# [SOLVED]maximum problem

#### DeusAbscondus

##### Active member
No matter how or what strategy I try, I can't get the optimum value for $x$ in the following equation:

$$V=4x^3-60x^2+200x$$ Let V=Volume of a rectangular prism, so:
$$V'=12x^2-60x+200$$ Set V'=0 to get turning point
$$12x^2-60x+200=0$$
The answer given in my text is when
$$x=2.11 \Rightarrow \text{ the maximum volume is }\approx 192.45cm^2$$

I can graph it to get this, but can't get it using the quadratic formula for some reason.

some help would be appreciated

Last edited:

#### CaptainBlack

##### Well-known member
No matter how or what strategy I try, I can't get the optimum value for $x$ in the following equation:

$$V=4x^3-60x^2+200x$$ Let V=Volume of a rectangular prism, so:
$$V'=12x^2-60x+200$$
$V'=12x^2-120x+200$

CB

#### MarkFL

Staff member
Take another look at the linear or second term of your derivative...

#### Amer

##### Active member
No matter how or what strategy I try, I can't get the optimum value for $x$ in the following equation:

$$V=4x^3-60x^2+200x$$ Let V=Volume of a rectangular prism, so:
$$V'=12x^2-60x+200$$ Set V'=0 to get turning point
$$12x^2-60x+200=0$$
The answer given in my text is when
$$x=2.11 \Rightarrow \text{ the maximum volume is }\approx 192.45cm^2$$

I can graph it to get this, but can't get it using the quadratic formula for some reason.

some help would be appreciated
$$V' = 12x^2 - 120 x + 200$$
$$12x^2 - 120x + 200 = 0$$
$$3x^2 - 30 + 50 = 0$$
$$x = \frac{30 \mp \sqrt{900 - 4(3)(50}}{6} = \frac{ 30 \mp 10 \sqrt{3}}{6}$$
study the sign of V' around it is zeros we will get it has a maximum at $$x = \frac{30 - 10\sqrt{3}}{6}$$

#### DeusAbscondus

##### Active member
No matter how or what strategy I try, I can't get the optimum value for $x$ in the following equation:

Thanks people; what a goose am I; had been staring at this way too long; now I've got it:
$$V=4x^3-60x^2+200x$$ Let V=Volume of a rectangular prism, so:
$$V'=12x^2-120x+200$$ Set V'=0 to get turning point
$$12x^2-120x+200=0$$ (simply by factoring by 4 then apply quadratic formula)
$$3x^2-30x+50=0$$
$$\Rightarrow \frac{30\pm\sqrt{900-600}}{6}$$
$$=\frac{30\pm\sqrt{300}}{6}\Rightarrow \frac{30-\sqrt{300}}{6}\approx2.11$$  