How would I solve this question?

Question

If 1800 cm^2 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

Since box is open top, it has 5 surfaces, one square base and 4 .

sides of rectanglular surfaces.

let side of the base square = s

let height = h

totalsurface area = area of base + 4(area of side)

=> s^2 + 4sh

so s^2 + 4sh = 1800

4sh = 1800 - s^2

h = (1800 - s^2)/4s

volume = area of base* h

V = s^2(h)

substituting h value

V = s^2(1800 -s^2)/4s

V = (1/4) s(1800 - s^2)

V = (1/4)[1800 s - s^3)

V = 450s - s^3/4

Volume will be maximum when dV/ds = 0

dV/ds = 450 - (1/4) 3s^2

dV/ds = 450 - (3/4)s^2

450 - (3/4)s^2 = 0

(3/4)s^2 = 450

s^2 = 450(4/3) = 600

s = sqrt(600) = 10sqrt(6)

substituting s value in h = (1800 - s^2)/4s

h = (1800 - 600)/4(10sqrt(6))

h = 1200/40 sqrt(6)

h = 30/sqrt(6) = 5 sqrt(6)

Volume = s^2*(h)

=>600(5sqrt(6))

=>3000sqrt(6)

=>7348.47 cm^3

Answer 2

Since the top is open, it's NOT a perfectly symmetrical problem.

Let base side = x, height = h

A = x² + 4xh = 1800 (constraint)
V = x²h

4xh = 1800 - x²
h = (1800 - x²)/4x

V = x²h = x²(1800 - x²)/4x = (1800x - x³)/4
V has an extremum at dV/dx = 0
(1800 - 3x²)/4 = 0
1800 = 3x²
x² = 600
x = √600 = 10√6 ≈ 24.49

h = (1800 - √600²)/4√600 = 1200/4√600 = 300/√600 = √300/√2 = √150 = 5√6 [mwana forgot the x in the denominator]

V_max = x²h = (√600²)(√150) = 3000√6 = 7348.5

Answer 3

The largest possible volume can be achieved with all sides having equal surface area. Take 1800, divide it by 6 (to make six sides for the box). This equals 300. Each side should have area 300 cm^2. Then cut that in two for the length and width of each side to be equal. Thus b=150, w=150, h=150 creating a volume of 150^3 cm^3

Answer 4

V= l x w x h

To find the perimeter you know that the perimeter of a square is 4a where 'a' is the length of 1 side:

So a box with the top open would be 5 squares so the perimeter is:

P= 5 x 4a
P = 20a

And we know that the P can't exceed 1800 cm²

So P = 20a = 1800
a = 1800/20
a = 90

V = 90 x 90 x 90
V = 729,000

Hope that was helpful! :)

Answer 5

a box with square base a and open base b
will have area a^2+4ab =1800
=> b=(1800-a^2)/4a and has a volume
V=a^2*b= a^2*(1800-a^2)/4a=(1800a-a^3)/4
for maximum dV/da =0
dV/da=(1800-3a^2)/4=0
=>1800-3a^2=0
=>600=a^2
but b=(1800-a^2)/4=(1800-600)/4=300
V =a^2*b=600*300= 180,000cm^3
For maximum ar

Answer 6

squares have maximum surface area. So if you build a box with perfect with 5 sides all of which are perfect squares, you will have max volume