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How much metal is required to make the water trough with a diameter of 2 and a height with 6.

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2 Answers

You are being asked to find the surface area of an open cylinder. To do this, you find the area of the different sections and add them together. You need to find the area around the side plus the area of the circle on bottom. For the water trough, there would not be a circle of metal on the top.

You know that the area of a circle is found using the formula pi times the square of the radius, πr2. Since the diameter is 2, the radius is one, and the area of the bottom of the trough is π. (That's supposed to be pi)

The sides of a cylinder may seem tricky at first, but think about taking the label off of a can. It's actually a rectangle. The formula for the area of a rectangle is length times width; the width of the can label is the height of the can; the length of the rectangle is the circumference of the can.

The formula for circumference is pi times the diameter of the circle, πd.

Circumference of the water trough is . So the amount of metal in the sides is 2π times 6, or 12π.

Add the bottom and the side together: 12π + π = 13π so the amount of metal needed is 13π square units.

 

Comments

The formula for the surface area of a cylinder is:

SA = 2pr2 + 2prh    

but the formula assumes there is a solid circle on each end of the cylinder. Since your trough would not have a circle on top, you could modify the formula by subtracting the area of one circle:

SA = (2pr2 + 2prh) - pr2

 = pr2 + 2prh

 

In the SA formula, it should be pi wherever it shows a p.

Shouldn't it be half the total surface area of a cylinder? I've always pictured a trough as a semi-cylinder, and it's stated in the original question that it's like half a cylinder.

The problem states that the trough has a height of 6. You could use 1/2 the SA of a cylinder with a height of 12. You're right, that's another way to approach this problem.

The most challenging part of this question was knowing what a trough was but since you mentioned half a cylinder, lets stick with that.  This question is basically asking for the surface area of the cylinder with one open end. 

The standard area of a cylinder A = the area of top circle + the area of the bottom circle + the area of the side.  The formula looks like this:

Cylinder Surface Area = 2*Pi*r2 + 2*Pi*r*h

Since this is a trough, there's an opening at the top and the formula therefore becomes:

Surface Area = Pi*r2 + 2*Pi*r*h

To help you solve this, here are some basics:

Pi is always equal to 3.14

Diameter (d) = 2*radius or d = 2*r

Given:  d = 2, h = 6

Calculate radius, r = d/2 = 1

I find it easier to deal with non-decimal numbers first so, enter just those values in the Surface Area equation shown below:

Surf. Area = (Pi)*12 + 2*(Pi)*1*6

                = Pi + 12*Pi

                = 13*Pi

Most porblems can be left as shown above but for practical purposes, if you need a real number, then substitute for Pi: 

Surf Area = 13*(3.14)

               = 40.82

 


 

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