Question

If 300 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000.

Answer 1

we have 300m^2 of material
this box has a square base but variable height y
so SA=x^2+4y. thus 300=x^2+4y
we know V=x^2y and by fdt we know a critcal point exists when when V'=0
we need y, 300=x^2+4y => 300-x^2=4y
and y=300/4-x^2/4
so V=x^2(75-x^2/4)=75x^2-1/4x^4
V'=150x-x^3=x(150-x^2)=0 if x=0 or x=sqrt150
Test these values, V=x^2(75-x^2/4)
Clearly, [V=5625]
and 300=x^2+4y=150+4(75-150/4)

Answer 2

Volume = 353cm³

Step-by-step explanation: Let one length of the square be L

Hence, Surface area of the cube = 6 X Area of one face

300 = 6 X L²

L² = 300/6 = 50

∴ L = √50 = 7.07 cm

Volume, L³ = 7.07 cm X 7.07 cm X 7.07 cm = 353 cm³