Problem 46 - Precalculus: Functions and Graphs

Question

A ladder 20 feet long leans against the side of a building, and the angle between the ladder and the building is \(22^{\circ}\) (a) Approximate the distance from the bottom of the ladder to the building. (b) If the distance from the bottom of the ladder to the building is increased by 3.0 feet, approximately how far does the top of the ladder move down the building?

Step-by-Step Solution

Answer

(a) 18.544 feet, (b) 0.231 feet.

Step 1: Understanding the Problem

The ladder forms a right triangle with the building and the ground. The ladder itself is the hypotenuse of this triangle, the distance from the ladder base to the building is the adjacent side, and the angle formed between the ladder and the building is given.

Step 2: Apply the Cosine Formula

To find the distance from the bottom of the ladder to the building (the adjacent side), apply the cosine formula: \[\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\]Plugging in the known values:\[\cos(22^\circ) = \frac{x}{20}\]where \(x\) is the distance from the bottom of the ladder to the building.

Step 3: Solve for Adjacent Side

Rearrange the equation from Step 2 to solve for \(x\):\[x = 20 \times \cos(22^\circ)\]Calculate using a calculator: \[x \approx 20 \times 0.9272 \approx 18.544 ext{ feet}\]So, the distance from the bottom of the ladder to the building is approximately 18.544 feet.

Step 4: Consider the Increase in Distance

If the distance from the bottom of the ladder to the building is increased by 3 feet, the new distance \(x'\) is:\[x' = 18.544 + 3 = 21.544 ext{ feet}\]This will change the angle of elevation of the ladder.

Step 5: Recalculate the Height with Increased Distance

Now, use the Pythagorean theorem to find the new height \(y'\) when the distance from the bottom is 21.544 feet:\[\text{hypotenuse}^2 = (\text{opposite})^2 + (\text{adjacent})^2\]\[20^2 = (y')^2 + 21.544^2\]Solve for \(y'\):\[(y')^2 = 400 - 464.556\]\[y' = \sqrt{400 - 464.556} \approx \sqrt{335.444} \approx 18.313 ext{ feet}\]

Step 6: Calculate the Change in Height

The change in height is the original height minus the new height:\[18.544 - 18.313 = 0.231 ext{ feet}\]So, the top of the ladder moves down approximately 0.231 feet.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry that relates the angles and sides of a right triangle. In our exercise, the cosine function helps us determine the distance from the base of the ladder to the building. We use the following formula to work it out: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]Here:

• \( \theta \) is the angle between the ladder and the building, given as \( 22^{\circ} \).
• The hypotenuse is the length of the ladder, which is 20 feet.
• The adjacent side is the distance we need to find to the building from the ladder's base.

Plugging in these values, you can solve for the adjacent side by rearranging the formula to find:\[ x = 20 \times \cos(22^{\circ}) \]Using a calculator or a trigonometric table, we approximate \( \cos(22^{\circ}) \) to 0.9272, which makes \( x \approx 18.544 \) feet. This computation highlights the practical utility of the cosine function in solving real-world problems involving right triangles.

Right Triangle

Right triangles are unique because they include a 90-degree angle, which allows us to use specific trigonometric functions like sine, cosine, and tangent to solve for unknown sides and angles. In this scenario, the ladder, ground, and the side of the building form a right triangle:

• The ladder acts as the hypotenuse, the longest side that is opposite the right angle.
• The adjacent side, which we've calculated using the cosine function, is the distance from the base of the ladder to the building.
• The opposite side is the height on the building where the ladder touches.

The right triangle allows the use of trigonometric identities to solve for sides when certain angles and lengths are known. Such triangles are an essential concept in geometry, playing a vital role not only in architectural and construction projects but also in problem-solving scenarios like ours.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics for dealing with right triangles. It states:\[ a^2 + b^2 = c^2 \]Where:

• \( a \) and \( b \) are the lengths of the triangle's shorter sides, referred to as legs.
• \( c \) is the length of the hypotenuse, the longest side.

In our problem, after increasing the base of the ladder by 3 feet, we use the Pythagorean theorem to find the new height of the ladder against the building:\[ 20^2 = y'^2 + (21.544)^2 \]By solving this equation, we calculate the new height and determine the difference as the ladder moves down the building. Such applications demonstrate the significance of the Pythagorean theorem in determining dimensions of spaces and objects practically.