If momentum (P ), area (A) and time (T ) are taken to be fundamental quantities, then energy has the dimensional formula

The correct option is (P¹ A¹/² T⁻¹).

Let us derive this step-by-step using dimensional analysis. We want to express energy (E) in terms of momentum (P), area (A), and time (T) as fundamental quantities.

First, let's write down the standard dimensions of the given quantities in the mass-length-time (M, L, T) system:

1. Energy (E): Since energy is work done (force × distance), its dimensional formula is:
[E] = [M L² T⁻²]

2. Momentum (P): Since momentum is mass × velocity, its dimensional formula is:
[P] = [M L T⁻¹]

3. Area (A): Its dimensional formula is:
[A] = [L²]
This implies that:
[L] = [A¹/²]

4. Time (T): Its dimensional formula is:
[T] = [T]

Now, let's express momentum (P) in terms of mass, length, and time: [P] = [M L T⁻¹].
We can rewrite energy (E) by grouping terms to relate it to momentum (P):
[E] = [M L² T⁻²] = [M L T⁻¹] × [L T⁻¹]

Substituting [P] for [M L T⁻¹], we get:
[E] = [P] × [L] × [T⁻¹]

Now, we substitute the dimensional formula of length (L) in terms of Area (A), which is [L] = [A¹/²]:
[E] = [P] × [A¹/²] × [T⁻¹]

Therefore, writing this in terms of powers of the fundamental quantities P, A, and T, we obtain the dimensional formula for energy as:
[E] = [P¹ A¹/² T⁻¹]