How to find h in terms of x?

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

5 Answers
Apr 23, 2018

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

Apr 23, 2018

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

Apr 23, 2018

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

Apr 23, 2018

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

Apr 23, 2018

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.