Respuesta :
the dimensions of the dumpster that will minimize its surface area
length l = 6.32
width w = 1.58
height h = 0.39
Length = l
width = w
height = h
given that length is four times its width
l = 4w
Now, the formula for the surface area of a cube is;
S = 2lh + 2lw + 2wh
substitute l = 4w in the above equation
S = 2(4w)h+2(4w)w+2wh
S = 8wh+8[tex]w^{2}[/tex]+2wh
S = 10wh + 8[tex]w^{2}[/tex]
formula for the volume of a cube is;
V = lwh
now substitute l=4w in the above equation
V = (4w)wh
V = 4[tex]w^{2}[/tex]h
h = 4[tex]w^{2}[/tex]/V
Put V/2w² for h in the surface area equation to get;
S= 8w² + 10w(V/4w²)
= 8w² + 5V/2w
given that volume V = 25.6 substitute that in the above equation
S = 8w² + 5(25.6)/2w
= 8w² + 128/2w
Since we want to minimize the surface area, let us find the first derivative of S to get;
S' = 16w - 128/2w²
At S' = 0
0 = 16w- 128/2w²
16w = 128/2w²
32w³ = 128
w³ = 128/32
w³ = 4
w = [tex]\sqrt[3]{4}[/tex]
w = 1.58 yds
since l = 4w substitute the value of w in that
l = 4(1.58) = 6.32
h = 4[tex]w^{2}[/tex]/V = 4([tex]1.58^{2}[/tex])/25.6
h = 0.39
the dimensions are
length l = 6.32
width w = 1.58
height h = 0.39
To learn more about Surface area:
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