Question

If 1000 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

The area of the base is x^2> The height is h. Each side of the box has area xh. There are 4 sides of the box so the total surface area of the box is x^2+4xh and that is equal to 1000. Solve that equation for h: x^2+4xh = 1000 h = (1000-x^2)/4x so the Volume = x^2[(1000-x^2)/4x] Simplify and get V = 250x-x^3/4 The volume will be a maximum when its first derivative is 0. V' = 250-3/4x^2 Set to 0 and solve. x=18.26 Now plug into the volume function to find the maximum volume: V=250(18.26)-(18.26)^3/4 V= 4564.35 - 1522.10 =3042.25

Answer 2

Answer:

Maximum volume of the box is 3042.91 cm³

Step-by-step explanation:

Area of the material available to make a box with square base and open top = 1000 square cm.

Total surface area of the box = 2(lh + bh) + l×b

Where l = length of the box

b = width of the box

h = height of the box

If l = b, Total surface area of the box = 2(lh + lh) + l×l

Surface area = 4lh + l² = 1000

l(4h + l) = 1000

4h + l = $\frac{1000}{l}$

h = $\frac{1000}{4l}-\frac{l}{4}$

Now volume of the square box = lbh

V = l²h

V = l²($\frac{250}{l}-\frac{l}{4}$)

V = 250l - $\frac{l^{3}}{4}$

Now to calculate the maximum volume we have to find the derivative of the volume.

$\frac{dV}{dl}=250-\frac{3l^{2} }{4}$

By equating derivative to zero,

250 - $\frac{3l^{2}}{4}$ = 0

l = $\sqrt{\frac{250\times 4}{3}}$

l = 18.26 cm

For l = 18.26, V = 250(18.26) - $\frac{(18.26)^{3}}{4}$

V = 4565 - 1522.09

  = 3042.91 cm³

Therefore, maximum volume of the box is 3042.91 cm³