Problem 93 - College Algebra

Question

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to \(r(t)=25 \sqrt{t+2},\) find the area of of the ripple as a function of time. Find the area of the ripple at \(t=2\) .

Step-by-Step Solution

Answer

The area of the ripple at \( t = 2 \) is \( 2500\pi \) square inches.

Step 1: Understanding the Problem

We are given a function for the radius of a ripple, \( r(t) = 25 \sqrt{t+2} \), and need to find the area of the ripple as a function of time. We also need to calculate this area at \( t = 2 \).

Step 2: Formula for Area of a Circle

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

Step 3: Express Area as a Function of Time

Substitute \( r(t) = 25 \sqrt{t+2} \) into the area formula: \[ A(t) = \pi (25 \sqrt{t+2})^2.\]

Step 4: Simplifying the Expression

Simplify the expression of \( A(t) \): \[ A(t) = \pi (625(t+2)) = 625\pi (t+2).\] Now we have the area as a function of time: \( A(t) = 625\pi (t+2) \).

Step 5: Calculate Area at Specific Time

Now, substitute \( t = 2 \) into the area function: \[ A(2) = 625\pi (2+2) = 625\pi \times 4 = 2500\pi.\]

Step 6: Final Area Conclusion

The area of the ripple at \( t = 2 \) is \( 2500\pi \) square inches.

Key Concepts

Functions

In the realm of calculus and mathematics, a function represents a relationship between a set of inputs and a set of possible outputs, where each input is connected to exactly one output. In this problem, we are dealing with a function that defines the radius of a circular ripple over time. Functions can be used to model a wide variety of real-world phenomena.

• Given function: The function provided here is \( r(t) = 25 \sqrt{t+2} \). This indicates how the radius of the ripple changes as time \( t \) changes.
• Function's structure: The function involves a square root, implying that as time increases, the rate of increase in the radius will diminish.

This function showcases how a simple mathematical expression can describe complex physical occurrences such as ripples on water.

Circular Ripple

A circular ripple is a wave that radiates outward from a central point, such as a raindrop hitting the surface of a body of water. The distance from the center to any point on the wave's edge is known as the radius. In our problem, the circular ripple's radius increases over time according to a given function.
When analyzing such ripples, it's important to consider:

• Symmetry: The shape of the ripple is perfectly circular, which simplifies calculating its area.
• Uniform Expansion: The ripple grows uniformly, meaning every point on the wavefront moves outward at the same rate as determined by the function \( r(t) \).

Understanding the properties of circular ripples helps in applying the mathematics needed to find characteristics such as the area.

Area of a Circle

The area of a circle is one of the fundamental concepts in geometry and calculus. It is given by the formula \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius. In the context of this exercise, the formula is used to determine how the area of a circular ripple changes over time.
Here's how we find the area as a function of time:

• Substitute the radius function \( r(t) = 25 \sqrt{t+2} \) into the area formula, replacing \( r \) with \( r(t) \).
• Simplify to find \( A(t) = \pi (25 \sqrt{t+2})^2 \), which further simplifies to \( A(t) = 625\pi (t+2) \).

Now, you have a function that directly links time to the growing area of the ripple.

Mathematical Modeling

Mathematical modeling involves representing real-world scenarios using mathematical language and symbols. It allows us to predict behaviors and outcomes based on mathematical principles. The problem at hand uses modeling to describe the dynamics of a ripple in a lake.
In this exercise, we:

• Start with a function \( r(t) \) for the radius, based on the real-life occurrence of ripples spreading on water.
• Use the function to express a further relationship between time and the area of the ripple, resulting in \( A(t) = 625\pi (t+2) \).
• Apply specific conditions, like \( t = 2 \), to find exact measurements, such as \( 2500\pi \), clarifying the size of the ripple at that time.

Mathematical modeling provides a structured way to grasp and predict how changes in time affect the physical properties of objects, like the area of a rippling circle.