Question

Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 11 inches wide and 17 inches long, and then folding up the sides. If the side lengths of her square cutouts are inches, then the volume of the box is given by V(x)=x(11-2x)(17-2x)

Answer 1

Answer:

Step-by-step explanation:

Refer the attached picture for complete question

a) V(x)=x(11-2x)(17-2x)

B(x)=140

Side and volume cannot be negative

So, x> 0

11-2x>0   and 17-2x>0

$x<\frac{11}{2}$ and $x< \frac{17}{2}$

Domain for V(x) :$0<x<\frac{11}{2}$

b)For finding the greatest Volume

$\frac{dV}{dx}=0$

$V(x)=x(11-2x)(17-2x)=4x^3-56x^2+187x$

$\frac{dV}{dx}=12x^2-112x+187=0\\x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\x=\frac{112\pm\sqrt{(112)^2-4(12)(187)}}{2(12)}$

x=7.155 , 2.177

Consider the graph

So, Volume at 7.155 is negative

Thus x≈2 gives the greatest volume

c)

Consider the graph .

It is increasing from $-1 \leq x \leq 2$ and $x \geq 7$

So, Graph increasing interval is [-1,2] ∪[7,∞)

d)

At the point of intersection, both graph has the same value .