11DIFFERENTIATION - Department of Mathematics

36. E FFECT OF A DVERTISING ON P ROFITS The relationship between
Cunningham Realty’s quarterly profits, P(x), and
the amount of money x spent on advertising per quarter
is described by the function
P(x) 1
8 x2 7x 30 (0 x 50)
where both P(x) and x are measured in thousands of
dollars. Use differentials to estimate the increase in
profits when advertising expenditure each quarter is increased
from $24,000 to $26,000.
37. E FFECT OF M ORTGAGE R ATES ON H OUSING S TARTS Astudy
prepared for the National Association of Realtors estimates
that the number of housing starts per year over
the next 5 yr will be
7
N(r)
1 0.02r 2
million units, where r (percent) is the mortgage rate.
Use differentials to estimate the decrease in the number
of housing starts when the mortgage rate is increased
from 12% to 12.5%.
38. S UPPLY-PRICE The supply equation for a certain brand
of transistor radio is given by
p s(x) 0.3 x 10
where x is the quantity supplied and p is the unit price
in dollars. Use differentials to approximate the change
in price when the quantity supplied is increased from
10,000 units to 10,500 units.
39. D EMAND-PRICE The demand function for the Sentinel
smoke alarm is given by
30
p d(x)
0.02x2 1
where x is the quantity demanded (in units of a thousand)
and p is the unit price in dollars. Use differentials to
estimate the change in the price p when the quantity
demanded changes from 5000 to 5500 units/wk.
40. S URFACE A REA OF AN A NIMAL Animal physiologists use the
formula
S kW 2/3
to calculate an animal’s surface area (in square meters)
from its weight W (in kilograms), where k is a constant
that depends on the animal under consideration. Sup-
11.7 DIFFERENTIALS 779
pose a physiologist calculates the surface area of a horse
(k 0.1). If the horse’s weight is estimated at 300 kg,
with a maximum error in measurement of 0.6 kg, determine
the percentage error in the calculation of the
horse’s surface area.
41. F ORECASTING P ROFITS The management of Trappee and
Sons, Inc., forecast that they will sell 200,000 cases of
their Texa-Pep hot sauce next year. Their annual profit
is described by
P(x) 0.000032x 3 6x 100
thousand dollars, where x is measured in thousands of
cases. If the maximum error in the forecast is 15%, determine
the corresponding error in Trappee’s profits.
42. F ORECASTING C OMMODITY P RICES Acertain country’s government
economists have determined that the demand
equation for soybeans in that country is given by
p f(x) 55
2x 2 1
where p is expressed in dollars per bushel and x, the
quantity demanded per year, is measured in billions of
bushels. The economists are forecasting a harvest of 1.8
billion bushels for the year, with a maximum error of
15% in their forecast. Determine the corresponding maximum
error in the predicted price per bushel of soybeans.
43. C RIME S TUDIES Asociologist has found that the number
of serious crimes in a certain city per year is described
by the function
500(400 20x)1/2
N(x)
(5 0.2x) 2
where x (in cents per dollar deposited) is the level of
reinvestment in the area in conventional mortgages by
the city’s ten largest banks. Use differentials to estimate
the change in the number of crimes if the level of reinvestment
changes from 20 cents/dollar deposited to 22
cents/dollar deposited.
44. F INANCING A H OME The Millers are planning to buy a
home in the near future and estimate that they will
need a 30-yr fixed-rate mortgage for $120,000. Their
monthly payment P (in dollars) can be computed using
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