4.4. Problem 60
I know there has been some confusion with 4.4.60., so I hope I can be of some assistance.
The way I did 60 is as fallows:
I first though of an empty sheet of metal and specific instructions (that being the problem.
We know that the can shall have two genereal shapes cut, in order to form it. These shapes would be circles for the top and base, and a rectangle which will be folded to consruct a cylinder to house the can's content. We know that the can's top must specifically be three tims as thick as the sides and bottom. So the rectangle must be the smae thickness as the base and the top must be the same material except three times as thick.
Now we can start thinking in mathematical reasoning.
Out of the material we should cut:
A Side: Rectangle (length * width)
A Base: Circle (pie*radius squared)
A top: 3 Circles because the top should be three times as thick as the base.(3*pie*radius squared)
Besides this we also know that the can should hold a volume of 355.
The volume of a cylinder is of course (pie*r^2)*height
Now lets get down to business:
The equation for the amount of metal used should be.
(3*pie*r^2) + (1*pie*r^2)+((2*pie*r)*height) = Amount of metal to use
As stated above there should be 3 circle cut: hence 3*pie*r^2
Also there should be 1 circle cut for the base:1*pie*r^2
These expressions give us the top and base.
The amount of metal cut for the side is both important and in question because the height of the rectangle is unknown. We do know that the other side of the rectangle is the circumference of a circlebecause the rectangle should be folded arond two circles to form a cylinder. Hence the expression for the width of the rectangle is 2*pie*r (expression fo the circumference of a circle).
We can solve for the height utalizing information that was given.
We are provided that the volume must be 355 and we know that the volume of a cylinder is (pie*r^2)*height.
There for we solve the equation for height and plug the expression into are equation for the amount of material to be cut.
You should get 355/(pie*r^2)
Plugging height into the equation for material you should get:
(3*pie*r^2) + (1*pie*r^2) + ((2*pie*r)*(355/(pie*r^2)) = Amount that should be cut
Now all that is left is two find the amount cut to minimize amount cut and meet the volume an thickness of top requirements. Simply put, graph the equation and find its maximun in a realistic window.
The way I did 60 is as fallows:
I first though of an empty sheet of metal and specific instructions (that being the problem.
We know that the can shall have two genereal shapes cut, in order to form it. These shapes would be circles for the top and base, and a rectangle which will be folded to consruct a cylinder to house the can's content. We know that the can's top must specifically be three tims as thick as the sides and bottom. So the rectangle must be the smae thickness as the base and the top must be the same material except three times as thick.
Now we can start thinking in mathematical reasoning.
Out of the material we should cut:
A Side: Rectangle (length * width)
A Base: Circle (pie*radius squared)
A top: 3 Circles because the top should be three times as thick as the base.(3*pie*radius squared)
Besides this we also know that the can should hold a volume of 355.
The volume of a cylinder is of course (pie*r^2)*height
Now lets get down to business:
The equation for the amount of metal used should be.
(3*pie*r^2) + (1*pie*r^2)+((2*pie*r)*height) = Amount of metal to use
As stated above there should be 3 circle cut: hence 3*pie*r^2
Also there should be 1 circle cut for the base:1*pie*r^2
These expressions give us the top and base.
The amount of metal cut for the side is both important and in question because the height of the rectangle is unknown. We do know that the other side of the rectangle is the circumference of a circlebecause the rectangle should be folded arond two circles to form a cylinder. Hence the expression for the width of the rectangle is 2*pie*r (expression fo the circumference of a circle).
We can solve for the height utalizing information that was given.
We are provided that the volume must be 355 and we know that the volume of a cylinder is (pie*r^2)*height.
There for we solve the equation for height and plug the expression into are equation for the amount of material to be cut.
You should get 355/(pie*r^2)
Plugging height into the equation for material you should get:
(3*pie*r^2) + (1*pie*r^2) + ((2*pie*r)*(355/(pie*r^2)) = Amount that should be cut
Now all that is left is two find the amount cut to minimize amount cut and meet the volume an thickness of top requirements. Simply put, graph the equation and find its maximun in a realistic window.
posted by shaun at 9:54 PM
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