Respuesta :
Answer:
The volume in such a package is 27648 in³
Step-by-step explanation:
Consider the provided information.
Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 144 in.
Let the dimension are x by x by y.
Where x is the variable for the square base package and y is the variable for length.
Thus l=x, b=x and h=y
Then the volume of the box is: [tex]V(x)=x^2y[/tex] (∵V=lbh)
The maximum combined length and girth is 144.
Therefore, [tex]4x+y=144[/tex]
[tex]y=144-4x[/tex]
Substitute the value of y in volume of the box.
[tex]V(x)=x^2(144-4x)[/tex]
[tex]V(x)=144x^2-4x^3[/tex]
[tex]V'(x)=288x-12x^2[/tex]
Substitute V'(x)=0.
[tex]0=288x-12x^2[/tex]
[tex]12x(24-x)=0[/tex]
[tex]x=0\ or\ x=24[/tex]
Now apply second derivative test.
[tex]V''(x)=288-24x[/tex]
[tex]V''(0)=288-24(0)>0[/tex] (Min)
[tex]V''(24)=288-24(24)<0[/tex] (Max)
Hence, the maximum volume is at x=24
If x=24 then [tex]y=144-4(24)=48[/tex]
Substituting value of x = 24 and y = 48 [tex]V(x)=x^2y[/tex] gives 27648.
Hence, the volume in such a package is 27648 in³
If parcel sent by priority mail may have a combined length and girth of no more than 144 in and rectangular package has a square cross section has possible largest volume is 27648.
What is volume ?
Amount of space occupied by a 3D object is known as volume.
Here given that parcel sent by priority mail may have a combined length and girth of no more than 144 in and rectangular package has a square cross section .
Let suppose
Length=Breadth =x
Height=y
Hence volume can be written as
[tex]V=x^{2}y[/tex]
Now combined length and girth can be written as
4x+y=144
y=144-4x
Hence
[tex]$\begin{aligned}&V(x)=x^{2}(144-4 x) \\&V(x)=144 x^{2}-4 x^{3} \end{aligned}$[/tex]
Now differentiating with respect to x
[tex]V^{\prime}(x)=288 x-12 x^{2}[/tex]
Now for maximum value
[tex]V^{\prime}(x)=0[/tex]
[tex]$\begin{aligned}&0=288 x-12 x^{2} \\&12 x(24-x)=0 \\&x=0 \text { or } x=24\end{aligned}$[/tex]
Now differentiating again with respect to x
[tex]$\begin{aligned}&V^{\prime \prime}(x)=288-24 x \\&V^{\prime \prime}(0)=288-24(0) > 0 \text { (Min) } \\&V^{\prime \prime}(24)=288-24(24) < 0 \text { (Max) }\end{aligned}$[/tex]
So maximum volume
[tex]V_{max}=V(24)=24^2(144-4\times24)\\\\V_{max}=576(48)\\\\V_{max}=27648[/tex]
If parcel sent by priority mail may have a combined length and girth of no more than 144 in and rectangular package has a square cross section has possible largest volume is 27648.
To learn more about volume visit :https://brainly.com/question/1972490
