To determine the rate at which oil is flowing from the site of the rig, we need to find the rate of change in the volume of the oil slick. The slick has an elliptical shape with a thickness of 7 inches, which is equivalent to $\frac{7}{12}$ feet, or approximately $0.5833$ feet.

Step 1: Volume of the Elliptical Slick

The volume $V$ of an elliptical cylinder is given by: $V = \text{Area of ellipse} \times \text{thickness}$ The area $A$ of an ellipse with a semi-major axis $a$ and semi-minor axis $b$ is: $A = \pi a b$ Thus: $V = \pi a b \times \text{thickness}$

Step 2: Substitute Known Values

Here:

• The major axis is 10 miles long, so $a = 5$ miles = $5 \times 5280 = 26400$ feet.
• The minor axis is 0.3 miles wide, so $b = 0.15$ miles = $0.15 \times 5280 = 792$ feet.
• The thickness is 0.5833 feet.

So, $V = \pi \times 26400 \times 792 \times 0.5833$

Step 3: Differentiate with Respect to Time

Now, we want to find $\frac{dV}{dt}$, the rate of change of the volume. Applying the product rule: $\frac{dV}{dt} = \pi \left( b \cdot \frac{da}{dt} + a \cdot \frac{db}{dt} \right) \times \text{thickness}$

We know:

• $\frac{da}{dt} = 30$ feet per hour (rate of increase of the semi-major axis),
• $\frac{db}{dt} = 6$ feet per hour (rate of increase of the semi-minor axis).

Thus: $\frac{dV}{dt} = \pi \left( 792 \cdot 30 + 26400 \cdot 6 \right) \times 0.5833$

Step 4: Calculate

1. Calculate $792 \cdot 30 = 23760$.
2. Calculate $26400 \cdot 6 = 158400$.
3. Sum the products: $23760 + 158400 = 182160$.
4. Multiply by $\pi$ and then by 0.5833:

$\frac{dV}{dt} \approx \pi \times 182160 \times 0.5833 \approx 333731 \text{ cubic feet per hour}$

Conclusion

The rate at which oil is flowing from the rig is approximately 333731 cubic feet per hour.

Would you like further clarification on this calculation, or have questions about related topics?

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Here are some related questions:

1. What is the difference between elliptical and circular area calculations?
2. How does thickness influence the volume rate in 3D shapes?
3. What are common applications of calculus in environmental science?
4. How do conversions affect precision in rate-related calculations?
5. Why does changing one axis impact the volume rate more than the other?

Tip: When solving rate problems, always identify units for consistency.