Find the fourth proportional of $$5\sqrt{24}, \, 3\sqrt{10}, \, 3\sqrt{15}$$ .

Correct option is A

Given:
The three given numbers are $$5\sqrt{24}, \, 3\sqrt{10}, \, 3\sqrt{15}$$​​
Formula Used:
​$$\text{If } a : b = c : d, \text{ then } d = \frac{b \times c}{a}$$​​
Solution:
Let the fourth proportional be x.
​$$\frac{5\sqrt{24}}{3\sqrt{10}} = \frac{3\sqrt{15}}{x} \\x = \frac{3\sqrt{10} \times 3\sqrt{15}}{5\sqrt{24}} \\x = \frac{3 \times 3 \times \sqrt{10} \times \sqrt{15}}{5 \times \sqrt{24}} \\x = \frac{9 \times \sqrt{150}}{5 \times \sqrt{24}} \\x = \frac{9 \times 5\sqrt{6}}{5 \times 2\sqrt{6}} \\x = \frac{45\sqrt{6}}{10\sqrt{6}} \\x = \frac{45}{10} \\x = 4.5$$​​
Thus, the fourth proportional is 4.5.