Marty wants to make a replacement part for a aeration machine that he is working on. He decides to take a rectangular sheet of metal with perimeter of 46cm and dimensions x cm nd y cm and roll it into a right circular cylinder as shown below. What values of x and y will give the largest volume?

To find the values of x and y that will give the largest volume, we need to first determine the relationship between the dimensions of the rectangular sheet and its volume as a right circular cylinder.

Let's start with the given information: the perimeter of the rectangular sheet is 46 cm. The formula for the perimeter of a rectangle is given by:

P = 2x + 2y,

where P is the perimeter, x and y are the dimensions of the rectangle.

In this case, we have 2x + 2y = 46.

Now, we can express the volume of the right circular cylinder formed by rolling the rectangular sheet as:

V = πr^2h,

where V is the volume, r is the radius, and h is the height of the cylinder.

To relate the dimensions of the rectangular sheet to the dimensions of the right circular cylinder, we need to consider the following:

The length of the rectangle, x, becomes the height of the cylinder, h.
The width of the rectangle, y, becomes the circumference of the cylinder, which is equal to 2πr.

Hence, we have x = h and y = 2πr.

Using the relationship between the dimensions, we can rewrite the perimeter equation in terms of r:

2(x + y) = 2x + 4πr = 46.

Simplifying this equation, we have:

x + 2πr = 23.

Now, we can express the volume in terms of r:

V = πr^2h.

Substituting x = h and y = 2πr, we have:

V = πr^2(x).

Since we know x + 2πr = 23, we can rewrite the volume equation in terms of r:

V = πr^2(23 - 2πr).

Now, we want to find the maximum value of V. To do this, we need to find the critical points of V, which occur when the derivative of V with respect to r is equal to zero.

To find the derivative, we differentiate V with respect to r:

dV/dr = π(2r(23 - 2πr) - r^2(2π)) = π(46r - 4πr^2 - 2πr^2).

Setting dV/dr equal to zero and simplifying, we have:

46r - 4πr^2 - 2πr^2 = 0.

Rearranging the equation and factoring out r, we get:

(46 - 2πr)(r - πr) = 0.

To find the critical points, we set each factor equal to zero and solve for r:

46 - 2πr = 0, => r = 46 / (2π),

r - πr = 0, => r(1 - π) = 0.

From the first equation, we can solve for r and calculate its value.

Using the second equation, r = 0 is not a valid solution because it would mean a cylinder with a radius of zero, which is not possible. Therefore, we ignore this solution.

Substituting the value of r into the equation x + 2πr = 23, we can solve for x:

x + 2π(46 / (2π)) = 23,

x + 46 = 23,

x = 23 - 46,

x = -23.

Since negative values do not make sense in this context, we disregard -23 as a valid solution for x.

Therefore, the values of x and y that will give the largest volume are x = 23 - 46 = -23 and y = 2π(46 / (2π)) = 46.