A manufacturer wishes to produce the most economical packaging for detergent.
The containers need to hold 25litres and must be cylindrical in shape. What
dimensions will ensure that the least amount of plastic is used to create the
containers. (Note: the volume and surface area of cylinders are V =πr2h and
S = 2πr2 + 2πrh respectively)
v=pi r^2 h = 25, so
h = 25/(pi r^2)
a = 2πr^2 + 2πrh
= 2πr^2 + 2πr*25/(πr^2)
= 2πr^2 + 50/r
da/dr = 4πr - 50/r^2
= (4πr^3 - 50)/r^2
since r≠0, da/dr=0 when 4πr^3 = 50
r = ∛(25 / 2π)
To determine the dimensions that will ensure the least amount of plastic is used to create the containers, we need to minimize the surface area of the cylindrical container while maintaining a volume of 25 liters.
Let's denote the radius of the base of the cylinder as 'r' and the height of the cylinder as 'h'.
We want to minimize the surface area, which is given by the formula S = 2πr^2 + 2πrh.
To find the dimensions that minimize this surface area, we can use calculus. The surface area function, S, is a function of two variables: r and h. We can express the surface area function as S(r, h) = 2πr^2 + 2πrh.
To apply calculus, we first need to express the function in terms of one variable. We know that the volume of the cylinder should be 25 liters, which is equivalent to 25000 cubic centimeters (since 1 liter = 1000 cubic centimeters).
The volume of a cylinder is given by the formula V = πr^2h. We can use this formula to express the height 'h' in terms of the radius 'r' and the given volume V: h = V / (πr^2). Substituting the volume, we have h = 25000 / (πr^2).
Now we can substitute this expression for 'h' into the surface area function to get a single-variable function S(r) = 2πr^2 + 2πr(25000 / (πr^2)) = 2πr^2 + 50000 / r.
To find the minimum of this function, we need to find where its derivative with respect to 'r' is equal to zero.
Taking the derivative of S(r) with respect to 'r', we have dS/dr = 4πr - 50000 / r^2. Setting this derivative equal to zero and solving for 'r', we get 4πr - 50000 / r^2 = 0.
Multiplying through by r^2, we have 4πr^3 - 50000 = 0.
Solving this equation for 'r', we get r^3 = 50000 / (4π), which simplifies to r^3 = 12500 / π.
Taking the cube root of both sides, we find r ≈ 12.6 cm (rounded to one decimal place).
Now that we have the value of 'r', we can substitute it back into our expression for 'h' to find the corresponding height 'h': h = 25000 / (π(12.6^2)), which evaluates to h ≈ 52.4 cm (rounded to one decimal place).
Therefore, the dimensions that will ensure the least amount of plastic is used to create the containers are approximately r = 12.6 cm and h = 52.4 cm.