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Question 1138953: exponential functions
The formula for the volume of a cylinder of radius r is V = π r^2h
a) Solve this equation for r. Write two versions, one in radical form and one in exponential form.
b) Determine the radius of a cylinder with a volume of 4356π cm^3 and height of 9 cm.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula for the volume of a cylinder is v = pi * r^2 * h
solve this formula for r.
divide both sides of the foermula by (pi * h) to get:
v / (pi * h) = r^2
flip sides in the equation to get:
r^2 = v / (pi * h)
take the square root of both sides of this equaiton to get:
r = plus or minus sqrt(v / (pi * h)).
sincer can only be positive, just take the positive square rootto get:
r = sqrt(v / (pi * h)).
that's the radical form of the solution.
since square root is equal to exponent of 1/2, the exponential form of this equation is:
r = (v / (pi * h)) ^ (1/2).
to confirm this is correct, do the following.
let r be equal to any number.
i used 5.
let h be equal to any number.
i used 10.
v = pi * r^2 * h becomes v = pi * 5^2 * 10 which becomes v = pi * 25 * 10 which becomes v = 250 * pi.
the formula for r is r = sqrt(v / (pi * h)) in radical form.
therefore, when v = 250 * pi and h = 10, the formula for r becomes r = sqrt((250 * pi) / (10 * pi).
simplify this to get r = sqrt(25) because 250/ 10 = 25 and pi/pi is equal to 1.
solve for r to get r = sqrt(25) = 5.
you worked your way up with radius equal to 5 and then you were able to work your way back from the volume to get radius = 5, so the formula for the reverse operation was good.
for part b, you want to find the radius of a cylinder with a volume of 4356 * pi and a height of 9.
the formula for r we'll use this time is the exponential form of:
r = (v / (pi * h)) ^ (1/2)
this formula becomes r = ((4356 * pi) / (pi * 9).
this can be shown as r = ((4356 / 9) * (pi / pi)).
simplify to get r = (484) ^ (1/2).
solve for r to get r = 22.
working your way back up, you get v = pi * r^2 * h becomes v = pi * 22^2 * 9 which becomes v = pi * 484 * 9 which becomes v = pi * 4356 * pi which confirms the solution is good.
in exponent form, roots are shown in the denominator of the exponent.
in other words, x ^ (a/b) is equal to the (b root of x) ^ a or is equal to the b root of (x ^ a).
consider the cube root of 27 ^ 6.
in exponential form, this would be 27 ^ (6/3).
that's equivalent to (the cube root of 27) raided to the 6th power, or equivalent to the cube root of (27 raised to the 6th power).
if (the cube root of 27) raised to the 6th power, you get 3 raised to the 6th power = 729.
if the cube root of (27 raised to the 6th power), you get cube root of 387420489 = 729.
all of this is additional information you may or may not be interested in.
the solution to your problems is:
a) Solve this equation for r. Write two versions, one in radical form and one in exponential form.
r = sqrt(v / (pi * h))
r = (v / (pi * h)) ^ (1/2)
b) Determine the radius of a cylinder with a volume of 4356π cm^3 and height of 9 cm.
the radius of the cylinder is equal to 22 cm.
here's a reference on radicals.
https://mathbitsnotebook.com/Algebra1/Exponents/EXFractional.html
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