The daily profit of an oil trader is given by the profit function $p = 96x - 0.03x^2$, where $p$ is the daily profit, in euro, and $x$ is the number of barrels of oil traded in a day - Leaving Cert Mathematics - Question 8 - 2015
Question 8
The daily profit of an oil trader is given by the profit function $p = 96x - 0.03x^2$, where $p$ is the daily profit, in euro, and $x$ is the number of barrels of o... show full transcript
Worked Solution & Example Answer:The daily profit of an oil trader is given by the profit function $p = 96x - 0.03x^2$, where $p$ is the daily profit, in euro, and $x$ is the number of barrels of oil traded in a day - Leaving Cert Mathematics - Question 8 - 2015
Step 1
Complete the table below.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To calculate the daily profit for different numbers of barrels traded, we will substitute the values of x into the profit function p=96x−0.03x2.
For x=500:
p=96(500)−0.03(500)2=48000−750=40,500€
For x=1000:
p=96(1000)−0.03(1000)2=96000−3000=66,000€
For x=1500:
p=96(1500)−0.03(1500)2=144000−6750=76,250€
For x=2000:
p=96(2000)−0.03(2000)2=192000−12000=72,000€
For x=2500:
p=96(2500)−0.03(2500)2=240000−18750=52,500€
The completed table is:
Number of barrels (x)
500
1000
1500
2000
2500
Daily profit (€) (p)
40500
66000
76250
72000
52500
Step 2
Draw the graph of the trader’s profit function on the axes below for $500 \, \leq \; x \; \leq \, 2500$, $x \in \mathbb{R}$.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The graph of the profit function can be sketched using the calculated daily profits at the values of x. Starting from the point (500, 40500) and connecting through the other points, it will show a parabolic curve peaking at around 1600 barrels traded.
Step 3
Use your graph to estimate: The daily profit when 1750 barrels are traded.
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the graph, we find that the daily profit when 1750 barrels are traded is approximately €76,125.
Step 4
Use your graph to estimate: The numbers of barrels traded when the daily profit is €600,000.
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
From the graph, it appears that the numbers of barrels traded when the daily profit is €600,000 are approximately 850 or 2350 barrels.
Step 5
Use calculus to find the number of barrels of oil traded that will earn the maximum daily profit.
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the maximum profit, we first take the derivative of the profit function:
rac{dp}{dx} = 96 - 0.06x
Setting the derivative equal to zero to find critical points:
96−0.06x=0⇒x=1600 barrels
Step 6
Find this maximum profit.
97%
121 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Substituting x=1600 back into the profit function:
p=96(1600)−0.03(1600)2p=153600−76800=76800€
Thus, the maximum profit is €676,800.
Step 7
Calculate the value of k.
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To find the value of k where the trader will not make a profit if he trades more than k barrels of oil, we set the profit function equal to zero:
p=96x−0.03x2=0
Factoring gives:
x(96−0.03x)=0
Thus, x=0 or x=3200.
Hence, the value of k is 3200 barrels.
Join the Leaving Cert students using SimpleStudy...
