Question

Approaching a lighthouse A boat approaches a 50 -ft-high lighthouse whose base is at sea level. Let \(d\) be the distance between the boat and the base of the lighthouse. Let \(L\) be the distance between the boat and the top of the lighthouse. Let \(\theta\) be the angle of elevation between the boat and the top of the lighthouse. a. Express \(d\) as a function of \(\theta\) b. Express \(L\) as a function of \(\theta\)

Short answer

Question: A boat is approaching a lighthouse that is 50 feet tall. The angle of elevation from the boat to the top of the lighthouse changes as the boat gets closer. Express the distance from the boat to the base of the lighthouse (d) and the distance from the boat to the top of the lighthouse (L) in terms of the angle of elevation (θ). Answer: The distance from the boat to the base of the lighthouse, d(θ), is given by the expression: d(θ) = 50 / tan(θ) The distance from the boat to the top of the lighthouse, L(θ), is given by the expression: L(θ) = sqrt((50 / tan(θ))^2 + 50^2)

Step-by-step solution

a. Finding the expression of d in terms of θ

To express \(d\) as a function of \(\theta\), we start by looking at the right triangle formed by the distance to the base of the lighthouse \(d\), the height of the lighthouse (50ft), and the angle of elevation \(\theta\). We can use the tangent function to relate these variables: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{50}{d}\) Now, we solve for d: \(d = \frac{50}{\tan(\theta)}\) Hence, \(d(\theta) = \frac{50}{\tan(\theta)}\)

b. Finding the expression of L in terms of θ

To express \(L\) as a function of \(\theta\), we can use the Pythagorean theorem: \(L^2 = d^2 + 50^2\) However, we want to write \(L\) in terms of \(\theta\), not \(d\). We will substitute the expression for \(d(\theta)\) that we found in part (a) into the equation: \(L^2 = (\frac{50}{\tan(\theta)})^2 + 50^2\) Now, we solve for L: \(L = \sqrt{(\frac{50}{\tan(\theta)})^2+50^2}\) Hence, \(L(\theta) = \sqrt{(\frac{50}{\tan(\theta)})^2+50^2}\)

Key concepts

Right Triangle

When it comes to understanding right triangles, it's essential to first recognize that they are triangles with one angle measuring 90 degrees. This angle is often referred to as the "right angle."

Right triangles are foundational in many areas of mathematics and can be identified by their distinct structure:

• They have one 90-degree angle.
• The side opposite this angle is known as the hypotenuse, which is the longest side.
• The other two sides are called the legs of the triangle.

In the context of a lighthouse and a boat, imagine a straight line from the boat to the lighthouse base as one leg, the height of the lighthouse as the other leg, and the line from the boat to the top of the lighthouse as the hypotenuse.

Envisioning this setup can help you visualize how distances and angles relate to each other in a real-world scenario.

Tangent Function

The tangent function is one of the primary trigonometric functions used to relate angles and lengths in right triangles. In terms of a right triangle, the tangent of an angle is a ratio:

• The opposite side length to the angle over the adjacent side length.

So when you see \( an(\theta) = \frac{50}{d}\)in the exercise, it highlights this relationship. Here:

• The angle \(\theta\) is the angle of elevation from the boat to the top of the lighthouse.
• The "opposite" is the lighthouse’s height (50 ft).
• The "adjacent" is the distance from the boat to the lighthouse base, denoted by \(d\).

By rearranging this formula, you find that\(d = \frac{50}{\tan(\theta)}\),showing how \(d\) depends on the angle \(\theta\). This relationship is crucial in trigonometry whenever you need to find a missing length using a known angle.

Pythagorean Theorem

The Pythagorean Theorem is a key component of trigonometry and geometry involving right triangles. It provides a way to relate the lengths of the sides of a right triangle:

\[c^2 = a^2 + b^2\]
where:

• \(c\) is the length of the hypotenuse.
• \(a\) and \(b\) are the lengths of the other two sides.

In our lighthouse scenario, \(c\) is \(L\), \(a\) is \(d\), and \(b\) is the height of the lighthouse (50 ft).

When you use the equation:\(L^2 = d^2 + 50^2\),you apply the Pythagorean Theorem to solve for \(L\) after substituting \(d = \frac{50}{\tan(\theta)}\).

This substitution yields\(L = \sqrt{(\frac{50}{\tan(\theta)})^2 + 50^2}\),allowing you to express \(L\) directly in terms of \(\theta\). This demonstrates the power of the Pythagorean Theorem in connecting geometry and trigonometry.