Part (a):<\/p>

Let the side of the equilateral triangle be 'a'. The median of an equilateral triangle is given by m = (√3/2)a. Given dm/dt = 2√3 cm/s. We need to find da/dt.

Differentiating m = (√3/2)a with respect to time t, we get:

dm/dt = (√3/2) * (da/dt)

Substituting dm/dt = 2√3:

2√3 = (√3/2) * (da/dt)

da/dt = (2 * 2√3) / √3 = 4 cm/s

Part (b):<\/p>

Let the two numbers be x and y. Given x + y = 5, so y = 5 - x. We want to minimize x³ + y³.

Let S = x³ + y³ = x³ + (5 - x)³

Differentiating S with respect to x:

dS/dx = 3x² + 3(5 - x)²(-1) = 3x² - 3(25 - 10x + x²) = 3x² - 75 + 30x - 3x² = 30x - 75

For minimum value, dS/dx = 0:

30x - 75 = 0 => x = 75/30 = 5/2

Then y = 5 - x = 5 - 5/2 = 5/2

The sum of the squares of these numbers is x² + y² = (5/2)² + (5/2)² = 25/4 + 25/4 = 50/4 = 25/2 = 12.5