AP Calculus: Optimizations Take-home Quiz. Name:______________________________ Due: 02/16/10
Directions: Show all work to receive credit. You must use calculus when answering each question.
1. Two positive numbers x, y have a sum of 20. What are their values if one number plus the square root of the
other is as large as possible? (10 points)
Primary Equation: Secondary Equation:
2. A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy duty
fencing selling for $3 a foot, while the remaining two sides will use standard fencing costing $2 a foot. If the
area to be fenced in is 150,000 square feet, find the dimensions of the rectangle that could be built with the
cheapest cost. What is the cost to the nearest cent? (10 points)
Primary Equation: Secondary Equation:
1000t
3. The number of flies around a dropped piece of bread at a picnic grows according to the formula P(t ) ,
10 t 2
where t is measured in hours. How long will it take for the number of flies to be at a maximum? What is the
maximum size of the population of flies to the nearest fly?
4. A pro golf operator has found that he can sell 500 sets a year of his top grade clubs at $300 per set. For each
$5.00 he drops the price, he will sell 10 more sets of clubs. What price would give him the largest income and
what would that income be? Start by filling in the chart below.
# of $5.00 Cost per # sets sold Income
drops in set
price
0
1
5
10
x
5. A man is on the bank of a river that is 1 mile wide. He wants to travel to a town on the opposite bank but 1 mile
upstream. He intends to row on a straight line to some point P on the opposite bank and then walk the
remaining distance along the bank. How far from the town should he row in order to reach his destination in
the shortest time if he can walk 5 mph and row 3mph?
1 mile (shoreline)
Town
P
1 mile River
Man
Extra Credit: A rectangle has its base on the x-axis and its upper two vertices on the parabola y 12 x 2 . What is the
largest area the rectangle can have, and what are its dimensions? (round to the nearest hundredth)