We are tasked with finding how much the height of the liquid will change after 277 microliters ($\mu L$) of water is added to the cylinder.

Given:

• Volume of water initially: $V_1 = 5.00 \, \text{mL} = 5000 \, \mu L$
• Initial height of water: $h_1 = 57.0 \, \text{mm}$
• Volume of water added: $\Delta V = 277 \, \mu L$

Step 1: Calculate the cross-sectional area of the cylinder

The height $h_1$ and the volume $V_1$ are related by the formula for the volume of a cylinder: $V_1 = A \times h_1$ Where:

• $A$ is the cross-sectional area of the cylinder (constant),
• $h_1$ is the height of the liquid,
• $V_1$ is the volume of the liquid.

Rearranging the formula to solve for $A$: $A = \frac{V_1}{h_1}$

Substitute the known values ($V_1 = 5000 \, \mu L$ and $h_1 = 57.0 \, \text{mm}$): $A = \frac{5000 \, \mu L}{57.0 \, \text{mm}} = \frac{5000 \, \text{mm}^3}{57.0 \, \text{mm}} \approx 87.72 \, \text{mm}^2$

Step 2: Calculate the change in height

The new volume of water after adding 277 $\mu L$ will be: $V_2 = V_1 + \Delta V = 5000 \, \mu L + 277 \, \mu L = 5277 \, \mu L$

The change in height $\Delta h$ is given by the relation: $\Delta h = \frac{\Delta V}{A}$

Substitute $\Delta V = 277 \, \mu L = 277 \, \text{mm}^3$ and $A = 87.72 \, \text{mm}^2$: $\Delta h = \frac{277 \, \text{mm}^3}{87.72 \, \text{mm}^2} \approx 3.16 \, \text{mm}$

Final Answer:

The height of the liquid will increase by approximately $3.16 \, \text{mm}$.

---

Would you like more details or have any questions about the steps? Here are some related questions:

1. How does the height change relate to the volume in different shaped containers?
2. How can we determine the diameter of the cylinder from the cross-sectional area?
3. What happens if we add 1.00 mL of water instead?
4. How would the result change if the cylinder were not perfectly vertical?
5. Can this problem be extended to non-cylindrical containers?

Tip: In physics and geometry problems, knowing how to manipulate formulas is key to switching between volume, area, and height relationships.