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.