An open-topped cylindrical pot is to have volume 125 cm3. Determine the minimum possible amount of material used in making this pot? Neglect the thickness of the material as well as possible wastage. Give your answer accurate to 2 decimal places.
let the radius of the can be r cm
and its height be h cm
V= πr^2h
125 = π(r^2)h
h = 125/(πr^2)
surface area = bottom + rectangle
= πr^2 + 2πrh
= πr^2 + 2πr(125/(πr^2)
= πr^2 + 250/r
d(surface area)/dr = 2πr - 250/r^2
= 0 for a min of surface area
2πr = 250/(r^2)
r^3 = 125/π
r = 3.414
put that back into surface area = ...
( I got appr. 109.84)
Well, if we neglect the thickness of the material and possible wastage, I guess we don't really need any material at all! So the minimum possible amount of material used in making this pot would be 0 cm³. Voilà!
To determine the minimum amount of material used in making the pot, we need to consider the volume and shape of the pot.
The volume of a cylinder can be calculated using the formula:
V = πr^2h
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
In this case, we are given that the volume is 125 cm^3, so we can set up the equation as:
125 = πr^2h
To minimize the amount of material used, we need to minimize the surface area. For a cylinder, the surface area can be calculated using the formula:
A = 2πrh + πr^2
To minimize the surface area, we need to minimize both the height (h) and the radius (r). Since we want to find the minimum amount of material used, we need to minimize the surface area, which is directly related to the amount of material used.
By substituting the volume equation (125 = πr^2h) into the surface area equation (A = 2πrh + πr^2), we can eliminate the height:
A = 2πr * (125 / πr^2) + πr^2
Simplifying this equation, we get:
A = 250 / r + πr^2
To minimize the surface area (or amount of material used), we need to find the minimum value of the function A = 250 / r + πr^2.
To find the minimum, we can take the derivative of A with respect to r and set it equal to zero:
dA/dr = -250 / r^2 + 2πr = 0
Solving this equation for r, we get:
-250 / r^2 = -2πr
r^3 = 125 / π
Taking the cube root of both sides, we find:
r = (125 / π)^(1/3)
Substituting this value of r back into the volume equation, we can find the corresponding height:
125 = π * [(125 / π)^(1/3)]^2 * h
Simplifying this equation, we get:
h = 125 / [(125 / π)^(2/3)]
Therefore, the minimum possible amount of material used in making the pot is obtained by calculating the surface area using the minimum values of r and h:
A = 2πr * h + πr^2
Substituting the values, we get:
A = 2π * (125 / π)^(1/3) * 125 / [(125 / π)^(1/3)] + π * [(125 / π)^(1/3)]^2
Simplifying this equation will give you the minimum possible amount of material used.
Please note that since the question asks for the answer accurate to 2 decimal places, you will need to substitute the value of π with an approximate value, such as 3.14, for the calculations.
To determine the minimum amount of material used in making the pot, we need to find the dimensions that will give the minimum surface area.
Let's assume the radius of the base of the cylindrical pot is 'r' cm, and the height of the pot is 'h' cm.
The volume of a cylinder is given by the formula:
V = π * r^2 * h
In this case, we know the volume V is given as 125 cm^3. So we have the equation:
125 = π * r^2 * h ---(1)
The surface area of a cylinder is given by the formula:
A = 2 * π * r * h + π * r^2
We need to minimize the surface area, so we differentiate the surface area function with respect to either 'r' or 'h' to find the critical points.
Differentiating the surface area function with respect to 'r', we get:
dA/dr = 2 * π * h + 2 * π * r = 2π(h + r)
Setting dA/dr = 0, we get:
2π(h + r) = 0
Since π is a positive constant, this implies h + r = 0, which is not possible for a positive height and radius. So, we cannot find a critical point by differentiating with respect to 'r' alone.
Therefore, we differentiate the surface area function with respect to 'h', giving us:
dA/dh = 2 * π * r
Setting dA/dh = 0, we get:
2πr = 0
Since π is a positive constant, this implies r = 0, which is not possible for a positive radius. So, we cannot find a critical point by differentiating with respect to 'h' alone.
Thus, we conclude that the minimum amount of material used in making the pot occurs at either the maximum or minimum points of the surface area function. In this case, since the surface area needs to be minimized, we look for a minimum point.
Since we cannot find a critical point by differentiating either with respect to 'r' or 'h' alone, we need to explore the boundary conditions. The given information does not provide any restrictions on the values of 'r' and 'h', so we need to consider the extreme cases.
Case 1: Let's consider the theoretical limit where the radius becomes infinitely small (r → 0). In this case, the height must become infinitely large (h → ∞), and the volume will be 125 cm^3.
Using equation (1), we can solve for 'h' when 'r' approaches 0:
125 = π * (0^2) * h
125 = 0 * h
Since anything multiplied by 0 is 0, we cannot determine the value of 'h' when 'r' becomes 0.
Case 2: Let's consider the theoretical limit where the height becomes infinitely small (h → 0). In this case, the radius must become infinitely large (r → ∞), and the volume will be 125 cm^3.
Using equation (1), we can solve for 'r' when 'h' approaches 0:
125 = π * r^2 * 0
Since anything multiplied by 0 is 0, we cannot determine the value of 'r' when 'h' becomes 0.
Since both extreme cases are not feasible, we need to find the values of 'r' and 'h' that optimize the surface area for the given volume.
Let's solve equation (1) for 'r' in terms of 'h':
r^2 = 125 / (π * h)
Taking the square root of both sides:
r = √(125 / (π * h))
Now, substitute this value of 'r' into the surface area formula:
A = 2 * π * (√(125 / (π * h))) * h + π * (√(125 / (π * h)))^2
Simplifying the expression:
A = 2 * π * √(125 * h / π) * h + π * (125 * h / π)
A = 2 * √(125 * π * h) * h + 125 * h
Now, we need to find the values of 'h' that minimize the surface area A.
To determine the minimum value of 'h', we can use calculus. We differentiate the surface area function with respect to 'h' and set it equal to zero to find the critical points.
dA/dh = 2 * √(125 * π * h) + 250 * √(π * h) = 0
After simplifying the equation, we have:
√(125 * π * h) + 125 * √(π * h) = 0
By solving this equation for 'h', we can find its possible minimum value.
However, since this involves solving a transcendental equation, there is no simple algebraic solution. We would have to use numerical methods like iteration or approximation methods like Newton's method to find the critical point.
Once we find the value of 'h' that minimizes the surface area, we can substitute it back into equation (1) to find the corresponding value of 'r'. The minimum amount of material used in making the pot can then be calculated by finding the surface area using the optimized values of 'r' and 'h'.