Explanation:
what is the problem you don't understand ?
to get the surface area of a cylinder, let us just imagine one in our minds :
its body is circular round, and it has 2 flat end areas.
these flat end areas are circles, and they have both the same size.
we know the area of a circle, right ?
Ac = pi×r²
with r being the radius of the circle.
also to be considered : the diameter of a circle is 2 times the radius (or the radius is half the diameter).
now to the "body" of the cylinder.
how can we get the area of the outside mantle is that body ?
well, imagine you have a very sharp knife, and you cut that cylinder open. we make a straight cut top-down from the rim of the upper circle to the lower circle.
and then we cut along the rims of both circles.
we end up with 2 circles (the top and bottom circles we discussed earlier) and a rectangle (if we flatten the now loose mantle on a surface).
that rectangle is on one side exactly the height of the cylinder.
and in the other side (the one that was connected to a circle) it is as long as the circumference of the circle (because it was originally going exactly 1 time around the circle).
and the circumference of a circle is
Cc = 2×pi×r
so, the full surface area of a cylinder is
2×circle area + rectangle
2×pi×r² + h×2×pi×r
now that we understand the background, let's deal with the questions :
1.
2×pi×6² + 10×2×pi×6 = 72pi + 120pi = 192pi =
= 603.1857895... cm²
2.
2×pi×3.5² + 25×2×pi×3.5 = 24.5pi + 175pi = 199.5pi =
= 626.7477344... in²
3.
2×pi×(9/2)² + 8.5×2×pi×9/2 = 40.5pi + 76.5pi = 117pi =
= 367.5663405... in²
4.
2×pi×(15/2)² + 10×2×pi×15/2 = 112.5pi + 150pi = 262.5pi =
= 824.6680716... cm²
5.
pi×r² = 25
r² = 25/pi
r = sqrt(25/pi) = 5×sqrt(1/pi)
2×pi×(25/pi) + 8×2×pi×(5×sqrt(1/pi)) = 50 + 80pi×sqrt(1/pi) =
= 191.7963081... square units (not specified)
6.
oh, an additional challenge.
the volume of a cylinder is
base area × height = pi×r² × h
pi×r²×25 = 1000
pi×r² = 40
r² = 40/pi
r = sqrt(40/pi) = 2×sqrt(10/pi)
2×pi×(40/pi) + 25×2×pi×(2×sqrt(10/pi)) = 80 + 100pi×sqrt(10/pi) =
= 640.4991216... cm²