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Referring to the previous two problems, discuss what effect (if any) the overland and under water costs have on the determination of $x$.

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

Oregon State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Let's go back and look at this river problem one more time. This is the river problem from problems 88 89. So I'm just going to re copy this and remind you what we have here. We have a power plant on the river and a factory on the other side, and our river is five miles long. So what we did was we try to find a way to put a line across the river to some point a and then complete the journey over to the factory. We wanted to minimize the cost. And the question here is what effect, if any, do the different costs have in determining X those over land and underwater costs? Does it make a difference? Well, yes, it does. Think about where we would place a if it is very, very expensive to go under the water. Let's say it's five times as expensive is over the land or six times or eight times. If the water is very expensive in relation to the overland, then you're going to minimize the water. You're gonna want to get across that water as quickly as possible. So you're going to make this section here from P. T A. B. As small as possible. Okay, you just want to minimize that that line segment their p A. Because the longer you're underwater, it's gonna start costing you more and more and more. On the other hand, let's say that it's cheap to run it underwater. Maybe you just have to string it across and it's easy. You don't have to clear the land, you just run it across. If the water is cheap, who can't read that? Sorry, let's first. Are we writing? If it's cheap to go across the water comparatively compared to the land, then you're gonna want to maximize that distance. P A. And in fact, if it's very, very cheap, why not just go right from P two f? Do it all under the water, and you don't have to even worry about being on land. Yeah, our case that we had in 88 89 one wasn't super expensive compared to the other. You know what? There was a difference, but there was still a point where we're gonna do some under the water, some on the land if it starts being a very, very big mismatch. in price, though, you're gonna either want to minimize or maximize that distance in the water to take advantage of wherever the cheaper price lies. So that's how those costs will relate to your final answer.

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