what should be the dimensions of the cylinder if you wanted it to have the least amout of surface area as possible?
the cylinders volume is 100ft cubed
To find the dimensions of a cylinder that has the least amount of surface area possible, we need to understand the relationship between the volume and surface area of a cylinder.
The surface area of a cylinder is given by the formula:
A = 2πr² + 2πrh,
where r is the radius of the circular base and h is the height of the cylinder.
The volume of a cylinder is given by the formula:
V = πr²h.
In this case, we are given that the volume of the cylinder is 100 ft³. So we can substitute this value into the equation:
100 = πr²h.
Since we want to minimize the surface area, we assume that the height h is fixed. Therefore, we can express the radius (r) in terms of h using the equation for the volume:
r² = (100 / (πh)).
Since we want to minimize the surface area, we need to minimize the radius. To do this, we set the height h to the maximum value, which is the height that makes the cylinder the most slender, namely the height will be equal to the double the radius (2r).
Substituting 2r for h, the equation becomes:
r² = (100 / (π * (2r))).
Simplifying the equation, we get:
r³ = (100 / (2π)).
To find the value of r, we can solve this equation by taking the cube root of both sides:
r = (100 / (2π))^(1/3).
Now that we have the value of r, we can find the value of h by substituting it into the equation for the volume:
100 = π * r² * h,
h = (100 / (π * r²)).
Finally, we can substitute the value of r into the equation for h to get the value of h.
Therefore, the dimensions of the cylinder that will result in the least surface area for a given volume of 100 ft³ can now be calculated using these equations.