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Ali is building a theater in the man-cave in his new house. He cannot recall the exact dimensions, but he knows that the length is 4 feet longer than the width, and the area is 525 square feet. Help Ali and the contractor solve for the length and width of his theater. Show all of your work and explain your answer.

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Given that sinu + sinq - (27/65) and tan[(u + q)/ 2] = (3/7), find cos[(u - q)/ 2]

Differentiate: 2arcsin(y/2)+√(4-y^2)

Determine the value of sin2x given that tan^2x=81/49 and Pi<x<3pi/2

The population of mice in Canterbury High School fluctuates from a minimum of 80 mice to a maximum of 420 mice. This population can be modelled as a function of time in months from Jan. 1st, by a cosine function. At the beginning of the year (January 1st), the population is at its least (80 mice) . After 3 months the population reaches its maximum (April 1st). On Canada Day (July 1st) the population goes back down to its minimum of 80. Three months later it reaches the maximum again (Oct. 1st). (And so on.)

(a) Draw a detailed graph of this situation for 1 year.

(b) Determine the equation of this function as a cosine function and a sine function that

describes the population of the mice in CAnterbury High School.

(c) Determine the time in months (to 1 decimal place) for one year, when the population is

above 165 mice.

find the domain of the indicated functions.express answers in both interval notation and inequality notation.

f (x)=4-9x+3x2

translate each algebraic definition of the following function into a verbal definition.

f (x)=2x2+5

g (x)=-2x+7

M (x)=5t-2

solve the following equation for 0«x«360

a) 6 cos²x + sin x - 4

b) 9 tan x + tan²x + 5 sec² x - 3

given sin α = 3/5 and cos β = -4/5 , α is an acute angle and β is an obtuse angle find ;

a) cot α + cosec β

b) sin 2α tan 2β

A poster is to have an area of 180 in with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?

A plane ﬂying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 300. At what rate is the distance from the plane to the radar station increasing a minute later?

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