Answer:
the answer is 24 ft^3. I just took the connexus quiz trust me!
Step-by-step explanation:
The maximum volume of the box is required.
The height of the box is 1.81 ft
The maximum volume of the box is [tex]24.19\ \text{ft}^3[/tex].
The given function is
[tex]V(h)=h(h-5)(h-6)\\\Rightarrow V(h)=h^3-11h^2+30h[/tex]
Differentiating with respect to [tex]h[/tex]
[tex]V'(h)=3h^2-22h+30[/tex]
Equating to zero
[tex]0=3h^2-22h+30\\\Rightarrow h=\frac{-\left(-22\right)\pm \sqrt{\left(-22\right)^2-4\times3\times30}}{2\times3}\\\Rightarrow h=5.52,1.81[/tex]
Double derivative of the function
[tex]V''(h)=6h-22[/tex]
Substituting [tex]h[/tex] values
[tex]V''(5.52)=6\times 5.52-22=11.22[/tex]
[tex]V''(1.81)=6\times 1.81-22=-11.14[/tex]
Since, [tex]V''(1.81)[/tex] is negative, the function will be maximum at [tex]h=1.81[/tex].
Checking domain [tex]1.81\in (0,6)[/tex]
The maximum volume is
[tex]V(1.81)=1.81^3-11\times 1.81^2+30\times 1.81=24.19\ \text{ft}^3[/tex]
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