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Math for Modelers
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Real exam 2023…..latest update
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1. Joanne sells silk-screened t-shirts at a community festival. Her Marginal cost to produce one t-shirt is
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$3.50. Her total cost to produce 60 t-shirts is $300, and she sells them for $7 each. Use Python to
graph this information and determine the number of t-shirts Joanne must produce and sell to break
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even.
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So the equation to solve is
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7x = 3.5x + 90
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Solve for x to find the breakeven number of shirts:
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3.5x = 90
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x = 25.7
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At least 26 shirts should be produced
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Python Graph:
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(25.7, 180)
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Python Code:
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# import the matplotlib module
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import matplotlib.pyplot
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from matplotlib.pyplot import *
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# import the numpy module
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import numpy
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# import linspace
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from numpy import linspace
# assign the variables
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x= linspace(0,50,100)
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figure()
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y= 3.5*x + 90
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# bring in the chart
# plot the y equation
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plot(x,y)
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# plot the z equation
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plot(x,z)
# add legend
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legend (('cost','revenue'),loc=2)
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# add chart title
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title('Breakeven Analysis T-Shirts')
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# show chart
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show()
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2. A Rent-A-Truck company plans to spend $14 million on 280 new vehicles. Each commercial van will
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cost $55,000, each small truck $20,000, and each large truck $70,000. Past experience shows that
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they will need twice as many vans as small trucks. How many of each vehicle can they buy?
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Variables
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V = vans, S = small truck, L = large truck
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New vehicle totals: V + S + L = 280
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Spend: 55,000V + 20,000S + 70,000L = 14,000,000
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Twice as many vans: V = 2S
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Equations in proper form:
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V + S + L = 280
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55,000V + 20,000S + 70,000L = 14,000,000
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V – 2S = 0
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Insert into Matrix and reduce using Echelon:
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V S L Z
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1 1
[ ]
55,000 20,000 70,000 14,000,000
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11 −2 0 0
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280
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R2 = R1 (-55,000) + R2
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R3 = R1 (-1) + R3
[ ]
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1 1 1 280
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0 −35,000 −15,000 −14,000,000
−3 −1 −280
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1
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R2 = R2 (-1/35,000)
[ ]
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1 1 1 280
0 1 −3/ 7
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40
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1 −3 −1 −280
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R1 = R2 (-1) + R1
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R3 = R2 (3) + R3
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Math for Modelers
ne
la
tz
Real exam 2023…..latest update
sh
Sp
ne
1. Joanne sells silk-screened t-shirts at a community festival. Her Marginal cost to produce one t-shirt is
la
$3.50. Her total cost to produce 60 t-shirts is $300, and she sells them for $7 each. Use Python to
graph this information and determine the number of t-shirts Joanne must produce and sell to break
sh
even.
Sp
ne
So the equation to solve is
la
7x = 3.5x + 90
tz
Solve for x to find the breakeven number of shirts:
sh
3.5x = 90
Sp
x = 25.7
ne
At least 26 shirts should be produced
la
Python Graph:
sh
Sp
ne
la
tz
sh
Sp
ne
la
sh
Sp
ne
(25.7, 180)
la
tz
sh
Sp
ne
la
sh
Sp
Python Code:
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# import the matplotlib module
a
import matplotlib.pyplot
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from matplotlib.pyplot import *
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sh
# import the numpy module
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import numpy
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# import linspace
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from numpy import linspace
# assign the variables
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Sp
x= linspace(0,50,100)
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Sp
,Sp Sp Sp Sp Sp Sp
la l
a la la la z= 7*x
figure()
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y= 3.5*x + 90
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# bring in the chart
# plot the y equation
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plot(x,y)
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# plot the z equation
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plot(x,z)
# add legend
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legend (('cost','revenue'),loc=2)
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# add chart title
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title('Breakeven Analysis T-Shirts')
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# show chart
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show()
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2. A Rent-A-Truck company plans to spend $14 million on 280 new vehicles. Each commercial van will
Sp
cost $55,000, each small truck $20,000, and each large truck $70,000. Past experience shows that
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they will need twice as many vans as small trucks. How many of each vehicle can they buy?
la
tz
Variables
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V = vans, S = small truck, L = large truck
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New vehicle totals: V + S + L = 280
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Spend: 55,000V + 20,000S + 70,000L = 14,000,000
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Twice as many vans: V = 2S
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Equations in proper form:
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V + S + L = 280
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55,000V + 20,000S + 70,000L = 14,000,000
la
V – 2S = 0
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sh
Insert into Matrix and reduce using Echelon:
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V S L Z
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1 1
[ ]
55,000 20,000 70,000 14,000,000
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11 −2 0 0
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280
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R2 = R1 (-55,000) + R2
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R3 = R1 (-1) + R3
[ ]
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Sp
1 1 1 280
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0 −35,000 −15,000 −14,000,000
−3 −1 −280
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1
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R2 = R2 (-1/35,000)
[ ]
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1 1 1 280
0 1 −3/ 7
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40
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1 −3 −1 −280
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Sp
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R1 = R2 (-1) + R1
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R3 = R2 (3) + R3
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