Respuesta :
Answer:
dimension is 20ft*20ft*10ft
Explanation:
let a,a and h are length, width and height of given tank with square base.
volume = a*a*h
[tex]4000ft^3 = a^2 *h[/tex]
[tex]h =\frac{4000}{a^2}[/tex]
surface area =aa +2(ah+ah)
[tex] = a^2 + 4ah[/tex]
[tex] = a^2 + 4r(\frac{4000}{a^2})[/tex]
[tex]=a^2 + (\frac{16000}{a})[/tex]
for weight to minimize, surface area is to be minimum i.e.
[tex] \frac{dA}{da} = 0[/tex]
[tex]\frac{dA}{da} = 2a -(\frac{16000}{a^2})[/tex]
[tex]2a = (\frac{16000}{a^2})[/tex]
[tex]a^3 = 8000[/tex]
a = 20ft
now
[tex]\frac{d^2A}{dr^2} = 2+(\frac{32000}{r^2})[/tex]
at a = 20 ft
[tex]h =\frac{4000}{20^2}[/tex]
h = 10ft
hence dimension is 20ft*20ft*10ft
Answer:
dimension of the rectangular tank = (20 ft × 20 ft × 10 ft)
Explanation:
volume of rectangle = 4000 ft³
volume of the tank = a² × h
surface area of the tank = 4 × a × h + a²
from the volume of the tank h = 4000/a²
now surface area becomes
[tex]S = a^2 + \dfrac{16000}{a}[/tex]
now ,
[tex]\frac{\mathrm{d} s}{\mathrm{d} a}= 2a - \dfrac{16000}{a^2}[/tex]
[tex]\frac{\mathrm{d} s}{\mathrm{d} a}= 0\\2a - \dfrac{16000}{a^2}=0\\a^3 = 8000\\a=20 ft [/tex]
h = 10 ft.
hence, the dimension of the rectangular tank comes out to be
(20 ft × 20 ft × 10 ft)
