Problem 66 - College Algebra

Question

Use the written statements to construct a polynomial function that represents the required information. 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 as a function of days is \( A(d) = 400\pi d^2 \).

Step 1: Understand the Problem

The problem involves a circle whose radius increases as a function of time. Specifically, the radius is increasing at a constant rate of 20 meters per day.

Step 2: Express the Radius as a Function of Time

Since the radius increases by 20 meters each day, after \( d \) days the radius \( r \) can be expressed as \( r(d) = 20d \).

Step 3: Recall the Formula for the Area of a Circle

The area \( A \) of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step 4: Substitute the Radius Function into the Area Formula

Substituting the radius function \( r(d) = 20d \) into the area formula gives us \( A(d) = \pi (20d)^2 \).

Step 5: Simplify the Expression

Simplifying \( A(d) = \pi (20d)^2 \), we get \( A(d) = \pi \, \cdot \, 400d^2 \). This simplifies to \( A(d) = 400\pi d^2 \).

Key Concepts

Radius and Area

When dealing with circular shapes, two key concepts arise quite often: radius and area. In our exercise, we encounter an oil slick that forms a circle whose radius expands over time.
The radius of a circle is the distance from its center to any point on the edge. It's a fundamental measure that helps us understand the size of the circle. In this problem, the radius grows by 20 meters each day; hence, after two days, the radius would be 40 meters, after three days, it would be 60 meters, and so on.
The area of a circle, on the other hand, gives us the amount of space enclosed within its boundary. We calculate it using the formula:

• Area: \[ A = \pi r^2 \] where \( A \) is the area and \( r \) the radius.

By understanding how both radius and area relate, we can express the overall spread of this oil slick in terms of changing radius over time. Here, we use the rate of radius change to find the function for area.

Expressing Functions

Expressing functions in mathematics allows us to model real-world situations and extract meaningful information from them. In our exercise, we express how the radius of an oil slick changes over time and how this information helps us determine the area covered by the slick.
A function consists of an input, a process, and an output. For example, in expressing the radius as a function of time, the number of days is the input \(d\), and the function is \(r(d) = 20d\). Here, the process is multiplying the number of days by 20, as the radius increases 20 meters per day. The output is the radius of the circle at any given time \(d\).
Next, by substituting this radius function into the area formula, we arrive at an expression for area in terms of days: \( A(d) = 400\pi d^2 \).
This function tells us how the area increases as days go by. Writing functions like this allows us to predict and analyze events in a structured manner.

Polynomial Expressions

Polynomial expressions are fundamental in many areas of mathematics and science for modeling change and growth. In the context of this problem, the expression for the area of our expanding oil slick provided an example of what a polynomial expression looks like.
Polynomials are expressions involving variables raised to whole number exponents and coefficients, such as \(Ax^n\). Our area expression, \( A(d) = 400\pi d^2 \), is a polynomial function in terms of \(d\), with \(d^2\) being the term and \(400\pi\) the coefficient. The degree of this polynomial is 2, as the highest power of \(d\) is 2.
Understanding polynomial expressions involves knowing:

• Terms: The individual parts of the expression separated by '+' or '-'.
• Coefficients: The numerical part of terms, like \(400\pi\) here.
• Degree: The highest exponent in the expression, indicating growth rate.

Polynomials provide a versatile way to express changes and predict future behaviors in various contexts.