A hemisphere of radius 7 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.
radius:
height:
it is like this u multiply ..... i don't know how to do this
To find the radius and height of the cylinder with maximum volume, we can express the problem in terms of a single variable and then use optimization techniques.
Let's denote the radius of the cylinder by r and the height of the cylinder by h.
First, we need to find a relationship between r and h. Since the top circular rim of the cylinder touches the hemisphere, we can form a right triangle with the radius of the hemisphere (7) as the hypotenuse and the height of the cylinder (h) as one leg. The other leg of the right triangle will be the radius of the cylinder (r).
Using the Pythagorean theorem, we have: r^2 + h^2 = 7^2
Next, we need to express the volume of the cylinder in terms of r and h. The volume of a cylinder is given by V = πr^2h.
Substituting r^2 = 7^2 - h^2 into the volume equation, we get: V = π(7^2 - h^2)h
Now, we can express V as a function of a single variable, h: V(h) = 49πh - πh^3
To find the maximum volume, we need to find the value of h that maximizes V(h). We can do this by taking the derivative of V(h) with respect to h and setting it equal to zero.
dV(h)/dh = 0
Simplifying the equation, we get: 49π - 3πh^2 = 0
Dividing both sides by π, we have: 49 - 3h^2 = 0
Rearranging the equation, we get: h^2 = 49/3
Taking the square root of both sides, we find: h = √(49/3) = 7/√3 = (7√3)/3
Now that we have the value of h, we can substitute it back into the equation r^2 + h^2 = 7^2 to find r^2.
r^2 = 7^2 - h^2 = 49 - (49/3) = 98/3
Taking the square root of r^2, we find: r = √(98/3) = √(98)/√(3) = 7√(2/3)
Therefore, the radius of the cylinder with maximum volume is 7√(2/3) and the height is (7√3)/3.