Radius is exactly 7*sqrt(2)/sqrt(3) cm, approximately 5.715 cm
Height is exactly 21/sqrt(3) cm, approx. 12.124 cm
Volume is exactly 686pi/sqrt(3) cm^3, approx. 1244.266 cm^3
Let's first calculate the height of the cylinder as a function of the radius of the cylinder. One of easier ways is to reduce the problem to one of inscribing a rectangle within a circle of radius 7. If you do that, you'll see that you can create a right triangle where the hypotenuse is the radius of the circle. One of the legs will be the desired radius of the cylinder, and the remaining leg will be half the height of the cylinder. Using the Pythagorean theorem, determining that height then becomes simple. So
h = 2*sqrt(7^2 - r^2)
h = 2*sqrt(49 - r^2)
Now the volume of a right circular cylinder is
V = pi*h*r^2
Substitute the formula for h, giving
V = pi*2*sqrt(49 - r^2)*r^2
V = 2*pi*r^2*sqrt(49 - r^2)
Looking at the equation, the maximum will happen with a r value somewhere between 0 and 7. We're not concerned with imaginary roots, or negative roots. Looking further at the equation, if we can maximize the value of r^2*sqrt(49-r^2), then we will maximize the volume of the cylinder (we can ignore the 2*pi part because it's a constant and eliminating that from consideration will make things easier to calculate). So
r^2*sqrt(49 - r^2)
Let's square it to get rid of the square root. After all, looking for the largest square will still give us the desired largest value. And I'm lazy.
F = r^4*(49 - r^2)
F = 49*r^4 - r^6
Since we're looking for a maximum, that will happen only where the slope of the function is 0. And the first derivative will give us the slope at each point of the function. So let's calculate the first derivative of that function. To do so, just multiply each coefficient by the exponent, then subtract 1 from the exponent.
F = 49*r^4 - r^6
F' = 196*r^3 - 6r^5
Now let's see where that function is 0.
F' = 196*r^3 - 6r^5
0 = 196*r^3 - 6r^5
6r^5 = 196r^3
6r^2 = 196
r^2 = 196/6 = 98/3
r^2 = 2*7*7/3
r = 7*sqrt(2)/sqrt(3)
r is approximately 5.715476066 cm
Now let's calculate the height.
h = 2*sqrt(49 - r^2)
h = 3*sqrt(49 - 98/3)
h = 3*sqrt(147/3 - 98/3)
h = 3*sqrt(49/3)
h = 3*7/sqrt(3)
h = 21/sqrt(3)
h is approximately 12.12435565 cm
The volume is
V=pi*21/sqrt(3)*(7*sqrt(2)/sqrt(3))^2
V=pi*21/sqrt(3)*(98/3)
V=pi*686/sqrt(3)
V=686pi/sqrt(3)
V is approximately 1244.266364 cm^3
