A steel factory wishes to make a cylindrical container, of thin metal, to hold 100cm³, using the least possible area of metal. If the outside surface is S cm² and the radius is r cm, show that S = 2πr² + 200r-1 and hence find the required radius and height for the container. (Leave π in your answer.)
let the height be 100 cm^3
πr^2 h = 100
h = 100/(πr^2)
S = 2 circles + the sleeve
= 2πr^2 + 2πrh
= 2πr^2 + 2πr(100/(πr^2))
= 2πr^2 + 200/r
you typed it wrong, it should have been
2πr^2 + 200 r^-1
dS/dr = 4πr - 200/r^2
= 0 for a min of S
4πr = 200/r^2
r^3 = 200/(4π) = 25/π
r = (25/π)^(1/3) , the cuberoot of (25/π)
h = 100/(πr^2)
= I will let you sub in the value of r
To find the surface area of the cylindrical container, we need to consider both the curved surface area and the area of the two circular ends.
Let's start by finding the curved surface area. The area of a cylinder's curved surface can be given by the formula:
Curved Surface Area = 2πrh
Here, h represents the height of the cylinder.
Next, we need to calculate the area of the two circular ends. The area of a circle can be given by the formula:
Circle Area = πr²
Since there are two circular ends, the total area of the two circular ends is:
Total Circular Ends Area = 2πr²
Now, to find the total surface area of the cylinder, we need to add the curved surface area and the total circular ends' area:
Total Surface Area = Curved Surface Area + Total Circular Ends Area
= 2πrh + 2πr²
= 2π(rh + r²)
We know that the volume of the cylinder is given as 100 cm³. The volume of a cylinder can be calculated using the formula:
Volume = πr²h
We can rearrange this formula to solve for h:
h = (Volume) / (πr²)
= 100 / (πr²)
= 100r^(-2) / π
Substituting this value of h in the equation for total surface area, we get:
Total Surface Area = 2π(rh + r²)
= 2π(r * (100r^(-2) / π) + r²)
= 2π(100r^(-1) + r²)
= 2πr² + 200r^(-1)
Therefore, we have verified that S = 2πr² + 200r^(-1).
To find the required radius and height for the container, we need to minimize the surface area. In this case, we can differentiate S with respect to r and equate it to zero to find the minimum value.
dS/dr = 4πr - 200r^(-2)
Setting dS/dr = 0, we have:
4πr - 200r^(-2) = 0
Rearranging the equation, we get:
4πr = 200r^(-2)
Dividing both sides by 4π, we get:
r = (200/4π)^(1/3)
= 5^(1/3) (leave π in the answer)
Now, we can substitute this value of r back into the equation for h:
h = 100r^(-2) / π
= (100/(π * (5^(1/3))^2))
= 4π / (5π)
= 4/5 (leave π in the answer)
Therefore, the required radius for the container is 5^(1/3) (leave π in the answer), and the height is 4/5 (leave π in the answer).
To find the surface area of the cylindrical container, we need to consider the area of the curved surface and the area of the two circular bases.
1. Let's start by finding the area of the curved surface:
The curved surface of a cylinder can be visualized as rolling out a rectangle. The length of this rectangle is equal to the height of the cylinder (h) and the width is equal to the circumference of the circular base.
2. The circumference of the circular base can be calculated using the formula 2πr, where r is the radius of the base.
3. The area of the curved surface will be equal to the product of the height (h) and the circumference (2πr). So, the area of the curved surface is 2πr * h.
4. Now, let's find the areas of the two circular bases:
The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius.
5. Since we have two circular bases, the total area of the two bases will be 2πr².
6. Finally, to find the total surface area (S), we can sum up the areas of the curved surface and the two bases:
S = 2πr * h + 2πr²
7. We are given that the container should hold 100 cm³ of volume. The volume of a cylinder is given by the formula V = πr²h. We can rearrange this formula to find h:
h = V / (πr²)
h = 100 / (πr²)
8. Substituting this value of h in the equation for S, we get the following expression in terms of r:
S = 2πr * (100 / (πr²)) + 2πr²
= 200 / r + 2πr²
Hence, we have shown that S = 2πr² + 200/r.
To find the required radius and height, we need to minimize the surface area. We can do this by differentiating the expression for S with respect to r and setting it equal to zero. However, since we just need the required values, we can solve this equation manually.
1. We need to make the area of the container as small as possible. To do this, we can minimize the expression S.
2. Differentiate S with respect to r:
dS/dr = d(2πr² + 200/r)/dr
= 4πr - 200/r²
3. To find the minimum, set dS/dr equal to zero:
4πr - 200/r² = 0
4. Multiply through by r² to get rid of the denominator:
4πr³ - 200 = 0
5. Rearrange and solve for r:
4πr³ = 200
r³ = 50 / π
r = (50 / π)^(1/3)
Now that we have the radius, we can substitute it back into the equation for the height:
h = 100 / (πr²)
= 100 / [π * ((50 / π)^(1/3))^2]
= 100 / [π * (50 / π)^(2/3)]
Hence, the required radius is (50 / π)^(1/3) and the required height is 100 / [π * (50 / π)^(2/3)].