A cylinder is inscribed in a right circular cone of height 4.5 and radius (at the base) equal to 5. What are the dimensions of such a cylinder which has maximum volume

The maximum volume of inscribed cylinder  when its radius is r  =  10 \3 units  and its height is  1.5 units.

Explanation:

The formula to find the volume of inscribed cylinder is: V  =  (πr^2).y

let 'y' is the height of the cylinder and 'r' be the radius of the cylinder.

Radius of the inscribed cylinder is r = 5-x.

and the value of y is y = (4.5)/5 * x, it is find by the formula of slope because we know the slope of the slanted side of the cone. We go up 4.5 units for every 5 units we move to the right.

Now,

V = π(5-x)^2 × (4.5/5) . x

V = (4.5/5 . π) × (25x + x^3 - 10x^2)

And now to find the maximum volume derivative of V with respect to 'x' will be equal to zero like: dV/dx = 0

dV/dx = (4.5/5 . π) × (25 + 3x^2 - 20x)

(4.5/5 . π) × (25 + 3x^2 - 20x) = 0

25 + 3x^2 - 20x = 0

Write the equation in sequence and then use the factorization to find x

3x^2 - 20x + 25 = 0

3x^2 - 15x - 5x + 25 = 0

3x(x-5) -5(x-5) = 0

(x-5) (3x-5) = 0

x = 5/3 and x = 5

So, we have two values of 'x'. But the value 5/3 gives us the point of maximum by graphing V with 'x'. Because when the value is between 0 to 5 then the left edge of the inscribed cylinder stays properly within the cone.

Hence, at x = 5/3,

y = (4.5)/5 * x =  (4.5)/5 * 5/3

y = 1.5 ≈ 2.

and r = 5 - x = 5 - 5/3 = 10/3.

The maximum volume of inscribed cylinder  when its radius is r  =  10 \3 units  and its height is  1.5 units.

well, you would divide 135 by 33 to get your answer. then that is how many seconds it would take. boardman should fly about 132 m or 4.090909090909091 seconds.

because growing process had stoped after 4-5 year