naknoi

This problem models pollution effects in the Great Lakes. We assume pollutants are flowinginto a lake at a constant rate I, and that water is flowing out at a constant rate F. We also assumethat the pollutants are uniformly distributed throughout the lake. If c(t) denotes the concentrationof pollutants at time t, then it satisfies the differential equation

c'(t)=-F/V(c)+I/V,

where V is the volume of the lake. We assume that rain and streams flowing into the lake keep thevolume of water in the lake constant

a) Solve this differential equation assuming c(0) = cÎ.

b) compute C∞ = lim c(t)

the lim is at t approaches ∞

Suppose you drop a stone of mass m from a great height in the earth's atmosphere, and the onlyforces acting on the stone are the earth's gravitational attraction (g is 9.8m/s2) and a retardingforce due to air resistance, which is proportional to the velocity v. Then, since F = ma and a = dv/dt,we have the differential equation m(dv/)dt= mg -kv (where k is some constant).

a) Find v as a function of time t.

b) Calculate lim v(t), the terminal velocity of the stone

t¢£

t¢£ means as t approaches infinity by the way

dy/dx = (ycosx)/(1+y^2) , y(0) = 1

dy/dx = √x/e^y

y=ln(cosx), 0 ≤ x ≤ π/3

∫ x/sqrt(3-x^4)

∫ r^2/(r+4)

A 10 g coin slides upward on a surface that is inclined at an angle of 14° above the horizontal. The coefficient of kinetic friction between the coin and the surface is 0.23; the coefficient of static friction is 0.31. Find the magnitude and direction of the force of friction under the following circumstances

Okay, so I know the answers to both of these, and this is how you get the answers...however, when I don't understand how to get b, because when I think that I might be inputting the wrong information or I am not understanding something correctly because I am not getting the same answer as the book, and I KNOW i am doing something wrong...but this is what the solution says.

Answer]

(a) kinetic friction Fr = N * k = mg*cos(14°) * k = 0.0219N

direction is against motion, downward;

(b) static friction force Fr = N * s = mg*cos(14°) * s = 0.0237N

direction is against the tendency of motion, upward

so i understand part A, and I got it right, however when I imput part b, i put in .01(9.81)*cos14*.31=.0295, but the answer says .0237, and I know I am doing it wrong because i had another problem like it and I am getting part b wrong on the one also. So, can someone please tell me how to get part b.

∫ x/√x^2+x+1

∫x^2/(3+4x-4x^2)^3/2

∫ x^2+1/(x^2-2x+2)^2

a. Determine the average values of x, x^2, x^3, and x^4 on the interval [1,2]

b. Determine the average values of sin(x), sin^2(x), sin^3(x), and sin^4(x) on the interval [0, π]

c. explain geometrically why the average values in a) increased, while those in b) decreased.

y=sin^2(x), y = cos^2(x), -π /4 ≤ x ≤ π/4

I know that I have to integrate this by parts. I let u=cos^-1x

dv=dx v=x, but i don't know what du should equal. Can someone help me please? thank you

g(x) =

3x

∫ (u^2-1)/u^2+1 du

2x

[Hint:

3x

∫ f(u)du =

2x

0

∫ f(u)du +

2x

3x

∫ f(u)du ]

0

i need help with this problem, preferably step by step.

∫ x^3√ x^2 + 1 dx

please show me step by step.

∫ x^3√ x^2 + 1 dx

If

5

∫ f(x)dx=12 and

1

5

∫ f(x)dx=3.6, find

4

4

∫ f(x)dx.

1

10

∫ |x-5| dx

0

A boat is cruising in a straight line at a constant speed of 2.8 m/s when it is shifted into neutral. After coasting 10.0 m the engine is engaged again, and the boat resumes cruising at the reduced speed of 1.6 m/s. Assume constant acceleration while coasting.

(a) How long did it take for the boat to coast the 10.0 m?

(b) What was the boat's acceleration while it was coasting?

(c) When the boat had coasted for 5 m, was its speed equal to, more than, or less than 0.6 m/s?

i don't really understand how to solve this.