1. **State the problem:** We need to estimate the value of $$S = \frac{5.86 + \sqrt{88.21} \times 11.35}{3.82}$$ correct to one significant figure for each number, then find the difference between the actual and estimated values. 2. **Estimate each number to one significant figure:** - $5.86 \approx 6$ - $\sqrt{88.21} \approx \sqrt{88} \approx 9$ (since $9^2=81$ and $10^2=100$) - $11.35 \approx 11$ - $3.82 \approx 4$ 3. **Calculate the estimate for $S$ using these approximations:** $$S \approx \frac{6 + 9 \times 11}{4} = \frac{6 + 99}{4} = \frac{105}{4}$$ 4. **Simplify the fraction:** $$\frac{105}{4} = 26.25$$ 5. **Calculate the actual value of $S$:** First calculate $\sqrt{88.21}$: $$\sqrt{88.21} = 9.39$$ Then calculate numerator: $$5.86 + 9.39 \times 11.35 = 5.86 + 106.56 = 112.42$$ Divide by denominator: $$S = \frac{112.42}{3.82} = 29.45$$ 6. **Find the difference between actual and estimated values:** $$29.45 - 26.25 = 3.20$$ 7. **Final answers:** - Estimated $S$ (one significant figure inputs): $26.25$ - Actual $S$: $29.45$ - Difference (to three decimal places): $3.200$ Note: The user’s original estimate and difference seem to have used different rounding or denominator values; here we strictly follow one significant figure rounding and exact calculations for clarity.