Pollution in a lake. consider a lake with a stream flowing out of it and a nearby factory dumping pollution into it. without worrying about the modeling details yet, the mass of pollution in the lake is represented by dp dt = − r/v p + f, where p is the mass of pollution in the lake (kg), r is the flow rate of the stream (m3/sec), v is the volume of the lake (m3) and f is the mass of pollution dumped per time by the factory (kg/sec). treat r, v and f as constants. use an integrating factor to solve the ode in general form. find the constant of integration if the initial condition is p(0) = a, and write the solution for p(t) in a clean, final form.

Step-by-step explanation:

We are given that pollution in a lake is given by the differential equation:

where, P(t) is the pollution at time t, r is the flow rate, V is the volume of lake and F is the mass of pollution dumped.

The given differential equation can be written as:

Comparing to linear differential equation:

,

we get,

Integrating factor:

Solution:

,

where C is the constant of integration.

Now, we are given that P(0) = A

putting these value in the above equation, we get,

Putting this value of C in equation, we get:

Dividing the equation by , we get:

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6   = 0.00000180

1.80 x 10-4   = 0.000180 (bigger, or "less small", than the former one)

so

1.8 x 10-2       = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.