Question

Two ladders of length \(a\) lean against opposite walls of an alley with their feet touching (see figure). One ladder extends \(h\) feet up the wall and makes a \(75^{\circ}\) angle with the ground. The other ladder extends \(k\) feet up the opposite wall and makes a \(45^{\circ}\) angle with the ground. Find the width of the alley in terms of \(a\) h, and/or \(k\). Assume the ground is horizontal and perpendicular to both walls.

Short answer

Question: Determine the width of the alley in terms of a, h, and/or k. Answer: w = a(cos(75) + cos(45))

Step-by-step solution

Identify which trig functions to use

Since we are given the lengths of the ladders and their angles with respect to the ground, we can use the sine function to find the horizontal distances between the foot and the top of the ladders which touch the wall. Let's denote the width of the alley by w.

Setup trigonometric equations

First, let's consider the ladder that forms a 75-degree angle with the ground. We can set up the sine equation as follows: sin(75) = \frac{h}{a} Next, we will consider the ladder that forms a 45-degree angle with the ground. We can set up the sine equation as follows: sin(45) = \frac{k}{a}

Solve for the horizontal distances

We need to find the horizontal distances between the foot and the top of the ladders which touch the wall in order to find the width of the alley. We can use the cosine function to do so. Let's denote the horizontal distance of the ladder with height h as x, and the horizontal distance of the ladder with height k as y. Then: cos(75) = \frac{x}{a} cos(45) = \frac{y}{a} Now we can solve for x and y in terms of a: x = a * cos(75) y = a * cos(45)

Determine the width of the alley

Since the feet of both ladders are touching, the width of the alley, w, is equal to the sum of the horizontal distances x and y. Therefore, we can write the equation for w as follows: w = x + y Now, substituting our expressions for x and y from Step 3, we get: w = a * cos(75) + a * cos(45) And so we have found the width of the alley as a function of a: w = a(cos(75) + cos(45))

Key concepts

Sine Function

The sine function is a fundamental part of trigonometry, linked to the angles of triangles. It specifically relates to right-angled triangles where the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Sine is quite useful in solving real-world problems like finding heights or lengths in indirect measurements, such as in this ladder problem.

• For an angle \( \theta \), the sine function is defined as: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• In the original exercise, the angles of 75° and 45° are used for sine calculations to relate the vertical heights \( h \) and \( k \) to the ladder's length \( a \).

These equations help us solve for unknown distances by rearranging the formula to find either the opposite side (height in this case) or the hypotenuse. By understanding the context and using the formula wisely, problems like these become straightforward.

Cosine Function

The cosine function is another critical concept in trigonometry. It is particularly useful when determining horizontal distances or shadows cast by objects. The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
It complements the sine function and often both are used together to solve for unknown dimensions in geometry or real-world scenarios like our ladder problem.

• For an angle \( \theta \), the formula is: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• In this situation, we use cosine to find the horizontal distance from the base of each ladder to the wall: \( x = a \cdot \cos(75^\circ) \) and \( y = a \cdot \cos(45^\circ) \).

These calculations are crucial because the sum of these horizontal distances defines the width of the alley where both ladders touch at the base. Understanding how to use the cosine function enables us to accurately compute distances parallel to the ground.

Angle of Elevation

The angle of elevation is a concept that describes the angle between the horizontal sightline and an object above the line of sight. It is an essential idea in trigonometry used to determine heights or distances when a clear line of sight exists.
In the ladder problem, this concept is applied to the angle each ladder makes with the ground. Observing angles of elevation helps us to set up the right trigonometric equations.

• An angle of elevation is measured from the earth's surface towards an object higher up.
• In our problem, the angles 75° and 45° imply that the ladders lean steeply and have less inclination respectively.
• Using these angles, combined with trigonometric functions like sine and cosine, allows the conversion of angled measurements into usable height and width parameters for solving the problem.

This understanding permits solutions for real-world applications, whether you are observing a ladder, hill, or building seen from the ground up.