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.
Expert Solution
To determine
To find: 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.
Answer to Problem 89AYU
Solution:
a. The speed at which the car is travelling as a function of the distance .
Required to come to a complete stop is .
Explanation of Solution
Given:
The distance function .
Calculation:
a. Given the distance function .
We can find the speed in terms of as
Expert Solution
To determine
To find: 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.
b. Verify that is the inverse of by showing that and .
Answer to Problem 89AYU
Solution:
b. Verified that and .
Explanation of Solution
Given:
The distance function .
Calculation:
b. Verify is the inverse by showing below.
Similarly, we can prove
Expert Solution
To determine
To find: 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.
c. Predict the speed that a car was traveling if the distance required to stop was 300 feet.
Answer to Problem 89AYU
Solution:
c. .
Explanation of Solution
Given:
The distance function .
Calculation:
c. The speed that a car was traveling if the distance required to stop was 300 feet.
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY