Respuesta :
Answer:
The maximum error in calculating the surface area of the box is [tex]168\:cm^2[/tex]
Step-by-step explanation:
The differential df of a function [tex]f=f(x,y,z)[/tex] is related to the differentials dx, dy, and dz by
[tex]\begin{equation*}df = f_x(x_0,y_0,z_0) dx + f_y(x_0,y_0,z_0)dy+f_z(x_0,y_0,z_0)dz\end{equation*}[/tex]
We can use this relationship to approximate small changes in f that result from small changes in x, y and z.
Let the dimensions of the box be [tex]l[/tex], [tex]w[/tex], and [tex]h[/tex] for length, width, and height, respectively.
The surface area of a box is the total area of each side and is given by
[tex]S=2(lw+wh+lh)[/tex]
The change in area can be written as:
[tex]\Delta S\approx dS = \frac{\partial S}{\partial l} dl+\frac{\partial S}{\partial w} dw+\frac{\partial S}{\partial h} dh[/tex]
From the information given the partial derivatives are evaluated at [tex]l =60[/tex], [tex]w=60[/tex], and [tex]h=90[/tex], and [tex]dl=dw=dh=0.2[/tex].
The partial derivatives are
[tex]\frac{dS}{dl}=2(w+h)=2(60+90)=300\\\\\frac{dS}{dw}=2(l+h)=2(60+90)=300\\\\\frac{dS}{dh}=2(l+w)=2(60+60)=240[/tex]
Substituting these in for [tex]dS[/tex],
[tex]dS = 300\cdot 0.2+300\cdot 0.2+240\cdot 0.2\\\\dS=2\cdot \:300\cdot \:0.2+240\cdot \:0.2\\\\dS=120+48=168[/tex]
Thus, the maximum error in calculating the surface area of the box is [tex]168\:cm^2[/tex]
