"At time t = 0, a single-engine military jet is flying due east at 12mi/min. At the same altitude and 208 mi directly ahead of the military jet, still at time t = 0, a commercial jet is flying due north at 8 mi/min. When are the two planes closest to each other? What is the minimum distance between them?"
2007-12-07 08:56:11 · 5 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
This is not a calculus problem. It's an algebraic problem searching for mininum value of a function. If the commerical jet is 208 mile from the military jet due east. Then the distance between the two jet is:
d = √[(208 - 12t)² + (8t)², or to simplify
d² = 4² [(52 - 3t)² + (2t)²] then
d² = 16(13t² - 312t + 52²) →
d² = 16(13t - 1872)² + 13312
Therefore, two planes are closest to each other after 144 min or 2.4 hours. The distance between them would be 115.4 miles.
XR
2007-12-07 09:27:59 · answer #1 · answered by XReader 5 · 1⤊ 0⤋
Draw a right triangle. The base is the line of flight of the military jet. The length of the base is 208 miles. On the extreme right, draw a vertical at right angles from the end of the 208 line. That will be the flight path of the commercial jet. Make it some arbitrary length.
Now draw the hypotenuse which is the line connecting the starting point of the military jet's travel up to the end of the arbitrarily long commercial jet line.
As the planes fly, the base will get smaller, and the opposite side will get longer.
At any time t, the length of the hypotenuse will be:
Dist = √[(208 - 12t)² + (8t)²]
Expand that expression, and then take the derivative
d Dist/dt and set it equal to zero. This will give you a value for t. The answer is t = 12 minutes and the minimum distance is 118.9 miles.
I had to rush, so check the arithmetic. The logic is correct.
2007-12-07 09:51:00 · answer #2 · answered by Joe L 5 · 0⤊ 0⤋
Choose the position of the military jet at t = 0 to be the origin of a coordinate system (x, y), with due east the positive x direction and due north the positive y direction.
The position of the military jet in this cordinate system is given by f(t) = (12t, 0), and the position of the commercial jet is g(t) = (208, 8t), where t is time in minutes after t = 0.
The distance d(t) between them at any time t is given by
d²(t) = (8t)² + (12t - 208)²
You want find the minimum of d . To do this you can find the minimum of d² (using calculus, set the derivative = 0, solve for t... ) then take the square root. You should get
√13,312 ≈ 115.4 mi
2007-12-07 09:23:47 · answer #3 · answered by a²+b²=c² 4 · 0⤊ 0⤋
sin(a million/x)-x*cos(a million/x)(a million/x^2) i think of it somewhat is actual utilising the product rule then the chain rule on the final area. "by-made of x circumstances sin(a million/x) plus x circumstances the by-made of sin(a million/x)"
2016-11-14 00:24:20 · answer #4 · answered by ? 4 · 0⤊ 0⤋
not enough info to solve correctly
2007-12-07 09:30:56 · answer #5 · answered by Ben C 2 · 0⤊ 0⤋