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Question 499276: A cylindrical tin of height h cm and radius r cm, has a surface area, including its top and bottom, A cm^2.
i) Write down an expression a A in terms of r, h and pi.
I got this to be A = 2pi r h + 2pi r^2
ii) A tin of height 6cm has surface area 54picm^2. What is the radius of the tin?
I'm not sure how to work this part out.
iii) Another tin has the same diameter as height. Its surface area is 150picm^2. What is its radius?
I'd really appreciate your help.
h = 6
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! question 1:
height = h
radius = r
surface area = the area of the sides of the can plus the area of top and bottom of the can.
area of the top and bottom of the can is equal to 2*(pi*r^2)
area of the side of the can is equal to h*(2*pi*r)
S = surface area of the can.
S = 2*pi*r^2 + 2*pi*r*h
question 2:
you are given that S = 54*pi square centimeters.
you are given that h = 6 centimeters.
you want to find the radius.
the formula used is the same formula you just derived in question number 1.
that formula is:
S = 2*pi*r^2 + 2*pi*r*h
you know h, so substitute for h in the equation to get:
S = 2*pi*r^2 + 2*pi*r*6
simplify to get:
S = 2*pi*r^2 + 12*pi*r
you know the value of S, so substitute for S in the equation to get:
54*pi = 2*pi*r^2 + 12*pi*r
divide both sides of the equation by pi to get:
54 = 2*r^2 + 12*r
subtract 54 from both sides of the equation to get:
2r^2 + 12r - 54 = 0
divide both sides of the equation by 2 to get:
r^2 + 6r - 27 = 0
this is a quadratic equation in standard form.
factor this equation to get:
(r + 9) * (r - 3) = 0
this equation is true if (r+9) = 0 or if (r-3) = 0 or if both are 0.
solve for r+9 = 0 to get r = -9
solve for r-3 = 0 to get r = 3
r can't be negative so your answer has to be r = 3.
let's see if that's true.
your original equation is:
S = 54*pi
the formula is:
S = 2*pi*r^2 + 2*pi*r*h
you now know that:
h = 6
r = 3
the formula becomes:
S = 2*pi*3^2 + 2*pi*3*6
simplify to get:
S = 2*pi*9 + 2*pi*18
simplify further to get:
S = 18*pi + 36*pi
simplify further to get:
S = 54*pi
the value of 3 for r is good.
your answer is:
r = 3 cm
question 3:
you are given that S = 150*pi square centimeters
same formula is used again.
formula is:
S = 2*pi*r^2 + 2*pi*r*h
h = height
r = radius
S = surface area
d = diameter
you are given that the diameter is equal to the height.
you get:
d = h
diameter is equal to twice the radius.
this leads to:
d = 2r
since h = d, this leads to:
h = 2r
we can substitute for h in the equation by replacing h with 2r to get:
S = 2*pi*r^2 + 2*pi*r*h becomes:
S = 2*pi*r^2 + 2*pi*r*2r
simplify this to get:
S = 2*pi*r^2 + 4*pi*r^2
these are now like terms so we can combine them to get:
S = 6*pi*r^2
we are given that S = 150 square cm.
we replace S with 150 to get:
150 = 6*pi*r^2
divide both sides of the equation by 6 to get:
25 = pi*r^2
divide both sides of the equation by pi to get:
25/pi = r^2
take the square root of both sides of the equation to get:
r = +/- sqrt(25/pi)
we can simplify this a little further to get:
r = +/- 5/sqrt(pi)
since r can't be negative, this then becomes:
r = 5/sqrt(pi)
to confirm this is a good number we start over with the additional information that r = 5/sqrt(pi)
our formula is, once again:
S = 2*pi*r^2 + 2*pi*r*h
we replace h with 2r to get:
S = 2*pi*r^2 + 2*pi*r*2r
we combine like terms to get:
S = 2*pi*r^2 + 4*pi*r^2
we know that r^2 = 25/pi, so we can replace r^2 with that to get:
S = 2*pi*25/pi + 4*pi*25/pi
the pies in the numerator and denominator cancel out and we have:
S = 2*25 + 4*25 which becomes S = 50 + 100 which becomes S = 150
That's the number we are looking for, so the value of r = 5/sqrt(pi) is good.
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