Respuesta :
The dimensions that maximize the surface area are(r,h) = (5.42,10.84).
The surface area of a cylindrical can is S(r,h)= 2πr² +2πrh
The capacity or volume of the rectangular box is V(r,h) =πr²h
V(r,h) = πr²h-1000,S(r,h)= 2πr² +2πrh-2π
Find first-order partial derivatives of V and S:
[tex]V_{r}[/tex] = 2πrh ,[tex]V_{h}[/tex] =πr² ,[tex]S_{r}[/tex] =4πr +2πrh,[tex]S_{h}[/tex]=2πr.
Find the values of x,y, and z and λ such that simultaneously satisfy the equation:
▽S =λ▽V, and V(r,h) =0, that is
4πr +2πh =λ(2πrh)⇒ λ=(4πr +2πh) /2πrh= [tex]\frac{2r +h }{rh}[/tex]
2πr = λ(πr²) ⇒λ=2πr/πr² = 2/r
πr²h-1000 =0
solving [tex]\frac{2}{r}[/tex] = [tex]\frac{2r +h }{rh}[/tex] ⇒2r²-2rh +hr =0⇒2r²-hr=0⇒r(2r-h)=0
⇒r=0,r=h/2.
Then,(r,h) =(h/2,h)
substitute this point in πr²h-1000=0, and solving
π [tex](h/2)^{2}[/tex]h -1000 = 0⇒ π×h³/4 =1000 ⇒1000×4×7×[tex]\frac{1}{22}[/tex]
h≈ 10.84
Then, r =[tex]\frac{h}{2}[/tex] = 5.42
The value of S(r,h) = S(5.42,10.84) =553.74
The dimensions that maximize the surface area are(r,h) = (5.42,10.84).
Quantity is defined as the distance occupied inside the barriers of an object in a three-dimensional area. it's also referred to as the capacity of the item. quantity is a degree of occupied 3-dimensional area.[1] it's far regularly quantified numerically through the use of SI-derived gadgets (which includes the cubic meter and liter) or through diverse imperial or US commonplace devices (consisting of the gallon, quart, and cubic inch). The definition of the period (cubed) is interrelated with the extent. The volume of a field is generally understood to be the capacity of the box; i.e., the amount of fluid (gas or liquid) that the box could hold, as opposed to the quantity of space the field itself displaces.
In ancient times, the extent is measured by the usage of similar-shaped natural packing containers and for a while, standardized packing containers. a few simple 3-dimensional shapes could have their quantity easily calculated with the usage of the arithmetic formulation. Volumes of greater complicated shapes can be calculated with essential calculus if a formula exists for the form's boundary.
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