Total surface area of a cylinder of height H & base radius r is double that of another cylinder of height h & base radius r. Prove h=(H-r)/2 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to remember that the total surface of cylinder is the sum of two times area of base and the surface area of lateral part such that:

TSA = 2A base + `h*(2pi*r)`

You need to evaluate TSA of cylinder that has the height H and the radius r such that:

`TSA_(H,r) = 2*pi*r^2 + 2H*pi*r`

You need to factor out `2pi*r`  such that:

`TSA_(H,r) = 2pi*r(r + H)`

The problem provides the information that `TSA_(H,r)`  is double of `TSA_(h,r).`

You need to evaluate `TSA_(h,r)`  such that:

`TSA_(h,r) = 2pi*r(r + h)`

If you multiply by 2 both sides yields:

`2TSA_(h,r) = 4pi*r(r + h)`

Since `TSA_(H,r) = 2TSA_(h,r),`  then you may substitute `TSA_(H,r)`  for  `2TSA_(h,r)`  such that:

`2pi*r(r + H) = 4pi*r(r + h)`

You need to divide by `2pi*r`  both sides such that:

`(r + H) = 2(r + h) `

You need to open the brackets such that:

`r + H = 2r + 2h`

You need to isolate 2h to one side such that:

`r + H - 2r = 2h`

`H - r = 2h`

You need to divide by 2 such that:

`h = (H-r)/2`

The last line proves what problem demands, hence `h = (H-r)/2`  under given conditions.

Top Answer

jkj1362's profile pic

jkj1362 | College Teacher | (Level 1) eNoter

Posted on

lucky092569,

the total surface area of a cylinder of hieght H & radius r is

2pie * r^2 + 2pie * r * H

The total surface area of a cylinder of height h & radius r is

2pie * r^2 + 2pie * r * h

Since the total surface area of the first cylinder is double that of the second cylinder,

2pie * r^2 + 2pie * r * H = 2*(2pie * r^2 + 2pie * r * h)

so, 2pie * r^2 + 2pie * r * H = 4pie * r^2 + 4pie * r * h

If we simplify this equation,

-2pie * r^2 = 4pie * r * h - 2pie * r * H

so, -2pie * r^2 = (2pie * r)*(2h - H)

Since the common factor of this equation on both sides of the equation is (2pie * r), we can eliminate this factor from both sides.

-r = 2h - H

If we solve this equation to h,

2h = H - r

hence, h = (H - r)/2

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