Answer:
36 x 18 x 18
Step-by-step explanation:
By Lagrange multiplier's method,
∇f(x,y,z)= λ∇g(x,y,z) & g(x,y,z)=k
considering x,y,z as three unequal sides of the box
f(x,y,z)=xyz
If x represents the length, then constraint condition is g(x,y,z)= x + 2(y+z) =108
We have the following three equations:
yz= λ -->eq(1) fx= λgx
xz=2λ -->eq(2) fy=λgy
xy= 2λ -->eq(3) fz=λgz
x+2y+2z=108-->eq(4)
Dividing eq(2) by eq(1), we'll have
x / y =2
=> x= 2y
Also, by using eq(2)and eq(3), we can represent y=z
By substituting 'x= 2y' and 'y=z' in eq(4), we have
eq(4)=>
2y+ 2(y+y)= 108
y=108/6
y=18
For x: x=2y=> 2(18)=>36
For z: z= y = 18
Therefore, the dimensions are 36 x 18 x 18
Answer:
Radius of package is 36/π inches
Height of package is 36 inches
Step-by-step explanation:
Here we have
Sum of height plus perimeter of package = 108 inches that is
2πr + h = 108 inches
Where:
Height of package = h
Radius of package = r
Volume of package = πr²h
However, h = 108 - 2πr
Therefore, the equation for the volume of the package is;
Volume of package, V = πr²(108 - 2πr) = πr²×108 - 2π²r³
Differentiating the above equation and equating to zero to find a maximum value, we have;
2πr×108 - 6r²π² = 0
2πr×108 = 6r²π²
36 = r·π
r = 36/π inches
h = 108 - 2πr = 108 - 72 = 36 inches
Radius of package, r = 36/π inches
Height of package, h = 36 inches.
