Find two mean proportional between 2 and 54.

Correct option is A

Formula for Mean Proportion: 

If $$a, x_1, x_2, b$$​ are in continued proportion, then:

$$\frac{a}{x_1} = \frac{x_1}{x_2} = \frac{x_2}{b}$$​​

Step-by-Step Solution:
#Step 1: Let the two mean proportion be $$x_1$$​ and $$x_2$$​ .
- Given: a = 2 , b = 54 .

#Step 2: Relationship between terms:

$$x_1 = \sqrt{a \cdot x_2}, \quad x_2 = \sqrt{x_1 \cdot b}$$​


#Step 3: Use $$x_1^3 = a \cdot b$$​ :

$$x_1^3 = 2 \cdot 54 = 108$$​


#Step 4: Solve for $$x_2$$​ :

$$x_2 = \frac{a \cdot b}{x_1^2} = \frac{2 \cdot 54}{6^2} = \frac{108}{36} = 18$$​


Final Answer:
The two mean proportionals are 6 and 18.

Correct Option: A) 6 and 18.