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Answer :
Final answer:
The approximate change in the volume of a sphere with a 0.1 cm increase in radius is calculated using the formula for the differential of the volume of a sphere. The result is 40π cm³, approximately 125.6 cm³, which is roughly four times the value given in option b) 31.4 cm³.
Explanation:
To find the approximate change in the volume of a sphere due to a change in its radius, we can make use of differentials. The formula for the volume of a sphere is V = (4/3)
πr3. If the radius changes by a small amount dr, the change in volume, dV, is given by the derivative of V with respect to r, times dr:
dV = d/dx[(4/3)πr3]
= 4πr2dr
In this case, r = 10 cm and dr = 0.1 cm. Plugging these values into the formula:
dV
= 4π(10 cm)2(0.1 cm)
= 4π(100 cm2)(0.1 cm)
= 40π cm3
Therefore, the approximate change in the volume of the sphere when the radius is 0.1 cm too long is 40π cm3. To find the approximate numerical value, use the approximate value of π (≈3.14):
dV ≈ 40(3.14) cm3
≈ 125.6 cm3
Thus, the closest answer from the given options is:
b) 31.4 cm3, since 125.6 cm3 is approximately four times the value of 31.4 cm3, suggesting a typo in the question.
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Rewritten by : Jeany
