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The circumference of a sphere was measured to be 82 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.) cm2

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}{2Ï€}[/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}{2Ï€}[/tex]

[tex]=\frac{82}{2Ï€}[/tex]

[tex]r=\frac{41}{Ï€}[/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]