Functions/Equations

ECON 1003 - Graded Tutorial 1

(15%)

Original Deadline for Submission 11th October 2018 at 8 pm (ECT)

Late Submission Deadline 12th October 2018 at 8 pm (ECT) (3% deduction)

Units: Functions, Equations and Inequalities

Submissions typed in WORD and saved in WORD or PDF format are preferred. However, written assignments can be properly scanned, saved and submitted.

All submissions must be typed in WORD and saved in the same format.

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧 𝟏

The profit function for a product is given by π(x) = −x3 + 28x2 − 57x − 450,

where x is the number of units produced and sold. If break-even occurs when 6

units are produced and sold:

i. Find the quadratic factor of π(x).

ii. Find a number of units other than 6 that gives break-even for the product.

Question 2

(b) A model for the number N of people in a sub-urban community who have heard a certain

rumour is N = P (1 − 1

e0.10d ) where P is the total population of the community and d

is the number of days that have elapsed since the rumour began. In a community of

2000 residents, how many days will elapse before 750 have heard the rumour? (To the nearest

whole number)

(b) The number of years n for a piece of machinery to depreciate to a known salvage value can be

found using the formula n = ln 𝑠 − ln 𝑖

ln(1 − d) where 𝑠 is the salvage value of the machinery,

𝑖 is its initial value, and d is the annual rate of depreciation.

(i) How many years will it take for a piece of machinery to decline in value from $100,000 to $10,000

if the annual rate of depreciation is (8 %)? (To 1 d. p)

(ii) What would the salvage value of the machinery be after 6.5 years, if its value if the annual

rate of depreciation increased to 15%? (to the nearest hundred)

Question 3

Period Price ($) Quantity Demanded (kgs) Quantity Supplied (kgs)

Week 1 150 9000 5000

Week 2 400 6000 10000

For the information given in the schedule above, clearly showing all workings ∶

a. Derive the demand curve (Pd = a + bQd).

b. Derive the supply curve. (Ps = γ + δQs)

c. Derive the equilibrium price and quantity.

Question 4

The price p (in dollars) and the quantity q sold of boxed lunches obey the demand curve

p = 100 − q

3

(i) Find a model that expresses the revenue R as a function of q.

(ii) State the domain of R?

(iii) What is the revenue if 120 lunches are sold?

(iv) What quantity q of lunches will maximizes revenue? What will be the maximum revenue?

(v) What price should the foodseller charge in order to maximize revenue?

Question 5

The car company has found that the revenue from sales of sedans is a function of

the unit price p, in dollars, that it charges. If the revenue R, in dollars, is

R(p) = 1900p − 0.5p2

(i) At what prices p is revenue zero?

(ii) For what range of prices will revenue exceed $1.2 million?

END OF TUTORIAL