If the mean proportional of A and 98 is 84, then find the value of A.

For quantities a and b, the mean proportional (also called the geometric mean) is the value m such that a, m, and b form a continued proportion, written a : m = m : b. This gives the identity m2 = a × b.

1. Given: the mean proportional of A and 98 is 84, so A, 84, and 98 are in continued proportion — A : 84 = 84 : 98.

2. This continued proportion gives the identity 842 = A × 98.

3. Compute 842: 84 × 84 = 7056.

4. Solve for A: A = 7056 ÷ 98 = 72.

Cross-check: the mean proportional of 72 and 98 is √(72 × 98) = √7056 = 84, which matches the given value — confirming A = 72.