SOLUTION: A school is building a new track for cycling teams. The track is to be elliptical,
with the ratio between the lengths of the major and minor axes to be 4:3. If the available
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-> SOLUTION: A school is building a new track for cycling teams. The track is to be elliptical,
with the ratio between the lengths of the major and minor axes to be 4:3. If the available
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Question 1185272: A school is building a new track for cycling teams. The track is to be elliptical,
with the ratio between the lengths of the major and minor axes to be 4:3. If the available
land is 200 yd long and 100 yd wide, what are the maximum lengths of the axes? Sketch
a graph of the track and indicate the location of the foci.
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A school is building a new track for cycling teams. The track is to be elliptical,
with the ratio between the lengths of the major and minor axes to be 4:3. If the available
land is 200 yd long and 100 yd wide, what are the maximum lengths of the axes? Sketch
a graph of the track and indicate the location of the foci.
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Let "a" be the major semi-axis of the ellipse; "b" be the minor semi-axis.
From the condition, we have
a = 4x; b = 3x, where x is the common factor
4x <= 100 yards (one half of 200 yards); (1)
3x <= 50 yards (one half of 100 yards). (2)
From restrictions (1) and (2)
x <= 25 yards;
x <= 16 yards.
According to the meaning of the variables, we must take the minimum value of (3) and (4) for x; so, we take
x = 16 yards.
Thus a = 4x = 64 = 66 yards.
b = 3x = 50 yards.
The distance from the center of the ellipse to the foci is
c = = = 44.095 yards.
