Math - Problems Involving Quadratic Functions

Question

PROBLEM A: Johnsen is standing on a 4ft-high wall and throw a ball into the air with an initial upward velocity of 40in/s. Its height h in inches after t seconds is given by the function h(t)=120t-〖16t〗^2.

a. What is the standard form of the quadratic function?

b. What height will the ball be when 3 seconds have passed

c. What is the ball’s maximum height from where Johnsen is standing?

d. Can it exceed the height of a bird that is flying 22ft above the ground?

PROBLEM B: The profit (in thousand pesos) of A&J Publishing Company is given by P(x) = 750 + 500x - x^2 , where x is the amount (in thousand pesos) the company spends on advertising.

a. What is the standard form of quadratic function?

b. What will be the company’s profit if they will spend P50,000.00 on advertising?

c. Find the amount, x, that the company has to spend to maximize its profit.

d. Find the maximum profit of the company.

Answer 1

h(t) -16t^2 + 40t + 48 is the appropriate quadratic to give the height in inches after t seconds

the constant term is the initial height = 4 feet = 4x12 = 48 inches

the linear term's coefficient is the initial velocity, 40 inches per second

the quadratic term's coefficient is the effect of gravity, -32 feet per second per second

h(3) = -16(9) +40(3) + 48 = -144 + 120 + 48 = 24 inches = 2 feet high after 3 seconds

maximum height is found by taking the derivative and setting it equal to zero

h'(t) = -32t + 40 =0

t =40/32 = 5/4 = 1 1/4 seconds to reach max height. plug that t value into the original equation to find maximum height

h(5/4) = -16(5/4)^2 + 40(5/4) + 48 = -25 +50 + 48 = 73 inches max height = 6 feet an 1 inch

No, it can't exceed the height of a bird at 22 feet. 22 inches, yes, but not 22 feet.

P(x) = 750 +500x - x^2

P(50 = 750 +500(50) - (50)^2 = 750 +25,000 - 2,500 = 25,750 - 2,500 = 23,250 thousand pesos profit

P'(x) = 500 -2x = 0

x = 250,000 output to maximize profits

P(250) = 750 +500(250) - (250)^2 = 750 + 125,000 -62,500 = 63,250 thousand pesos is max profit