### Video Transcript

The rate of change in the area π΄ of a metallic plate with respect to time due to heating is given by the relation dπ΄ by dπ‘ is equal to 0.036π‘ squared plus 0.038π‘, where the area π΄ is in square meters and the time π‘ is in minutes. Given that π΄ is equal to 67 square meters when π‘ is equal to eight minutes, find, correct to the nearest two decimal places, the area of the plate just before heating.

In this question, weβre given a problem involving the change in area of a metallic plate due to heating. The area of this metallic plate is given by the function π΄ of π‘, where π΄ is measured in square meters and π‘ is measured in minutes. And we need to find the area of the plate just before heating. Since π‘ is equal to zero minutes will be when the heating starts, this is just π΄ evaluated at zero. However, we canβt just substitute π‘ is equal to zero into the function π΄ of π‘. This is because weβre not given the function π΄ of π‘. Instead, weβre told that the derivative of π΄ with respect to π‘ is equal to 0.036π‘ squared plus 0.038π‘.

Therefore, we need to use the derivative of this function with respect to time to find the original function. We need to find its antiderivative. And we can recall the indefinite integrals give us the most general antiderivative. In particular, π΄ of π‘ is going to be equal to the indefinite integral of dπ΄ by dπ‘ with respect to π‘. And there is one thing worth pointing out. Since weβre finding the most general antiderivative, this will be only up to a constant of integration πΆ. To calculate this indefinite integral, letβs start by substituting our expression for dπ΄ by dπ‘ into the integral. We get the indefinite integral of 0.036π‘ squared plus 0.038π‘ with respect to π‘. And this is the indefinite integral of a polynomial. And we can evaluate this term by term by using the power rule for integration.

This tells us for any real constants π and π, where π is not negative one, the integral of π times π₯ to the πth power with respect to π₯ is equal to π times π₯ to the power of π plus one divided by π plus one plus the constant of integration πΆ. We add one to the exponent of our variable and divide by this new exponent. And weβll apply this term by term. Letβs start by integrating the first term. We add one to the exponent to get a new exponent of three and then divide by this new exponent. We get 0.036π‘ cubed over three. And we could add a constant of integration here. However, we can just combine all of the constants of integration into one constant of integration at the end of our expression.

So now letβs integrate our second term. Weβll do this by rewriting π‘ as π‘ to the first power. We add one to our exponent of π‘ to get a new exponent of two and divide by this new exponent. We get 0.038π‘ squared over two. And then remember, we add our constant of integration πΆ. And now we can simplify this equation by evaluating the coefficients. We get that π΄ of π‘ is equal to 0.012π‘ cubed plus 0.019π‘ squared plus πΆ. And now we see we canβt just answer our question by substituting π‘ is equal to zero into the equation because we have an unknown value of πΆ. This means weβre going to need to determine the value of πΆ. And we can do this by using information weβre given in the question.

Weβre told that the area of the metallic plate is equal to 67 square meters when π‘ is equal to eight minutes. In terms of our function, this is saying that π΄ evaluated at eight is equal to 67. Therefore, we can use this to evaluate our equation at π‘ is equal to eight. Substituting π‘ is equal to eight, we have π΄ evaluated at eight is 67. And this will be equal to 0.012 times eight cubed plus 0.019 times eight squared plus πΆ. And now we can rearrange this to find the value of πΆ. If we do this and evaluate, we see that πΆ is equal to 59.64. And we can substitute this value of πΆ back into our equation for π΄ of π‘. This gives us that π΄ of π‘ is 0.012π‘ cubed plus 0.019π‘ squared plus 59.64.

And now we can answer our question. We need to substitute π‘ is equal to zero into this equation. However, when we do this, our first term will have a factor of zero and our second term will have a factor of zero. This will just leave us with the constant value of πΆ. Therefore, π΄ evaluated at zero is 59.64. But remember, this represents a physical quantity. Itβs the initial area of the metallic plate before we start eating. And weβre told that this area is measured in square meters, so we can give this a unit. And this is already given to two decimal places.

Therefore, weβve shown the initial area of the metallic plate before we started heating, to two decimal places, is 59.64 square meters.