, the height of the tree is approximately 36 feet.

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