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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Here are the solutions to the problems: Problem 1: a) Find the donation given in 2012. Step 1: Identify the initial donation, annual increase rate, and number of years. The initial donation in 2010 is P = \7,000. The annual increase rate is r = 8\% = 0.08. The number of years from 2010 to 2012 is n = 2012 - 2010 = 2 years. Step 2: Use the compound growth formula to calculate the donation in 2012. The formula for the amount after n years is A = P(1 + r)^n. A_2012 = \7,000 (1 + 0.08)^2 A_2012 = \7,000 (1.08)^2 A_2012 = \7,000 × 1.1664 Step 3: Calculate the final donation amount. A_2012 = \8,164.80 The donation given in 2012 is \8,164.80. b) Find the total amount of money donated from 2010 to 2013. Step 1: Calculate the donation for each year from 2010 to 2013. Donation in 2010 (A_2010) = \7,000 Donation in 2011 (A_2011) = \7,000 (1.08)^1 = \7,000 × 1.08 = \7,560 Donation in 2012 (A_2012) = \7,000 (1.08)^2 = \7,000 × 1.1664 = \8,164.80 Donation in 2013 (A_2013) = \7,000 (1.08)^3 = \7,000 × 1.259712 = \8,817.984 Step 2: Sum the donations for all four years. Total amount = A_2010 + A_2011 + A_2012 + A_2013 Total amount = \7,000 + \7,560 + \8,164.80 + \8,817.984 Step 3: Calculate the total sum. Total amount = \31,542.784 Step 4: Round the total amount to two decimal places for currency. Total amount = \31,542.78 The total amount of money donated from 2010 to 2013 is \31,542.78. Problem 2: a) Determine how far apart X and Y will be after 3 hours. Step 1: Identify the speeds of the cars and the time traveled. Speed of car X (S_X) = 60 km/h Speed of car Y (S_Y) = 80 km/h Time (t) = 3 hours The cars travel in opposite directions. Step 2: Calculate the distance traveled by each car. Distance traveled by X (D_X) = S_X × t = 60 km/h × 3 h = 180 km Distance traveled by Y (D_Y) = S_Y × t = 80 km/h × 3 h = 240 km Step 3: Calculate the total distance apart. Since they travel in opposite directions from a common point, their total distance apart is the sum of the distances each traveled. Total distance apart = D_X + D_Y = 180 km + 240 km = 420 km Alternatively, using relative speed: Relative speed = S_X + S_Y = 60 km/h + 80 km/h = 140 km/h Total distance apart = Relative speed × Time = 140 km/h × 3 h = 420 km Cars X and Y will be 420 km$ apart after 3 hours. Got more? Send 'em 📸