proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at the endpoints of these arcs form a regular polygon.

Test the series for convergence or divergence.

∞∑n=1(−1)nn4n

Identify bn.

A 600​-room hotel can rent every one of its rooms at $90 per room. For each​ $1 increase in​ rent,

3 fewer rooms are rented. Each rented room costs the hotel​ $10 to service per day. How much should the hotel charge for each room to maximize its daily​ profit? What is the maximum daily​ profit?

Differentiate.

1a) p(x) = 4th root of (2x-3/x^3)

b) q(x)= (2 sinx)/(1-cosx)

c)r(x)= sin(csc^3(x^4))

d) U(x)= ((cube root of x^2) sinx -2x-3)/sq. rt. x

A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.5 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. Find the work done. (Use 9.8 m/s2 for g.) Show how to approximate the required work by a Riemann sum. (Let x be the height in meters above the ground. Enter xi* as xi.).

Sketch the region enclosed by the given curves and find its area.

a) y=4/x,y=4x,y=(1/4)x,x>0

b) x=y²−4y,x=2y−y²

Someone explain and show how finding a subspace works and knowing how it is one with a matrix example.

In the figure below you see an ellipse which is enclosing a rectangle

The equation of ellipse is given by

x^2/4 + y^2 = 1

Find the length(L) and width (W) of the the rectangle which will maximize its area,( A). What is max(A) ?

Note: Cant upload the figures but i think the equation of ellipse is enough. the rectangle should be fitting inside the ellipse.

1. Sugary drink tax/soda taxand Mammography are different levels of prevention, please categorize them from the three options below and explain why.
   1. Primary level of prevention.
   2. Secondary level of prevention.
   3. Tertiary level of prevention.

2. Explain (Sugary drink tax/soda tax):

   Explain (Mammography):

The temperature , u(x,t), in a metal rod of length L satisfies
          del u/ del t = k del squared u / del x squared limit 0 less than or equal to, x less than or equal to L , t greater than or equal to 0

The ends of the rod at x=0 and x=L , are maintained at a constant temperature T not 0 , so that the boundary conditions are
                u(0, t) =0    u(L, t) = 0
The initial temperature distribution is
      u(x,0) = 4 sin 2 pi x/ L - 6sin (3 pi x/L) +12 sin (5 pi x / L)
Find the temperature, u(x, t).

An investment pays 8% interest compounded continuously. If money is invested steadily at the rate of

​$16,000​, how much time is required until the value of the investment reaches $160000?

2) Given f'(t)=-0.5t-e^-2t, compute f(5)-f(3)

3) Find the area under the given curve over the indicated interval.

y= 6x^2+x+3e^x/3; x=1 to x=5

Next, let’s suppose a rancher wants to fence off a rectangular shaped enclosure that has two identical sections. You can think of this as a rectangle with an additional fence dividing the rectangle in half. For the sake of this question, he wants that additional fence running North-South (up-down). He still has 2400 feet of fencing that he can use. What are the dimensions that gives the enclosure the most area?

(a) Use the sketch of the pen below for this question. Appropriately label the relevant information in the sketch. Remember, East-West is right-left, North-South is up-down.

(b) Based on the sketch above, what equation is being maximized?

(c) Based on the sketch above, what equation represents the given constraint?

(d) Find the dimensions of the enclosure that gives the largest area.

(e) How much fence is used on the East-West sides? How much fence is used on the North-South sides? What is the ration of the amount of fence used on the East-West sides to the amount of fence used on the North-South sides?

Problem 7. Consider the line integral Z C y sin x dx − cos x dy. (Please show all work)

a. Evaluate the line integral, assuming C is the line segment from (0, 1) to (π, −1).

b. Show that the vector field F = is conservative, and find a potential function V (x, y).

c. Evaluate the line integral where C is any path from (π, −1) to (0, 1).

Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 100 bacteria.

(a) What is the population after 15 hours?
(b) What is the population after t hours?
(c) Find the population after 20 hours.
(d) Find the time at which the population reaches 50,000.

Given that f "= - 12 (x-2) ^ 2 + 4, estimate the error obtained by approximating the integral of f (x) on the interval [1.5,2.5] with n = 4, using trapezoids.

Find the domain of the vector function r(t)= <sent,lnt,1/(x-2).

find the equation of the plane that passes through the point (-1,3, -8) and is parallel to the plane 3x-4y-6y = 9

find the equation of the line parallel to the plane 2x + y + z = 8 that passes through the point (1,2,3)

Please answer this 4 i need them quickly