Section 2.4 Optimization
1. Biomedical: Pollen Count The average pollen count in NYC on days x of the pollen
season is
P(x) = 8x – 0.2x 2 (for 0 < x < 40)
On which days is the pollen count highest?
P is the continuous and (0,40) is a open interval. Find the critical values. (When P’ (x) = 0)
P’ (x) = 8 -0.4x = 04(20 – x)
The critical value is x = 20.
The pollen count is highest on the twentieth day.
OR use your calculator. and
maximum.
2. GENERAL: Fuel Economy
The fuel economy (in miles per gallon) of an average American midsized car is
E(x) = -0.01x 2 + 0.62x + 10.4, where x is the driving speed (in miles per hour, 20 ≤ x ≤ 60)
At what speed is the fuel economy greatest?
E is continuous and [20,60] is a closed interval. Find the critical values. (When E’ (x) = 0)
E’ (x) = -0.02x + 0.62
The critical value is x = 31
Fuel economy is greatest at 31 mph.
OR use your calculator. and
maximum.
3. BUSINESS: Copier Repair
A copier company finds that copies are x years old require on average,
f)x) = 1.2x 2 - 4.7x + 10.8
repairs annually for 0 ≤ x ≤ 5. Find the year that requires the least repairs, rounding your
answer to the nearest year.
f is continuous and [0,5] is a closed interval. Find the critical values. (When f’ (x) = 0)
f’(x) = 2.4x – 4.7
The critical value is 4.7/2.4 = 1.96
The second year requires the least repairs.
OR use your calculator. and
minimum.
4. GENERAL: Driving and AGE Studies have shown that the number of accidents a driver
has, varies with the age of the drive, and is highest for very young and very old drivers.
The number of serious accidents for drivers of age x during 2003 was approx.
f(x) = 0.013x 2 – 1.25x + 48 for 16 ≤ x ≤ 85.
Find the age that has the least accidents, rounding to the nearest year.
OR use your calculator. and
minimum.
5. ENVIRONMENTAL SCIENCE: Pollution Two chemical factories are discharging toxic
waste into a large lake, and the pollution level at a point x, miles from factory A towards
factory B is
P(x) = 3x2 -72x + 576 parts per million (for 0 ≤ x ≤ 50.)
Find where the pollution is the least.
P is continuous and [0,50] is a closed interval. Find the critical values. (When P’ (x) = 0)
P’(x) = 6x – 72
The critical value is x = 12
Pollution is the least 12 miles away from factory A towards factory B.
OR use your calculator. and
minimum.
6. BUSINESS: Maximum Profit
Country Motorbikes Incorporation finds that is costs $200 to produce each motorbike,
and that fixed costs are $1500 per day. The price function is P(x) = 600 – 5x, where p
is the price (in dollars) at which exactly x motorbikes will be sold. Find the quantity
Country Motorbikes should produce and the price it should charge to maximize profit.
Also find maximum profit.
C(x) = 200x +1500
p(x) = 600-5x
R(x) = (600 – 5x)x = 600x – 5x 2
P(x) = R(x) – C(x) = 600x – 5x 2 – (200x + 1500) = - 5x 2 + 400x – 1500
To maximize profit, we consider P(x) = 0
P’(x) = -10x + 400 = 0
x = 40
To maximize profit, 40 motorbikes should be sold. The price for 40 bikes is
p(40) = 600 – 5(40) = $400
The maximum profit is P(40) = R(40) – C(40)
= 400(40 – [200(40) + 1500] = $6500
OR
6. BUSINESS: Maximum Profit
Country Motorbikes Incorporation finds that is costs $200 to produce each motorbike,
and that fixed costs are $1500 per day. The price function is P(x) = 600 – 5x, where p
is the price (in dollars) at which exactly x motorbikes will be sold. Find the quantity
Country Motorbikes should produce and the price it should charge to maximize profit.
Also find maximum profit.
P(x) = - 5x 2 + 400x – 1500
OR use your calculator to find
maximum profit.
7. GENERAL: Parking Lot Design
A company wants to build a parking lot along the side of one of its building using
800 feet or fence. If the side along the building needs no fence, what are the dimensions
of the largest possible parking lot?
Building
Parking lot
Let x = the side perpendicular to the building and let y = the side parallel to the building.
Since only three sides will be fences, 2x + y =800
y = 800 – 2x
We with to maximize A = xy = x(800 -2x), so we take the derivative.
A = x(800 -2x) = 800x = 2x 2
A’ = 800 -4x = 0
x = 200
y = 800 – 2x = 800 -400 = 400
The side parallel to the building is 400 feet and the side perpendicular to the building
is 200 feet.
OR
7. GENERAL: Parking Lot Design
A company wants to build a parking lot along the side of one of its building using
800 feet or fence. If the side along the building needs no fence, what are the dimensions
of the largest possible parking lot?
Building
Parking lot
maximize A = x(800 -2x) on your calculator.
8. A farmer wants to make three identical rectangular enclosures along a straight river,
as in the diagram shown below. If he has 1200 yards of fence, and if the sides along the river
need no fence what should be the dimensions of each enclosure if the total are is to be
maximized?
River
y + 4x = 1200
xy = A
y = 1200 – 4x
A = (1200 – 4x)x
= 1200 – 4x 2
A’ = 1200 – 8X = 8(150 –x) = 0
x = 150
y = 1200 – 4(150) = 600
Since there are three identical enclosures the four sides perpendicular to the river are 150
yards long and the side parallel to the river is 600 yards long.
OR
8. A farmer wants to make three identical rectangular enclosures along a straight river,
as in the diagram shown below. If he has 1200 yards of fence, and if the sides along the river
need no fence what should be the dimensions of each enclosure if the total are is to be
maximized?
River
Or use your calculator to maximize area A = (1200 – 4x)x
9. GENERAL: Tax Revenue - Suppose that the relationship between the tax rate t on
imported shoes and the total sales S (in millions of dollars) is given by the function
below. Find the tax rate t that maximizes revenue for the government.
S (t)  8  15 3 t
Revenue = tax rate times total sales S(t)

R (t)  t 8  15 3 t
  8t  15 t
The tax rate that maximizes revenue is 6.4%.
4 3
10. BUSINESS: Exploring a Profit Maximization Problem - Let a business have the following
functions:
C(x) = 200 (20 + 2x) This implies that each item cost $200 and the number of units
purchased is (20 + 2x)
R(x) = (20 + 2x)(400 – 10x) This implies that the number of units sold is (20 + 2x) and
the price each is (400 – 10x).
Maximize the profit.
P (x) = R – C = (400-10x)(20+2x) – (200 (20 + 2x) ) = (8000 + 600x – 20x 2 ) – (4000 + 400x)
= 4000 + 200x – 20x 2
The maximum profit is $4500.