Question

How to find h in terms of x?

Can someone please explain to me how to do question 31 a? Thank!

Answer 1

`h = 1000/(2pix) - x`

Explanation:

for `31a`, you need the formula for the total surface area of a cylinder.

the total surface area of a cylinder is the same as the total of both circular surfaces (top and bottom) and the curved surface area.

the curved surface area can be considered as a rectangle (if it were to be rolled out). the length of this rectangle would be the cylinder's height, and its width would be the circumference of a circle on the top or bottom.

the circumference of a circle is `2pir`.
height is `h`.

curved surface area = `2pirh`.

the area of a circle is `pir^2`.

area of top and bottom circles: `2pir^2`

the total surface area of the cylinder is `2pirh + 2pir^2`, or `2pir(h+r)`.

we are given that the total surface area of the cylinder is `1000cm^2`.

this means that `2pir(h+r) = 1000`.

then, `h + r = 1000/(2pir)`

`h = 1000/(2pir) - r`

in this question, the radius is actually denoted as `x`, so `h` in terms of `x` would be

`h = 1000/(2pix) - x`

Answer 2

`h = 500/{pi x} + x`

Explanation:

The radius of the base is `x`. The circumference of the base must be `2pi x`.

So the surface area of the curved face is `2pi x h`. From the description it sounds like we're to include the surface are of the end caps as well, there are two, each area `pi x^2`.

So the total surface area is

`1000 = 2 pi x h + 2 pi x^2`

`pi x h = 500 - pi x^2`

`h = 500/{pi x} - x`

Answer 3

The surface area of a cylinder is:

`A = 2pixh+2pix^2`

We are given that `A = 1000" cm"^2`

`1000" cm"^2 = 2pixh+2pix^2`

Flip the equation:

`2pixh+2pix^2=1000" cm"^2`

Multiply both sides by `1/(2pix)`:

`h+x=(1000" cm"^2)/(2pix)`

Subtract x from both sides of the equation:

`h=(1000" cm"^2)/(2pix) -xlarr` this is h in terms of x

Answer 4

`h=500/(pix)-x`

Explanation:

The surface area is made up of the two circles and the rectangular body
The circles area is `pix^2` so double this `=>` `2pix^2`
The height of the rectangle is `h` and the width of the rectangle is the circumference of the cylinder.
Circumference`=piD=2xpi`
The area of the rectangle `=2xpixxh`
We are given the surface area is `1000cm^2`
So `2pix^2+2pixh=1000`

`2pix(x+h)=1000`

`x+h=1000/(2pix)`

`x+h=500/(pix)`

`h=500/(pix)-x`

Answer 5

`h`= `[1000-2pix^2]/[2pix]`, i.e, `h=1000/[2pix]-x`.

Explanation:

The total surface area of the cylinder will be the area of its two circular ends plus the area of the outside of the cylinder.

Area of one end=`pir^2`. Area of outside of cylinder=`2pirh`

So the total area of the cylinder is `2pir^2` +`2pirh`. we are given that the radius `r`=`x`, so ,

Total area of the cylinder is `2pix^2+2pixh`=`1000` and making `h` the subject of this equation gives the above answer. Hope this was helpful.