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.
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Question 5.1.3 The distance from Burgersfort to Cape Town is 1 750 km. The cyclists are expected to take how many days for their trip if they do not overcome any challenges along the way? Show your calculations. Step 1: Identify the pattern of distances covered per day. The distances between rests form an arithmetic progression: 100 km, 150 km, 200 km, ... The first term (a) is 100 km. The common difference (d) is 150 - 100 = 50 km. We assume each segment between rests represents a day's travel. Step 2: Write the formula for the sum of an arithmetic series. The total distance (S_N) covered after N days is given by: S_N = (N)/(2)(2a + (N-1)d) Step 3: Substitute the known values into the formula. We have S_N = 1750 km, a = 100 km, and d = 50 km. 1750 = (N)/(2)(2(100) + (N-1)50) 1750 = (N)/(2)(200 + 50N - 50) 1750 = (N)/(2)(150 + 50N) Step 4: Simplify and solve the equation for N. 1750 = N(75 + 25N) 1750 = 75N + 25N^2 Rearrange into a standard quadratic equation: 25N^2 + 75N - 1750 = 0 Divide the entire equation by 25 to simplify: N^2 + 3N - 70 = 0 Factor the quadratic equation: (N+10)(N-7) = 0 This gives two possible solutions for N: N+10 = 0 N = -10 N-7 = 0 N = 7 Step 5: Choose the valid solution. Since the number of days cannot be negative, we take the positive value. N = 7 The cyclists are expected to take 7 days for their trip. Last free one today — make it count tomorrow, or type /upgrade for unlimited.