How to find h in terms of x?

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How to find h in terms of x?

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

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Answer 1 · 2018-04-23T18:56:42

$h = \frac{1000}{2 π x} - x$ $h = 1000/(2pix) - x$

Explanation:

for $31 a$ $31a$ , 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 $2 π r$ $2pir$ .
height is $h$ $h$ .

curved surface area = $2 π r h$ $2pirh$ .

the area of a circle is $π r^{2}$ $pir^2$ .

area of top and bottom circles: $2 π r^{2}$ $2pir^2$

the total surface area of the cylinder is $2 π r h + 2 π r^{2}$ $2pirh + 2pir^2$ , or $2 π r (h + r)$ $2pir(h+r)$ .

we are given that the total surface area of the cylinder is $1000 c m^{2}$ $1000cm^2$ .

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

then, $h + r = \frac{1000}{2 π r}$ $h + r = 1000/(2pir)$

$h = \frac{1000}{2 π r} - r$ $h = 1000/(2pir) - r$

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

$h = \frac{1000}{2 π x} - x$ $h = 1000/(2pix) - x$

Answer 2 · 2018-04-23T18:56:57

$h = \frac{500}{π x} + x$ $h = 500/{pi x} + x$

Explanation:

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

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

So the total surface area is

$1000 = 2 π x h + 2 π x^{2}$ $1000 = 2 pi x h + 2 pi x^2$

$π x h = 500 - π x^{2}$ $pi x h = 500 - pi x^2$

$h = \frac{500}{π x} - x$ $h = 500/{pi x} - x$

Answer 3 · 2018-04-23T18:57:59

The surface area of a cylinder is:

$A = 2 π x h + 2 π x^{2}$ $A = 2pixh+2pix^2$

We are given that $A = 1000 {cm}^{2}$ $A = 1000" cm"^2$

$1000 {cm}^{2} = 2 π x h + 2 π x^{2}$ $1000" cm"^2 = 2pixh+2pix^2$

Flip the equation:

$2 π x h + 2 π x^{2} = 1000 {cm}^{2}$ $2pixh+2pix^2=1000" cm"^2$

Multiply both sides by $\frac{1}{2 π x}$ $1/(2pix)$ :

$h + x = \frac{1000 {cm}^{2}}{2 π x}$ $h+x=(1000" cm"^2)/(2pix)$

Subtract x from both sides of the equation:

$h = \frac{1000 {cm}^{2}}{2 π x} - x \leftarrow$ $h=(1000" cm"^2)/(2pix) -xlarr$ this is h in terms of x

Answer 4 · 2018-04-23T19:10:53

$h = \frac{500}{π x} - x$ $h=500/(pix)-x$

Explanation:

The surface area is made up of the two circles and the rectangular body
The circles area is $π x^{2}$ $pix^2$ so double this $\Rightarrow$ $=>$ $2 π x^{2}$ $2pix^2$
The height of the rectangle is $h$ $h$ and the width of the rectangle is the circumference of the cylinder.
Circumference $= π D = 2 x π$ $=piD=2xpi$
The area of the rectangle $= 2 x π \times h$ $=2xpixxh$
We are given the surface area is $1000 c m^{2}$ $1000cm^2$
So $2 π x^{2} + 2 π x h = 1000$ $2pix^2+2pixh=1000$

$2 π x (x + h) = 1000$ $2pix(x+h)=1000$

$x + h = \frac{1000}{2 π x}$ $x+h=1000/(2pix)$

$x + h = \frac{500}{π x}$ $x+h=500/(pix)$

$h = \frac{500}{π x} - x$ $h=500/(pix)-x$

Answer 5 · 2018-04-23T19:23:31

$h$ $h$ = $\frac{1000 - 2 π x^{2}}{2 π x}$ $[1000-2pix^2]/[2pix]$ , i.e, $h = \frac{1000}{2 π x} - x$ $h=1000/[2pix]-x$ .

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= $π r^{2}$ $pir^2$ . Area of outside of cylinder= $2 π r h$ $2pirh$

So the total area of the cylinder is $2 π r^{2}$ $2pir^2$ + $2 π r h$ $2pirh$ . we are given that the radius $r$ $r$ = $x$ $x$ , so ,

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

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How to find h in terms of x?

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

5 Answers

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