Optimization

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CoreyyV

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I need help solving this. I only need to write the constraint equation and objective function (don't have to find the minimum cost) using variables x and y. The question states that I'm framing a door that's in the shape of a rectangle with a semicircle on top (no framing along the dotted line). I plan to use 20 ft of framing and the cost of framing for the straight edges is $2/ft and for the circular edge $5/ft. The rectangle has width of "x" and height of "y". Obviously the objective is to minimize cost and the constraint is that there's only 20ft of framing, but I don't know how to use the variables to create a function/equation for the constraint and objective.
(I've attached a photo of the shape of the rectangle if that helps as well).
 

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What have you tried?

Using x and y as you defined them (very good!) can you find an equation for the perimeter (ie framing) of the door? What is the perimeter of a semi-circle?

The cost equation is not that bad. c(x,y) = cost of the 3 sides plus the cost of the semi-circle = $2/ft*(the length of the 3 sides) + $5/ft(the length of the semi-circle).

Of course the attachment of the picture of the door helps, otherwise we do not know the definition of x and y. Of course you could have left out the attachment if you defined x and y.
 
Jomo said:
What have you tried?

Using x and y as you defined them (very good!) can you find an equation for the perimeter (ie framing) of the door? What is the perimeter of a semi-circle?

The cost equation is not that bad. c(x,y) = cost of the 3 sides plus the cost of the semi-circle = $2/ft*(the length of the 3 sides) + $5/ft(the length of the semi-circle).

Of course the attachment of the picture of the door helps, otherwise we do not know the definition of x and y. Of course you could have left out the attachment if you defined x and y.
Click to expand...
Well for the perimeter, I think that I would have to add the 2 sides (y) + the bottom (x) and then the circumference of the semicircle on top. I know the circumference of a semicircle is pi*r + the diameter, but since the diameter, in this question, is the dotted line, and am told not to take it into account. So would the equation for the perimeter of the semicircle in this circumstance just be: pi*r? But I don't know how to interpret this into the equation. Especially the semicircle. Would it be classified as "x"?
 
Jomo said:
What have you tried?

Using x and y as you defined them (very good!) can you find an equation for the perimeter (ie framing) of the door? What is the perimeter of a semi-circle?

The cost equation is not that bad. c(x,y) = cost of the 3 sides plus the cost of the semi-circle = $2/ft*(the length of the 3 sides) + $5/ft(the length of the semi-circle).

Of course the attachment of the picture of the door helps, otherwise we do not know the definition of x and y. Of course you could have left out the attachment if you defined x and y.
Click to expand...
Would the constraint equation be: 2y+x+pi*r*x=20?
 
hello I have this same question. What exactly do you think the constraint and objective is, given the information he has provided..
 
Jomo said:
What is r? And why is r being multiplied by x?

Also, the circumference of a semi-circle never includes the dotted line.
Click to expand...
I thought r was the radius, so i thought that since the equation for the circumference of a full circle is: 2pi*r, the equation for a semicircle would be that multiplied by 1/2 (so I came up with pi*r). And since the diameter is the same length as the width (x), my thought process was since there are two sides, a bottom and a top that's a semicircle, I would just add those together to get the circumference (hence why I typed: 2y+x+pi*r*x).

The reason I thought that the circumference of a semicircle included that dotted line (the diameter) is because I was using the terms of circumference and perimeter interchangeably (which may have been my mistake) and thought that the equation for the perimeter of a semicircle was pi*r+d.
 
Jomo said:
What is r? And why is r being multiplied by x?

Also, the circumference of a semi-circle never includes the dotted line.
Click to expand...
I see what you mean by me adding r. It seems irrelevant in the terms of this question. So then would it be something like: 2y+x+(x/2)pi=20 or 2y+2x+(x/2)? Since all the radius is is 1/2 the diameter, and in this case, the diameter is the same length as x, to add the perimeter of the semicircle would be (x/2)*pi?

Also I think because of this, the objective function would be: cost= 2(2y)+2(2x)+5(pi*(x/2)) --> cost= 4y+4x+5(x/2)pi---> cost= x(4+(5/2)pi)+4y
 
CoreyyV said:
I see what you mean by me adding r. It seems irrelevant in the terms of this question. So then would it be something like: 2y+x+(x/2)pi=20 or 2y+2x+(x/2)? Since all the radius is is 1/2 the diameter, and in this case, the diameter is the same length as x, to add the perimeter of the semicircle would be (x/2)*pi?

Also I think because of this, the objective function would be: cost= 2(2y)+2(2x)+5(pi*(x/2)) --> cost= 4y+4x+5(x/2)pi---> cost= x(4+(5/2)pi)+4y
Click to expand...
Which one: 2y+x+(x/2)pi=20 or 2y+2x+(x/2)?
 
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