Calculate Water Height Y in a 150 cm³ Rectangular Prism with 50 ml Liquid

To solve this problem, we'll follow these steps:

• Step 1: Calculate the base area of the rectangular prism.
• Step 2: Use the base area to find the height of the water, $Y$.

Now, let's work through each step:

Step 1: The volume of the rectangular prism is given as 150 cm$^3$. Let's denote the height of the prism without any liquid as $H$, and the gap from the water line to the top of the prism is 3 cm. Thus, the total height is $H - 3$. The water volume given is 50 cm$^3$, which means the volume occupied by the air is:

$150 - 50 = 100$ cm$^3$.

Since the gap is 3 cm (the height of air column), the base area $A$ can be calculated as:

$A \times 3 = 100$.

This implies $A = \frac{100}{3} \approx 33.\overline{3}$ cm$^2$.

Step 2: Now, to find $Y$ (the height of water):

We use the formula for the volume of water in the prism:

$A \times Y = 50$,

where $A = 33.\overline{3}$ cm$^2$. Therefore,

$Y = \frac{50}{33.\overline{3}} = 1.5$ cm.

Therefore, the solution to the problem is $Y = 1.5 \, \text{cm}$.