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Suppose the power station in the previous exercise is moved one mile inland find the point $A$ at which the power line enters the river.

$x \approx 16$ miles

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Oregon State University

McMaster University

University of Michigan - Ann Arbor

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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00:42

Distance Across River. Giv…

01:22

Given the data in Exercise…

01:19

01:53

Calculate the distance acr…

09:02

A power line is to be inst…

06:27

01:26

You are in a boat 2 miles …

for this problem. We are going to revisit the power station problem from 88. But we're going to make a few changes. We still have a river, and the river is still going to be five miles wide. Right? We still have a factory located here on the bottom. Right? But now we've taken this power plant, and we've moved it away from the river. So now we're one mile inland. This is not to scale, obviously. Okay, So between the power plant and the factory, if I go straight down the river, that is 20 miles. So now a has changed. According to this problem, A is the point at which the power line enters the river. So I'm gonna be going some distance. A Actually, we do this in green. That's my wire that is gonna be on the land. And then from a to f will be underwater. Now, I kinda drew this halfway. It's a straight line is not necessarily a could be anywhere from directly below the power plant to directly above the factory. So we're going to call this distance x the distance along the river from the power plant to where the line is going toe enter the river and go under under the river there. Which means that this piece here is going to be 20 minus X and X could be anywhere from 0 to 20 miles on. Our goal is to minimize the cost. So we're going to find a function of the cost in terms of that distance X just like in the last problem, we have to costs on the land when, when I'm putting the wire on the land, that cost is $50 per mile. When I'm under the water, my cost rises, which makes sense. It's harder to put wire under the water than it is on land and there is $80 a mile. So what? I do the cost? I would have to do this in two pieces. First, look at the land that is this section from P. T. A. And I'm going to call that section M. So I have a way to refer to it. There is a right triangle there, so I can say that one squared plus x squared equals m squared. And if I want to solve for em, I could just take the square root of both sides. So on my cost function, the distance between P and A is the square root of one plus x squared. And I'm gonna multiply that by 50 because that's how much it costs for each one of those land miles. Now to that, I need to add the water mileage. We're going to do the same thing. I'll just call this l and again, Ellis. The hypotheses have a right triangle. I have 20 minus X squared. That's with one leg, five squared. That's the second leg, and that equals l squared. And again, if I just want l. I'm gonna take the square root of both sides so that distance is the square root of 20 minus X squared plus 25. And for each of those water miles, it's going to cost me $80. So that is my function. Later in calculus, you'll learn ways to find, um, you know, using calculus ways to find the best X to minimize these costs for right now that we're gonna estimated using a spreadsheet. So I've got to put together a spreadsheet where I measure X, which is that distance? Um, how far along the riverbank I'm gonna put a and cost and I've plugged in this formula into my spreadsheet. So if you look, I start with X equals zero right below the power plant, and it costs almost $1700 as I move is X gets larger, you can see my prices are coming down. So I'm gonna look for where it stops, decreasing and starts increasing again. And as I go through, you can see I've got you could like six. It's still going down 89 10 miles, still decreasing all the way to X equal 16. You could see right there. As soon as I cross over to 16.1 miles, that price starts to rise again. So 16 is my best estimate. So I'm going to say that X will be approximately 16 miles downstream. That's the position to minimize my cost.

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