A single-engine airplane carries enough fuel for an 8-hour flight. After the airplane has been flying for 1 h, the fuel tank begins to leak at a rate that would empty the tank in 12 h. How long after the leak begins does the plane have until it runs out of fuel?
The rest of the fuel can last 7 hours in flight, and will leak in 12 hours.
So each hour, the plane uses 1/7th of what's left, and leaks 1/12th of what's left, for a total of 19/84th of what's left.
It will therefore last 84/19 hours.
To find out how long the plane has until it runs out of fuel after the leak begins, we need to first determine the amount of time that has passed since the leak started.
Let's assume that the leak started x hours after the airplane began flying.
Since the airplane carries enough fuel for an 8-hour flight, the total amount of fuel in the tank initially is sufficient for 8 hours.
However, after 1 hour of flying, the fuel tank has been depleted by 1/8th, leaving 7/8th of the fuel remaining.
Now, the leak rate is such that it would empty the tank in 12 hours. So, every hour the leak is active, it reduces the fuel level by 1/12th.
Therefore, the equation representing the fuel level after x hours can be written as:
Initial fuel level - fuel depletion after 1 hour - fuel depletion due to the leak = Remaining fuel level
1 - (1/8) - (x/12) = 7/8
To solve this equation for x and find out how long after the leak started the plane runs out of fuel, we can follow these steps:
1. Multiply both sides of the equation by 24 to eliminate fractions:
24 * [1 - (1/8) - (x/12)] = 24 * (7/8)
Simplifying,
24 - 3 - 2x = 21
2. Combine like terms:
21 - 24 + 3 + 2x = 0
Simplifying,
2x = 6
3. Divide both sides of the equation by 2:
x = 3
According to the calculation, the plane has 3 hours until it runs out of fuel after the leak begins.