This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the calculations for the given measurements: Question 3 Part A: Calculate (10.2 ± 0.01) × (3.1 ± 0.005). Let the two measurements be A = x_1 ± x_1 = 10.2 ± 0.01 and B = x_2 ± x_2 = 3.1 ± 0.005. We need to find the product P = A × B. Step 1: Calculate the product of the mean values. P_mean = x_1 × x_2 = 10.2 × 3.1 = 31.62 Step 2: Calculate the relative uncertainties for each measurement. ( x_1)/(x_1) = (0.01)/(10.2) ≈ 0.00098039 ( x_2)/(x_2) = (0.005)/(3.1) ≈ 0.00161290 Step 3: Calculate the relative uncertainty of the product using the formula for multiplication: ( P)/(P_mean) = sqrt((( x_1)/(x_1))^2 + (( x_2)/(x_2))^2) ( P)/(P_mean) = sqrt((0.00098039)^2 + (0.00161290)^2) ( P)/(P_mean) = sqrt(0.00000096116 + 0.00000260140) ( P)/(P_mean) = sqrt(0.00000356256) ≈ 0.00188747 Step 4: Calculate the absolute uncertainty of the product. P = P_mean × (( P)/(P_mean)) = 31.62 × 0.00188747 ≈ 0.05969 Step 5: Round the uncertainty to one significant figure and the mean value to the same decimal place. Rounding P = 0.05969 to one significant figure gives 0.06. The uncertainty is in the hundredths place. Therefore, P_mean = 31.62 should be rounded to the hundredths place. The final result is 31.62 ± 0.06. 31.62 ± 0.06 Question 4: Calculate (19.02 ± 0.03) - (19.01 ± 0.3). Let the two measurements be A = x_1 ± x_1 = 19.02 ± 0.03 and B = x_2 ± x_2 = 19.01 ± 0.3. We need to find the difference D = A - B. Step 1: Calculate the difference of the mean values. D_mean = x_1 - x_2 = 19.02 - 19.01 = 0.01 Step 2: Calculate the absolute uncertainty of the difference using the formula for addition/subtraction: D = sqrt(( x_1)^2 + ( x_2)^2) D = sqrt((0.03)^2 + (0.3)^2) D = sqrt(0.0009 + 0.09) D = sqrt(0.0909) ≈ 0.301496 Step 3: Round the uncertainty to one significant figure and the mean value to the same decimal place. Rounding D = 0.301496 to one significant figure gives 0.3. The uncertainty is in the tenths place. Therefore, D_mean = 0.01 should be rounded to the tenths place, which is 0.0. The final result is 0.0 ± 0.3. 0.0 ± 0.3 That's 2 down. 3 left today — send the next one.