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