Business Applications of Differentiation (Continuation)
Example 6:
A manufacturer determines that when [tex]$x$[/tex] units of a particular commodity are produced, the profit generated will be
[tex]P(x)=-400 x^2+6800 x-12000$[/tex]
Ghana cedi. At what rate is profit changing with respect to the level of production [tex]$x$[/tex] when;
(a) 7 units are produced?
(b) 9 units are produced?
Example 7:
It is estimated that [tex]$x$[/tex] months from now, the population of a certain community will be
[tex]P(x)=x^2+20 x+8000$[/tex]
(a) At what rate will the population be changing with respect to 15 months from now?
Answer (2)
Find the derivative of the profit function P ( x ) = − 400 x 2 + 6800 x − 12000 , which is P ′ ( x ) = − 800 x + 6800 .
Evaluate P ′ ( 7 ) to find the rate of change of profit when 7 units are produced: P ′ ( 7 ) = 1200 Ghana cedi per unit.
Evaluate P ′ ( 9 ) to find the rate of change of profit when 9 units are produced: P ′ ( 9 ) = − 400 Ghana cedi per unit.
Find the derivative of the population function P ( x ) = x 2 + 20 x + 8000 , which is P ′ ( x ) = 2 x + 20 , and evaluate P ′ ( 15 ) to find the rate of change of population 15 months from now: P ′ ( 15 ) = 50 people per month. The rates of change are 1200 , − 400 , and 50 .
Explanation
Problem Setup We are given the profit function P ( x ) = − 400 x 2 + 6800 x − 12000 and the population function P ( x ) = x 2 + 20 x + 8000 . We need to find the rate of change of profit with respect to production level at x = 7 and x = 9 , and the rate of change of population with respect to time at x = 15 .
Finding the Derivative of Profit Function First, let's find the derivative of the profit function P ( x ) with respect to x . This will give us the rate of change of profit with respect to the level of production. Using the power rule, we have: P ′ ( x ) = d x d ( − 400 x 2 + 6800 x − 12000 ) = − 800 x + 6800
Rate of Change at x=7 (a) Now, let's find the rate of change of profit when 7 units are produced. We evaluate P ′ ( x ) at x = 7 :
P ′ ( 7 ) = − 800 ( 7 ) + 6800 = − 5600 + 6800 = 1200 So, when 7 units are produced, the profit is changing at a rate of 1200 Ghana cedi per unit.
Rate of Change at x=9 (b) Next, let's find the rate of change of profit when 9 units are produced. We evaluate P ′ ( x ) at x = 9 :
P ′ ( 9 ) = − 800 ( 9 ) + 6800 = − 7200 + 6800 = − 400 So, when 9 units are produced, the profit is changing at a rate of -400 Ghana cedi per unit. This means the profit is decreasing.
Finding the Derivative of Population Function Now, let's find the derivative of the population function P ( x ) with respect to x . This will give us the rate of change of population with respect to time. Using the power rule, we have: P ′ ( x ) = d x d ( x 2 + 20 x + 8000 ) = 2 x + 20
Rate of Change at x=15 (a) We want to find the rate at which the population will be changing with respect to time 15 months from now. We evaluate P ′ ( x ) at x = 15 :
P ′ ( 15 ) = 2 ( 15 ) + 20 = 30 + 20 = 50 So, 15 months from now, the population will be changing at a rate of 50 people per month.
Final Answers In summary: (a) The rate of change of profit when 7 units are produced is 1200 Ghana cedi per unit. (b) The rate of change of profit when 9 units are produced is -400 Ghana cedi per unit. (c) The rate of change of population 15 months from now is 50 people per month.
Examples
Understanding rates of change through differentiation is crucial in business. For instance, a company can use the derivative of a cost function to determine the marginal cost, which is the cost of producing one additional unit. Similarly, in marketing, the derivative of a sales function can help predict how sales will change with increased advertising. These insights enable businesses to make informed decisions about production levels, pricing strategies, and marketing efforts, ultimately optimizing profitability and resource allocation.
The profit function's rate of change at 7 units is 1200 Ghana cedi per unit and at 9 units is -400 Ghana cedi per unit. The population growth rate 15 months from now is 50 people per month. These derivatives provide insights on both profit changes due to production levels and population changes over time.
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