To find the area of a triangle with sides 0.24 m, 28 cm, and 32 cm, we first need to convert all measurements to a consistent unit. Let's convert the 0.24 m to centimeters: 0.24 m = 24 cm. The sides now are 24 cm, 28 cm, and 32 cm. We use Heron's formula to find the area of the triangle.

Heron's formula states that the area of a triangle with sides a, b, and c is given by:

Area = √(s × (s - a) × (s - b) × (s - c))

where s is the semi-perimeter of the triangle:

s = (a + b + c) / 2

Inserting our values:

s = (24 + 28 + 32) / 2 = 42

Now, apply Heron's formula:

Area = √(42 × (42 - 24) × (42 - 28) × (42 - 32))

Calculate the terms:

• 42 - 24 = 18
• 42 - 28 = 14
• 42 - 32 = 10

Substitute these back into the formula:

Area = √(42 × 18 × 14 × 10)

Calculating inside the square root gives:

• 42 × 18 = 756
• 756 × 14 = 10584
• 10584 × 10 = 105840

Finally, calculate the square root:

Area = √105840

We recognize that the expression can be rewritten to include a perfect square:

Area = √(15 × 7056)

Since √7056 = 84, we get:

Area = 84√15 cm²

Thus, the area of the triangle is 84√15 cm², which corresponds to the correct answer 84√15 cm².