1.Assume that it costs Apple approximately C ( x ) = 28,900 + 100 x + 0.01 x 2 d

Question

1.Assume that it costs Apple approximately

C(x) = 28,900 + 100x + 0.01x2

dollars to manufacture x 30-gigabyte video iPods in a day.† How many iPods should be manufactured in order to minimize average cost?
iPods per day

What is the resulting average cost of an iPod? (Give your answer to the nearest dollar.) HINT [See Example 1.]

2.The cost pf controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is a possible model:

C(q) = 4,000 + 94q2

where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm (in dollars) of this reduction. What level of reduction corresponds to the lowest average cost per pound of pollutant? (Round your answer to two decimal places.)

pounds of pollutant per day

What would be the resulting average cost to the nearest dollar?
$ per pound

3.You are building a right-angled triangular flower garden along a stream as shown in the figure.

The fencing of the left border costs $4 per foot, while the fencing of the lower border costs $1 per foot. (No fencing is required along the river.) You want to spend $48 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose? [The area of a right-triangle is given by

A = xy/2.]

left border    

ft

bottom border    

ft

garden area    

ft2

4. Assume that the demand for tuna in a small coastal town is given by

p =

650,000 /

q1.5

where q is the number of pounds of tuna that can be sold in a month at p dollars per pound. Assume that the town's fishery wishes to sell at least 5,000 pounds of tuna per month.

(a) How much should the town's fishery charge for tuna in order to maximize monthly revenue? HINT [See Example 3, and don't neglect endpoints.] (Round your answer to the nearest cent.)
p = $ per lb

(b) How much tuna will it sell per month at that price?
q = lb

(c) What will be its resulting revenue? (Round your answer to the nearest dollar.)
$ per month

left border    

ft

bottom border    

ft

garden area    

ft2

Explanation / Answer

Q.1. Ans -

C(x) = 28,900 + 100x + 0.01x2

Average cost is the cost function divided by x;

so average cost A = 28,900 / x + 100 + 0.01 x.

Take the derivative:

A' = - 28,900 / x2 + 0.01

For average cost to be minimum, A' = 0

So, - 28,900 / x2 + 0.01 = 0

0.01 = 28,900 / x2

So, x2 = 28,900 / 0.01 = 2890000

So, x = 1700

Thus, 1700 iPods should be manufactured in order to minimize average cost.

Average cost A = 28,900 / x + 100 + 0.01 x

Average Cost if 1700 iPods per day will be,

A = 28,900 / 1700 + 100 + 0.01 (1700)

= 17 + 100 + 17

= 134

Therefore, $134 is the resulting average cost of an iPod.

Q.2. Ans -

C(q) = 4,000 + 100 q2

Average cost is the cost function divided by q;

so average cost A = 4,000 / q + 100 q.

Take the derivative:

A' = - 4,000 / q2 + 100

For average cost to be minimum, A' = 0

So, - 4,000 / q2 + 100 = 0

100 = 4,000 / q2

So, q2 = 4,000 / 100 = 40

So, q = 6.32

Thus, 6.32 pounds of pollutant per day reduction corresponds to the lowest average cost per pound of pollutant.

Average cost A = 4,000 / q + 100 q

Average Cost if 6.32 pounds of pollutant per day reduction will be,

A = (4,000 / 6.32) + 100 (6.32)

= 632.91 + 632

= 1264.91

Therefore, $1264.91 would be the resulting average cost.

Q.3. Ans -

Let x be the left side and y be the bottom side.

The left border costs $4 per foot and the bottom border costs $1 per foot.

So, 4x + y = 48

Gives y = 48 - 4x

Area, A = xy / 2

Putting y = 48 - 4x in A = xy / 2 , we get

A = x (48 – 4x) / 2

A = (48x – 4x2 ) / 2

A = 24x – 2x2

Factoring this to find x intercepts.

24x – 2x2 = x (24 – 2x) = 0

So, x = 0 or x = 12

Maximum A will be the midpoint of the X intercept value.

So, maximum value of x is 12 / 2 = 6

When x = 6, y = 48 - 4x = 48 – (4*6) = 48 - 24 = 24

Therefore, maximum area = xy / 2

                                           = 6 * 24 / 2

                                           = 3 * 24

                                           = 72

Therefore,

Left border         = 6 ft

Bottom border   = 24 ft

Garden area      = 72 ft2

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