Use symmetry and the region area formula under the parametric curve to calculate the area of the region enclosed by the astroid.

x= a cos^3(t)

y= a sin^3(t)

0 < t < 2pi

a= 3

The Digital Star Company makes two different models of computer tablets, which are assembled on two different assembly lines. Line 1 can assemble 30 units of the Star model and 40 units of the Prostar model per hour, and Line 2 can assemble 150 units of the Star model and 40 units of the Prostar model per hour. The company needs to produce at least 420 units of the Star model and 400 units of the Prostar model to fill an order. (a) Write the inequalities that describe the production constraints on the number of each type of tablet needed to fill the order. (Let x be the number of hours that Line 1 is run, and let y be the number of hours that Line 2 is run.

5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.

i) If we know a solution y = φ(x) of this equation, then any other solution

can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown

function which satisfies a certain linear equation. Using the fact that

φ and y both solve the above Riccati equation, find the differential

equation that v satisfies.

ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.

iii) Use the method of part(i) to find the general solution of the equation

in (ii).

1. The time rate of change of a rabbit population P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 and after 8 months there are 196 rabbits. How many rabbits will be two years later (find P(t) for t = 24 months)?

In each case, check that { v1,...vn} is a basis for R^n, and express the given vector b as a linear combination of the basis vectors.

(a). v1=(2,3), v2=(3,5). b=(3,4)

(b) v1=(1,0,3), v2=(1,2,2), v3=(1,3,2). b=(1,1,2)

(c) v1=(1,0,1), v2=(1,1,2), v3=(1,1,1). b=(3,0,1)

An xy trail M is greedy if every edge incident to y is contained in M. Let H be an Eulerian graph, and let x be a vertex in H. Prove that every greedy trail starting from x is a Eulerian circuit iff every cycle in H contains x.

A mass weighing 16 pounds stretches a spring 8/3
feet. The mass is initially released from rest from a point 3 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 1/2
the instantaneous velocity. Find the equation of motion x(t)
if the mass is driven by an external force equal to f(t) = 10 cos(3t).
(Use g = 32 ft/s2
for the acceleration due to gravity.)

Evaluate the iterated integral.

use the four step process to find f'(x) and the find f'(4) f'(6) and f '(8)

f(x)= 10 square root x+2

a. f'(x)

b. f'(4)

c. f'(6)

d. f'(8)

A company produces a special new type of TV. The compant has fixed costs of $471,000 and it costs $1200 to produce each Tv. The company projects that if it charged $2300 for the TV it will sell 700. If the company wants to sell 750 the price must be $200. Assume a linear demand.

what price should the company charge to earn a profit of $679,000

It would need to charge..?