To determine how much the water level will rise when 325 cubic centimeters of water are poured into a half-full rectangular tank with given dimensions, we can follow these steps:
1. Calculate the Area of the Tank's Base:
- The base of the tank is a rectangle with dimensions [tex]\(10 \text{ cm} \times 5 \text{ cm}\)[/tex].
- The area of the base ([tex]\(A\)[/tex]) is:
[tex]\[
A = \text{length} \times \text{width} = 10 \text{ cm} \times 5 \text{ cm} = 50 \text{ cm}^2
\][/tex]
2. Determine the Initial Volume of Water in the Tank:
- The tank is initially half full.
- The total height of the tank is [tex]\(20 \text{ cm}\)[/tex].
- Since the tank is half full, the height of the water is:
[tex]\[
\text{initial height} = \frac{\text{total height}}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm}
\][/tex]
- The volume ([tex]\(V_{\text{initial}}\)[/tex]) of water initially in the tank is calculated as:
[tex]\[
V_{\text{initial}} = \text{base area} \times \text{initial height} = 50 \text{ cm}^2 \times 10 \text{ cm} = 500 \text{ cm}^3
\][/tex]
3. Volume of Water Added:
- The volume of water being added is given as [tex]\(325 \text{ cm}^3\)[/tex].
4. Calculate the Final Volume of Water in the Tank:
- The final volume ([tex]\(V_{\text{final}}\)[/tex]) of water in the tank after adding the water is:
[tex]\[
V_{\text{final}} = V_{\text{initial}} + \text{added volume} = 500 \text{ cm}^3 + 325 \text{ cm}^3 = 825 \text{ cm}^3
\][/tex]
5. Determine the New Height of the Water Level:
- With the final volume known, we can find the new height ([tex]\(h_{\text{new}}\)[/tex]) of the water level in the tank:
[tex]\[
h_{\text{new}} = \frac{V_{\text{final}}}{\text{base area}} = \frac{825 \text{ cm}^3}{50 \text{ cm}^2} = 16.5 \text{ cm}
\][/tex]
6. Calculate the Rise in Water Level:
- The rise in the water level is the difference between the new height and the initial height:
[tex]\[
\text{rise in water level} = h_{\text{new}} - \text{initial height} = 16.5 \text{ cm} - 10 \text{ cm} = 6.5 \text{ cm}
\][/tex]
Therefore, the water level will rise by [tex]\(6.5 \text{ centimeters}\)[/tex] when 325 cubic centimeters of water are poured into the tank.
