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Sedbober [7]
3 years ago
14

A box of volume 180 m3 with square bottom and no top is constructed out of two different materials. The cost of the bottom is $4

0/m2 and the cost of the sides is $30/m2 . Find the dimensions of the box that minimize total cost. (Let s denote the length of the side of the square bottom of the box and h denote the height of the box.)
Mathematics
1 answer:
zysi [14]3 years ago
5 0

Answer:

Side of square bottom=6.46 m

Height of box=4.31 m

Step-by-step explanation:

Let s be the side bottom

Height of box=h

Volume of box=180m^3

Volume of box=lbh=s^2h

s^2h=180

h=\frac{180}{s^2}

Cost of bottom=$40 per square  m

Cost of sides =$30 per square  m

Total cost=C=40s^2+30(4sh)

C=40s^2+120s\times \frac{180}{s^2}

C(s)=40s^2+\frac{21600}{s}

Differentiate w.r.t s

C'(s)=80s-\frac{21600}{s^2}

C'(s)=0

80s-\frac{21600}{s^2}=0

80s=\frac{21600}{s^2}

s^3=\frac{21600}{80}=270

s=(270)^{\frac{1}{3}}=6.46

Again,differentiate w.r.t s

C''(s)=80+\frac{43200}{s^3}

Substitute s=6.46

C''(6.46)=80+\frac{43200}{(6.46)^3}=240.2>0

Hence, the cost is minimum at s=6.46

h=\frac{180}{s^2}=\frac{180}{(6.46)^2}=4.31

Side of square bottom=6.46 m

Height of box=4.31 m

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