In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Vehicle Stopping Distance Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function d ( r ) = 6.97 r − 90.39 (a) Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop. (b) Verify that r = r ( d ) is the inverse of d = d ( r ) by showing that r ( d ( r ) ) = r and d ( r ( d ) ) = d . (c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Vehicle Stopping Distance Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function d ( r ) = 6.97 r − 90.39 (a) Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop. (b) Verify that r = r ( d ) is the inverse of d = d ( r ) by showing that r ( d ( r ) ) = r and d ( r ( d ) ) = d . (c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using
to represent a function, an applied problem might use
to represent the cost
of manufacturing q units of a good. Because of this, the inverse notation
used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as
will be
. So
is a function that represents the cost
as a function of the number
of units manufactured, and
is a function that represents the number
as a function of the cost
. Problems 91-94 illustrate this idea.
Vehicle Stopping Distance Taking into account reaction time, the distance
(in feet) that a car requires to come to a complete stop while traveling
miles per hour is given by the function
(a) Express the speed
at which the car is traveling as a function of the distance
required to come to a complete stop.
(b) Verify that
is the inverse of
by showing that
and
.
(c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.
Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment,
capacity, and so on. The Dayton plant has weekly costs that can be expressed as a f
a function of the number of gloves produced
TCD(X)=x²-x*4
where x is the weekly production volume in thousands of units and TCDX)Is the cost in thousands of dollars. The Hamiton plant's weekly production costs are given by
TCH(Y) = y² + 2Y+8
where Y is the weekly production volume in thousands of units and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 9,000 gloves per week at the lowest possible
cost.
(a) Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility.
min
st.
X, Y 20
(b) Use Excel Solver or LINGO to find the solution to your mathematical model to determine the optimal number of…
Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment, capacity, and so on. The Dayton plant has weekly costs
that can be expressed as a function of the number of gloves produced:
TCD(X) = x²-x+5,
where X is the weekly production volume in thousands of units, and TCD(X) is the cost in thousands of dollars. The Hamilton plant's weekly production costs are given by:
TCH(Y)²+2Y+3,
where Y is the weekly production volume in thousands of units, and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 8,000 gloves per week at the lowest possible cost.
a. Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility. If the constant is "1" it must be entered in the box. For subtractive or negative numbers use a minus sign even if there
is a + sign…
Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment, capacity, and so on. The Dayton plant has weekly
costs that can be expressed as a function of the number of gloves produced
TCD(X) = X² X + 3
where X is the weekly production volume in thousands of units and TCD(X) is the cost in thousands of dollars. The Hamilton plant's weekly production costs are given by
TCH(Y)
y² + 2Y + 9
where Y is the weekly production volume in thousands of units and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 5,000 gloves per week at the lowest possible cost.
(a) Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility.
min
s.t.
=
X, Y Z 0
= 5
(b) Use Excel Solver or LINGO to find the solution to your mathematical model to determine the optimal…
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