two trains leave a station at the same time. One travels north on a track at 30
ID: 3341686 • Letter: T
Question
two trains leave a station at the same time. One travels north on a track at 30 mph. The second travels east on a track at 45 miles per hour. How fast are they traveling away from one another in miles per hour when the northbound train is 60 miles from the station?
Estimate the area of the ellipse given by the equation 4x2+y2=25 as follows:
The area of the ellipse is 4 times the area of the part of the ellipse in the first quadrant ( x and y positive).
Estimate the area of the ellipse in the first quadrant by solving for y in terms of x. Estimate the area under the graph of y by dividing the interval [0,2.5] into four equal subintervals and using the left endpoint of each subinterval. Be sure you draw a picture.
Don't forget to multiply your estimate for the area of the part of the ellipse in the first quadrant by 4 to get the entire area.
The area of the ellipse (using the above method) is approximately
Explanation / Answer
x^2 + y^2 = L^2 where L is the distance between the trains
x = 3 y / 2 at any time t since the trains start together
x dx/dt + y dy/dt = L dL/dt
dL/dt = (x dx/dt + y dy/dt) / L
When y = 60 x = 90
L = (60^2 + 90^2)^1/2 = 108
dL/dt = (90 * 45 + 60 * 30) / 108 = 54 mph
Only one question at a time please, but for the elipse
just draw a straight line from x = 2.5 to y = 5 and then divide
the triangle into the requested rectangles and add the areas.
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