To estimate the maximum error in the surface area of a sphere using differentials, we start with the measurements provided. The circumference (C) of a sphere can be related to its radius (r) by the formula: C = 2\pi r.
1. **Find the radius using the circumference:**
– Using the given circumference: C = 78 cm,
– We can solve for r:
\[ r = \frac{C}{2\pi} = \frac{78}{2\pi} \approx 12.43 \text{ cm} \]
2. **Calculate the surface area (A) of the sphere:**
– The formula for the surface area of a sphere is: \[ A = 4\pi r^2. \]
– Substituting the value of r:
\[ A = 4\pi (12.43)^2 \approx 1543.25 \text{ cm}^2. \]
3. **Use differentials to estimate the maximum error:**
– The differential of the surface area with respect to the radius can be derived from the surface area formula:
– \[ dA = 8\pi r \, dr. \]
– Here, dr represents the possible error in the radius, which arises from the error in the circumference measurement. The error in circumference (dC) is given as 0.5 cm.
– We can relate the change in circumference to the change in radius using the derivative of the circumference formula: \[ dr = \frac{dC}{2\pi}. \]
– Therefore, substituting dC:
\[ dr = \frac{0.5}{2\pi} \approx 0.0797 \text{ cm}. \]
4. **Calculate the differential for surface area:**
– Now substituting r and dr back into the differential formula for A:
\[ dA = 8\pi (12.43) (0.0797) \approx 2.49 \text{ cm}^2. \]
5. **Conclusion:**
– The estimated maximum error in the calculated surface area of the sphere is approximately 2.49 cm². This implies that the actual surface area could vary by this amount in either direction due to the possible error in measuring the circumference.