Now that we can set up these problems, we are ready to solve them. For each of the previous examples and problems, a desired revenue will be given. We will set that revenue equal to the revenue equation. This will be a quadratic equation. Some of these equations will be solved by factoring, others by the quadratic formula. Some problems will have more than one solution.
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.
What price
should be charged if the revenue needs to be $2240?
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.
What should
the monthly rent be if the rental company needs $20,800 each month in revenue?
If x = 3,
the monthly rent will be 1000 + 200(3) = $1600. If x = 8, the monthly rent will
be 1000 + 200(8) = $2600.
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 price.
What should
the price of bananas be for weekly sales to be $140? How many bananas (in
pounds) will be sold at this price (these prices)?
(The revenue
will be in terms of cents, so $140 becomes 14,000 cents.)
If x = 2,
the price per pound will be 45 – 5(2) = 35 cents. The number of pounds sold
each week will be 300 + 50(2) = 400. If x = 1, the price per pound will be 45 –
5(1) = 40 cents and the number of pounds sold each week will be 300 + 50(1) =
350.
A music
storeowner sells 60 newly released CDs per day when the cost is $12 per CD. For
each $1.50 decrease in the price, the store will sell an additional 16 CDs per
week.
Let x represent the number of $1.50
decreases in the price.
What should
the price of bananas be for weekly sales to be $140? How many bananas (in pounds)
will be sold at this price (these prices)?
(The revenue
will be in terms of cents, so $140 becomes 14,000 cents.)
If x = 2, the price per pound will be 45 – 5(2)
= 35 cents. The number of pounds sold each week will be 300 + 50(2) = 400. If x = 1, the price per pound will be 45 – 5(1)
= 40 cents and the number of pounds sold each week will be 300 + 50(1) = 350.
A music
storeowner sells 60 newly released CDs per day when the cost is $12 per CD. For
each $1.50 decrease in the price, the store will sell an additional 16 CDs per
week.
Let x represent
the number of $1.50 decreases in the price.
What should
the price be if the storeowner needs revenue of $810 per week for the sale of these
CDs? How many will be sold at this price (these prices)?
When x = 1.25,
the price should be 12 – 1.50(1.25) = $10.13 and the number sold would be 60 +
16(1.25) = 80. If x = 3, the price should be 12 – 1.50(3) = $7.50 and the
number sold would be 60 + 16(3) = 108.
Practice
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. What should the rent be for the revenue to be $20,400?
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. What does the price need to be so that
weekly milk sales reach $11,475?
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. What should the price be so that $2450 worth of popcorn is
sold? How many buckets will be sold at this price (these prices)?
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. What would the shop have to charge in order for the
daily revenue from oil changes to be $1120? How many oil changes will the shop
perform each day?
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 are sold. If the manager wants $378
weekly revenue from onion ring sales, what should she charge for onion rings?
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 experi- ence leads him to
believe that for each $2 increase in the price of the shoe, one less pair would
be sold each week. What price would result in $3648 weekly sales?
Solutions
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