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The problem describes water leaking from an inverted conical tank, where water is also being pumped in. To solve this, we need to determine the rate at which water is being pumped into the tank, given certain conditions.

### Step-by-Step Solution

1. **Identify given dataThe problem describes water leaking from an inverted conical tank, where water is also being pumped in. To solve this, we need to determine the rate at which water is being pumped into the tank, given certain conditions.

### Step-by-Step Solution

1. **Identify given **
   - Rate at which water is leaking out of the tank: \(-12700.0 \, \text{cm}^3/\text{min}\) (negative because it's leaking).
   - Height of the tank: \(10.0 \, \text{m} = 1000 \, \text{cm}\).
   - Diameter of the tank at the top: \(3.5 \, \text{m} = 350 \, \text{cm}\), giving the radius \(r = 175 \, \text{cm}\).
   - Water level is rising at a rate: \(\frac{dh}{dt} = 16.0 \, \text{cm/min}\) when the height of the water is \(h = 1.5 \, \text{m} = 150 \, \text{cm}\).

2. **Relationship between height and radius using similar triangles:**
   The radius \(r\) of the water's surface and the height \(h\) of the water are proportional to the dimensions of the tank:
   \[
   \frac{r}{h} = \frac{175}{1000}
   \]
   Thus, the radius \(r\) of the water surface at height \(h\) is:
   \[
   r = \frac{175}{1000}h = 0.175h
   \]

3. **Volume of water in the cone:**
   The volume \(V\) of the water in a cone is given by:
   \[
   V = \frac{1}{3} \pi r^2 h
   \]
   Substituting \(r = 0.175h\):
   \[
   V = \frac{1}{3} \pi (0.175h)^2 h = \frac{1}{3} \pi (0.030625h^2) h = \frac{1}{3} \pi (0.030625h^3)
   \]
   Simplifying:
   \[
   V = \frac{\pi}{3} \times 0.030625h^3 = 0.0102083\pi h^3
   \]

4. **Differentiate \(V\) with respect to time:**
   To find the rate of change of volume with respect to time (\(\frac{dV}{dt}\)), apply the chain rule:
   \[
   \frac{dV}{dt} = \frac{d}{dt} \left( 0.0102083\pi h^3 \right) = 0.0102083 \pi \times 3h^2 \times \frac{dh}{dt}
   \]
   Substituting \(h = 150 \, \text{cm}\) and \(\frac{dh}{dt} = 16.0 \, \text{cm/min}\):
   \[
   \frac{dV}{dt} = 0.0102083 \pi \times 3 \times 150^2 \times 16.0
   \]
   Let me calculate this value.The rate of change of the volume of water, \(\frac{dV}{dt}\), is approximately \(34635.95 \, \text{cm}^3/\text{min}\).

5. **Determine the rate at which water is being pumped into the tank:**

We know from the problem that water is being pumped in at a rate \(R\), while it is leaking out at a rate of \(12700 \, \text{cm}^3/\text{min}\). Therefore, the relationship between the rates is:
\[
R - 12700 = \frac{dV}{dt}
\]
Substituting \(\frac{dV}{dt} = 34635.95 \, \text{cm}^3/\text{min}\):
\[
R - 12700 = 34635.95
\]
Solving for \(R\):
\[
R = 34635.95 + 12700 = 47335.95 \, \text{cm}^3/\text{min}
\]

Thus, the rate at which water is being pumped into the tank is approximately \(47335.95 \, \text{cm}^3/\text{min}\).

---

Would you like a more detailed explanation or have any other questions? Here are five related questions to consider:

1. How does the relationship between height and radius change if the tank's dimensions are modified?
2. What if the water is leaking at a different rate? How would that affect the solution?
3. How can we express the time it will take for the tank to fill completely?
4. How would the solution change if the tank's shape were cylindrical rather than conical?
5. How can we incorporate units' conversion into the problem, like converting cm\(^3\)/min to liters/min?

**Tip:** Always check units carefully, especially in problems involving geometry and rates, to ensure consistency throughout the solution.

Water is leaking out of an inverted conical tank at a rate of 12700.0 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 10.0 meters and the diameter at the top is 3.5 meters. If the water level is rising at a rate of 16.0 centimeters per minute when the height of the water is 1.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Learn how to calculate the rate at which water is being pumped into an inverted conical tank. This problem involves related rates, geometry, and differentiation. Given the tank dimensions and the rate of water leaking, determine the pump rate when the water level is rising at 16 cm/min at a height of 1.5 meters.

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