Problem 29 - Beginning and Intermediate Algebra

Question

Oil spilled from a ship off the coast of Alaska with the oil spreading out in a circle across the surface of the water. The radius of the oil spill is given by \(r(t)=4 t\) where \(t\) is the number of minutes after the leak began and \(r(t)\) is in feet. The area of the spill is given by \(A(r)=\pi r^{2}\) where \(r\) represents the radius of the oil slick. Find each of the following and explain their meanings. (IMAGE CANT COPY) a) \(r(5)\) b) \(\quad A(20)\) c) \(A(r(t))\) d) \(A(r(5))\)

Step-by-Step Solution

Answer

a) \(r(5)=20\) feet: The oil spill has a radius of 20 feet, 5 minutes after the leak began. b) \(A(20)=400\pi\) square feet: The area of the oil spill when the radius is 20 feet is \(400\pi\) square feet. c) \(A(r(t))=16\pi t^2\): The area of the oil spill as a function of time is given by \(16\pi t^2\) square feet. d) \(A(r(5))=400\pi\) square feet: The area of the oil spill is \(400\pi\) square feet, 5 minutes after the leak began.

Step 1: 1. Calculate \(r(5)\)

To find the radius 5 minutes after the leak began, plug in \(t=5\) into the equation \(r(t) = 4t\): \(r(5) = 4(5) = 20\,\text{feet}\) Meaning: The radius of the oil spill is 20 feet, 5 minutes after the leak began.

Step 2: 2. Calculate \(A(20)\)

To find the area of the spill when the radius is 20 feet, plug in \(r=20\) into the equation \(A(r) = \pi r^2\): \(A(20) = \pi (20)^2 = 400\pi\, \text{square feet}\) Meaning: The area of the oil spill is \(400\pi\, \text{square feet}\) when its radius is 20 feet.

Step 3: 3. Calculate \(A(r(t))\)

To find the function for the area of the spill depending on the time, substitute the function \(r(t) = 4t\) into the equation \(A(r) = \pi r^2\): \(A(r(t)) = \pi (4t)^2 = 16\pi t^2\, \text{square feet}\) Meaning: The area of the oil spill is \(16\pi t^2\, \text{square feet}\) at time \(t\) minutes.

Step 4: 4. Calculate \(A(r(5))\)

To find the area of the spill 5 minutes after the leak began, substitute the value we calculated in step 1, \(r(5) = 20\), into the equation \(A(r) = \pi r^2\): \(A(r(5)) = A(20) = 400\pi\, \text{square feet}\) Meaning: The area of the oil spill is \(400\pi\, \text{square feet}\) 5 minutes after the leak began.

Key Concepts

Radius Calculation

When dealing with spreading phenomena, like an oil spill, understanding the radius is crucial. The radius (\( r(t) \)) indicates how far outward the spill has spread from the original point. In this exercise, the radius is a function of time. Specifically, the function is given by \( r(t) = 4t \), where \( t \) is time in minutes.

• This function tells us that every minute, the spill's radius increases by 4 feet.
• So, for example, after 5 minutes, plugging in \( t = 5 \) results in \( r(5) = 4 \times 5 = 20 \text{ feet} \).

In simpler terms, the oil is spreading at a rate of 4 feet per minute. By understanding this, you can predict how large the affected area will grow over time.

Area of a Circle

To measure the extent of an oil spill, calculating the area is essential. The area of a circle is determined using the formula \( A(r) = \pi r^2 \).

• This formula tells us that the area depends on the radius squared, meaning any change in radius has a substantial impact on the area.
• For instance, if the radius is 20 feet, the area would be \( A(20) = \pi(20)^2 = 400\pi \text{ square feet}\).

This indicates the total surface area covered by the spill. As the radius increases, the area increases rapidly, showing how quickly an oil spill can cover large parts of the water surface.

Mathematical Modeling

Mathematical modeling provides a way to represent real-world phenomena, like oil spills, using mathematical functions. Here, two key equations provide insights.

• The radius function \( r(t) = 4t \) models how the radius changes over time.
• The area function for the radius, \( A(r) = \pi r^2 \), translates the radius into the area of the spill.

By combining these, we derive a new model: \( A(r(t)) = 16\pi t^2 \).

• This function shows the total area as it changes over time, dependent on the duration since the spill began.
• For example, after 5 minutes, this becomes \( A(r(5)) = 400\pi \text{ square feet}\).

Mathematical modeling helps predict future states of the spill, aiding in effective response and planning efforts.