Write A Function C(x)

[mathdad]

A rectangular area adjacent to a river is fenced in; no fence is needed on the river side. The enclosed area is 2000 square feet. Fencing for the side parallel to the river is $10 per foot, and fencing for the other two sides is $2 per foot. The four corner posts are $35 each. Let x be the length of one of the sides perpendicular to the river. Write a function C(x) that described the cost of the project?

Below is the textbook breakdown of the solution in brackets. My questions follow each bracket.

[A rectangular area adjacent to a river is fenced in; no fence is needed on the river side. The enclosed area is 2000 square feet. Let x be the length of one of the sides perpendicular to the river. Then (2000/x) is the length of the side parallel to the river.]

How did the above words lead to (2000/x)? Does it have anything to do with A = L•x?

[Fencing for the side parallel to the river is $10 per foot.The cost is 10(2000/x) or (20000/x).]

Does the fraction (2000/x) actually mean a measure per linear foot? So, 10 dollars per foot means to multiply by (2000/x).
Correct?

[Fencing for the other two sides is $2 per linearfoot. The cost is 2(2x) = 4x.]

Where does (2x) come from? Does x here represent the width of the rectangle?

[The four corner posts are $35 each. The cost is (4)(35) = 140 dolloars.]

This I understand.

[Write a function C(x) that describes the cost of the project. We now add everything up.]

C(x) = (20000/x) + 4x + 140

One question about the domain not requested by the author.

The domain is D = {x|x cannot be 0}. Yes?

 

[Otis]

... Let x be ... sides perpendicular to the river. Then (2000/x) is ... side parallel to the river ... (2000/x)? ... anything to do with A = L•x?

Yes. The rectangular area formula shows a product. The area is the product of the two dimensions.

If the dimensions are x and y, then:

x∙y = 2000

and (as with any non-zero product of two numbers) you can always use division and write:

y = 2000/x

x = 2000/y

... [Is] the fraction 2000/x [measured in feet]?

Yes. When area is reported in square feet, then each dimension must be measured in feet. Only feet×feet results in feet² (square feet).

So, 10 dollars per foot means to multiply by (2000/x). Correct?

Yes. The reasoning is the same as with 4(35), where you multiplied [number of objects] by [$/object] to get [number of $].

[total ft]/1 × $/ft = [total $]/1

Unit analysis helps, when working through questions like those.

Does x here represent the width of the rectangle?

Whenever you forget the definition of a symbol (whether it was given or you defined it yourself), then read the definition again.

The exercise statement tells us: "Let x be the length of one of the sides perpendicular to the river."

The exercise doesn't use names 'length' and 'width'. If you desire to introduce those terms, then you need to tell the audience whether you define width to be the side parallel to the river or the sides perpendicular.

Fencing for the other two sides is $2 per linearfoot. The cost is 2(2x) ... Where does (2x) come from?

Looking at a diagram or quick sketch helps to organize given information.

There are two sides perpendicular to the river. They told us each measures x feet, so:

x + x = 2x

and that's the combined length of the two sides perpendicular to the river.

C(x) = (20000/x) + 4x + 140

That is a typo.

You may omit the grouping symbols, too, as the order of operations doesn't change without them.

... One question about the domain not requested by the author.

The domain is D = {x | x cannot be 0}. Yes?

No -- you've written that x can be any number of feet other than zero. In this exercise, it does not make sense to measure the rectangle using negative numbers.

Also, this exercise pertains to a real-world scenario; hence, the domain has to make sense in the real world. The smallest and largest values in the domain must be reasonable, positive numbers.

For example, the river lot might accommodate fenced-in areas where the sides perpendicular to the river could be any length between 60 and 120 feet (inclusive).

D = {60 ≤ x ≤ 120}

You're free to make up any domain, in this exercise, as long as it makes sense.

?

 

[mathdad]

I appreciate the time you have taken to answer my questions.