Respuesta :
Answer:
The maximum error in calculating the surface area of the box is [tex]158.4 \:cm^2[/tex].
Step-by-step explanation:
Differentials are infinitely small quantities. Given a function [tex]y=f(x)[/tex] we call [tex]dy[/tex] and [tex]dx[/tex] differentials and the relationship between them is given by,
[tex]dy=f'(x) \:dx[/tex]
If the dimensions of the box are [tex]l[/tex], [tex]w[/tex] and [tex]h[/tex], its surface area is [tex]A=2(wl+hl+hw)[/tex] and
[tex]dA=\frac{\partial A}{\partial l}dl+\frac{\partial A}{\partial w}dw+\frac{\partial A}{\partial h}dh\\\\dA=2(w+h)dl+2(h+l)dw+2(w+l)dh[/tex]
We are given that [tex]|\Delta l|\leq 0.2[/tex], [tex]|\Delta w|\leq 0.2[/tex], and [tex]|\Delta h|\leq 0.2[/tex].
To find the largest error in the surface area, we therefore use [tex]dl=0.2, dw=0.2, dh=0.2[/tex] together with [tex]l=97, w=67,h=34[/tex]
[tex]dA=2(w+h)dl+2(h+l)dw+2(w+l)dh\\\\dA=2(67+34)\cdot 0.2+2(34+97)\cdot 0.2+2(67+97)\cdot 0.2\\\\dA=158.4 \:cm^2[/tex]
An error of 0.2 cm in each dimension could lead to an error of [tex]158.4 \:cm^2[/tex] in the calculated surface area.
