Questions: You're flying from Joint Base Lewis-McChord (JBLM) to an undisclosed location 216 km south and 134 km east. Mt. Rainier is located approximately 56 km east and 40 km south of JBLM. If you are flying at a constant speed of 800 km / hr, how long after you depart JBLM will you be the closest to Mt. Rainier?

Solution Steps

To find the time when you are closest to Mt. Rainier, we need to calculate the distance between your flight path and Mt. Rainier as a function of time, and then find the time at which this distance is minimized. The flight path can be represented parametrically, and the distance to Mt. Rainier can be expressed using the distance formula. We then find the minimum of this distance function.

The position of the plane as a function of time \( t \) (in hours) can be expressed as: \[ \text{plane\_position} = \left( \frac{67t}{175}, \frac{108t}{175} \right) \] The position of Mt. Rainier is given by: \[ \text{mt\_rainier\_position} = (56, 40) \]

The squared distance \( D^2 \) between the plane and Mt. Rainier is given by: \[ D^2 = \left( \frac{67t}{175} - 56 \right)^2 + \left( \frac{108t}{175} - 40 \right)^2 \]

To find the time \( t \) when the distance is minimized, we solve the equation: \[ \frac{d(D^2)}{dt} = 0 \] The solution yields: \[ t = \frac{1412600}{16153} \approx 87.5 \text{ hours} \]

To convert the time to minutes, we use the conversion factor \( 1 \text{ hour} = 60 \text{ minutes} \): \[ \text{time\_in\_minutes} = \frac{37080750}{16153} \approx 22950.0 \text{ minutes} \]

Final Answer