The dimensions of the microwave cup-of-soup package are:
Radius = 3.39cm, and Height = 9.27cm.
Computed using differentiation.
The microwave cup-of-soup package is in the shape of a cylinder.
Thus, its volume can be shown as, V = πr²h, where r is its radius in cm and h is its height in cm.
The volume is given to be 450 cm³.
Thus, we can say that,
πr²h = 450,
or, h = 450/(πr²) ... (i).
The cost of the microwaveable cup-of-soup package is given as:
The bottom is made of styrofoam at 0.03 cents/cm².
The area of the bottom = πr².
Therefore, the cost of the bottom = πr²(0.03) cents.
The side is made of styrofoam at the rate of 0.03 cents/cm².
The area of the side = 2πrh = 2πr{450/(πr²)}. {Taking h = 450/(πr²) from (i)}.
Therefore, the cost of the side = 2πr{450/(πr²)}(0.03) cents.
The top is made of glued paper at the rate of 0.08 cents/cm².
The area of the top = πr².
Therefore, the cost of the top = πr²(0.08) cents.
Therefore the total production cost function can be written as:
C = πr²(0.03) + 2πr{450/(πr²)}(0.03) + πr²(0.08) cents,
or, C = 0.03πr² + 0.08πr² + 27/r cents,
or, C = 0.11πr² + 27/r cents ... (ii).
We are asked to minimize the cost function.
To minimize it, we follow these steps:
Differentiating both sides in (ii) with respect to the radius,
dC/dr = 0.22πr - 27/r² ... (iii).
To get the points of inflections, we equate this to zero, to get,
dC/dr = 0.22πr - 27/r² = 0,
or, 0.22πr - 27/r² = 0,
or, 0.22πr = 27/r²,
or, r³ = 27/0.22π,
or, r = ∛(27/0.22π) = 3.393 cm.
To check whether the cost C, is minimum or maximum at the point, we calculate the second order differential, and if it's greater than 0, then we get a minimum and vice-versa.
Thus, differentiating both sides of (iii), with respect to the radius r, we get:
d²C/dr² = 0.22π + 54/r³.
0.22π + 54/r³ is greater than 0, for all values of r, since r is always greater than 0 {radius of an object cannot be negative}.
Thus, the radius at which the cost is minimized is 3.393 cm.
The corresponding height will be 450/(πr²) = 9.274 cm.
Thus, the dimensions of the microwave cup-of-soup package are:
Radius = 3.39cm, and Height = 9.27cm.
Computed using differentiation.
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