Problem 5: 1998-1999

    During taping for the latest edition of Circus of the Stars, Angela Lansbury is shot out of a cannon. The firing goes completely awry and sends her on a collision course with a 747. As they converge, Angela and Boeing travel at right angles to each other (see diagram). Angela is 200 miles away from the point of impact and traveling at 600 mph; the plane is 150 miles from impact and traveling at 450 mph.

    (a)    At what rate is the distance between the two decreasing?

    (b)    How much time is there until they hit?

    (c)    Would it be appropriate to entitle the segment "Getting Sucked into a Jet Engine, She Wrote"?


Solution:
    (a)    I will use the following diagram to help me out here:
    Notice that a is the distance from Angela to the collision, p is the distance from the plane to the collision, and s is the distance between the two. I am looking for ds/dt, so I'll begin by using the Pythagorean Theorem as the relationshop between all the variables. Therefore, a2 + p2 = s2. Now, I take the derivative with respect to t, getting 2a(da/dt) + 2p(dp/dt) = 2s(ds/dt). At this point, we can divide everything by 2 and solve for ds/dt to get an equation you can plug the specific information into:
Notice that the rates of change are negative (since the sides they represent are decreasing in size). Also, we found side s by using the Pythagorean Theorem.

    (b)    To figure out how long it will take for the two bodies to collide, we notice that the distance between the two (250 miles according to the Pythagorean Theorem as described above) is decreasing at a rate of -750 mph (according to part (a) above). How long will it take that distance to disappear? Fall back on the formula you knew since the dawn of time ... d = rt. Yes! Distance equals rate times time! Yes the overall problem is difficult, but all we want to know is how long it takes a length of 250 miles to completely dissapear at a given rate! So, if d = rt, then t = d/r. Therefore, t = 250/-750, or 1/3 hours (20 minutes). Ignore the negative sign since negative time makes about as much sense as disco.

    (c)    I suppose so, but even a better title would be "Dringo, the Hapless Monkey." Proof is omitted.


Home Problem Fun Calculus Stuff Kelley's Books Superbowl of Calculus