2. • Shana works at a flower shop on the weekend.
• The owner told her she needs to sell at least 5
dozen roses, but no more than 20 dozen.
• Additionally, she needs to sell at least 1 dozen
carnations, but no more than 10 dozen.
• Up to 24 dozen flowers arrive in each shipment.
THE SETUP
3. THE OBJECTIVE
If they make…
• $25 for each dozen roses sold and
• $15 for each dozen carnations sold,
what combination of sales
would bring in the most profit?
4. Identifying the …
what do we need to know?
Since we are asked for the combination that would
provide the most profit, we’re looking for the
number of roses (by dozen) and number of
carnations (by dozen) Shana should aim to sell to
make the most money possible given her resources.
Let’s let x = number of roses (by dozen)
and y = number of carnations (by dozen)
5. The number of roses should be at least 5,
but no more than 20.
“At least” means you can go at or above that
number but not below… x ≥ 5,
and “no more than” means you can be at that
number or below but not above… 𝑥 ≤ 20.
When put together it is 5 ≤ 𝑥 ≤ 20.
CONSTRAINT 1
7. The number of carnations should be at least 1,
but no more than 10.
“At least” means you can go at or above that
number but not below… 𝑦 ≥ 1,
and “no more than” means you can be at that
number or below but not above… 𝑦 ≤ 10.
When put together it is 1 ≤ 𝑦 ≤ 10.
CONSTRAINT 2
9. Up to 24 dozen flowers arrive in each shipment.
That means the total amount of roses & carnations
(by dozen) together cannot go over 24.
However, it is possible for the total to = 24.
This would mean x + y ≤ 24
CONSTRAINT 3
11. Graph all the constraints together to find the feasible region
(what is possible given our constraints)
12. The region shaded by all 3 constraints is feasible. We
check the vertices (corner points) of the region because 1 of
them will provide the most profit.
(5,10) (14,10)
(20,4)
(20,1)(5,1)
13. OBJECTIVE FUNCTION
Shana’s goal is to make the most profit. If she
makes $25 for each dozen roses and $15 for each
dozen carnations…
Profit = 25x + 15y
14. Now we just need to check each of the vertices of
the feasible region by substituting them into the
objective function.
x y Profit = 25x + 15y
5 1 25(5)+15(1)=140
5 10 25(5)+15(10)=275
14 10 25(14)+15(10)=500
20 4 25(20)+15(4)=560
20 1 25(20)+15(1)=515
15. CONCLUSION
Shana should sell 20 dozen roses
and 4 dozen carnations.
That combination will maximize
the profit that is possible with the
given constraints.
