Problem 66 - College Algebra

Question

An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of \(d\), the number of days elapsed.

Step-by-Step Solution

Answer

The area is \(A(d) = 400\pi d^2\).

Step 1: Understand the Problem

The problem involves an oil slick expanding in a circular shape. We know that the radius of this circle increases by 20 meters each day. We need to express the area of this circle as a function of the number of days, \(d\).

Step 2: Determine the Radius as a Function

The initial radius of the circle can be considered as increasing from zero with time. Let the radius on day \(d\) be \(r(d)\). Given that the radius increases by 20 meters per day, the function for the radius can be expressed as \(r(d) = 20 \times d\).

Step 3: Use the Area Formula for a Circle

The formula for the area \(A\) of a circle is \(A = \pi r^2\). Since we have the radius as a function of time \(d\), substitute \(r(d) = 20d\) into the area formula to get the area as a function of \(d\).

Step 4: Substitute the Radius Function into the Area Formula

Substitute \(r(d) = 20d\) into \(A = \pi r^2\). This gives:\[A(d) = \pi (20d)^2\].

Step 5: Simplify the Area Function

Simplify the expression for the area:\[A(d) = \pi (20d)^2 = \pi \, \times \, 400d^2 = 400\pi d^2\].This is the function for the area of the circle in terms of the number of days elapsed, \(d\).

Key Concepts

Circular Motion

Circular motion in mathematics and physics is often linked with objects that have a repetitive, rotational movement about a central point. In daily life, understanding circular motion helps us comprehend how planets orbit stars or how wheels rotate.
In our exercise, the oil slick expands uniformly as a circle, a perfect example of circular motion properties at work. Unlike standard circular motion, this example focuses more on the expansion aspect rather than rotation, yet the concept is similar. The radius of the oil slick circle increases with time.
Key properties of circular objects include:

• Radius: Straight line from center to any point on the circle's edge.
• Diameter: Twice the radius, passing through the circle's center.
• Circumference: Total distance around the circle.
• Area: Inside surface of the circle, calculated as \(\pi r^2\).

In this problem, we are particularly interested in how the circle grows as the radius increases linearly with time.

Rate of Change

Rate of change is a fundamental concept in calculus and algebra that helps us understand how a quantity changes over time. Whenever you hear about speed, acceleration, growth, or decay, you're dealing with a rate of change.
In this exercise, the rate of change is applied to the increase in the radius of the oil slick. The radius is increasing at a constant rate of 20 meters per day. This implies that the change in radius, or \(\Delta r\), is 20 meters daily. This consistent increase forms a linear function, expressed as \(r(d) = 20d\).
Recognizing and understanding rates of change allow us to predict future values by calculating how a variable will move over a specified period. It's crucial for real-life situations like estimating distances traveled or predicting growth over time.

Area Formulas

Area formulas are essential tools in geometry, allowing us to find the space inside 2D shapes. For circles, the area formula is \(A = \pi r^2\), where \(r\) is the radius. This formula is vital to determine how much space the oil slick occupies when assessed over time.
In this problem, we take the area formula and integrate our radius function, \(r(d) = 20d\), into it to see how the area changes over days. By substituting the radius function, we have:

• \(A(d) = \pi (r(d))^2 = \pi (20d)^2\)
• After simplification, \(A(d) = 400\pi d^2\)

This final equation \(400\pi d^2\) shows how the area grows exponentially with time, due to the square of the radius factor. Area formulas, like this one, provide insight into spatial dynamics and help in visualizing growth across dimensions as time progresses.