Question

There are two two numbers whose product is 320 and difference of their cubes is to the cube of their differences has ratio 61:1. Find the numbers

Answer 1

Answer:

x * y = 320 => y = 320/x

x^3 - y^3 / (x - y)^3 = 61/1

x^3 - y^3 = 61 * (x - y)^3

(x - y) * (x^2 + xy + y^2) = 61 * (x - y)^3

x^2 + xy + y^2 = 61 * (x - y)^2

x^2 + xy + y^2 = 61 * (x^2 - 2xy + y^2)

x^2 + xy + y^2 = 61x^2 - 122xy + 61y^2

60x^2 - 123xy + 60y^2 = 0

put y = 320/x, we get

60x^2 - 123*x*320/x + 60*(320/x)^2 = 0

60x^2 - 123*320 + 60*(320/x)^2 = 0

Divide by 60, we get

x^2 - 656 + 102400/x^2 =0

Multiply by x^2, we get

x^4 - 656x^2 + 102400 = 0

put x^2 = a, we get

a^2 - 656a + 102400 = 0

factorise method used, we get

a^2 - 256a - 400a + 102400 = 0

(a - 256) * (a - 400) = 0

a = 256 or a = 400

resubstitude x^2 = a, we get

x^2 = 256 or x^2 = 400

take squareroot

x = 16 or x = 20

Required Numbers are 16, 20

Step-by-step explanation: