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2.9

#3

For the given cost function

C(x)=62500+500x+x2C(x)=62500+500x+x2 find:

a) The cost at the production level 1400

b) The average cost at the production level 1400

c) The marginal cost at the production level 1400

d) The production level that will minimize the average cost

e) The minimal average cost

#4

For the given cost function C(x)=16√ x +x28000C(x)=16x+x28000 find

a) The cost at the production level 1750

b) The average cost at the production level 1750

c) The marginal cost at the production level 1750

d) The production level that will minimize the average cost.

e) The minimal average cost.

#5

Using the given cost function C(x)=1750+540x+0.2x2C(x)=1750+540x+0.2×2, where CC is total

cost and xx is the number of units produced.

The demand function p(x)=1620p(x)=1620, where pp is the price charged per unit xx.

Find the production level that will maximize profit.

#6

A baseball team plays in a stadium that holds 52000 spectators. With the ticket price

at $11 the average attendence has been 20000. When the price dropped to $8, the

average attendence rose to 26000. Assume that attendence is linearly related to

ticket price.

What ticket price would maximize revenue? $

#7

A manufacture has been selling 1100 television sets a week at $540 each. A market

survey indicates that for each $23 rebate offered to a buyer, the number of sets sold

will increase by 230 per week.

a) Find the demand function p(x)p(x), where xx is the number of the television sets

sold per week.

p(x)=p(x)=

b) How large rebate should the company offer to a buyer, in order to maximize its

revenue? $

c) If the weekly cost function is 99000+180×99000+180x, how should it set the size of the

rebate to maximize its profit? $

#8

For the given cost function

C(x)=14400+800x+x2C(x)=14400+800x+x2 find:

a) The production level that will minimize the average cost

b) The minimal average cost

#9

A farmer finds that if she plants 55 trees per acre, each tree will yield 80 bushels of

fruit. She estimates that for each additional tree planted per acre, the yield of each

tree will decrease by 2 bushels. How many trees should she plant per acre to

maximize her harvest?

trees

Give your answer rounded to the nearest whole number.

#10

The function P(x)=−1.75×2+475x−9000P(x)=-1.75×2+475x-9000 gives the profit when xx units

of a certain product are sold. Find

a) the profit when 80 units are sold

dollars

b) the average profit per unit when 80 units are sold

dollars per unit

c) the rate that profit is changing when exactly 80 units are sold

dollars per unit

d) the rate that profit changes on average when the number of units sold rises from

80 to 160.

dollars per unit

e) The number of units sold when profit stops increasing and starts decreasing.

(Round to the nearest whole number if necessary.)

units

#11

Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment

from a supplier and then sell it to customers in their store. The

function C(x)=1x+120000x−1+40000C(x)=1x+120000x-1+40000 models their total inventory

costs (in dollars) as a function of xx the lot size for each of their orders from the

supplier. The inventory costs include such things as purchasing, processing, shipping,

and storing the equipment.

What lot size should Glorious Gadgets order to minimize their total inventory costs?

(NOTE: your answer must be the whole number that corresponds to the lowest

cost.)

What is their minimum total inventory cost?

#12

Suppose the Sunglasses Hut Company has a profit function given

by P(q)=−0.03q2+6q−36P(q)=-0.03q2+6q-36, where qq is the number of thousands of pairs of

sunglasses sold and produced, and P(q)P(q) is the total profit, in thousands of dollars,

from selling and producing qq pairs of sunglasses.

A) Find a simplified expression for the marginal profit function. (Be sure to use the

proper variable in your answer.)

MP(q)=MP(q)=

B) How many pairs of sunglasses (in thousands) should be sold to maximize profits?

(If necessary, round your answer to three decimal places.)

Answer:

thousand pairs of sunglasses need to be sold.

C) What are the actual maximum profits (in thousands) that can be expected? (If

necessary, round your answer to three decimal places.)

Answer:

thousand dollars of maximum profits can be expected.

#13

Suppose a company’s revenue function is given by R(q)=−q3+200q2R(q)=-q3+200q2 and its

cost function is given by C(q)=350+12qC(q)=350+12q, where qq is hundreds of units

sold/produced, while R(q)R(q) and C(q)C(q) are in total dollars of revenue and cost,

respectively.

A) Find a simplified expression for the marginal profit function. (Be sure to use the

proper variable in your answer.)

MP(q)=MP(q)=

B) How many items (in hundreds) need to be sold to maximize profits? (Round your

answer to two decimal places.)

Answer:

hundred units must be sold.

2.11

#1

Suppose that x4+y4=82×4+y4=82.

(1) Use the method of implicit differentiation to find dydxdydx.

dydx=dydx=

(2) Find the equation of the tangent line at the point (x,y)=(−3,1)(x,y)=(-3,1).

The equation is y=y=

#2

Use implicit differentiation to determine dydxdydx given the equation x4+y2=−10×4+y2=10.

dydx=dydx=

#3

Given the equation below, find dydxdydx.

48×6+9x48y+y7=5848×6+9x48y+y7=58

dydx=dydx=

Now, find the equation of the tangent line to the curve at (1, 1). Write your answer

in mx+bmx+b format

y=y=

#4

Let AA be the area of a circle with radius rr. If drdt=2drdt=2, find dAdtdAdt when r=4r=4.

#5

A spherical snowball is melting in such a way that its radius is decreasing at rate of

0.2 cm/min. At what rate is the volume of the snowball decreasing when the radius is

12 cm. (Note the answer is a positive number).

cm^3min

Hint: The volume of a sphere of radius r is V=43πr3V=43πr3

#6

When air expands adiabatically (without gaining or losing heat), its pressure PP and

volume VV are related by the equation PV1.4=CPV1.4=C where CC is a constant.

Suppose that at a certain instant the volume is 550550 cubic centimeters and the

pressure is 7979 kPa and is decreasing at a rate of 1515 kPa/minute. At what rate in

cubic centimeters per minute is the volume increasing at this instant?

cm^3min

(Pa stands for Pascal — it is equivalent to one Newton/(meter squared); kPa is a

kiloPascal or 1000 Pascals. )

#7

A company’s revenue from selling x units of an item is given as R=1800x−1x2R=1800x1x2. If sales are increasing at the rate of 50 units per day, how rapidly is revenue

increasing (in dollars per day) when 860 units have been sold?

dollars per day

#8

A company selling widgets has found that the number of items sold, x, depends upon

the price, p at which they’re sold, according the equation x=30000√3p+1x=300003p+1

Due to inflation and increasing health benefit costs, the company has been

increasing the price by $2 per month. Find the rate at which revenue is changing

when the company is selling widgets at $200 each.

dollars per month

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