212 Chapter 3: Applications of Differentiation
3.7 Optimization Problems
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find two positive numbers whose product is 181 and whose sum is a minimum.
a.
b.
c.
d.
e.
____ 2. Find two positive numbers such that the sum of the first and twice the second is 56
and whose product is a maximum.
a.
b. 28 and 14
c.
d.
e.
____ 3. Find the length and width of a rectangle that has perimeter meters and a maximum
area.
a. 12 m; 12 m.
b. 16 m; 9 m.
c. 1m; 23 m.
d. 13 m; 11 m.
e. 6 m; 18 m.
____ 4. Find the length and width of a rectangle that has an area of 968 square feet and whose
perimeter is a minimum.
a.
b.
c.
d.
e.
____ 5. Find the point on the graph of the function that is closest to the point
. Round all numerical values in your answer to four decimal places.
a.
b.
3.7 Optimization Problems 213
c.
d.
e.
____ 6. Find the point on the graph of the function that is closest to the point
.
a.
b.
c.
d.
e.
____ 7. A rectangular page is to contain square inches of print. The margins on each side
are 1 inch. Find the dimensions of the page such that the least amount of paper is used.
a.
b.
c.
d.
e.
____ 8. Determine the dimensions of a rectangular solid (with a square base) with maximum
volume if its surface area is 529 square meters.
a.
Dimensions:
b.
Dimensions:
c.
Dimensions:
d.
Dimensions:
e.
Dimensions:
214 Chapter 3: Applications of Differentiation
____ 9. A Norman window is constructed by adjoining a semicircle to the top of an ordinary
rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the
total perimeter is 38 feet.
a.
b.
c.
d.
e.
____ 10. A rectangle is bounded by the x– and y-axes and the graph of (see figure).
What length and width should the rectangle have so that its area is a maximum?
3.7 Optimization Problems 215
a.
b.
c.
d.
e.
____ 11. A solid is formed by adjoining two hemispheres to the ends of a right circular
cylinder. The total volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that
produces the minimum surface area. Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 12. The sum of the perimeters of an equilateral triangle and a square is 19. Find the
dimensions of the triangle and the square that produce a minimum total area.
a.
b.
c.
d.
e.
____ 13. A sector with central angle is cut from a circle of radius 10 inches, and the edges of
the sector are brought together to form a cone. Find the magnitude of such that the volume of the
cone is a maximum.
a.
b.
c.
d.
e.
216 Chapter 3: Applications of Differentiation
____ 14. Assume that the amount of money deposited in a bank is proportional to the square of
the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 36%.
Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)
a.
b.
c.
d.
e.
____ 15. A farmer plans to fence a rectangular pasture adjacent to a river (see figure). The
pasture must contain 720,000 square meters in order to provide enough grass for the herd. No fencing
is needed along the river. What dimensions will require the least amount of fencing?
a. x = 600 and y = 1200
b. x = 1000 and y = 720
c. x = 1200 and y = 600
d. x = 720 and y = 1000
e. none of the above
3.7 Optimization Problems 217
3.7 Optimization Problems
Answer Section