Math 165 – Quiz on 4.6, Applied
Optimization – solutions
Problem 1 Distance Space capsule M-INI is orbiting the mothership
NEWTON on an elliptical orbit with equation
√
y = ± 10 − 2x2
NEWTON is positioned at (1, 0). You want to send a teleporter with supplies to M-INI.
Which point(s) on the space capsule’s orbit are closest to the mothership?
Make sure to
a) name the quantity you want to maximize or minimize for this problem,
b) name the input variable, and give its domain,
c) find critical points of the target function (quantity to optimize) and determine whether they are indeed a maximum or minimum.
Solution a) We need to minimize the distance z of points (x, y) on the orbit
from (1, 0). b) We choose the
of the point (x, y) as input vari√
√ x-coordinate
able. The domain for x is [− 5, 5] (outside of this interval, y is undefined).
c) We express z as a function of x and y,
p
z = (x − 1)2 + y 2 .
Using the orbit equation, we express z as a function of x alone,
p
p
√
z(x) = (x − 1)2 + 10 − 2x2 = 10 − 2x − x2 = 11 − (x + 1)2 .
(1)
To find critical points, we compute
−(x + 1)
z 0 (x) = p
11 − (x + 1)2
and this is zero exactly for x = −1, with sign pattern +−. So at x = −1,
we have a maximum, and local minima are at the endpoints. Evaluating z
at the endpoints gives
√
√
z( 5) =
5−1
√
√
z(− 5) =
5+1
√
√
so√the minimum is at x = 5. The only point on the orbit with x = 5 is
( 5, 0).
The plot below shows the positions of MINI when its distance from NEWTON is maximal (with x = −1).