The number of square meters in the total surface area of a right circular cylinder, including the top and bottom, is equal to the number of cubic meters in its volume. If the radius of the cylinder is five times its height, what is its volume?

1 Answer
Feb 28, 2018

See a solution process below:

Explanation:

We know r = 5hr=5h for the cylinder

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The formula for the volume of a cylinder is:

V = pir^2hV=πr2h

We can substitute 5h5h for rr giving:

V = pi(5h)^2hV=π(5h)2h

V = pi25h^2hV=π25h2h

V = pi25h^3V=π25h3

V = 25pih^3V=25πh3

The formula for the surface area of a cylinder is:

A = 2pirh + 2pir^2A=2πrh+2πr2

Again, we can substitute 5h5h for rr giving:

A = (2pi5h xx h) + 2pi(5h)^2A=(2π5h×h)+2π(5h)2

A = 10pih^2 + 2pi25h^2A=10πh2+2π25h2

A = 10pih^2 + 50pih^2A=10πh2+50πh2

A = (10 + 50)pih^2A=(10+50)πh2

A = 60pih^2A=60πh2

Because the Area is equal to the volume we can equate the two and solve for hh:

60pih^2 = 25pih^360πh2=25πh3

(60pih^2)/(25pih^2) = (25pih^3)/(25pih^2)60πh225πh2=25πh325πh2

(60color(red)(cancel(color(black)(pih^2))))/(25color(red)(cancel(color(black)(pih^2)))) = (color(red)(cancel(color(black)(25pi)))h^3)/(color(red)(cancel(color(black)(25pi)))h^2)

60/25 = h^3/h^2

(5 xx 12)/(5 xx 5) = (h^2 xx h)/h^2

(color(red)(cancel(color(black)(5))) xx 12)/(color(red)(cancel(color(black)(5))) xx 5) = (color(red)(cancel(color(black)(h^2))) xx h)/color(red)(cancel(color(black)(h^2)))

12/5 = h

h = 12/5

The we can calculate r as r = 5 xx 12/5 = 12

We can substitute these back into the formula for the volume of a cylinder and calculate V:

V = pir^2h becomes:

V = pi xx 12^2 xx 12/5

V = pi xx 144 xx 12/5

V = pi xx 1728/5

V = pi xx 345.6

V = 345.6pi

Approximating pi with 3.14 gives:

V = 345.6pi ~= 345.6 xx 3.14 ~= 1085.184