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A dentist's office is being carpeted. The cost to install the new carpet is [tex]$100[/tex] plus [tex]$12[/tex] per square foot of carpeting.

(a) Determine a linear function that predicts the cost [tex]y[/tex] (in dollars) to carpet an [tex]x[/tex]-square foot office.

(b) Use this function to determine the cost (in dollars) to carpet 297 square feet of floor space.

Answer :

Sure! Let's work through the problem step-by-step.

(a) Determine a linear function that predicts the cost y (in dollars) to carpet an x-square foot office.

The problem states that there is a fixed installation cost of [tex]$100, and an additional cost of $[/tex]12 per square foot for the carpeting. To write a linear function for the total cost [tex]\( y \)[/tex], we can use the formula:

[tex]\[ y = \text{fixed cost} + \text{(cost per square foot)} \times x \][/tex]

Substituting the given values:

[tex]\[ y = 100 + 12x \][/tex]

So the linear function is [tex]\( y = 100 + 12x \)[/tex].

(b) Use this function to determine the cost (in dollars) to carpet 297 square feet of floor space.

Now that we have the function [tex]\( y = 100 + 12x \)[/tex], we can find the cost for 297 square feet by substituting [tex]\( x = 297 \)[/tex] into the equation:

[tex]\[ y = 100 + 12(297) \][/tex]

First, calculate the product:

[tex]\[ 12 \times 297 = 3564 \][/tex]

Then, add the fixed cost:

[tex]\[ y = 100 + 3564 = 3664 \][/tex]

Therefore, the cost to carpet 297 square feet of floor space is [tex]\( \$3664 \)[/tex].

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