Answer:
x = 6.54 cm
h = 3.27 cm
Step-by-step explanation:
Volume of open top box
V = 140 cm³
Dimensions of the box
It is a base square box then area of the base of side x is
A(b) = x²
And we will call h the height of the box then
V = 140 ⇒ 140 = x²*h ⇒ h = 140/ x²
We have to calculate the area of the 4 sides
Area of one side is As = x*h ⇒ total area of 4 sides = 4 x* 140/x²
Ats = 560/x
Then Total area of the box is
A(t) = Area of the base + Total area of sides
A(x) = x² + 560/x
Taking derivatives on both sides of the equation we get:
A´(x) = 2x - 560/x²
A´(x) = 0 ⇒ 2x - 560/x² = 0 ⇒ 2x³ - 560 = 0
x³ = 280 ⇒ x = 6.54 cm
And h h = 140/ (6.54)² ⇒ h = 140/ 42.77 h = 3.27 cm
