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Question 1174416: A 10-feet tall, 10-feet wide truck is approaching a tunnel on a one-way road. The arch at the tunnel’s
entrance forms the upper half of an ellipse with a height of 15 feet at the center and a base of 12 feet wide.
Consider that the center is at the origin, will the truck be able to fit through the arch?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's analyze the ellipse describing the tunnel's entrance.
**1. Equation of the Ellipse:**
* The ellipse has a horizontal major axis (base of 12 feet) and a vertical minor axis (height of 15 feet).
* The center is at the origin (0, 0).
* The semi-major axis (horizontal) is a = 12 / 2 = 6 feet.
* The semi-minor axis (vertical) is b = 15 feet.
* The equation of the ellipse is: (x^2 / a^2) + (y^2 / b^2) = 1
* Substituting the values of a and b: (x^2 / 36) + (y^2 / 225) = 1
**2. Truck Dimensions:**
* Truck height: 10 feet
* Truck width: 10 feet
**3. Checking if the Truck Fits:**
* Since the truck is 10 feet wide, we need to check the tunnel's height at x = 10 / 2 = 5 feet (half the truck's width).
* Plug x = 5 into the ellipse equation and solve for y:
* (5^2 / 36) + (y^2 / 225) = 1
* (25 / 36) + (y^2 / 225) = 1
* (y^2 / 225) = 1 - (25 / 36) = (36 - 25) / 36 = 11 / 36
* y^2 = 225 * (11 / 36)
* y = sqrt(225 * 11 / 36) = (15 / 6) * sqrt(11) = (5 / 2) * sqrt(11)
* y ≈ (5 / 2) * 3.3166 ≈ 8.2915 feet
* Since the tunnel is only the upper half of the ellipse, we keep the positive value of y.
* The tunnel's height at 5 feet from the center is approximately 8.2915 feet.
**4. Comparing Heights:**
* The truck is 10 feet tall.
* The tunnel's height at 5 feet from the center is about 8.2915 feet.
* Since 10 feet > 8.2915 feet, the truck is taller than the tunnel at the edges of the truck.
* Therefore, the truck will not fit through the arch.
**Conclusion:**
The truck will not be able to fit through the arch.
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