Answer to Question #258356 in Calculus for Wah

Question

3)

You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides

If 𝑙 = 200𝑚𝑚 and 𝑤 = 150𝑚𝑚 calculate:

a) The value of x which will give the maximum volume

b) The maximum volume of the box

c) Comment of the value obtained in part b.

4) The gain of an amplifier is found to be

𝑮 = 𝟐𝟎 𝐥𝐨𝐠(𝟏𝟎𝑽𝒐𝒖𝒕):

The tasks are to find equations for:

a) 𝑑𝐺/𝑑𝑉𝑂𝑢𝑡

b) 𝑑^2𝐺/𝑑𝑉𝑂𝑢𝑡 2

Answer 1

3.

The volume of the box can be written in the form: $V(x)=(L - 2\cdot x)\cdot(W- 2\cdot x)\cdot x$

Lengths and width of the box decreased that is of sheet metal by $x$ from each corner, and height of the box is equal $x$. We bring $V(x)$ to a simple form: $V(x)=4\cdot x^3 -2\cdot (L+W)\cdot x^2+ L\cdot W\cdot x$

To find maximum volume one compute the derivative of volume with respect to $x$

$V^{'}_x=12\cdot x^2 - 4\cdot (L+W)\cdot x+ L\cdot W$ and define the root of the equation $V^{'}_x=0$ :

$x_{1,2}=(2\cdot(L+W)\pm\sqrt{4(L+W)^2-12\cdot L\cdot W} )/12=\frac{1}{6}(L+W\pm\sqrt{L^2+W^2-L\cdot W})$

$x_1=88.38 ;\space x_2=28.29$

The first value cannot be implemented. It is clear that the box will succeed only if $x<W/2$

The second value corresponds to the maximum volume shown in the figure.

Answer: $x=28.29 mm; max V=379037.81 mm^2$

4.

$(a) d G / d V o u t=20 * 1 * 10 /(10 * Vout * \ln 10)=20 /( Vout * \ln 10) \\(b) d^{2} G / d V o u t^{2}=(-20 * d(\text { Vout } * \ln 10) / d V \text { out }) /(\text { Vout } * \ln 10)^{2} \\=-20 * \ln 10 /\left((\ln 10)^{2} * \text { Vout }^{2}\right) =-20 /\left(\ln 10 * V o u t^{2}\right)$