Let's solve the problem step by step to find the value of $m$.

Step 1: Find the Radius of the Sphere

The total surface area $A$ of a sphere is given by: $A = 4 \pi r^2$

Given: $A = 315 \text{ cm}^2$ $\pi = 3.15$

We set up the equation: $4 \pi r^2 = 315$ $4 \times 3.15 \times r^2 = 315$ $12.6 r^2 = 315$

Solving for $r^2$: $r^2 = \frac{315}{12.6} = 25$ $r = \sqrt{25} = 5 \text{ cm}$

Step 2: Calculate the Volume of the Sphere

The volume $V$ of a sphere is given by: $V = \frac{4}{3} \pi r^3$

Using $r = 5$ cm and $\pi = 3.15$: $V = \frac{4}{3} \times 3.15 \times 5^3$ $V = \frac{4}{3} \times 3.15 \times 125$ $V = \frac{4}{3} \times 393.75$ $V = 525 \text{ cm}^3$

Step 3: Determine the Initial Volume of Liquid in the Tank

The edge length of the cubical tank is 10 cm, so its total volume $V_{\text{tank}}$ is: $V_{\text{tank}} = 10^3 = 1000 \text{ cm}^3$

Let the initial volume of the liquid be $V_{\text{liquid}}$, which is $m\%$ of the tank's capacity: $V_{\text{liquid}} = \frac{m}{100} \times 1000 = 10m \text{ cm}^3$

Step 4: Calculate the Final Volume of the Liquid in the Tank

After dropping the sphere, the volume of the liquid increases by the volume of the sphere: $V_{\text{final}} = 10m + 525 \text{ cm}^3$

Step 5: Calculate the Initial Height of the Liquid in the Tank

The initial height $h$ of the liquid is given by the volume of the liquid divided by the base area of the tank. Since the tank is a cube with edge length 10 cm, the base area $A_{\text{base}}$ is: $A_{\text{base}} = 10 \times 10 = 100 \text{ cm}^2$

So the initial height $h$ is: $h = \frac{V_{\text{liquid}}}{A_{\text{base}}} = \frac{10m}{100} = 0.1m \text{ cm}$

Step 6: Determine the Final Height Increase

The height increase is 175% of the initial height: $\Delta h = 1.75 \times h = 1.75 \times 0.1m = 0.175m \text{ cm}$

The final height $h_{\text{final}}$ is: $h_{\text{final}} = h + \Delta h = 0.1m + 0.175m = 0.275m \text{ cm}$

Step 7: Calculate the Final Volume Based on the Height Increase

The final volume of the liquid based on the new height is: $V_{\text{final}} = h_{\text{final}} \times A_{\text{base}} = 0.275m \times 100 \text{ cm}^2 = 27.5m \text{ cm}^3$

Step 8: Set Up the Equation and Solve for $m$

We know that the final volume is also given by: $V_{\text{final}} = 10m + 525$

Thus, we set up the equation: $27.5m = 10m + 525$

Solving for $m$: $27.5m - 10m = 525$ $17.5m = 525$ $m = \frac{525}{17.5} = 30$

Thus, the value of $m$ is $30$.

Would you like further details or have any questions?

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Relative Questions:

1. How is the radius of a sphere derived from its surface area?
2. What is the volume formula for a sphere, and how is it applied?
3. How do you determine the volume of a cube?
4. What does the term "175% increase in height" imply in this context?
5. How does the volume of a liquid in a cubical tank relate to the height of the liquid?
6. What are the steps to solve the problem if the tank had a different edge length?
7. How would you calculate the volume change if the sphere were a different shape, such as a cylinder?
8. Why is it important to ensure all units are consistent when performing these calculations?

Tip: When solving geometry problems involving volume, always check to ensure the units are consistent throughout the calculation to avoid errors.