Assuming that x>0, use the method of reduction of order to find a second solution to

x^2y''−3xy'+4y=0

Given y1(x)=x^2

Find the point of intersection of the tangent lines to the curve r(t) = 5 sin(πt), 2 sin(πt), 6 cos(πt) at the points where t = 0 and t = 0.5. (x, y, z) =

Curvilinear integral of the function f (x, y) = x² + y² on a (3,0) centered and 3 radius circle.
a)Calculate the curvilinear integral by expressing the curve in parametrically.
b)Calculate the curvilinear integral by expressing the curve in polar coordinates.
c)Calculate the curvilinear integral by expressing the curve in cartesian coordinates.

Let V be the set of all ordered triples of real numbers. For u = (u1, u2, u3) and v = (v1, v2, v3), we define the following operations of addition and scalar multiplication on V :

u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2, ku3).

For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then

u + v = (1 + 2, 0 + 1 − 1, 3 + 1 − 2) = (3, 0, 2) and 2u = (2 · 1, 2 · 0, 2 · 3) = (2, 0, 6).

Complete the following:

(a) Calculate (1, 1, 1) + (2, 2, 2).

(b) Show that (0, 0, 0) 6= 0.

(c) What is 0?

(d) State a vector space axiom that fails to hold. Give an example to justify your claim.

You are looking at a hot air balloon that sinks vertically at a speed of 4.5 m / s.

You stand a horizontal distance A from the point in the ground where the balloon will land.

You choose how far away you are. For calculations, you can choose a distance between 100 m and
700 m from the balloon landing site.

There is a straight line between you and the top of the balloon that forms an angle θ with the ground. The ground is horizontal.

Calculate how fast this angle decreases when the balloon is at a 200 m above the starting point.

Question 1. Solve the following 1. If tan θ = 4 where 0 ≤ θ ≤ π 2 , find sin θ, cos θ,sec θ, csc θ, cot θ.

2. If α = 3π 4 , find exact values for sec α, csc α,tan α, cot α.

Question 2. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle. a. 225◦ b. 300◦ c. 135◦ d. 210◦

Question 3. The point P is on the unit circle. If the x-coordinate of P is 1/5, and P is in quadrant IV, find the y-coordinate.

Question 4. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Question 5. A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36◦ and that the angle of depression to the bottom of the tower is 23◦ . How tall is the tower?

What level of output would generate a net income of $15,000 if a product sells for $24.99, has unit variable costs of $9.99, and total fixed costs of $55,005?

An unmanned reconnaissance drone is flying over enemy territory using a preprogrammed flight pattern for a total flight time of 137 minutes. Its height? (in feet) can be modeled by the function H(t)=1/10t²-23t+1622, where t is the number of minutes of flight time since takeoff.?? ?A.) Due to the? terrain, the? enemy's radar cannot track the drone when it is below 322 feet. For what values of t will the drone be exactly 322 feet? high? Earliest time? = ?? Latest time? = *******Not sure where the question marks came from but the question reads "Due to the terrain, the enemy's radar cannot track the drone when it is below 322 ft. For what values of "t" will the drone be exactly 322 ft? It's asking for the earliest and latest times.*******

Minimize c = 8x − 8y subject to x/7≤ y

y ≤ 2x/3

x + y ≥ 10

x + 2y ≤ 35

x ≥ 0, y ≥ 0.

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least 490 mg of calcium, 23 mg of iron, and 60 mg of vitamin C, whereas Tom's meals should contain at least 440 mg of calcium, 18 mg of iron, and 50 mg of vitamin C. Bob has also decided that the meals are to be prepared from three basic foods: Food A, Food B, and Food C. The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin C are met for each patient's meals.

Contents (mg/oz)
Calcium Iron Vitamin C
Food A 30 1 2
Food B 25 1 5
Food C 20 2 4

Susan's meals:

Food A     oz
Food B     oz
Food C     oz

Tom's meals:

Food A     oz
Food B     oz
Food C     oz

If there are 3 known functions, namely:
X₁ + 2X₂ + 3X₃ = 6
2X₁ - 2X₂ + 5X₃ = 5
4X1 - X₂ - 3X₃ = 0
Use the Jacobian determinant to see whether there is a functional freedom function for each pair. Determine the values of X1, X2 and X3 in the above equation?

Find the perimeter of the curve for one full rotation.

x=6cost−2cos3t

y=6sint−2sin3t

Please provide a detailed explanation of this problem. show the necessary formulas.

Find the centroid (¯x,¯y) of the region bounded by:
y=3x² + 7x,   y=0,   x=0,   and   x=6

Thanks.

The manufacturer of a brand of mattresses will make x hundred units available in the market when the unit price is

p = 150 + 60e^0.05x

dollars.

(a) Find the number of mattresses the manufacturer will make available in the market place if the unit price is set at $350/mattress. (Round your answer to the nearest integer.)


(b) Find the producers' surplus if the unit price is set at $350/mattress. (Round your answer to the nearest dollar.)
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