Download EXERCISES 5 MA1100 (1) Determine the equation of the line which

EXERCISES 5
MA1100
(1) Determine the equation of the line which goes through the two points (-1, 3) and (1, 2).
(2) Plot the line with slope −1 and y-intercept (0, 2). Find the equation of this line.
(3) Write the equation of the line 2y + 5x + 10 = 0 in the form y = m x + b. Determine the number of units y changes
as x increases by 1 unit. Plot the line.
(4) Given the linear supply function P = 300 + 5 Q where P is the total price of bottles of perfume supplied, and Q
is the quantity of bottles supplied: (i) Graph the linear supply function. What is the meaning of the horizontal and
vertical intercepts? (ii) How many bottles are supplied if each bottle costs e 20? (iii) How high is the bill if 1000
bottles are supplied?
(5) The linear demand equation for a crate of potatoes is given by Q = 210 − 3.5 P , where P is the price in e and
Q the number of crates ordered. (i) What is the price per crate if 70 are ordered? (ii) How many crates are ordered
if the price per crate is e 30?
(6) A distributor supplies 100 DVDs for the price of e 3 per DVD, and he supplies 125 DVDs for the price of e 2.50
per DVD. Determine the equation of the linear supply function. What is the price per DVD if 150 DVDs are supplied?
(7) A canteen has fixed costs of e 1500 per week. A meal costs e 5 to make, and is sold for e 9. (i) Find the equation
for total cost and plot the graph. (ii) Calculate the cost of producing 100 meals per week. (iii) How many meals are
made if the total costs are e 2300? (iv) Calculate the profit when 500 meals are sold per week.
EXERCISES 6
(1) A widow receives e 5, 600 interest per year from e 50, 000 invested in two bonds, one paying 10% and the other
12%. How much does she have invested in each bond?
(Ans.: e 30, 000 at 12%; e 20, 000 at 10%)
(2) With e 20, 000 to invest, how much should one invest at 16%, and how much at 12% to earn 15% on the total
investment?
(Ans.: e 15, 000 at 16%; e 5, 000 at 12%)
(3) Determine whether the following are demand or supply functions: (i) P = 60 − 3.5Q; (ii) Q = 40 − 0.5P ;
(iii) P = 5 + 1.2Q .
(4) A student has e 140 per month to spend on football matches and going to the cinema. If a match ticket costs
e 30 and a cinema ticket e 10, write down the budget constraint equation. Plot the corresponding line.
(5) Find the point of intersection of the given two lines: (i) y = x and y = 3 − x. (ii) 5x − 2y = 15 and 15x − 45 = 6y.
(iii) 3y + 2x = 5 and 2 − 6y = 4x.
(6) A supplier is known to supply a quantity of goods Q = 30 when the price is P = e 25. If the quantity supplied
increases by 3 for each increase in price of e 1, determine the equation of the supply function.
(7) The demand and supply functions for earrings are given by Pd = 800 − 2 Q and Ps = 8 Q − 40. (i) Calculate the
equilibrium price and quantity. (ii) Calculate the excess supply when P = 720. (iii) Calculate the excess demand
when P = 560.
EXERCISES 7
(1) Suppose demand and supply functions of a good are given by: Qd = 920 − 8 P and Qs = −120 + 2 P . (i)
Calculate equilibrium price and quantity. (ii) Calculate excess demand when P = 90. (iii) Calculate excess supply
when P = 105. (iv) Calculate the profit made on the black market if a price ceiling of P = 65 is imposed.
(2) Suppose demand and supply price of a good are given by: Pd = 50 − 1.5 Q and Qs = −11 + 0.5 P . (i) Calculate
equilibria. (ii) If a tax of 15.05 per unit is imposed, calculate the new equilibrium. (iii) How are taxes distributed
amongst supplier and buyer?
(3) Suppose demand and supply price of a good are given by: Pd = 124 − 4.5 Q and Ps = 33 + 2 Q. (i) Calculate
equilibria. (ii) If a tax of 30 per unit is imposed, calculate the new equilibrium. (iii) How are taxes distributed
amongst supplier and buyer?
(4) Suppose demand and supply price of organic turkey are given by: Pd = 80 − 0.4 Q and Ps = 20 + 0.4 Q. (i)
Calculate equilibria. (ii) If a subsidy of e 4 is introduced per bird, calculate the new equilibrium. (iii) How does the
subsidy benefit supplier and buyer?
(5) The demand function for a good is P = 2400 − 8 Q, where P is the price per unit and Q the number of units
sold. Find the equation for total revenue. Find the two values of Q at which the total revenue is zero. What is the
value of Q that maximizes revenue?
EXERCISES 8
(1) Differentiate the following functions
(i) f (x) = x3 (x2 − 3x)
(v) y = ln(x2 )
(ii) y =
x2 −1
x3 +x
(vi) u(x) = ex
(2) Use implicit differentiation to find
(i) y 2 x3 + x2 = 1
dy
dx
(iii) g(x) = (x5 + x2 )4
2
+1
(vii) h(t) =
t2 ln(t)
t4 +1
(iv) h(x) = x2 e−x
(viii) y = ln(x +
1
x
+ ex )
for the following
(ii) y − ln(x2 − 1) − y 3 x3 = 2
(iii) ln(y) + x2 e−y = 100
(3) A firm’s fixed costs are e 1,000 and variable costs are V C = 3 Q2 .
(i) Write down the equation for total costs T C, and compute T C when Q = 20.
(ii) Find the equation for marginal costs M C, and compute marginal costs when Q = 25.
(4) Suppose a supply function is given by Q = 820 + 2P + 0.3P 2 .
(i) Calculate the price elasticity of supply when the price is P = 30.
(ii) Estimate the percentage change in supply if the price drops by 3%.
(5) Suppose P is the monthly fee for membership at a club, and demand is given by
P = 300 e−0.005 Q .
(i) Find the equation for price elasticity d , and compute d when Q = 150.
(ii) Suppose there are 200 members: How much is each member’s fee? If the fee increases by 5%, use elasticity to
approximate the percentage change in demand.
EXERCISES 9
(1) Suppose total revenue is T R = 2500 Q − 12.5 Q2 and total cost is T C = Q3 − 5 Q2 + 250 Q + 500. Determine the
quantity Q that maximizes profit, and compute the maximal profit.
(2) Determine the production level that maximizes profit for a company with total cost and demand price given by
T C = 84 + 1.26 Q − .01 Q2 + .00007 Q3
and P = 3.5 − .01 Q .
(3) A company needs to run a pipeline from an oil rig 25 km out to sea to a storage tank 5 km inland. The shoreline
runs north-south and the tank is 8 km south of the rig. Assume it costs e 50, 000 per km to construct the underwater
pipeline, and e 20, 000 per km to construct the pipeline on land. The pipeline will be built in a straight line from the
rig to a selected point on the shoreline, and from there in a straight line to the tank. What point on the shoreline
will minimize the cost of the pipeline?
(4) A magazine has 5000 subscribers paying e 36 a year. A survey indicates that there will be 100 new subscribers for
every price drop of e 1 in the subscription. What annual rate should the publisher charge to maximize the revenue?
(5) The cost of operating a truck independent of the driver’s wages is (.13 + v/500) Euro per kilometer, where v is
the average speed in kilometers per hour. The driver earns e 9.80 per hour. What average speed should the truck
maintain on a 600 km trip to minimize the costs?
Mid-Term: 25/02/15 @ 6:00 - 8:00 pm: WGB G05 (BCOM1) and WGB 1.07 (BCOM1
Intl + BFM1 + Visiting). Course web-page: http://euclid.ucc.ie/pages/staff/mk/MA1100/ma1100.html