Question 251476: If the diameter of a cylindrical can is increased by 30 percent, by approximating what percentage should the height be increased to triple the volume of the can ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if the diameter of the can is increased by 30%, then the radius of the can is also increased by 30%.
consider.
d2 = d1 * 1.3 = 1.3*d1
d1 = r1*2
d2 = 1.3*d1 = 1.3*2*r1
r2 = d2/2 = 1.3*2*r1/2 = 1.3*r1
volume of the can is equal to pi*r^2*h
increase the radius by 30% and you get volume of the can is equal to pi*(1.3*r)^2*h.
in order for the volume of the enlarged radius can to equal 3 times the volume of the original can, the following equation needs to be satisfied.
let x = the amount the height of the can has to be increased.
then:
let v[1] = the volume of the original can.
let v[2] = the volume of the can with a 30% increase in the length of the radius and an unknown increase in the height of the can.
let x = the unknown increase in the height of the can.
v[1] = pi*r^2*h
v[2] = pi*(1.3*r)^2*x*h = 3 * v[1] = 3 * pi*r^2*h
this results in:
pi * (1.3*r)^2 * x * h = 3 * pi * r^2 * h
you want to solve for x.
divide both sides of this equation by h to get:
pi * (1.3*r)^2 * x = 3 * pi * r^2
divide both sides of this equation by (pi * (1.3*r)^2 to get:
x = 3 * pi * r^2 / (pi * (1.3*r)^2)
(1.3*r)^2 becomes (1.3)^2 * r^2. substitute in equation to get:
x = 3 * pi * r^2 / (pi * (1.3)^2 * r^2)
pi and r^2 cancel out from numerator and denominator to get:
x = 3 / (1.3)^2 = 1.775147929
to make the equations equivalent, the height needs to be multiplied by 1.775147929 which is the same as (3/(1.3)^2).
substitute in the original equation to confirm that this is true.
the original equation is:
pi*(1.3*r)^2*x*h = 3 * pi*r^2*h
replace x with (3/(1.3)^2) to get:
pi*(1.3*r)^2*(3/(1.3)^2)*h = 3 * pi*r^2*h
simplify by removing parentheses to get:
pi * (1.3)^2 * r^2 * 3 * h / (1.3)^2 = 3 * pi * r^2 * h
(1.3)^2 in numerator and denominator cancel out to get:
pi * r^2 * 3 * h = 3 * pi * r^2 * h
except for the order in which the terms are presented, the expressions on each side of the equal side are identical confirming that the value of x = (3/(1.3)^2) is correct.
your answer is:
the height needs to be multiplied by (3/(1.3)^2) = 1.775147929 in order for the volume of the resulting can to be tripled, assuming that the diameter has been increased by 30%.
example:
let r = 15
let h = 22
volume of the original can is pi*r^2*h = pi*(15)^2*22 = 1550.88364
multiply radius by 1.3 to get r = 19.5
multiply height by (3/(1.3)^2) to get h = 39.05325444
volume of the enlarged can is pi*r^2*h = pi*(19.5)^2*39.05325444 = 46652.65091.
46652.65091 / 1550.88364 = 3 confirming the value of x is good.
the answer to the question is that the height should be increased by approximately 77.51%.
the original height is x.
the new height is 1.775147929 times x
new height minus old height times 100% equals the percent increase.
1.775147929 * x - x = .775147929 * x * 100% equals 77.5147929% * x = ~ 77.51% * x.
~ means approximately after rounding to the nearest hundredth of a percent.
the height needs to be increased by approximately 77.51%.
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