Question

Environmental Concerns The spread of oil leaking from a tanker is in the shape of a circle. If the radius \(r\) (in feet) of the spread after \(t\) hours is \(r(t)=200 \sqrt{t},\) find the area \(A\) of the oil slick as a function of the time \(t .\)

Short answer

The area is 40000\( \pi \)t.

Step-by-step solution

- Understand the problem

The problem involves finding the area of an oil slick from an oil spill, given that the spread is circular and the radius changes over time.

- Write the formula for the area of a circle

The area of a circle is given by the formula: \[ A = \pi r^2 \]

- Substitute the given radius function

We are given the radius as a function of time: \[ r(t) = 200 \sqrt{t} \]Substitute this into the area formula: \[ A(t) = \pi (200 \sqrt{t})^2 \]

- Simplify the expression

Simplify the expression by squaring the radius: \[ A(t) = \pi (200)^2 (\sqrt{t})^2 = \pi \times 40000 \times t = 40000\pi t \]

- Write the final function

The final function for the area of the oil slick as a function of time is: \[ A(t) = 40000\pi t \]

Key concepts

radius function

The radius function describes how the radius of the oil spill changes over time. In our exercise, the radius of the spill is given by the function: \(r(t) = 200 \sqrt{t}\).
This function means that as time passes, the radius of the oil spill increases. Here:

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