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To solve this problem, we need to calculate the force exerted by water pressure on the dam using hydrostatic pressure principles. The force exerted by water pressure on the dam can be found using the formula:

\[
F = \rho g \int_a^b (h(y) \cdot L(y)) \, dy
\]

where:
- \(\rho\) is the density of water (\(1000 \, \text{kg/m}^3\)),
- \(g\) is the acceleration due to gravity (\(9.8 \, \text{m/s}^2\)),
- \(h(y)\) is the depth of water at a given height \(y\) (measured from the top of the water),
- \(L(y)\) is the width of the dam at height \(y\) (determined by the curve).

Given:
- The curve representing the shape of the dam is \(y = 0.6x^4\).
- The water level is 38 meters below the top of the dam, so the top of the water is at \(y = 160 - 38 = 122\).
- The water surface is at \(y = 122\) and the bottom of the dam is at \(y = 0\).

Now let's calculate the force:

1. **Depth function \(h(y)\):** Since water depth at height \(y\) from the base of the dam is \(122 - y\), the depth function is \(h(y) = 122 - y\).

2. **Width function \(L(y)\):** The width of the dam at height \(y\) is given by solving \(y = 0.6x^4\) for \(x\). This gives \(x = \left(\frac{y}{0.6}\right)^{\frac{1}{4}}\). Since the dam is symmetric, \(L(y) = 2x = 2 \left(\frac{y}{0.6}\right)^{\frac{1}{4}}\).

3. **Setting up the integral:**

\[
F = 1000 \times 9.8 \int_0^{122} (122 - y) \cdot 2 \left(\frac{y}{0.6}\right)^{\frac{1}{4}} \, dy
\]

Let's calculate this integral to find the force \(F\).It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the calculations manually if you'd like! 

Would you like me to walk you through the steps to calculate the force exerted by water pressure on the dam, or is there another specific question you'd like help with? Let me know how you'd like to proceed!

Here are a few related questions that might help deepen your understanding:
1. How does changing the water density affect the force exerted on the dam?
2. What impact does altering the shape of the dam (the curve equation) have on the pressure distribution?
3. How would the force change if the water level increased by 10 meters?
4. What is the significance of the width function \(L(y)\) in the force calculation?
5. How would you find the point of maximum pressure on the dam?

**Tip:** Remember that hydrostatic pressure increases linearly with depth, so the force exerted on a submerged surface grows rapidly as depth increases.

A dam on a river is built so that the wall facing the water is shaped like the region above the curve y = 0.6x^4 and below the line y = 160. The water level is 38 meters below the top of the dam. Find the force (in Newtons, to one decimal place) exerted on the dam by water pressure. Water has a density of 1000 kg/m^3, and the acceleration of gravity is 9.8 m/s^2.

Learn how to calculate the force exerted by water pressure on a dam using hydrostatic pressure principles and calculus. This problem involves the definite integral of a function representing the dam's shape and water depth, suitable for college-level students.

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