Volume and Surface area of cylinders?

Lateral area is 2*Pi*r*H, while total area is 2*Pi*r*H + 2*Pi*r^2 = 2*Pi*r*(H + r). Their ratio is equal to H/(H + r) = 1/2. Solve for r and find that r = H. So total area is 4*Pi*r^2 = 616

Volume is Pi*r^2*H = Pi*r^3 = Pi*(616/4Pi)^3/2

First of the formula findind the surface area of a cylinder is 2 area of the circles + the circumference( height). then the formula of the volume is pi r^2(height) 1)SURFACE AREA 2(pi 7^2) + 2pi(7) times 10 98pi+ 140pi=238pi S= 238picm^2 VOLUME pi7^2 times 10=490pi V=490picm^3 2)SUFACE AREA 2(pi5^2) + 2pi(5) times 25 50pi+250pi=300pi S=300picm^2 Volume pi5^2 times 25=625pi V=625picm^3

Let the cylinder have a radius of r and a height of h.

The lateral surface is equivalent to the area of the rectangle that you get when you peel the label off a can of beans. ie., circumference of the cylinder times height.

Lateral area = 2 pi r h

Total area = lateral area plus area of top plus area of bottom = (2 pi r h) + (pi r^2) + (pi r^2)

So the equations you have is

1) from ratio:

(2 pi r h) * 2 = (2 pi r h) + (2 pi r^2)

2) (2pi r h) + (2 pi r^2) = 616

From 1) 2pi r^2 = 2 pi r h

hence r = h

plut into 2) and solve for r (or h)

Now calculate the volume as:

pi r^2 * h = pi r^3 and plug in what you calculated above.

Okay so think of a cylinder. And we are thinking of surface area. So what would you have if you look at each surface separately.

So you have two circles (one is the top and one is the bottom).

Then the middle part is just a rectangle rolled out right?

Okay so what is TOTAL surface are?

2 bases + rectangle piece

2 (pi r^2) + rectangle piece

what is the area of the rectangle piece? it's the height multiplied by the circumference of the circle.

so rectangle piece is = (2pi r)(h)

so total surface area is

2 (pi r^2) +(2pi r)(h)

they tell you that the total surface area is 616 so set this equal to 616

2 (pi r^2) +(2pi r)(h) = 616

now what is lateral surface are? It's just the surface area of the rectangular piece which is (2pi r)(h).

Now they tell you that this surface area is half of the total surface area (that's what ratio 1:2 means).

so set this equal to half of 616 which is 308

(2pi r)(h)=308

So now you have two equations:

2 (pi r^2) +(2pi r)(h) = 616

(2pi r)(h)=308

take the second equation and solve for h.

(2)(pi)(r)(h)=308

divide both sides by 2*pi*r

you get h= 154/(pi*r)

Now you have the value of h. plug this into the first equation to solve for r.

2 (pi r^2) +(2pi r)(h) = 616

2 (pi r^2) +(2pi r)(154/(pi*r)) = 616

2 (pi r^2) +308 = 616

2 (pi r^2) = 308

(pi r^2)=154

r^2= 154/pi

r= sqrt(154/pi)

Okay so now you have r and h. So we can write the formula for volume

V= base* height

V= pi*r^2 *h

V= pi[ sqrt(154/pi)]^2 (154/(pi*r))

V= pi*(154/pi)(154/(pi*r)

V= 23716/ pi r

V= 23716/ pi (sqrt(154/pi))