# MAXIMIZING REVENUE WORD PROBLEMS INVOLVING QUADRATIC EQUATIONS

Problem 1 :

A company has determined that if the price of an item is \$40, then 150 will be demanded by consumers. When the price is \$45, then 100 items are demanded by consumers.

(a) Find the price-demand equation, assuming that it is linear.

(b) Find the revenue function.

(c) Find the number of items sold that will give the maximum revenue. What is the maximum revenue ?

(d) What is the price of each item when maximum revenue is achieved ?

Solution :

Let x be the price and y be the demand.

Price  =  independent variable and demand  =  dependent variable. Since the relationship between price and demand is linear, we can form a equation.

(40, 150) and (45, 100)

Slope  =  (y2 - y1) / (x2 - x1)

Slope  =  (100 - 150) / (45 - 40)

Slope  =  -50 / 5

Slope  =  -10

Price demand equation :

(y - y1)  =  m(x - x1)

(y - 150)  =  -10(x - 40)

y - 150  =  -10x + 400

y  =  -10x + 400 + 150

y  =  -10x + 550

Price of item per unit.

(b)  Revenue function

Revenue  =  Units Sold x Sales Price

=  (-10x + 550) ⋅ x

R(x)  =  -10x2 + 550x

(c)  To find the number of units sold to get the maximum revenue, we should find "y" coordinate at the maximum point.

x coordinate at maximum  =  -b/2a

 x  =  550/20x  =  55/2x  =  27.5 y  =  -10 (27.5) + 550y  =  -275 + 550y  =  275

when 275 units sold, we can get the maximum revenue.

(d)

R(27.5)  =  -10(27.5)2 + 550(27.5)

=  -7562.5 + 15125

=  \$7562.5

The maximum revenue is \$7562.5.

Problem 2 :

A deli sells 640 sandwiches per day at a price of \$8 each. A market survey shows that for every \$0.10 reduction in price, 40 more sandwiches will be sold.

(a) Find the linear price-demand function.

(b) Find the revenue equation.

(c) How many sandwiches should be sold to maximize the revenue ?

(d) How much should the deli charge for a sandwich in order to maximize its revenue ?

Solution :

 Number of sandwiches640680720 Cost per sandwich\$8\$7.9\$7.8

(a)  Let x the number of sandwiches and y be the cost per sandwich.

Price demand function :

(640, 8)  and (680, 7.9)

m  =  (7.9 - 8)/(680 - 640)

m  =  -0.1/40

m  =  - 0.0025

(y - y1)  =  m(x - x1)

(y - 8)  =  -0.0025(x - 640)

y  =  -0.0025x + 1.6 + 8

y  =  -0.0025x + 9.6

(b)

Revenue  =  Units Sold x Sales Price

=  (-0.0025x + 9.6) x

R(x)  =  -0.0025x2 + 9.6x

(c)  We should find the number of sandwiches to be sold  out to maximize the revenue.

x  =  -b/2a

x  =  9.6/2(0.0025)

x  =  1920 (number of sandwiches)

To get the maximum revenue, 1920 sandwiches to be sold out.

(d)  Cost per sandwich

y  =  -0.0025x + 9.6

x  =  1920

y  =  -0.0025(1920) + 9.6

y  =  -4.8 + 9.6

y  =  4.8

Deli has to charge \$4.8 for a sandwich in order to maximize its revenue.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6