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1. Determine the unit tangent vector (T), the unit normal vector (N), and the curvature of ?(?) = 2? ? + ?^2 ? – 1/3 ?^3 k at t = 1.

Suppose that a triangle has sides of length a = 15 cm, b = 37 cm, and c = 29 cm, then α = _______, β = ______., and γ =______ . Carefully round your answers to the nearest tenth of a degree, do not include units in your typed answers.

Solve the system below having the initial values

x(0) = 1 = y(0)

x’ = x + 2y

y’ = -5x + 3y

Please use solving systems of linear des by elimination. Thank You

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1. A plane through the origin is perpendicular to the plane 2? − ? − ? = 5 and parallel to the line joining the points A (1, 2, 3) and (4, -1, 2). Find its equation.

A farmer decides to enclose a rectangular​ garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 60 ft of​ fence? What should the dimensions of the garden be to give this​ area?

The maximum area that the farmer can enclose with 60ft is _____ sq feet

The larger dimension of the garden to give this area is ______ and the smaller dimension is _____

Use the method of this section to solve the linear programming problem.

The minimum is C =  

at (x, y, z) =

.

why is the reflection of y=-(x+2)² +3 the same as the reflection of -y=(x+2)²-3

I know they're the same algebraically but which vertical shift to do -3 or +3 ?? and how do I determine that

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1. Show that the lines ?/1 = y+3/ 2 = z+1/3 and x-3/2 = y/1 = z-1/-1 intersect by finding their point of intersection. Find the equation of the plane determined by these lines. Find parametric equations for the line that is perpendicular to the two lines and passes through their point of intersection.

A farmer wants to fence in a rectangular pasture of 1.5 million square feet and then divide the pasture in half with a fence parallel to the sides of the rectangle. How can this be done in such a way as to minimize the amount of fencing required?

Analyze and sketch the curve y = x3 − 6x2 + 9x. A full analysis includes the following:

critical points, intervals of increase/decrease, concavity, points of inflection, domain and range, intercepts. Organize your work in chart form.

1. How much of each a 10% gold alloy and a 24% gold alloy should be mixed together to make 40 grams of a 22% gold alloy?

2. A cold water faucet can fill the bath tub in 12 minutes, and a hot water faucet can fill the bath tub in 18 minutes. The drain can empty the bath tub in 24 minutes. If both faucets are on and the drain is open, how long would it take to fill the bath tub?

3. A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed?

A hotel has 210 units. All rooms are occupied when the hotel charges ​$90 per day for a room. For every increase of x dollars in the daily room​ rate, there are x rooms vacant. Each occupied room costs $ 36 per day to service and maintain. What should the hotel charge per day in order to maximize daily​ profit?

A milk truck carries milk with density 64.6 lb/ft3 in a horizontal cylindrical tank with diameter 4 ft. (a) Find the force exerted by the milk on one end of the tank when the tank is full. (Round your answer to the nearest integer.) lb (b) What if the tank is half full? (Round your answer to one decimal place.) lb.

The distance between the parallel planes

ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0

is

D =

.

Find equations of the planes that are parallel to the plane

x + 2y − 2z = 1

and three units away from it. (Enter your answers as a comma-separated list of equations.)