To determine the height of the water level in the cylindrical vessel when the cubical tank's water is transferred, follow these steps:
1. Calculate the Volume of the Cubical Tank:
Since the side length of the cubical tank is given as 12 cm, we use the formula for the volume of a cube:
[tex]\[
\text{Volume of a cube} = \text{side}^3
\][/tex]
Substituting the given side length:
[tex]\[
\text{Volume of the cubical tank} = 12^3 = 12 \times 12 \times 12 = 1728 \text{ cm}^3
\][/tex]
2. Volume of Water Transferred:
The total volume of water being transferred from the cubical tank to the cylindrical vessel is 1728 cm³.
3. Calculate the Height of the Water Level in the Cylindrical Vessel:
The formula for the volume of a cylinder is:
[tex]\[
\text{Volume of a cylinder} = \pi \times r^2 \times h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height. We are given:
[tex]\[
r = 3.5 \text{ cm}
\][/tex]
We need to find [tex]\( h \)[/tex], the height of the water level. Rearrange the volume formula to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{\text{Volume of the cylinder}}{\pi \times r^2}
\][/tex]
Substitute the known values ([tex]\(\text{Volume of water} = 1728 \text{ cm}^3\)[/tex] and using [tex]\(\pi = 3.14\)[/tex]):
[tex]\[
h = \frac{1728}{3.14 \times (3.5)^2}
\][/tex]
First, calculate [tex]\((3.5)^2\)[/tex]:
[tex]\[
(3.5)^2 = 3.5 \times 3.5 = 12.25
\][/tex]
Now substitute this back into the height formula:
[tex]\[
h = \frac{1728}{3.14 \times 12.25} = \frac{1728}{38.465}
\][/tex]
Perform the division:
[tex]\[
h \approx 44.92 \text{ cm}
\][/tex]
Thus, the height of the water level in the cylindrical vessel will be approximately [tex]\( 44.92 \)[/tex] cm.
