Is there a non-integral choice for x, making the installation cheaper? "You work for the Silver Satellite Company. You need to determine the cost of running cable from a connection box to a new cable...

Is there a non-integral choice for x, making the installation cheaper?

"You work for the Silver Satellite Company. You need to determine the cost of running cable from a connection box to a new cable household, the Smith family, whose house is at the end of a two-mile long driveway off a nearby highway. The nearest connection box is along the highway, but 5 miles fromt the driveway. Because the Smith house is surrounded by farmland that they own, it would be possible to run the cable over land to the house dirrectly from the connection box.

It costs the company $10 per mile to install along the highway, and $14 per mile to install off the highway.

 

If x were to represent the distance in miles the cable runs along the highway before reaching the driveway, I got the function: f(x) = (x*10)+28, whereas 10 is the cost per mile along the highway, and 28 is the total cost of the 2-mile driveway. I don't know if this is right, but I think it is.

So, I graphed the function I got and it was a straight line, so I'm starting to doubt whether or not it is correct. But either way, if it is, how do I find the "non-integral choice for x" that would make installation cheaper?

Any help is appreciated! Thank you ouo

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tiburtius eNotes educator | Certified Educator

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I'm guessing both highway and driveway are streight lines and that driveway is perpendicular to highway wich would mean that Smiths house, connection box and driveway-highway crossroads make vertices of right angle triangle.

Now you have two options:

1. Run cable over 5 mi of highway + 2 mi of driveway

`cost=5 cdot 10 + 2 cdot 14 =78`

2. Run cable streight from connection box to the house that is over hypotenuse of triangle. Let us first calculate the length of hypotenuse `c = sqrt(2^2 + 5^2) =sqrt(29)`

`cost = 14c = 14sqrt(29) approx 75.3923`

In case 2 cost is lower implying that the company should run the cable directly from connection box over the farmland to Smiths house.

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