Revenue
A common business application of quadratic equations occurs when raising a price results in lower sales or lowering a price results in higher sales. The obvious question is what to charge to bring in the most revenue. This problem is addressed in Algebra II and Calculus. The problem addressed here is finding a price that would bring in a particular revenue.
The problem involves raising (or lowering) a
price by a certain number of increments and sales decreasing (or increasing) by
a certain amount for each incremental change in the price. For instance,
suppose for each increase of $10 in the price, two customers are lost. The
price and sales level both depend on the number of $10 increases. If the price
is increased by $10, two customers are lost. If the price is increased by $20,
2(2) = 4 customers will be lost. If the price is increased by $30, 2(3) = 6
customers will be lost. If the price does not change, 2(0) = 0 customers will
be lost. The variable will represent the number of incremental increases (or
decreases) of the price.
The revenue formula is R = PQ where R represents
the revenue, P represents the price, and Q represents the number sold. If the
price is increased, then P will equal the current price plus the variable times
the increment. If the price is decreased, then P will equal the current price
minus the variable times the increment. If the sales level is decreased, then Q
will equal the current sales level minus the variable times the incremental
loss. If the sales level is increased, then Q will equal the current sales
level plus the variable times the incremental gain.
Examples
A department
store sells 20 portable stereos per week at $80 each. The manager believes that
for each decrease of $5 in the price, six more stereos will be sold.
Let x represent the number of $5 decreases
in the price. Then the price will decrease by 5x:
P = 80 – 5x.
The sales
level will increase by six for each $5 decrease in the price—the sales level
will increase by 6x:
Q = 20 + 6x.
R = PQ becomes R = (80 – 5x)(20 + 6x).
A rental
company manages an office complex with 16 offices. Each office can be rented if
the monthly rent is $1000. For each $200 increase in the rent, one tenant will
be lost.
Let x represent the number of $200 increases
in the rent.
P
= 1000 + 200x Q = 300 + 50x R = (1000 + 200)(16 – x)
A grocery
store sells 300 pounds of bananas each day when they are priced at 45 cents per
pound. The produce manager observes that for each 5-cent decrease in the price
per pound of bananas, an additional 50 pounds are sold.
Let x represent the number of 5-cent
decreases in the rent.
P = 45 – 5x Q = 300 + 50x R =
(45 – 5x)(300 + 5x)
(The revenue will be in cents
instead of dollars.)
A music
storeowner sells 60 newly released CDs per day when the price is $12 per CD.
For each $1.50 decrease in the price, the store will sell an additional 16 CDs
each week.
Let x represent the number of $1.50
decreases in the price.
P = 12.00 – 1.50 Q = 60 + 16x R = (12.00 – 1.50)(60
+ 16x)
Practice
Let x represent
the number of increases/decreases in the price.
1. The
owner of an apartment complex knows he can rent all 50 apartments when the
monthly rent is $400. He thinks that for each $25 increase in the rent, he will
lose two tenants.
P
= _____________
Q
= _____________ R = _____________
2. A
grocery store sells 4000 gallons of milk per week when the price is $2.80 per
gallon. Customer research indicates that for each $0.10 decrease in the price,
200 more gallons of milk will be sold.
P
= _____________
Q
= _____________ R = _____________
3. A
movie theater’s concession stand sells an average of 500 buckets of popcorn
each weekend when the price is $4 per bucket. The manager knows from experience
that for every $0.05 decrease in the price, 20 more buckets of popcorn will be
sold each weekend.
P
= _____________
Q
= _____________ R = _____________
4. An
automobile repair shop performs 40 oil changes per day when the price is $30.
Industry research indicates that the shop will lose 5 customers for each $2
increase in the price.
P
= _____________
Q
= _____________ R = _____________
5. A
fast food restaurant sells an average of 250 orders of onion rings each week
when the price is $1.50 per order. The manager believes that for each $0.05
decrease in the price, 10 more orders will be sold.
P
= _____________
Q
= _____________ R = _____________
6. A
shoe store sells a certain athletic shoe for $40 per pair. The store averages
sales of 80 pairs each week. The store owner’s past experience leads him to
believe that for each $2 increase in the price of the shoe, one less pair would
be sold each week.
P
= _____________
Q
= _____________ R = _____________
Solutions
