Mathematics, 10.08.2021 06:10 emcarrell

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.

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Answer from: Geo777

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:

$\displaystyle \frac{dA}{dt} = 0.3 A$

Where t is measured in hours.

And substance B is represented by the equation:

$\displaystyle \frac{dB}{dt} = 1$

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:

$\displaystyle dB = 1 dt$

Now, we can take the integral of both sides:

$\displaystyle \int 1 \, dB = \int 1 \, dt$

Integrate. Remember the constant of integration!

$\displaystyle B(t) = t + C$

Since B(0) = 5:

$\displaystyle B(0) = 5 = (0) + C \Rightarrow C = 5$

Hence:

B(t) = t + 5

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

$\displaystyle dA = 0.3A \, dt$

We will have to divide both sides by A:

$\displaystyle \frac{1}{A}\, dA = 0.3\, dt$

Now, we can take the integral of both sides:

$\displaystyle \int \frac{1}{A} \, dA = \int 0.3\, dt$

Integrate:

$\displaystyle \ln|A| = 0.3 t + C$

Raise both sides to e:

$\displaystyle e^{\ln |A|} = e^{0.3t + C}$

Simplify:

$\displaystyle |A| = e^{0.3t} \cdot e^C = Ce^{0.3t}$

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

By definition:

$\displaystyle A(t) = \pm C e^{0.3t} = Ce^{0.3t}$

Since A(0) = 3:

$\displaystyle A(0) = 3 = Ce^{0.3(0)} \Rightarrow C = 3$

Therefore, the growth model of substance A is:

$A(t) = 3e^{0.3t}$

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

$\displaystyle A(t) = B(t)$

Substitute:

$\displaystyle 3e^{0.3t} = t + 5$

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.

Calculus!

The volume of a substance, A, measured in cubic centimeters increases according to the ex

Answer from: Quest

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2

Answer from: Quest

Abit confused about what you mean, but i think you might be referring to slope intercept form: y = mx + bin that case b, or the y intercept is already given.(0,-1)find the slope by using the formulay2 - y1/x2 - x1the final slope will be 4/2, equivalent to 2.now your equation should look like thisy = 4/2x -1sorry if this didn’t you out!

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