QUESTION 1

A company sells square carpets for $3 per square foot. It has a simplied manufacturing process for which all the carpets each week must be the same size, and the length must be a multiple of a half foot. It has found that it can sell 100 carpets in a week when the carpets are 2 ft by 2 ft, the minimum size. Beyond this, for each additional foot of length and width, the number sold goes down by 6. What size carpets should the company sell to maximize its revenue? What is the maximum weekly revenue? Write the equation for the revenue, R, the company will earn as function of the length, x, of the carpet squares sold. R(x) = The carpet size that maximizes revenue has a length of V The maximum weekly revenue is $ (Type integers or decimals rounded to two decimal places as needed.) Find an equation of the curve whose tangent line has a slope of f'(x) = 72446 given that the point ( - 1, - 4) is on the curve. The function f(x) satisfying f'(x) = 7x_ "5 and f( 1): _ 4 is f(x) = (Type an exact answer.) A company incurs debt at a rate of D'(t) = 102(t + 12)4/t2 + 24t dollars per year, where t is the amount of time (in years) since the company began. By the 25th year the company had accumulated $1,458,700 in debt. (a) Find the total debt function. (b) How many years must pass before the total debt exceeds $2,920,000? (a) The total debt function is D(t) = (Use integers or fractions for any numbers in the expression.) (b) In years the total debt will exceed $2,920,000. (Round to three decimal places as needed.) Suppose the position of an object moving in a straight line is given by s(t} = t3 - t2 + 8. Find the instantaneous velocity when t= 5. The instantaneous velocity is D at t= 5. (Simplify your answer.)