If the sum of the length, breadth, and height of a rectangular parallelepiped is 24 cm and the length of its diagonal is 15 cm, what is its total surface area?

Let length = $l$, breadth = $b$, height = $h$

Given:

$l + b + h = 24$
Diagonal = 15

So,
$l^2 + b^2 + h^2 = 15^2 = 225$

Step 1: Use identity

$(l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl)$

Substitute:
$24^2 = 225 + 2(lb + bh + hl)$

$576 = 225 + 2(lb + bh + hl)$

Step 2: Solve

$576 - 225 = 2(lb + bh + hl)$

$351 = 2(lb + bh + hl)$

$lb + bh + hl = 175.5$

Step 3: Total Surface Area

TSA = $2(lb + bh + hl)$

TSA = $2 \times 175.5 = 351$

Final Answer: 351 cm²