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
You're on a roll — Step 1: Determine the nominal area of the triangle. The base b = 12 cm and the height h = 8 cm. The nominal area A_0 is given by the formula A = (1)/(2)bh. A_0 = (1)/(2) × 12 cm × 8 cm = 6 cm × 8 cm = 48 cm^2 Step 2: Determine the absolute error in the base and height. Since the measurements are to the nearest centimeter, the maximum possible absolute error for each measurement is half of the smallest unit, which is 1 cm. Absolute error in base, b = (1)/(2) cm = 0.5 cm. Absolute error in height, h = (1)/(2) cm = 0.5 cm. a) Step 3: Calculate the absolute error in the area. For a product of two quantities, A = kbh, the maximum absolute error A is given by: A = A_0 ( ( b)/(b_0) + ( h)/(h_0) ) A = 48 cm^2 ( 0.5 cm12 cm + 0.5 cm8 cm ) A = 48 ( (1)/(24) + (1)/(16) ) To add the fractions, find a common denominator, which is 48. A = 48 ( (2)/(48) + (3)/(48) ) A = 48 ( (2+3)/(48) ) A = 48 ( (5)/(48) ) A = 5 cm^2 The absolute error in calculating the area of the triangle is 5 cm^2. b) Step 4: Calculate the percentage error in the area. The percentage error is given by the formula: Percentage Error = Absolute ErrorNominal Area × 100\% Percentage Error = ( A)/(A_0) × 100\% Percentage Error = 5 cm^248 cm^2 × 100\% To calculate (5)/(48): 5 ÷ 48 ≈ 0.104166... Percentage Error ≈ 0.104166... × 100\% Percentage Error ≈ 10.4166...\% Rounding to 1 decimal place: Percentage Error ≈ 10.4\% The percentage error in the area, to 1 decimal place, is 10.4\%. What's next?