A raindrop hitting a lake makes a circular ripple. The radius, in inches, grows as a function of time, in minutes, according to two times the square root of 𝑡 plus one. Find the area of the ripple as a function of time, and then determine the area of the ripple at three seconds.
As the raindrop hits, it creates a circle that is expanding as a function of time. Before we can find that, we need to think about what we know about every circle. It has a center, and the distance from the center to any point on the outside of the circle is the radius. We also know that the area of a circle is equal to 𝜋 times the radius squared. In this case, the radius is a function of time because the radius is changing. That means our radius will be equal to this function, two times the square root of 𝑡 plus one. 𝑟 equals two times the square root of 𝑡 plus one, where 𝑡 is a measure of time in seconds for the time after the raindrop hit the lake. And the radius is being measured in inches.
To find out what the formula for finding the area is, we need to plug in what we know about the radius into the area formula. The area of this circle will be equal to 𝜋 times two times the square root of 𝑡 plus one squared. Now, we need to be really careful in distributing the square. When we do that, we’ll get 𝜋 times two squared times the square root of 𝑡 plus one squared. Two squared equals four. We can bring down the 𝜋. And if we take the square root of 𝑡 plus one and we square it, we’ll just have 𝑡 plus one. But we need to make sure it stays in parentheses. Since the four 𝜋 is being multiplied by 𝑡 as well as one, the area of this circle can be found by taking four 𝜋 and multiplying it by the time in seconds plus one.
We might write it like this to show that the area, in this case, is a function of time. The area with respect to 𝑡 is equal to four 𝜋 times 𝑡 plus one. And now that we have a formula with respect to time, we’re curious about how big the circle is after three seconds. So we plug in three for our formula. Three plus one is four. Four times four is 16, and then bring down the 𝜋. The area of the circle after three seconds is 16𝜋. We know the radius is measured in inches. But that we’re dealing with area, so we’ll have inches squared. After three seconds, the area of the circle is 16𝜋 inches squared.