Geometry Help!?!

Question

A cylindrical can with an open top has a diameter thats is 4 times its height.

a.) Write the surface area of the can in terms of its height, h.
b.) Write the volume of the can in terms of its height, h.

Answer 1

Hi kristin,

 

Let's Imagine the cylinderical can which has diameter D, and height h, and radius r. Diameter is 4 times its height, So we have D=4*h. Diameter is always twice the radius, So mathematically it would be D=2*r. The surface area of the cylindrical can as the word describes is the area of its surface. So it's the perimeter of its base times its height plus the area of its bottom base(since it is open top cylinder) which is the area of one circle. The area of the circle is pi*[(radius)^2]. and the perimeter of the circle is 2*pi*r. Let me know with this description what should the surface area be eventually.

b)The volume is basically how much it can contain. So for the cylinder, it would be the area of its base which is circle times its height. The area of a circle as said previously is pi*[(radius) ^2]. With this description could you tell me what the volume would be.

 

Thanks,

Ehsan

Answer 2

1.  The surface area of a cylinder is the area of the circular top,

     plus the area of the circular bottom, plus the surface area

    of the side... 

    In this case we can disregard the surface area of the can top

    because the cylinder has an open top.

 

The surface area of the bottom is (pi)r²

 

The surface area of the side is the circumference of the cylinder multiplied by the height

SA_side = 2(pi)rh

 

Thus the surface area of the cylinder is  (pi)r²+2(pi)rh

 

Note that we are told the diameter is 4 times the height or d=4h

Since d = 2r we have 2r=4h

So r = 2h

 

Substituting that back into our formula we have:  SA = (pi)(2h)²+2(pi)(2h)h

                                                                             = 4(pi)h² + 4(pi)h²

                                                                          SA = 8(pi)h²

 

2.  The volume of a cylinder is given by V=(pi)r²h

     Again substitute r=2h into the formula, giving:

 

     V = pi(2h)²h

     V=4(pi)h³

 

Note that the open top does not effect the volume since volume is a measure

of the quantity of material the cylinder can contain.  The lack of a top does not

change that.