Answer :
To determine the total cost for expanding the patio, we are given that the initial cost of the patio is \[tex]$832, and the cost to expand the patio is \$[/tex]13 per square foot. We need to create a function that represents the total cost if the patio is expanded by [tex]\(x\)[/tex] square feet.
1. Understand the cost structure:
- The fixed cost (which doesn’t change) is \[tex]$832. This is the original quote for the patio without any added square feet.
- The variable cost (which changes based on the size of expansion) is \$[/tex]13 for every additional square foot.
2. Establish the function:
- Recognize that each additional square foot costs \$13, so for [tex]\(x\)[/tex] square feet, the cost is [tex]\(13 \times x\)[/tex].
- Since this is added to the fixed initial cost, the total cost function is linear.
3. Write the equation:
- The equation for this linear relationship will be in the form of [tex]\(f(x) = mx + b\)[/tex], where:
- [tex]\(m\)[/tex] is the cost per additional square foot, which is 13.
- [tex]\(b\)[/tex] is the initial cost, which is 832.
Therefore, the function that describes the total cost of the patio after expanding it by [tex]\(x\)[/tex] square feet is:
[tex]\[ f(x) = 13x + 832 \][/tex]
This equation shows that the total cost depends linearly on the number of square feet [tex]\(x\)[/tex] by which the patio is expanded.
1. Understand the cost structure:
- The fixed cost (which doesn’t change) is \[tex]$832. This is the original quote for the patio without any added square feet.
- The variable cost (which changes based on the size of expansion) is \$[/tex]13 for every additional square foot.
2. Establish the function:
- Recognize that each additional square foot costs \$13, so for [tex]\(x\)[/tex] square feet, the cost is [tex]\(13 \times x\)[/tex].
- Since this is added to the fixed initial cost, the total cost function is linear.
3. Write the equation:
- The equation for this linear relationship will be in the form of [tex]\(f(x) = mx + b\)[/tex], where:
- [tex]\(m\)[/tex] is the cost per additional square foot, which is 13.
- [tex]\(b\)[/tex] is the initial cost, which is 832.
Therefore, the function that describes the total cost of the patio after expanding it by [tex]\(x\)[/tex] square feet is:
[tex]\[ f(x) = 13x + 832 \][/tex]
This equation shows that the total cost depends linearly on the number of square feet [tex]\(x\)[/tex] by which the patio is expanded.
