The difference between two positive numbers is 4 and the difference between their cubes is 316. Find :

(i) their product

(ii) the sum of their squares.

Let two positive numbers be x and y with x > y.

Given,

The difference between two positive numbers is 4 and the difference between their cubes is 316.

x - y = 4 and x³ - y³ = 316.

(i) Given,

⇒ x - y = 4

Cubing both sides we get :

⇒ (x - y)³ = 4³

⇒ x³ - y³ - 3xy(x - y) = 64

⇒ 316 - 3xy × 4 = 64

⇒ 316 - 12xy = 64

⇒ 12xy = 316 - 64

⇒ 12xy = 252

⇒ xy = $\dfrac{252}{12}$ = 21.

Hence, product of numbers = 21.

(ii) Given,

x - y = 4

Squaring both sides we get :

⇒ (x - y)² = 4²

⇒ x² + y² - 2xy = 16

⇒ x² + y² - 2 × 21 = 16

⇒ x² + y² - 42 = 16

⇒ x² + y² = 16 + 42

⇒ x² + y² = 58.

Hence, sum of squares = 58.