two numbers differ by 40 and their arithmetic mean exceeds their positive geometric mean by 2
Answer (1)
Answer:The two numbers are 81 and 121.Step-by-step explanation:Let a and b be the two numbers, with b being greater than a (b>a).The two numbers differ by 40:b - a = 40Then, their arithmetic mean (AM) will be:AM = (a + b)/2Their geometric mean (GM) will be:GM = √(ab)Now, the arithmetic mean exceeds the geometric mean by 2:AM - GM = 2Substituting the previous equations:(a+b)/2 - √ab = 2(a+b) - 2√ab = 4a + b - 4 = 2√ab(a+b-4)^2 = 4abSubstituting b = 40 + a(a+40+a-4)^2 = 4a(40+a)(2a+36)^2= 4a(40+a)4a^2 +144a + 1296 = 4a(40+a)a^2 + 36a + 324 = 40a + a^24a = 324a = 81b - 81 = 40b = 121Hence, a=81 and b=121
