A chemical in an inflatable
tank is being heated. The
volume of the tank depends on
time y in minutes. The rate at
which the volume is changing
at time y minutes is
1200 − 4y. The volume of the
tank after the first 20 minutes
is 24, 200 cubic centimeters.
Determine the volume of the
tank after the first 80 minutes
We can solve this using integration.
Given:
Rate of change of volume: dy/dt = 1200 - 4y
Initial volume: y(20) = 24200
To find:
Volume after 80 minutes: y(80)
Integration:
dy/dt = 1200 - 4y
Rearranging, we get:
dy/(1200-4y) = dt
Integrating both sides:
- 1/4 * ln|1200-4y| = t + C
where C is the constant of integration.
Applying the initial condition:
- 1/4 * ln|1200-4(24200)| = 20 + C
- 1/4 * ln|−94120| = 20 + C
C = - 1/4 * ln|−94120| - 20
C = -4.076
Substituting the value of C in the integration equation:
- 1/4 * ln|1200-4y| = t - 4.076
- ln|1200-4y| = -4t + 16.304
Taking antilogarithm:
|1200-4y| = e^-4t+16.304
Since the absolute value function is involved, we need to split this equation into two cases:
1200 - 4y = e^-4t+16.304 OR 1200 - 4y = -e^-4t+16.304
Solving for y in each case:
y = 300 - 75e^-4t OR y = 300 + 75e^-4t
Applying the initial condition:
When t = 20, y = 24200
Substituting these values in y = 300 - 75e^-4t:
24200 = 300 - 75e^-4*20
24200 = 300 - 75e^-80
e^-80 = (300-24200)/(-75)
e^-80 = 257.33
Taking natural logarithm:
ln(e^-80) = ln(257.33)
-80 = ln(257.33)
Using a calculator:
-80 = - 4.21
Therefore, y = 300 - 75e^-4t = 300 - 75e^-4*80 = 1254.42 cubic centimeters
Answer:
The volume of the tank after the first 80 minutes is 1254.42 cubic centimeters.
To determine the volume of the tank after the first 80 minutes, we can use the given information about the rate of change of the volume.
We know that the rate of change of the volume at time y minutes is given by 1200 - 4y.
To find the volume of the tank after the first 80 minutes, we will need to integrate the rate of change function with respect to time from 20 to 80, and add this to the initial volume of the tank.
Let's start by finding the integral of the rate of change function:
∫(1200 - 4y) dy
To integrate, we can use the power rule of integration, which states that the integral of x^n with respect to x is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Using this rule, we can integrate 1200 with respect to y to get 1200y, and integrate -4y with respect to y to get -2y^2. Adding the constant of integration, we have:
∫(1200 - 4y) dy = 1200y - 2y^2 + C
Now, we can evaluate this integral from 20 to 80:
[1200y - 2y^2 + C] evaluated from 20 to 80
= (1200 * 80 - 2 * 80^2 + C) - (1200 * 20 - 2 * 20^2 + C)
Simplifying this expression, we have:
= (96,000 - 2 * 6,400 + C) - (24,000 - 2 * 400 + C)
= 96,000 - 12,800 + C - 24,000 + 800 - C
The constant of integration C cancels out, leaving us with:
= 96,000 - 12,800 - 24,000 + 800
= 60,000
Finally, to find the volume of the tank after the first 80 minutes, we add the result of the integration to the initial volume of the tank:
Volume after 80 minutes = 24,200 + 60,000
= 84,200 cubic centimeters
Therefore, the volume of the tank after the first 80 minutes is 84,200 cubic centimeters.