# Integration by separation of variables: algebraic rearrangement?

##
A lake contains 5,000,000 million litres of unpolluted water. A river flows into the lake at 100,000 litres per day. Due to polluters, the river flowing in contains 5 grams per litre of pollutant. A river flows out of the lake at 100,000 litres per day. Find an expression for the amount of pollutant in the lake.

I have:

#(dp)/dt = 500,000-p/50#

#p=25,000,000+e^(-t/50+c)#

Answer says #p=25,000,000(1-e^(-t/50))#

A lake contains 5,000,000 million litres of unpolluted water. A river flows into the lake at 100,000 litres per day. Due to polluters, the river flowing in contains 5 grams per litre of pollutant. A river flows out of the lake at 100,000 litres per day. Find an expression for the amount of pollutant in the lake.

I have:

Answer says

your answer is almost correct. Needs to get rid of c only, as explained below.

Your derivation is ok. Only thing left is to determine the constant of integration c.

For this apply the initial condition that at t=0, p=0 (there was no pollution initially)

By signing up, you agree to our Terms of Service and Privacy Policy

Integration by separation of variables is a technique used to solve certain types of first-order ordinary differential equations. The process involves isolating variables on one side of the equation and integrating with respect to each variable separately. This typically requires algebraic rearrangement of terms to group variables together and constants on one side of the equation. Once the variables are separated, integration can be performed to find the solution. The resulting solution often involves an implicit equation or may require further manipulation to solve for a specific variable explicitly. Overall, algebraic rearrangement is a crucial step in the integration by separation of variables method.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- The profit (in dollars) from the sale of x lawn mowers is P(x) = 50x - 0.05x^2 - 450. How to find the marginal average profit at a production level of 20 mowers?
- How do you find the arc length of #y=ln(cos(x))# on the interval #[pi/6,pi/4]#?
- What is the arc length of the curve given by #f(x)=x^(3/2)# in the interval #x in [0,3]#?
- The differential equation below models the temperature of a 95°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. How to solve the differential equation?
- What is the arclength of #f(x)=x^2/(4-x^2)^(1/3) # in the interval #[0,1]#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7