NZ carbon and coal #5
using equations and desmos
NZ coal and carbon uses two freight trains to deliver coal and timber to their depot.
Train A can carry 2 Tonnes of coal and 3 Tonnes of timber per trip.
Train B can carry 1 Tonne of coal and 6 Tonnes of timber per trip.
Each day the trains need to deliver at least 100 Tonnes of coal and 240 Tonnes of timber.
The maintenance crew has told management that the trains cannot make any more than 60 trips per day (each train)
Train A can carry 2 Tonnes of coal and 3 Tonnes of timber per trip.
Train B can carry 1 Tonne of coal and 6 Tonnes of timber per trip.
Each day the trains need to deliver at least 100 Tonnes of coal and 240 Tonnes of timber.
The maintenance crew has told management that the trains cannot make any more than 60 trips per day (each train)
Getting the equations from the text.
1) 2x + y ≥ 100 this represents the amount of coal
2) 3x + 6y ≥ 240 this represents the amount of timber
3) x ≤ 60 this represents the number of trips on train A
4) y ≤ 60 this represents the number of trips per day on train B
DESMOS
I will now enter the equations into DESMOS remembering to 'spin' around the inequality symbols.
I have changed the colour swatches for easier reading but this is not essential. The points are identified which make up the border of the feasibility region.
Next I need to add the equations to the graph and label the feasibility region. Don't forget to label the x and y axis Train A and Train B! I did.
I will now enter the equations into DESMOS remembering to 'spin' around the inequality symbols.
I have changed the colour swatches for easier reading but this is not essential. The points are identified which make up the border of the feasibility region.
Next I need to add the equations to the graph and label the feasibility region. Don't forget to label the x and y axis Train A and Train B! I did.
going for merit
Given the cost equation = 90x + 60y
1) calculate the number of trips per day that each train has to take to minimise costs.
2) what is the minimum cost?
Believe it or not you have done most of the hard work already! So here is how we calculate the costs by substituting the values of the coordinates of A,B,C and D into the cost equation above.
point A(20,60) cost = 90 x 20 + 60 x 60 = $5400 see how that was achieved?
point B(60,60) cost = 90 x 60 + 60 x 60 = $9000
point C(60,10) cost = 90 x 60 + 60 x 10 = $6000
point D(40,20) cost = 90 x 40 + 60 x 20 = $4800
The point that minimises fuel costs is D(40,20)
To minimise the fuel costs, NZ Carbon and coal should run 40 trips a day for train A and 20 trips a day for train B. The daily fuel cost is at a minimum of $4800
The cost of production changes such that the new fuel cost equation is
F = 140x + 70y
How does the increase in costs affect the number of trips the freight trains can take?
A(20,60) = 140 x 20 + 70 x 60 = $7000
B(60,60) = 140 x 60 + 70 x 60 =$12600
C(60,10) = 140 x 60 + 70 x 10 = $9100
D(40,20) = 140 x 40 + 70 x 20 = $7000
New cost $7000 - $4800 = an increase of $2200.00
The company now has two options that give the lowest fuel cost. It can either run 20 trips with train A and 60 trips with train B each day or 40 trips with train A and 20 trips with train B each day. The minimum cost is $7000.
F = 140x + 70y
How does the increase in costs affect the number of trips the freight trains can take?
A(20,60) = 140 x 20 + 70 x 60 = $7000
B(60,60) = 140 x 60 + 70 x 60 =$12600
C(60,10) = 140 x 60 + 70 x 10 = $9100
D(40,20) = 140 x 40 + 70 x 20 = $7000
New cost $7000 - $4800 = an increase of $2200.00
The company now has two options that give the lowest fuel cost. It can either run 20 trips with train A and 60 trips with train B each day or 40 trips with train A and 20 trips with train B each day. The minimum cost is $7000.
excellence
For Excellence you need to notice the there are two minimum values achieved, at points A and D.
The equation for that line is 2x + y ≥ 100, now find its gradient by rearranging it.
y ≥ -2x + 100 so the gradient = -2
The new cost function 140x + 70y has the same gradient as 2x + y ≥ 100 above.
70y ≥ -140x + c don't worry about c
y = 1140/70 + c
y = -2x + c so the gradient is -2 as above.
What does this mean?
Any integer (whole) coordinates along the line 2x + y ≥ 100 for 20 ≤ x ≤ 40, will minimise the cost. This means the company can run the trains in different combinations, not limited to 40 trips for train A and 20 trips for train B. For example the company could run 21 trips for train A and 38 trips for train B. When they increase train A’s trip number by 1, train B’s trip will decrease by 2 (see desmos graph).
(20, 40), (21, 38), (22, 36),……..(39, 22) (40, 20)
The daily fuel cost will be at minimum of $ 7000.
The equation for that line is 2x + y ≥ 100, now find its gradient by rearranging it.
y ≥ -2x + 100 so the gradient = -2
The new cost function 140x + 70y has the same gradient as 2x + y ≥ 100 above.
70y ≥ -140x + c don't worry about c
y = 1140/70 + c
y = -2x + c so the gradient is -2 as above.
What does this mean?
Any integer (whole) coordinates along the line 2x + y ≥ 100 for 20 ≤ x ≤ 40, will minimise the cost. This means the company can run the trains in different combinations, not limited to 40 trips for train A and 20 trips for train B. For example the company could run 21 trips for train A and 38 trips for train B. When they increase train A’s trip number by 1, train B’s trip will decrease by 2 (see desmos graph).
(20, 40), (21, 38), (22, 36),……..(39, 22) (40, 20)
The daily fuel cost will be at minimum of $ 7000.
video section
video 1
desmos
video 2
desmos