Please help answer the following 4 questions Find a function

Please help answer the following 4 questions.

Find a function f whose graph is a parabola with the given vertex and that passes through the given point. vertex (-1, 7); point (-2, -2) f(x) If a ball is thrown directly upward with a velocity of 36 ft/s, its height (in feet) after t seconds is given by y 36t 16 What is the maximum height attained by the ball? (Round your answer to the nearest whole number. A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x 100x 0.5x2, where the revenue R(x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? units at A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence? (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field. (Enter your answers as a comma- separated list.) (b) Find a function that models the area of the field in terms of one of its sides. A(x (c) Use your model to solve the problem, and compare with your answer to part (a). Maximum area occurs at the following values smaller dimension larger dimension

Solution

1) standard equation of parabola is given by

y = a(x-h)^2 + k

where, h,k is the vertex

vertex given = (-1,7) and point (-2,-2)

plugging the values in the standard equation

-2 = a ( -2 + 1) ^2 + 7

solving for a

-2 =a + 7

a = - 9

therefore, equation of parabola is

f(x) = -9 (x+1)^2 + 7

2) height after t seconds is given by

y = 36t - 16t^2

maximum height is given by y coordinate of the vertex

x coordinate of the vertex is given by x = -b/2a

b = 36 , a = -16

x = -36 / -16*2 = 1.125

plugging x = 1.125 in the equation to get maximum height

y = 36(1.125) - 16(1.125)^2

y = 20.25 feet

maximum height attained by ball = 20 feet

3) revenue function is given by

R(x) = 100x - .5x^2

maximum revenue occurs at x = -b/2a = -100/ 2*-.5 = $ 100

maximum revenue = 100(100) - .5(100)^2 = 5000 units