Question

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

Answer 1

Answer:

The Maximum error is  $26cm^2$

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.) $cm^2$

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

 $dC = 2\pi dr$

$dr =\frac{dC}{2π}$

$=\frac{0.5}{2\pi}$ (∵ dC = 0.5 cm)

Since Area  $A = 4\pi r^2$ square units

Differentiating with respect to r

$dA = 8\pi rdr$  square units

Since given C = 82 cm.and also C=2πr  we have

$r=\frac{C}{2π}$

$=\frac{82}{2π}$

$r=\frac{41}{π}$

From that $dA = 8\pi rdr$

Substituting the values of r and dr in the above equation we get

$dA = 8\pi (\frac{41}{\pi}) (\frac{0.5}{2\pi})$

$=4(41)\times (\frac{0.5}{3.14})$

$=26.1146$

$dA=26cm^2$

∴  Maximum error is  $26cm^2$