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Linear Algebra
a floor manager is going to install two types of machine, small and large. the following table shows the number of operators and the space requirements for each machine:
Small Large Maximum available
Number of operators 5 4 40
Space in m2 2 4 28
i) Taking x to represent the number of small machines and y to represent the number of large machines, write down two inequalities in x and y and illustrate these on a graph.
ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, calculate the number of each type of machine that should be installed in order to have a maximum profit. What is the profit?
Small Large Maximum available
Number of operators 5 4 40
Space in m2 2 4 28
i) Taking x to represent the number of small machines and y to represent the number of large machines, write down two inequalities in x and y and illustrate these on a graph.
ii) If the profit on each small machine is €120 per day and the profit on each large machine is €200 per day, calculate the number of each type of machine that should be installed in order to have a maximum profit. What is the profit?
Linear Algebra
A bookstore placed two orders with a publisher. The first order was for 12 copies of a history text
and 4 copies of a geography text, for a total cost of $1060. The second order was for 6 copies of the
history text and 4 copies of the geography text for a total cost was $640.
a) What is the cost of one copy of the history text?
b) What is the cost of one copy of the geography text?
and 4 copies of a geography text, for a total cost of $1060. The second order was for 6 copies of the
history text and 4 copies of the geography text for a total cost was $640.
a) What is the cost of one copy of the history text?
b) What is the cost of one copy of the geography text?
Linear Algebra
Wizard Mobile offers customers a choice of several monthly plans. The two least expensive ones are
Plan A and Plan B. Plan A, Pay As You Go, has no fixed monthly charges, and each minute costs 40
cents. Plan B charges $30 a month, includes 100 free minutes, and each additional minute costs 50
cents.
(a) For plan A: write a formula describing the monthly costs as a function of x, the number of
minutes the phone is used. Graph your equation, carefully labeling the axes. What is the interpretation of the y-intercept and the slope of this function?
(b) For plan B: write a formula describing the monthly costs as a function of x, the number of
minutes the phone is used. Graph the function on the same axes as the first function, carefully
labeling the axes. What is the interpretation of the y-intercept of this function?
(c) Give an example of the number of minutes used in a month (bigger than 0) for which plan A is
cheaper than plan B.
(d) Give an example of the number of minutes for which plan B is cheaper than plan A.
(e) Do the graphs of the two functions intersect? If so, what is the interpretation of the point(s) of
intersection? Explain.
Plan A and Plan B. Plan A, Pay As You Go, has no fixed monthly charges, and each minute costs 40
cents. Plan B charges $30 a month, includes 100 free minutes, and each additional minute costs 50
cents.
(a) For plan A: write a formula describing the monthly costs as a function of x, the number of
minutes the phone is used. Graph your equation, carefully labeling the axes. What is the interpretation of the y-intercept and the slope of this function?
(b) For plan B: write a formula describing the monthly costs as a function of x, the number of
minutes the phone is used. Graph the function on the same axes as the first function, carefully
labeling the axes. What is the interpretation of the y-intercept of this function?
(c) Give an example of the number of minutes used in a month (bigger than 0) for which plan A is
cheaper than plan B.
(d) Give an example of the number of minutes for which plan B is cheaper than plan A.
(e) Do the graphs of the two functions intersect? If so, what is the interpretation of the point(s) of
intersection? Explain.
Linear Algebra
10. A concrete company transports concrete from three plants 1, 2 and 3, to three construction sites A, B and C. Plant 1, 2 and 3 can supply 300, 300 and 100 tons per week respectively. While the requirements for site A, B and C are 200, 200 and 300 tons respectively. The cost of transporting one ton from plant 1 to site A, B and C is 4, 3 and 8 respectively, from plant 2 to A, B and C is 7, 5 and 9 respectively and from plant 3 to A, B and C is 4, 5 and 5 respectively. Provide recommendation to both the supplier and the user.
Linear Algebra
1) Translate the verbal description into a system of equations then solve.
(a) Find two numbers whose sum is 102 and one number is twice the other number.
(b) The sum of three times the first number and the second number is one. The first number minus the second number is seven.
2) Solve the system of equations
(a) y=−x
5x−7y=6
(b) 2x+3y=15
5x+4y=−1
(c) y=−x
5x−7y=6
please & thank you
(a) Find two numbers whose sum is 102 and one number is twice the other number.
(b) The sum of three times the first number and the second number is one. The first number minus the second number is seven.
2) Solve the system of equations
(a) y=−x
5x−7y=6
(b) 2x+3y=15
5x+4y=−1
(c) y=−x
5x−7y=6
please & thank you
Linear Algebra
competitors race distance of 28 m
Competitor 1 ant
Comp 2 snake
Comp 3 frog
use problem-solving strategies find the following information to synthesise mathematical models that represents the following situations. Ensure that each procedure and decision made is justified in a clear, concise and logical manner.
Three mathematical functions, using d for distance as the dependant variable and t for time (in minutes) as the independent variable are to be generated:
• One linear function in the form d=mt
• One quadratic function in the form d=at2 + bt
• One cubic function in the form d = pt3 + qt2 + rt
Each function must describe a competitor’s race performance.
What happens at any point in the race can be predicted mathematically from these functions.
You are to choose values for the constants, m, a,b, p, q & r so following conditions are satisfied:
All take at least 25 mins to finish race but no longer than 35 mins
all start at same time
deadheat between 2 but not 3
during race must be 3 overtakings
Competitor 1 ant
Comp 2 snake
Comp 3 frog
use problem-solving strategies find the following information to synthesise mathematical models that represents the following situations. Ensure that each procedure and decision made is justified in a clear, concise and logical manner.
Three mathematical functions, using d for distance as the dependant variable and t for time (in minutes) as the independent variable are to be generated:
• One linear function in the form d=mt
• One quadratic function in the form d=at2 + bt
• One cubic function in the form d = pt3 + qt2 + rt
Each function must describe a competitor’s race performance.
What happens at any point in the race can be predicted mathematically from these functions.
You are to choose values for the constants, m, a,b, p, q & r so following conditions are satisfied:
All take at least 25 mins to finish race but no longer than 35 mins
all start at same time
deadheat between 2 but not 3
during race must be 3 overtakings
Linear Algebra
True or False. Give reason
A Two-dimensional linear programming problem can have at most 2 optimal solutions.
A Two-dimensional linear programming problem can have at most 2 optimal solutions.
Linear Algebra
Solve the set of linear equations by the matrix method : a+3b+2c=3 , 2a-b-3c= -8, 5a+2b+c=9. Solve for c
Linear Algebra
1. Show that: [ x l m 1]
[ a x n 1] = (x-a)(x-b)(x-c)
[ a b x 1]
[ a b c 1]
2. Show that: [ 1+ (a)^2 - (b)^2 2b -2b ]
[ 2ab 1 - (a)^2 + (b)^2 2a ]
[ 2b -2(a)^2 1 - (a)^2 - (b)^2 ]
is a perfect cube.
{please note that these are determinants, the first one is a 4X4 determinant and the second one is a 3X3 determinant}
[ a x n 1] = (x-a)(x-b)(x-c)
[ a b x 1]
[ a b c 1]
2. Show that: [ 1+ (a)^2 - (b)^2 2b -2b ]
[ 2ab 1 - (a)^2 + (b)^2 2a ]
[ 2b -2(a)^2 1 - (a)^2 - (b)^2 ]
is a perfect cube.
{please note that these are determinants, the first one is a 4X4 determinant and the second one is a 3X3 determinant}
Linear Algebra
a) Find the inverse of the matrix
−
−
=
1 2 2
1 2 4
1 1 1
A
using Gauss-Jordon method.
−
−
=
1 2 2
1 2 4
1 1 1
A
using Gauss-Jordon method.
