Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

The front of an A-frame cottage has the shape of an isosceles triangle. It stands 28 feet high and the angle of elevation of its roof is \(70^{\circ} .\) What is the width of the cottage at its base?

Step-by-Step Solution

Answer

To solve for the width of an A-frame cottage when given its height and roof angle, apply the tangent function of trigonometry. After breaking the isosceles triangle into two identical right triangles, the tangent of the angle equates to the triangle's height divided by half of the base width. Solve this expression for the half-width, then double it to obtain the full width.

Step 1: Analyze the Shape and Assign Parameters

Recognize the shape as an isosceles triangle and subsequently assign its parameters. The height here is 28 feet (H), the angle of elevation of the roof is \(70^{\circ}\) (θ), and the width to be discovered (W) forms the base of the triangle.

Step 2: Slice the Triangle into Two Right Triangles

As we are dealing with an isosceles triangle, we can cut it along the height to create two congruent right triangles. This results in splitting the base into two equal halves, each end of the base becoming the adjacent side of its corresponding right triangle. Now, our width W is actually 2 times the length of this adjacent side (a).

Step 3: Apply Trigonometry

We can now apply trigonometry, citing the Tangent function specifically. The Tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle to the side adjacent to the angle. Which gives \(tan(θ) = \frac{H}{a}\) where θ is the angle, H is the height, and a is the length of the adjacent side. We are given \(tan(70^{\circ})\), and we can substitute the known value of H (28 feet) into our equation.

Step 4: Solve for a

Solving the equation \(tan(70^{\circ}) = \frac{28}{a}\) for a, we obtain \(a = \frac{28}{tan(70^{\circ})}\). Calculate this to find the length of a.

Step 5: Calculate the Width of the Cottage

Recall that the total width (W) is 2 times the length of the adjacent side (2a). So, perform this calculation to obtain the width of the cottage at its base.

Key Concepts

Angle of Elevation

The angle of elevation is a term used to describe the angle between the horizontal plane and the line of sight when looking up at an object. In the context of our exercise, it refers to the 70-degree angle formed by the roof of the A-frame cottage when viewed from the ground level.
The angle of elevation is crucial to solving problems involving heights and distances because it helps us use trigonometric relationships effectively. In this problem, the 70-degree angle of elevation allows us to explore the shape and resolve it into more manageable, right triangles.

Right Triangle

Recognizing a right triangle is essential when working with trigonometry problems because they make it easier to apply mathematical functions. In this particular problem, we initially encounter an isosceles triangle. However, by slicing it along its height, we can create two congruent right triangles.
These right triangles share the isosceles triangle's height as their opposite side, and each right triangle's hypotenuse forms part of the line running up the roof of the cottage. Understanding this transformation is the first step in solving for unknowns using trigonometric functions.

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. It's a crucial tool for solving problems involving angles and distances.
For this exercise, we're using trigonometry to relate the given angle of elevation and height to the unknown base width we're trying to calculate. Specifically, we use trigonometric ratios like tangent, sine, and cosine to connect these elements. Here, we narrow our focus on the tangent function, which connects the opposite and adjacent sides of the triangles we created.

Tangent Function

The tangent function is one of the primary trigonometric ratios used to relate the angles and sides of a right triangle. It is defined as the ratio of the side opposite the angle to the side adjacent to the angle in a right triangle. For our problem, this is expressed as:

• \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

In the exercise, we applied the tangent function by utilizing the given angle of elevation \( \theta = 70^{\circ} \) and the height of the triangle (28 feet) to find the length of the adjacent side.
Once we calculated the adjacent side, we simply doubled it to find the base of the isosceles triangle, ultimately solving for the width of the cottage.