Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/166

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Third Step. Differentiate with respect to the time.
Fourth Step. Make a list of the given and required quantities.
Fifth Step. Substitute the known quantities in the result found by differentiating (third step), and solve for the unknown.


EXAMPLES

1. A man is walking at the rate of 5 miles per hour towards the foot of a tower 60 ft. high. At what rate is he approaching the top when he is 80 ft. from the foot of the tower?

Example 2 illustration.
Example 2 illustration.
Solution. Apply the above rule.
First step. Draw the figure. Let x = distance of the man from the foot and y = his distance from the top of the tower at any instant.
Second step. Since we have a right triangle,
.
Third step. Differentiating, we get
, or,
(A) , meaning that at any instant whatever
(Rate of change of y) = (rate of change of x).
Fourth step. miles an hour,
  = 5 × 5280 ft. an hour.
  = ?
  = 100.
Fifth step. Substituting back in (A),
  = ft. per hour
  = 4 miles per hour. Ans.

2. A point moves on the parabola in such a way that when x = 6, the abscissa is increasing at the rate of 2 ft. per second. At what rates are the ordinate and length of arc increasing at the same instant?

Plot of a parabola.

Solution First step. Plot the parabola.
Second step.
Third step. , or,
(B) .
This means that at any point on the parabola
(Rate of change of ordinate) = (rate of change of abcissa).