CALCULUS OPTIOMIZATION PRACTICE PROBLEMS WORKSHEET

Problem 1 :

If a business employs x workers to manufacture furniture then the profit made by the business given by

P(x) = -3x3 + 6084x - 5000 dollars per week

a) How many employees should the business have to maximize the profit ?

b) What is the maximum profit ?

Solution

Problem 2 :

Two numbers have the sum of 10. What is the minimum value that the sum of their cubes could be ?

Solution

Problem 3 :

What is the least possible value of the sum of a positive number and its reciprocal ?

Solution

Problem 4 :

A rectangle sits on the x-axis under the graph of y = 9 - x2

a) Find the coordinate of P.

b) Write down a formula for the area of the rectangle in terms of x.

c)  Find the maximum possible area of the rectangle.

Solution

Problem 5 :

A beam with a rectangular cross section is to be cut from a log of diameter of 1 m.

The strength of the beam varies in proportion to the width and to the square of the depth.

a) If the beam is x m and y m deep, write down the equation connecting x and y.

b) The strength S of the beam is given by width times the square of depth. Write down the formula for S in terms of x only.

c)  Find the dimensions of the strongest possible beam that can be cut from the log.

Solution

1)  a)  the number of employees should have in the business is 26.

b)  Maximum profit = 100456

2)  Minimum at x = 5, minimum value = 250

3)  the least value is 2.

4)  a)  The coordinate P is (x, 9 - x2)

b)  A(x) = 18x - 2x3

c)  We will get maximum possible area when x = 3. So, the maximum area is 123 square units

5) a)  x+ y2 = 1

b)  Strength = x (1 - x2)

c)  the dimension is 0.577 m and 0.816 m.

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