Answer:
The volume of Solid A is 120 cm³.
Step-by-step explanation:
Given the area of the solids are in the ratio: 4 : 9
Let their actual areas be 4x² and 9x². We have assumed the common ratio as x² for easier computation.
We have area: [tex]$ A_A : A_B = $[/tex] 4x² : 9x²
This implies, the length should have been of the ratio of the square root of these numbers.
Therefore, the ratios of the lengths is: [tex]$ \sqrt{4x^2} : \sqrt{9x^2} $[/tex]
[tex]$ L_A : L_B = 2x : 3x $[/tex]
Now, volume is cube of lengths. Therefore, the ratio of volumes would be:
[tex]$ V_A : V_B = 8x^3 : 27x^3 $[/tex]
Given [tex]$ V_B = 405 cm^3 $[/tex]
[tex]$ \implies 27x^3 = 405 \hspace{1mm} cm^3 $[/tex]
[tex]$ \implies x^3 = \frac{405}{27} = 15 $[/tex]
Hence, the volume of solid A, [tex]$ V_A = 8x^3 = 8 \times 15 = \textbf{120} \hspace{1mm} cm^3 $[/tex].
Hence, the answer.
