To solve the problem of finding the volume of the larger tank, given the ratio of the lengths of the corresponding sides and the volume of the smaller tank, follow these steps:
Step 1: Understand the given ratio of the corresponding sides.
The ratio of the corresponding sides of the smaller tank to the larger tank is given as 3:5. This means that every dimension (length, width, and height) of the small tank is scaled up by this ratio to get the dimensions of the larger tank.
Step 2: Use the properties of similar figures to determine the ratio of the volumes.
For similar objects, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (lengths, widths, or heights). Therefore, the volume ratio can be calculated by cubing the linear ratio:
[tex]\[
\text{Volume Ratio} = \left(\frac{\text{side of small tank}}{\text{side of large tank}}\right)^3 = \left(\frac{3}{5}\right)^3
\][/tex]
Step 3: Calculate the volume ratio.
[tex]\[
\frac{3}{5} = 0.6
\][/tex]
[tex]\[
\text{Volume Ratio} = (0.6)^3 = 0.216
\][/tex]
Step 4: Use the volume ratio to find the volume of the larger tank.
Given that the volume of the smaller tank is 8.1 cubic meters, we use the volume ratio to find the volume of the larger tank. The volume of the smaller tank is a fraction of the volume of the larger tank according to the volume ratio:
[tex]\[
\frac{\text{Volume of Smaller Tank}}{\text{Volume Ratio}} = \text{Volume of Larger Tank}
\][/tex]
[tex]\[
8.1 \div 0.216 = \text{Volume of Larger Tank}
\][/tex]
Step 5: Perform the division to find the volume of the larger tank.
[tex]\[
8.1 \div 0.216 = 37.5 \text{ cubic meters}
\][/tex]
Therefore, the volume of the larger tank is [tex]\( \boxed{37.5} \)[/tex] cubic meters.
