### Video Transcript

The area ๐ด of a lamina is changing at the rate d๐ด by d๐ก equals ๐ to the power of negative 0.7๐ก 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 weโve been given information about the rate of change of the area. Thatโs d๐ด by d๐ก, in other words, the derivative of ๐ด with respect to ๐ก. Now, we know that integration and differentiation are the reverse of one another. So we can find an expression for ๐ด by integrating d๐ด by d๐ก with respect to ๐ก.

Now, what will happen is this will give us a general solution. And weโll need to use the fact that the starting area is 60 square centimeters to find a particular solution to this equation. But to begin, weโll simply integrate the expression for d๐ด by d๐ก with respect to ๐ก. Thatโs the indefinite integral of ๐ to the power of negative 0.7๐ก with respect to ๐ก.

Now, here we can recall the general result for the integral of ๐ to the power of ๐๐ฅ with respect to ๐ฅ for real constant values of ๐ด. Itโs one over ๐ times ๐ to the power of ๐๐ฅ plus a constant of integration ๐ถ. Now, in our example, we can see we can let ๐ be equal to negative 0.7 or negative seven-tenths. This means when we integrate ๐ to the power of negative 0.7๐ก, we get one over negative seven-tenths times ๐ to the power of negative 0.7๐ก. And of course we need that constant of integration ๐ถ. And we recall that to divide by a fraction, we simply multiply by the reciprocal of that fraction.

Now, letโs 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 weโve found a general equation for ๐ด. Itโs ๐ด is equal to negative 10 over seven times ๐ to the power of negative 0.7๐ก plus our constant of integration ๐ถ.

Now, recall we actually want to find the area of the lamina after 30 seconds, in other words, when ๐ก is equal to 30. So weโll begin by finding the value of our constant. And weโll use the fact that the starting area was 60 square centimeters. In other words, when ๐ก is equal to zero, ๐ด is equal to 60. Substituting these values into our equation, and we get 60 equals negative ten-sevenths times ๐ to the power of zero plus ๐ถ. But of course ๐ to the power zero is one. So we have 60 equals negative ten-sevenths plus ๐ถ.

Letโs clear some space and solve our equation for ๐ถ. Weโre 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 weโre easily able to add ten-sevenths to both sides of our equation to solve for ๐ถ. So we find ๐ถ is equal to four hundred and thirty sevenths. And so weโve found a particular equation for the area, given this information about its starting area. ๐ด is equal to negative ten-sevenths times ๐ to the power of negative 0.7๐ก plus 430 over seven.

Now, remember, we want an exact expression for the area of the lamina after 30 seconds. So we substitute ๐ก equals 30 into this equation. Weโre not going to type this into our calculator. Remember, weโre looking to find an exact expression. So instead, weโll simply evaluate negative 0.7 times 30. Negative 0.7 times 30 is negative 21. And of course weโre 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 ๐ to the power of negative 21 plus 430 over seven square centimeters.