If you start with a 12x16 rectangle and cut square with side length x, when you bend the sides you'll have an inner rectangle with sides $12-2x$ and $16-2x$ , and a height of x.

So, the volume will be given by the product of the dimensions, i.e.

$(12-2x)(16-2x)x = 4x^3-56x^2+192x$

The derivative of this function is

$12x^2-112x+192$

and it equals zero if and only if

$12x^2-112x+192=0 \iff x = \dfrac{14\pm 2\sqrt{13}}{3}$

If we evaluate the volume function at these points, we have

$f\left(\dfrac{14-2\sqrt{13}}{3}\right) = \dfrac{64}{27}(35+13\sqrt{13})> f\left(\dfrac{14-2\sqrt{13}}{3}\right) = -\dfrac{64}{27}(13\sqrt{13}-35)$

So, the maximum volume is given if you cut a square with side length

$x=\dfrac{14-2\sqrt{13}}{3}\approx 2.7$

5 0

3 years ago

A number k is less than 3 units from 10. <br> What's the absolute value inequality???

Brrunno [24]

It would be: k+3 = 10

subtracting 3 from both sides,

k+3-3 = 10-3

k=7

6 0

3 years ago

The ages of several trees on school property are shown below: 3 years, 8 years, 10 years, 14 years, 11 years What is the mean ag

almond37 [142]

3+8+10+14+11= 46
46/(how many trees) 5= 9.2
9.2 years is your answer.

5 0

3 years ago

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