Download Error propagation 2.The measurement of the circumference of a

Error propagation
1. The thickness of the shell is a = 0.7 centimeter. Use differentials to approximate the volume of the shell.
2.The measurement of the circumference of a circle is found to be 62 centimeters, with a possible error of 0.9
centimeter.
(a) Approximate the percent error in computing the area of the circle. (Round your answer to two decimal
places.)
(b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area
cannot exceed 1%. (Round your answer to one decimal place.)
Business and economics applications (Appendix F- Larson's book)
Definition (demand function): The number of units x that consumers are willing to purchase at a given
PRICE is defined as the demand function.
4. A business (in New York) sells 2000 hamburgers per month at a price of $10 each. It is estimated that
monthly sales will increase by 250 hamburgers for each $0.25 reduction in price. Assume there is a linear
relation between the price and the number hamburgers sold.
a) Find the demand function corresponding to this estimate.
b) Based on the fact that the firm wants both sales and the price to be positive, what is the appropriate domain of the
demand function p(x)?
b) Find the increase in revenue per hamburger (marginal revenue) for monthly sales of 5000 hamburgers.
c) Find the increase in revenue per hamburger (marginal revenue) for monthly sales of 7000 hamburgers.
d) What sales price p yields maximum revenue?
5. In marketing an item, a business has discovered that the demand for the item is represented by
p=50 . The cost C (in dollars) of producing x items is given by C=0.5x +500.
√
a) Find the revenue function R
b) Find the profit function P
c) Find the price per unit that yields a maximum profit.
d) Now assume that cost C=800+0.04x+0.0002x2.
i) Find the average cost function ̅
ii) Find the production level that minimizes that average cost.
6. The annual inventory cost C for a manufacturer is given below, where Q is the order size when the inventory is replenished. Find the change in
annual cost when Q is increased from 347 to 348, and compare this with the instantaneous rate of change when Q = 347. (Round your answers to two
decimal places.)
C = 1,010,000/ Q +8.9 Q
7.A business has a cost of C = 0.5x + 300 for producing x units. The average cost per unit is given by the following. Find the limit of the average
cost as x approaches infinity.
8.Suppose the yield V (in millions of cubic feet per acre) for a stand of timber at age t (in years) is V = 7.9e−(44.1)/t.
(a) Find the limiting volume of wood per acre as t approaches infinity.
(b) Find the rates at which the yield is changing when t = 20 years and t = 60 years. (Round your answers to three decimal places.)
9. A price p (in dollars) and demand x for a product are related by
.
If the price is increasing at a rate of 2 dollars per month when the price is 10 dollars, find the rate of change of the demand
8.The profit P for a company is P = 100xe−x/200, where x is demand and P is measured in millions of dollars. At the instant that x = 110, demand is
increasing at a rate of 5 units/month. Find the rate of change of profit at that instant. Round your answer to 3 decimal places.
10.The revenue R for a company selling x units is R = 1000x – 0.2x2. Use differentials to approximate the change in revenue if sales increase from x
= 2000 to x = 2100 units.
11.Find the number of units x that produces a maximum revenue R.
R = 3,000,000x/(0.01x2 + 625)
12.Find the number of units x that produces a maximum revenue R.
R = 36x2/3 – 4x
13.Use the cost function to find the value of x at which the average cost is a minimum.
C = 5x2 + 8x + 80 dollars
For that value of x, what are the marginal cost and the average cost?
16.A commuter train carries 960 passengers each day from Macon to Atlanta. It costs 5 dollars per person to ride the train. Market research reveals
that 80 more people would ride the train for each 25 cent decrease in the fare. Assume a linear relationship between the fare and the number of
riders. What fare should be charged in order to collect the largest possible revenue?
17.An offshore oil well is 4 kilometers off the coast. The refinery is 6 kilometers down the coast (see figure). Laying pipe in the ocean is twice as
expensive as on land. What path should the pipe follow in order to minimize the cost?
19.The measurement of the edge of a cube is measured to be 12 inches, with a possible error of 0.04 inch. Use differentials to approximate the
possible error and the relative error in computing the following values. (Round your answers to three decimal places.)
(a) the volume of the cube : possible error ± ; relative error ±
(b) the surface area of the cube : possible error ± ; relative error ±