Answer :
The dimensions of a box with the minimal surface area are now known: X, Y, and Z are all 16 cm in length.
Define minimization of a function?
- We are aware that we are being asked to determine the minimal value for a function whenever we are approached for a minimum value.
- To accomplish this, we need to identify the function that needs to be reduced (in this case, the surface area) then obtain its first derivative.
- The values which minimize the function are those obtained by setting the very first derivative to zero.
For the stated question-
The box's volume as well as the surface area formula are the first things we write down after receiving them:
Volume = 4096 xyz
Surface area = 2(xy, xz, and yz)
Surface area = x + y + 2z.
To make the problem simpler, we can rephrase a single variables on the basis of the remaining variables:
z = 4096/xy
This surface area function is now expressed in terms of x and y.
f(x,y, 4096/xy) = 2xy + 2(x + y)(4096/xy)
= 2xy + 8192/y + 8192/x
The first derivative of each variable is required in order to reduce a function. Now, by treating other variable as a constant, we can calculate the partial of the function's derivative for x and y:
df/dx = 2y - 8192/x²
df/dy = 2x - 8192/y²
We set these initial derivatives to zero in attempt to minimize a function:
df/dx = 0 = 2y - 8192/x²
df/dy = 0 = 2x - 8192/y²
yx² = 4096
y²x = 4096
To determine that x = y, we then set both equations equal to one another.
yx² = y²x
y = x
Using this knowledge, we solve for y, that is also equal to x, by plugging y into the preceding equation:
yy² = 4096
y = 16
x = y
x = 16
X and Y are both equal to 16. With this knowledge, we can enter the values into the z equation to determine:
z = 4096/xy
z = 4096/ 16² = 16
Thus, the dimensions of box with the smallest possible surface area are now known:
X, Y, and Z are all 16 cm in length.
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