Calculus! The volume of a substance, A, measured in cubic centimeters increases according to the exponential growth model dA/dt = 0.3A, where t is measured in hours. The volume of another substance, B, also measured in cubic centimeters increases at a constant rate of 1 cm^3 per hour according to the linear model dB/dt = 1. At t = 0, substance A has a volume A(0) = 3 and substance B has size B(0) = 5. At what time will both substances have the same volume?

Would it be correct to write the growth model of substance B as x + 5? And how could I write the growth model of substance A? Thank you in advance, and sorry for the poor formatting.

The two substances will have the same volume after approximately 3.453 hours.

Step-by-step explanation:

The volume of substance A (measured in cubic centimeters) increases at a rate represented by the equation:

Where t is measured in hours.

And substance B is represented by the equation:

We are also given that at t = 0, A(0) = 3 and B(0) = 5.

And we want to find the time(s) t for which both A and B will have the same volume.  

You are correct in that B(t) is indeed t + 5. The trick here is to multiply both sides by dt. This yields:

Now, we can take the integral of both sides:

Integrate. Remember the constant of integration!

Since B(0) = 5:

Hence:

We can apply the same method to substance A. This yields:

We will have to divide both sides by A:

Now, we can take the integral of both sides:

Integrate:

Raise both sides to e:

Simplify:

Note that since C is an arbitrary constant, e raised to C will also be an arbitrary constant.

By definition:

Since A(0) = 3:

Therefore, the growth model of substance A is:

To find the time(s) for which both substances will have the same volume, we can set the two functions equal to each other:

Substitute:

Using a graphing calculator, we can see that the intersect twice: at t ≈ -4.131 and again at t ≈ 3.453.

Since time cannot be negative, we can ignore the first solution.

In conclusion, the two substances will have the same volume after approximately 3.453 hours.

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50