The area 𝐴 of a lamina is changing at the rate d𝐴 by d𝑡 equals 𝑒 to the power of negative 0.7𝑡 square centimeters per second, starting from an area of 60 square centimeters. Give an exact expression for the area of the lamina after 30 seconds.
The key to answering this question is to spot that we’ve been given information about the rate of change of the area. That’s d𝐴 by d𝑡, in other words, the derivative of 𝐴 with respect to 𝑡. Now, we know that integration and differentiation are the reverse of one another. So we can find an expression for 𝐴 by integrating d𝐴 by d𝑡 with respect to 𝑡.
Now, what will happen is this will give us a general solution. And we’ll need to use the fact that the starting area is 60 square centimeters to find a particular solution to this equation. But to begin, we’ll simply integrate the expression for d𝐴 by d𝑡 with respect to 𝑡. That’s the indefinite integral of 𝑒 to the power of negative 0.7𝑡 with respect to 𝑡.
Now, here we can recall the general result for the integral of 𝑒 to the power of 𝑎𝑥 with respect to 𝑥 for real constant values of 𝐴. It’s one over 𝑎 times 𝑒 to the power of 𝑎𝑥 plus a constant of integration 𝐶. Now, in our example, we can see we can let 𝑎 be equal to negative 0.7 or negative seven-tenths. This means when we integrate 𝑒 to the power of negative 0.7𝑡, we get one over negative seven-tenths times 𝑒 to the power of negative 0.7𝑡. And of course we need that constant of integration 𝐶. And we recall that to divide by a fraction, we simply multiply by the reciprocal of that fraction.
Now, let’s think of negative seven over 10 as negative seven-tenths. And we see that this is equal to one times negative 10 over seven, which is simply negative 10 over seven. And so we’ve found a general equation for 𝐴. It’s 𝐴 is equal to negative 10 over seven times 𝑒 to the power of negative 0.7𝑡 plus our constant of integration 𝐶.
Now, recall we actually want to find the area of the lamina after 30 seconds, in other words, when 𝑡 is equal to 30. So we’ll begin by finding the value of our constant. And we’ll use the fact that the starting area was 60 square centimeters. In other words, when 𝑡 is equal to zero, 𝐴 is equal to 60. Substituting these values into our equation, and we get 60 equals negative ten-sevenths times 𝑒 to the power of zero plus 𝐶. But of course 𝑒 to the power zero is one. So we have 60 equals negative ten-sevenths plus 𝐶.
Let’s clear some space and solve our equation for 𝐶. We’re going to rewrite 60 as 420 over seven. Now, that comes from the fact that we can write 60 as 60 over one and then multiply the numerator and the denominator by seven. Then we’re easily able to add ten-sevenths to both sides of our equation to solve for 𝐶. So we find 𝐶 is equal to four hundred and thirty sevenths. And so we’ve found a particular equation for the area, given this information about its starting area. 𝐴 is equal to negative ten-sevenths times 𝑒 to the power of negative 0.7𝑡 plus 430 over seven.
Now, remember, we want an exact expression for the area of the lamina after 30 seconds. So we substitute 𝑡 equals 30 into this equation. We’re not going to type this into our calculator. Remember, we’re looking to find an exact expression. So instead, we’ll simply evaluate negative 0.7 times 30. Negative 0.7 times 30 is negative 21. And of course we’re working in square centimeters. So we can say that the exact expression for the area of the lamina after 30 seconds is negative ten-sevenths 𝑒 to the power of negative 21 plus 430 over seven square centimeters.