Respuesta :
Answer:
The Maximum error is [tex]26cm^2[/tex]
Step-by-step explanation:
Given that the circumference of a sphere was measured to be 82 cm with a possible error of 0.5 cm
To find :
(a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.) [tex]cm^2[/tex]
Given circumference of a sphere=82 cm and possible error =0.5 cm
Error of the surface area is dA
Error of circumference is dC = 0.5 cm
We know that the formula for Circumference of a sphere is C=2πr units
Differentiating with respect to r
[tex]dC = 2\pi dr[/tex]
[tex]dr =\frac{dC}{2π}[/tex]
[tex]=\frac{0.5}{2\pi}[/tex] (∵ dC = 0.5 cm)
Since Area [tex]A = 4\pi r^2[/tex] square units
Differentiating with respect to r
[tex]dA = 8\pi rdr[/tex] square units
Since given C = 82 cm.and also C=2πr we have
[tex]r=\frac{C}{2π}[/tex]
[tex]=\frac{82}{2π}[/tex]
[tex]r=\frac{41}{π}[/tex]
From that [tex]dA = 8\pi rdr[/tex]
Substituting the values of r and dr in the above equation we get
[tex]dA = 8\pi (\frac{41}{\pi}) (\frac{0.5}{2\pi})[/tex]
[tex]=4(41)\times (\frac{0.5}{3.14})[/tex]
[tex]=26.1146[/tex]
[tex]dA=26cm^2[/tex]
∴ Maximum error is [tex]26cm^2[/tex]
