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Determine

[i] the output level that maximises profit and

[ii] the maximum profit.

[f] A business’ supply function is p(q) = 0.59 +4 and its demand function is

p(q) = 25 – 0.2q, where up to 40 items can be produced and sold.

Calculate:

the output level at equilibrium, and

(ii) the market clearing price.

621

The following problems test your preliminaries:

[a] The daily demand q in gallons for bulk ice-cream at Riversdale Dairy is a linear function of the selling price p. When p = $6.30,9 = 70, and

when p = $6.00,9 = 100. Assume q is never less than 50 gallons and never more than 400 gallons. Find the price p as a linear function of q.

[b] Stavros’s Taxidermy Company can prepare up to 20 skin pelts per week. The demand function for those skin pelts is

p = 50g + 80

Find revenue R as a function of q.

[c] Fixed costs for the production of ice-cream at Riversdale Dairy are $100 per day and the variable cost per gallon is

v = -0.0054 + 4.3, where 0 595 860

Find the total cost C as a function of q.

[d] The revenue of a shoe producing factory per week is

R = 1.472 +44

Its weekly total costs are

C = 195 – 12q + 1.2q?

Find the weekly profit function P(q) for this shoe company.

On which interval will the factory be profitable, i.e. determine a suitable domain for the profit function?

[e] If the daily profit that Lucky Mike’s hot dog stand can earn is

P(q) = -0.2q2 + 169 – 195, where 0 54 5200

Q2

A group of enterprising Math 121 students decide to make and sell earrings over the

internet to earn some extra money for tuition during the summer holidays. When they

price her earrings at $20 per pair, they do not get any orders. So they reduce their

price to $15 per pair and they get 11 orders per day. Towards the end of summer,

they decide to reduce their price further so that they can sell off their inventory. They

notice that when they reduce their price to $5 per pair, they get so many orders per

day, they cannot keep up with mailing out the orders.

Assuming that the demand function, p, for their earrings is a linear-to-linear rational

function of the number of pairs sold, q, find this rational function.

Q3

Find of the following functions:

[a] y = (x + 2)x+1

Note: An expression of type y = f(x)}(x) can only be differentiated using logarithmic

differentiation.

x+4

[b] y =

x-1

[c] y = 67′ + log3(2×2)

Q4

Find the following integrals:

[a] S8 (2x – 3)dx

[b] / x?exdx

[c] / Inx dx

Q5

The demand function of Oaken Red Wine is modelled as

10p – 10q + pq = 300

where p is the price per bottle and q is the number of bottles that can be sold at that

price. When the price is set at $20 per bottle, 10 bottles of the wine can be sold per

day. If the price per bottle increases at the rate of 10 cents per day, calculate the rate

of change of demand for the wine.

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