SOLUTION: I am in College Algebra but the question seemed to fit in this category...The base and sides of a container are made of wood panels. The container does not have a top. The base and

Algebra ->  Surface-area -> SOLUTION: I am in College Algebra but the question seemed to fit in this category...The base and sides of a container are made of wood panels. The container does not have a top. The base and      Log On


   

Question 1099839: I am in College Algebra but the question seemed to fit in this category...The base and sides of a container are made of wood panels. The container does not have a top. The base and sides are rectangular. The width is x cm. The length is 4 times the width. The volume is 600 cm^3. Determine the minimum surface area to two decimal places.
Found 2 solutions by ankor@dixie-net.com, ikleyn:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
The container does not have a top.
The base and sides are rectangular.
The width is x cm. The length is 4 times the width.
The volume is 600 cm^3.
Determine the minimum surface area to two decimal places.
:
let h = the height of the box
Given
x = the width
4x = the length
Volume
4x * x * h = 600
4x^2 * h = 600
h = 600%2F%284x%5E2%29
cancel the 4
h = 120%2Fx%5E2
:
Surface area
S.A. =(4x*x) + 2(4x*h) + 2(x*h)
S.A. = 4x^2 + 8xh + 2xh
S.A. = 4x^2 + 10xh
replace h with 120%2Fx%5E2
S.A. = 4x^2 + 10x*120%2Fx%5E2
Cancel x
S.A. = 4x^2 + 1200%2Fx
plot the equation y = 4x^2 + (1200/x), where y = the surface area
+graph%28+300%2C+200%2C+-6%2C+10%2C+-100%2C+1000%2C+4x%5E2%2B%281200%2Fx%29%2C+339%29+
You can see minimum surface area occurs when x = 5.2 cm. Green: y=339
Find the minimum surface area
S.A. = 4(5.2^2) + 1200%2F5.2
S.A. = 108.16 + 230.77
S.A = 338.93 sq/cm

Answer by ikleyn(52784) About Me  (Show Source):
You can put this solution on YOUR website!
.
Since the tutor @ankor@dixie-net.com made several mistakes in his solution, I re-write and edit it in correct way. 

The container does not have a top.
The base and sides are rectangular. 

The width is x cm. The length is 4 times the width.
The volume is 600 cm^3.
Determine the minimum surface area to two decimal places. 

:
let h = the height of the box

Given

x = the width
4x = the length

Volume
4x * x * h = 600
4x^2 * h = 600
h = 600%2F%284x%5E2%29
cancel the 4
h = 150%2Fx%5E2                   <<<---=== I replaced 120 by 150
:
Surface area
S.A. =(4x*x) + 2(4x*h) + 2(x*h)
S.A. = 4x^2 + 8xh + 2xh
S.A. = 4x^2 + 10xh
replace h with 150%2Fx%5E2
S.A. = 4x^2 + 10x*150%2Fx%5E2
Cancel x
S.A. = 4x^2 + 1500%2Fx

plot the equation y = 4x^2 + (1500/x), where y = the surface area
+graph%28+300%2C+200%2C+-6%2C+10%2C+-100%2C+1000%2C+4x%5E2%2B%281500%2Fx%29%29+

Taking the derivative y' = 8x - 1500/x^2 = %288x%5E3-1500%29%2Fx%5E2,    <<<---=== I re-wrote this part

you will find the minimum surface area occurs when x = root%283%2C%281500%2F8%29%29= 5.72  

Find the minimum surface area
S.A. = 4%2A%285.72%5E2%29+%2B+1500%2F5.72

Answer  S.A. = 393.1 cm^2.