The overall cost of carpet and installation from a particular company is represented by the function, where [tex]$x$[/tex] represents the square footage of the carpet.
[tex]$P(x)=\$ 2.35 x+\$ 104.00$[/tex]
What is the average rate of change over the interval [tex]$[2,000,2,500]$[/tex]?
A. [tex]$\$[/tex]0.43 per square foot
B. [tex]$\$[/tex]4.81 per square foot
C. [tex]$\$[/tex]2.35 per square foot
D. [tex]$\$[/tex]1.04 per square foot
Answer (2)
Calculate P ( 2500 ) = 2.35 × 2500 + 104.00 = 5979 .
Calculate P ( 2000 ) = 2.35 × 2000 + 104.00 = 4804 .
Compute the average rate of change using the formula: 2500 − 2000 P ( 2500 ) − P ( 2000 ) = 500 5979 − 4804 .
The average rate of change is 2.35 dollars per square foot.
Explanation
Understanding the Problem We are given the cost function P ( x ) = 2.35 x + 104.00 , which represents the overall cost of carpet and installation, where x is the square footage of the carpet. We need to find the average rate of change of this function over the interval [ 2000 , 2500 ] .
Average Rate of Change Formula The average rate of change of a function P ( x ) over an interval [ a , b ] is given by the formula: b − a P ( b ) − P ( a ) In our case, a = 2000 and b = 2500 .
Calculating P(2500) First, we need to calculate P ( 2500 ) :
P ( 2500 ) = 2.35 ( 2500 ) + 104.00 = 5875 + 104 = 5979 So, P ( 2500 ) = 5979 .
Calculating P(2000) Next, we need to calculate P ( 2000 ) :
P ( 2000 ) = 2.35 ( 2000 ) + 104.00 = 4700 + 104 = 4804 So, P ( 2000 ) = 4804 .
Calculating Average Rate of Change Now, we can calculate the average rate of change: 2500 − 2000 P ( 2500 ) − P ( 2000 ) = 2500 − 2000 5979 − 4804 = 500 1175 = 2.35 The average rate of change is 2.35 .
Final Answer The average rate of change of the cost function P ( x ) over the interval [ 2000 , 2500 ] is $2.35 per square foot.
Examples
Imagine you're planning to install carpet in your new house. The average rate of change helps you understand how much the total cost increases for each additional square foot of carpet you install. For instance, if the average rate of change is $2.35 per square foot, you can quickly estimate the additional cost for carpeting a larger area. This concept is useful for budgeting and making informed decisions about home improvement projects, ensuring you have a clear understanding of how costs scale with the size of the area you're covering.
The average rate of change of the cost function over the interval [ 2000 , 2500 ] is $2.35$ per square foot. Therefore, the answer is option C: $2.35 per square foot.
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