Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

The spread of a contaminant is increasing in a circular pattern on the surface of a lake. The radius of the contaminant can be modeled by \(r(t)=5.25 \sqrt{t},\) where \(r\) is the radius in meters and \(t\) is time in hours since contamination. (a) Find a function that gives the area \(A\) of the circular leak in terms of the time \(t\) since the spread began. (b) Find the size of the contaminated area after 36 hours. (c) Find when the size of the contaminated area is 6250 square meters.

Step-by-Step Solution

Answer

Therefore, the function that gives the area of the circular leak in terms of time since the spread began is \(A(t)=27.5625t\, \pi\) square meters. The contaminated area after 36 hours is approximately 2980.96 square meters. The time when the size of the contaminated area is 6250 square meters is approximately 72.02 hours.

Step 1: Find the Area Function

To find the area \(A\) of the circular leak, we need to substitute the given \(r(t)\) function into the area of a circle formula \(A = \pi r^2\). That gives us \(A(t) = \pi [ r(t) ]^2 = \pi[ 5.25\sqrt{t} ]^2\). This simplifies to \(A(t)=27.5625t\, \pi\) square meters.

Step 2: The Size of Contaminated Area After 36 Hours

Now, to find the size of the contaminated area after 36 hours, we will substitute \(t=36\) into the \(A(t)\) function. This gives us \(A(36)=27.5625(36)\, \pi \approx 2980.96\) square meters.

Step 3: Solve for Time When the Contaminated Area is 6250 square meters

For the time when the contaminated area is 6250 square meters, we equate \(A(t)\) to 6250 and solve for \(t\). This gives us \(6250 = 27.5625t\, \pi\). If we solve this equation for \(t\), we get \(t \approx 72.02\) hours.

Key Concepts

Area of a Circle

Understanding the area of a circle is vital in various fields of science and mathematics. It’s the amount of two-dimensional space that occupies the interior of a circle and is given by the formula

\[ A = \pi r^2, \]
where \(A\) represents the area and \(r\) is the radius of the circle. In the context of our exercise, where a contaminant spreads over the surface of a lake, calculating the area helps in determining how much of the lake's surface is affected over time.

Radial Growth Function

A radial growth function models how the radius of a circle changes over time. In real-world applications, like the spread of a contaminant, the radial growth function can take various forms. In our exercise, the function is given by

\( r(t) = 5.25 \sqrt{t}, \)
which implies that the radius grows in proportion to the square root of time. This form of a growth function helps in understanding and predicting the spread's dynamics, as it quantifies the changing radius at any given time

Applying the Radial Growth Function

When we want to track the contaminant's spread, we apply this function to find the radius at specific times, inserting the time variable \(t\) into our radial function.

Solving for Time in Area Function

When working with the area of the circular spread of a contaminant, sometimes we know the area and need to find out how long it took to reach that size. This process is called solving for time in the area function. In our exercise, we derived an area function based on the given radial growth function, which is

\( A(t) = 27.5625t\pi. \)
Given an area, we set up an equation with the known area on one side and our area function on the other. The goal is to isolate \(t\) to find out the exact moment in time the contaminant's spread reached a specific size. For example, to find out when the contaminant's area is 6250 square meters, we solve

\( 6250 = 27.5625t\pi \)
for \(t\). This application is crucial in environmental studies or any context where timing is an essential factor.