Respuesta :
Answer:
28.51 cubic units
Step-by-step explanation:
From the given information;
The surface area is 56 and not 5656
Let assume that:
the length of the base = a &
the height of the box = h
∴
The volume of the box = a²h &
The surface area (A) = 2a² + 4ah
A = 2a² + 4ah
56 = 2a² + 4ah
56 - 2a² = 4ah
[tex]h = \dfrac{56-2a^2}{4a}[/tex]
The volume (V) = a²h
[tex]V = a^ 2( \dfrac{56-2a^2}{4a}) \\ \\ V = \dfrac{56a -2a^3}{4} \\ \\ V = \dfrac{2a(28- a^2)}{4} \\ \\ V = \dfrac{(28-a^2)a}{2}[/tex]
Taking the maximum of this function by assuming v = [0, [tex]\sqrt{8}[/tex]]
[tex]V = \dfrac{a(28-a^2)}{2}[/tex]
[tex]V(0) = \dfrac{0(28-0^2)}{2}=0[/tex]
[tex]V(\sqrt{8}) = \dfrac{\sqrt{8}(28-(\sqrt{8})^2)}{2} \\ \\ \implies \sqrt{8} \dfrac{(28-8)} {2} \\ \\ = 10 \sqrt{8}[/tex]
For the critical point V' = 0
[tex]V = \dfrac{a(28-a^2)}{2} \\ \\ V = \dfrac{28a}{2}- \dfrac{a^3}{a} \\ \\ V = 14 a - \dfrac{a^3}{2} \\ \\ V' = 14 - \dfrac{3a^2}{2} \\ \\ 0 = 14 - \dfrac{3a^2}{2} \\ \\ 14 =\dfrac{3a^2}{2}\\ \\ a^2 = 14 \times \dfrac{2}{3} \\ \\ a^2 = \dfrac{28}{3} \\ \\ a = \sqrt{\dfrac{28}{3}} \\ \\ a = \sqrt{9.333} \\ \\ a = 3.055[/tex]
Thus, side of the base (a) = 3.055 units
Recall that:
height [tex]h = \dfrac{56-2a^2}{4a}[/tex]
[tex]h = \dfrac{56-2(3.055)^2}{4(3.055)}[/tex]
h = 3.0551 units
The maximum volume now = a²h
= (3.055)²(3.0551)
= (9.333)(3.0551)
= 28.51 cubic units
Answer:
x ( side of the square base) = 43,42 ul
h = ( the height f the box ) = 0,17 ul
Step-by-step explanation:
V = Ab * h where Ab is an area of the base and h is the height of the box
as is this case s a square base box then Ab = x² ( s the side of the square.
The surface area is ( box without lid): area of the base x² plus 4 lateral areas each one equal to x*h
So x² + 4 x*h = 5656 u²
Then 4*x*h = 5656 - x²
h = 5656 - x² )/ 4*x
V(x) = x² ( 5656 - x² )/ 4*x
V(x) = x ( 5656 - x² ) / 4
Tacking derivatives on both sides of the equation :
V´(x) = (1/4 ) * [ (5656 - x² ) - x *2*x]
V´(x) = (1/4 ) * ( 5656 - 3*x² )
v¨(x) = 0 (1/4 ) * ( 5656 - 3*x² ) = 0 3*x² = 5656
x² = 1885,33
x = √ 1885,33
x = 43,42 ul
And h = 5656 - x² )/ 4*x
h = [( 5656 - (43,42)²] /4 * 5656
h = 3770,70 / 22624
h = 0,17 ul
