Problem 10 - Algebra 2 and Trigonometry

Question

City firefighters are told that they can use their 25 -foot long ladder provided the measure of the angle that the ladder makes with the ground is at least \(15^{\circ}\) and no more \(\operatorname{than} 75^{\circ}\) . a. If \(\theta\) represents the measure of the angle that the ladder makes with the ground in radians, what is a reasonable set of values for \(\theta ?\) Explain. b. Express as a function of \(\theta,\) the height \(h\) of the point at which the ladder will rest against a building. c. Graph the function from part b using the set of values for \(\theta\) from part a as the domain of the function. d. What is the highest point that the ladder is allowed to reach?

Step-by-Step Solution

Answer

a. \(\frac{\pi}{12} \leq \theta \leq \frac{5\pi}{12}\); b. \(h(\theta) = 25 \cdot \sin(\theta)\); c. Increasing graph from \(\frac{\pi}{12}\) to \(\frac{5\pi}{12}\); d. About 24.15 feet.

Step 1: Convert Angle Restrictions to Radians

The ladder must be at an angle between 15° and 75° with the ground. We start by converting these angle restrictions into radians. The formula to convert degrees to radians is: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]So, for 15°: \[15^{\circ} \times \frac{\pi}{180} = \frac{\pi}{12}\, \text{radians}\]And for 75°: \[75^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{12}\, \text{radians}\]Thus, the reasonable range for \(\theta\) is \( \frac{\pi}{12} \leq \theta \leq \frac{5\pi}{12} \).

Step 2: Define Height as a Function of Angle

The height \(h\) at which the ladder reaches the building is determined by the sine of the angle \(\theta\), since the sine function relates the opposite side (height) to the hypotenuse (ladder length). Given the ladder length is 25 feet, we write:\[h(\theta) = 25 \cdot \sin(\theta)\]This function expresses the height \(h\) of the point against the building as a function of the angle \(\theta\).

Step 3: Graph the Height Function

With the function \(h(\theta) = 25 \cdot \sin(\theta)\) and the domain \( \frac{\pi}{12} \leq \theta \leq \frac{5\pi}{12} \), we need to sketch its graph.The sine function is increasing within this interval, meaning the graph will start low at \(\theta = \frac{\pi}{12}\) and increase to a maximum at \(\theta = \frac{5\pi}{12}\). The endpoints can be calculated:- At \(\theta = \frac{\pi}{12}\), \(h(\theta) = 25 \cdot \sin\left(\frac{\pi}{12}\right)\).- At \(\theta = \frac{5\pi}{12}\), \(h(\theta) = 25 \cdot \sin\left(\frac{5\pi}{12}\right)\).Plot the points and draw the curve.

Step 4: Find the Maximum Height

The maximum height occurs at the maximum allowable angle, \(\theta = \frac{5\pi}{12}\). Calculate:\[h\left(\frac{5\pi}{12}\right) = 25 \cdot \sin\left(\frac{5\pi}{12}\right)\]Using the sine value:\[sin\left(\frac{5\pi}{12}\right) \approx 0.9659\]Thus,\[h\left(\frac{5\pi}{12}\right) \approx 25 \times 0.9659 = 24.15 \text{ feet}\]

Key Concepts

Angle Conversions

To ensure safety when positioning a ladder, it's crucial to use the correct angle. In this exercise, the ladder's angle with the ground must be more significant than \(15^{\circ}\) but not exceed \(75^{\circ}\).
However, sometimes, we need this in radians for calculations involving trigonometric functions. This conversion is straightforward. We use the formula:

• Radians = Degrees \(\times \frac{\pi}{180}\)

For \(15^{\circ}\), we get \(\frac{\pi}{12}\) radians. Similarly, for \(75^{\circ}\), the conversion yields \(\frac{5\pi}{12}\) radians. Therefore, the reasonable range for the angle \(\theta\) is between \(\frac{\pi}{12}\) and \(\frac{5\pi}{12}\). This conversion helps us when we deal with sine functions in the next steps.

Ladder Safety Angle

A proper safety angle ensures that ladders are both stable and effective. The safety angle is generally accepted to be between \(15^{\circ}\) and \(75^{\circ}\), as too steep or too shallow angles can pose hazards.
At shallow angles, the ladder might slide, while a steep angle might make it topple over. This specified safety range ensures that we use the ladder effectively to reach the desired height against a building without risking fall or injury. By adhering to this range, the maximum structural integrity of the ladder is ensured, optimizing safety during use.

Function of Sine

The sine function is vital in determining how high a ladder reaches on a building at a given angle. When a ladder leans against a building, it forms a right-angled triangle with the ground and the wall. The height where the ladder touches the wall is the opposite side of the triangle, and the ladder itself is the hypotenuse.
Using the function of sine, which is the ratio of the opposite side to the hypotenuse, you can calculate the height \(h\):

• \(h(\theta) = 25 \cdot \sin(\theta)\)

This formula shows the direct relationship between the ladder's angle and the height it reaches on the wall, with 25 feet being the ladder's length. As the angle increases within the safety range, the height reached also increases.

Maximum Height Calculation

To find the highest point the ladder can safely reach, we utilize the sine function at the maximum allowable angle \(\theta = \frac{5\pi}{12}\) radians. Here's how it's done:

• First, compute the sine of the angle, which is approximately \(0.9659\).
• Then, multiply this value by the ladder length to get \(h\left(\frac{5\pi}{12}\right) = 25 \times 0.9659 ≈ 24.15\) feet.

Thus, at the steepest safe angle, the ladder can reach approximately 24.15 feet up the wall. This calculation ensures that the ladder not only stays within the safe angle range but also maximizes the height it can safely reach against the building, without compromising stability.