Answer:
x=5.37 cm
Step-by-step explanation:
we know that
The volume of the box is
[tex]V=Bh[/tex]
where
B is the area of the base of the box
h is the height of the box
Box 1
we have that
The area of the base is
[tex]B=x^2\ cm^2[/tex]
[tex]h=(x+5)\ cm[/tex]
The volume of the Box 1
is equal to
[tex]V_1=x^2(x+5)\ cm^3[/tex]
[tex]V_1=(x^3+5x^2)\ cm^3[/tex]
Box 2
we have that
The area of the base is
[tex]B=(x+1)^2\ cm^2[/tex]
[tex]h=(x+2)\ cm[/tex]
The volume of the Box 2
is equal to
[tex]V_2=[(x+1)^2(x+2)]\ cm^3[/tex]
[tex]V_2=[(x^2+2x+1)(x+2)]\ cm^3[/tex]
[tex]V_2=(x^3+2x^2+x+2x^2+4x+2)\ cm^3[/tex]
[tex]V_2=(x^3+4x^2+5x+2)\ cm^3[/tex]
Equate the equation of Volume 1 to the equation of Volume 2
[tex](x^3+5x^2)=(x^3+4x^2+5x+2)[/tex]
[tex](5x^2)=(4x^2+5x+2)[/tex]
[tex]5x^2-4x^2-5x-2=0[/tex]
[tex]x^2-5x-2=0[/tex]
Solve the quadratic equation by graphing
using a graphing tool
The solution is x=5.37 cm
see the attached figure
