Assignment 33 52

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course Mth 158

Resubmission 4/21/13 Around 5:30PM

031. *   31 

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Question: *  4.4.10 / 7th edition 4.1.78 (was 4.1.60). A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

 

 

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Your solution:

 2x + w = 2000, w = -2x + 2000

A = xw = x(-2x + 2000) = -2x^2 + 2000x

 A(- b/(2a)) = A(500) = -2(500)^2 + 2000(500)

 -500,000 + 1,000,000 = 500,000m^2

 

 

confidence rating #$&*: 2

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Given Solution:

* *  If the farmer uses 2000 meters of fence, and fences x feet along the highway, then he have 2000 - x meters for the other two sides. So the dimensions of the rectangle are x meters by (2000 - x) / 2 meters.

 

The area is therefore x * (2000 - x) / 2 = -x^2 / 2 + 1000 x.

 

The graph of this function forms a downward-opening parabola with vertex at x = -b / (2 a) = -1000 / (2 * -1/2) = 1000.

 

At x = 1000 the area is -x^2 / 2 + 1000 x = -1000^2 / 2 + 1000 * 1000 = 500,000, meaning 500,000 square meters.

 

Since this is the 'highest' point of the area vs. dimension parabola, this is the maximum possible area. **

 

STUDENT QUESTION:

 

I got the formula for the area but don't understand how you get a parabola out of that.

 

INSTRUCTOR RESPONSE:

 

There are two overall steps to solving this problem. The first is to get an expression for the area. The second is to find the maximum possible value of the expression.
You have the first step, which gives you the equation A = 1000 x - x^2 / 2.
The second step is to find the maximum of your expression.
To find the maximum of the expression 1000 x - x^2 / 2, you can consider the graph of the function.
The expression can be written as - x^2 / 2 + 1000 x. This is in the form of a quadratic expression of form 
a x^2 + b x + c witha = -1/2, b = 1000 and c = 0.
As you know from your previous work a quadratic function has a graph which is a parabola, and the vertex of a parabola is either its highest or lowest point (depending on whether it opens downward or upward).
The vertex of the parabola occurs when x = - b / (2 a). Substituting b = 1000 and a = -1/2 we find that
x_vertex = - 1000 / (2 * (-1/2) ) = 1000. The corresponding y coordinate isy = -1/2 x_vertex^2 + 1000 * x_vertex = -1/2 * 1000^2 + 1000 * 1000 = 500 000.
Since a = -1/2, which is negative, the parabola opens downward. This makes the vertex the highest point, so the value of the function is maximized at the vertex.

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Self-critique (if necessary):

 I did it a different way, but I still got the correct answer. I think I see how the given solution does it.

 

 

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Self-critique Rating: 3

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Question: *  Extra Problem / 4.4.33 / 7th edition 4.1.101 (was 4.1.80). A rectangle has one vertex on the line y=10-x, x>0, another at the origin, a third on the positive x-axis, and a fourth vertex on the positive y-axis. Find the largest area that can be enclosed by the rectangle.

 

 

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Your solution:

 I sketched the rectangle on y and x axes, put one point on the origin (0 , 0) and sketched the line of y = 10 - x using its x and y intercepts to see where the points where that vertex could be placed. I don't know how to find the point that would give the largest area. If I did, I could use the distance formula to get the distances from the x axis to either the point on the y axis or the point from y = 10 - x, and from the origin to whatever point on the x axis. So I am not sure what to do.

 

 

confidence rating #$&*: 1

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Given Solution:

* *  The dimensions of the rectangle are x and y = 10 - x. So the area is

 

area = x ( 10 - x) = -x^2 + 10 x.

 

The vertex of this rectangle is at x = -b / (2 a) = -10 / (2 * -1) = 5.

 

Since the parabola opens downward this value of x results in a maximum area, which is

 

-x^2 + 10 x = -5^2 + 10 * 5 = 25. **

 

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Self-critique (if necessary):

 Okay, I think I was making it more complicated. I don't see how you can use a parabola to find the area of a rectangle, though. Is the parabola used to find the ""vertex of the rectangle"" meaning the highest y point of the rectangle???? Could you explain this in a little bit more detail please?

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The vertex is the vertex of the parabolic graph. The graph and the rectangle are two different things.

The area of the rectangle is x ( 10 - x ) = -x^2 + 10 x. This equation happens to have a parabolic graph, but you wouldn't draw the parabola on the same set of axes as the rectangle. The parabola gives you information about the rectangle, but it isn't geometrically related to the rectangle.

Lots of models consist of quadratic functions. The simplest price vs. revenue function is quadratic, with a parabolic graph, but there's no literal parabola involved in the pricing strategy.

In the present example, the height of the parabola corresponding to x represents the area of the rectangle you get if you locate one vertex of your rectangle at the point (x, 10 - x). You want the maximum possible area, so you look for the highest point on your graph, which corresponds to the vertex of the parabola.************

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Self-critique Rating: 1

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#*&!

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I see now what happened to your file. It got 'absorbed' into that of another student.

I remembered responding to the question and easily found it in my log of responses, back on the 13th.

Sorry for the confusion.

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