Problem 27 - Calculus Single Variable

Question

A variable \(u=f(x, y)\) is said to be jointly proportional to \(x\) and \(y\) if \(f(x, y)=k x y\) for some constant \(k\). The area of a sector of a circle is jointly proportional to its central angle and to the square of the radius \(r\) of the circle. What is the area \(A(r, \alpha)\) if the degree measure of the central angle of the sector is \(\alpha\) ? (Deduce the proportionality constant by using the value of \(A\) that corresponds to \(\alpha=360^{\circ} .\) )

Step-by-Step Solution

Answer

The area of the sector is \(A(r, \alpha) = \frac{\pi}{360} \cdot \alpha \cdot r^2\).

Step 1: Understand Joint Proportionality

Since the area of a sector, \(A\), is jointly proportional to its central angle, \(\alpha\), and the square of the radius, \(r\), the equation can be written as \(A(r, \alpha) = k \cdot \alpha \cdot r^2\) for some constant \(k\).

Step 2: Consider the Full Circle

When \(\alpha = 360^{\circ}\), the sector forms a complete circle. The area of a full circle is known to be \(\pi r^2\). Therefore, we can use this information to find \(k\).

Step 3: Set Up Equation for Full Circle

Substitute \(\alpha = 360^{\circ}\) into the joint proportionality equation: \(\pi r^2 = k \cdot 360 \cdot r^2\).

Step 4: Solve for the Constant \(k\)

Divide both sides by \(r^2\) to isolate \(k\) on one side: \(\pi = 360k\). Solve for \(k\) by dividing both sides by 360: \(k = \frac{\pi}{360}\).

Step 5: Write the General Formula for Sector Area

Substitute \(k = \frac{\pi}{360}\) back into the joint proportionality equation, giving \(A(r, \alpha) = \frac{\pi}{360} \cdot \alpha \cdot r^2\). This is the formula for the area of a sector in terms of the central angle and radius.

Key Concepts

Proportionality Constant

Proportionality constants are crucial when dealing with joint proportionality. They form the backbone of the relationship between variables in such expressions.
When we say something is jointly proportional to multiple variables, this simply means it changes in response to the multiplication of those variables, guided by a constant.In this exercise's context, the area of the circle sector is dependent on both the central angle and the square of the radius. The proportionality constant helps in discovering how these two variables scale the area.To find this constant, we set up an equation based on known information. Here, we use the full area of the circle, formed when the central angle is 360 degrees. Knowing this allows us to express the constant in terms of familiar values, particularly \(\pi\), as every full circle's area is known to be \(\pi r^2\). This provides a basis for determining that our proportionality constant \(k\), in this case, would be \(\frac{\pi}{360}\).
This constant is essential. It acts as the factor bridging the relationship between the central angle, radius, and the resulting sector area.

Circle Sector Area

The circle sector area is part of the total area of a circle, determined by the circle's radius and the specific central angle. Imagine a pizza slice; the sector area can be visualized as the area of one slice compared to the whole pizza.
The formula derived is \(A(r, \alpha) = \frac{\pi}{360} \alpha \cdot r^2\), which elegantly breaks down the sector area in relation to elements of the whole circle.
Here's why it makes sense:

• \(r^2\) captures the difference in scaling with changes in radius.
• The constant \(\frac{\pi}{360}\) reflects the transition between the angle and the full circle's area.
• It shows that for different angles \(\alpha\), we consider that angle as a fraction of a full circle (360°).

Using these elements, calculating the area of a sector is simplified. You just plug in the values for the radius and the central angle to find the desired area.

Central Angle

Understanding the central angle is vital when calculating the sector area. The central angle \(\alpha\) is simply the angle formed at the circle's center by radii extending to the perimeter.
Think of it as the "degrees" of pizza slice you choose to take out of the entire pizza!Each central angle directly determines how much of the circle's area is being considered within the sector.
Larger central angles encompass more of the circle, while smaller angles consider less.In the formula \(A(r, \alpha) = \frac{\pi}{360} \alpha \cdot r^2\), the central angle \(\alpha\) directly affects the sector's area. Its measure signifies how this portion stacks relative to a full circle (360°):

• A full circle encompasses the entire circle, seen when \(\alpha = 360°\).
• If the angle is halved to 180°, the sector formed is half the area of the circle.

The concept further illustrates why understanding angles is critical—in evaluating how they interact with the circle's radius, helping you correctly apply joint proportionality principles.