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's the solution to Question 3: a) To find the actual length of the road, we use the given map scale and convert the units. Step 1: Calculate the actual length in centimeters. The scale is 1 : 50\,000, meaning 1 cm on the map represents 50\,000 cm in reality. Actual length = 8.4 cm × 50\,000 Actual length = 420\,000 cm Step 2: Convert the actual length from centimeters to kilometers. Given 1 km = 100\,000 cm. Actual length in km = 420\,000 cm100\,000 cm/km Actual length in km = 4.2 km The actual length of the road is 4.2 km. b) To find the actual area of the lake, we use the area scale factor, which is the square of the length scale factor. Step 1: Calculate the area scale factor. The length scale factor is 50\,000. Area scale factor = (Length scale factor)^2 = (50\,000)^2 Area scale factor = 2\,500\,000\,000 Step 2: Calculate the actual area in square centimeters. Actual area = 2.5 cm^2 × 2\,500\,000\,000 Actual area = 6\,250\,000\,000 cm^2 Step 3: Convert the actual area from square centimeters to square kilometers. Since 1 km = 100\,000 cm, then 1 km^2 = (100\,000 cm)^2 = 10\,000\,000\,000 cm^2. Actual area in km^2 = 6\,250\,000\,000 cm^210\,000\,000\,000 cm^2/km^2 Actual area in km^2 = 0.625 km^2 The actual area of the lake is 0.625 km^2. c) To verify the given areas, we will convert the actual area to square centimeters and compare it with the map area using the area scale factor. Step 1: Convert the actual area from square meters to square centimeters. Given actual area is 125\,000 m^2. Since 1 m = 100 cm, then 1 m^2 = (100 cm)^2 = 10\,000 cm^2. Actual area in cm^2 = 125\,000 m^2 × 10\,000 cm^2/m^2 Actual area in cm^2 = 1\,250\,000\,000 cm^2 Step 2: Calculate the actual area based on the map area and the area scale factor. The map area is 0.5 cm^2. The area scale factor is (50\,000)^2 = 2\,500\,000\,000. Calculated actual area = 0.5 cm^2 × 2\,500\,000\,000 Calculated actual area = 1\,250\,000\,000 cm^2 Step 3: Compare the calculated actual area with the given actual area. The calculated actual area (1\,250\,000\,000 cm^2) matches the given actual area (1\,250\,000\,000 cm^2). Therefore, the statement is verified. 3 done, 2 left today. You're making progress.