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A power station is on one side of a straight river which is five miles wide, and a factory is on the other side, 20 miles up-river. A power line is to be run from the power station, under the river to some point $A,$ which is $x$ miles upriver from $P$, and then over land to the factory (see Fig. 27 ). If it costs $\$ 80$ per mile to run the line under water and $\$ 50$ per mile to run it overland, estimate $x,$ the point on the other side of the river where the power line comes out of the river, if the total cost of the power line is to be as small as possible (a sketch of the grap-or a table may be useful)

$x \approx 4$ miles.

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho 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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04:28

A power station is on one …

01:51

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Power Line A power station…

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Dayton Power and Light, In…

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09:04

for this problem. We're looking at a power station that sitting on a river. So let's begin by giving a little sketch of the situation. I have a power line or a power station. P It is on a five mile wide river, so five miles to go straight across and 20 miles downstream. I have a factory, okay? And I wanna lay a line from the power station to the factory in the most cost efficient way possible. So I'm going to have some point. I'm gonna call this a and that's gonna be at some random point here on the on the river bank, and I'm gonna be able to stretch my line from the power point for the power station to point a and then from a over to the factory. Now, I don't know where a is, so I'm going to call this X X is how many miles I am from that point directly across the power station. Since the whole distances 20 that section from Aid F is gonna be 20 minus x. It could be anywhere from zero, which means that comes straight across the river all the way over to 20 which means I go directly from the power plant to the factory, and I went to minimize the cost. Well, how much does it cost? We were told that underwater it's going to cost $80 per mile. On land, though, it's cheaper, which makes sense. It's easier toe string line on land than under the river. On land. It's $50 per mile. So let's see what the cost is. Let's see if we can make ah cost function in terms of X. We're gonna put this in terms of X, which is that distance from the power station directly across over to a well, let's look at those pieces first. Let's look at the water piece. We'll call this. I'll just call this m. Just have something to call it that is a right triangle so I can use the Pythagorean theorem to find em. That tells me that one side squared 25 plus the other side squared equals M squared. Or if I take the square root M equals 25 plus X squared, so the cost is going to be 25 plus X square. Take that square root. That's how many miles it is. That's the distance, and each one of those miles is going to cost $80. So that's the cost to go under the water under land, each of those miles is going to be $50 a mile, and the total mileage is gonna be 20 minus X. Whatever X is if we want to get rid of the parentheses we can. It's not entirely necessary at this stage, but for some people, it does make it a little bit easier to see. So there's my equation farther along and calculus, we'll learn how were how to actually find the value of X. But right now, we're just gonna estimate it. So what I've done is I've gone to a spreadsheet and I've plugged in values of X, starting at zero. I'm gonna go every 10th of a mile, just get a good, clear understanding of what's happening. And then I looked at the cost that goes along with that ex. So as you can see here, starting at zero if if I go straight across the river, my cost is $1400. As X starts to increase, my cost start coming down and it comes down until this point right here. When X equals four at four miles, that is the smallest value, Um, that we can have for the cost. It starts going back up again. So it looks like, according to our estimate here, that somewhere around four miles is the optimal place to put our point a thio to minimize the cost from our power station.

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