Example 42 - A car starts from a point P at time t = 0 seconds - Minima/ maxima (statement questions) - Geometry questions

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  1. Chapter 6 Class 12 Application of Derivatives
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Example 42 A car starts from a point P at time t = 0 seconds and stops at point Q. The distance x, in metres, covered by it, in t seconds is given by 𝑥=﷐𝑡﷮2﷯﷐2−﷐𝑡﷮3﷯﷯. Find the time taken by it to reach Q and also find distance between P and Q Given distance 𝑥 = t2 ﷐2−﷐𝑡﷮3﷯﷯ Let v be the velocity of the car at t second 𝑣 = change in distance w.r.t t i.e. 𝑣 = ﷐𝑑𝑥﷮𝑑𝑡﷯ 𝑣 = ﷐𝑑﷐﷐𝑡﷮2﷯﷐2 − ﷐𝑡﷮3﷯﷯﷯﷮𝑑𝑡﷯ 𝑣 = ﷐𝑑﷐2﷐𝑡﷮2﷯− ﷐﷐𝑡﷮3﷯﷮3﷯﷯﷮𝑑𝑡﷯ 𝑣 = 4t – t2 Putting v = 0 ⇒ 4t – t2 = 0 ⇒ t﷐4−𝑡﷯=0 So, t = 0 & t = 4 Hence, the car is not moving at t = 0 & t = 4 second Now, car is not moving (i.e. v = 0) at point P as well as at point Q Thus the car will reach the point Q after 4 seconds ⇒ Distance PQ = Distance travelled in 4 seconds i.e. 𝑥 = t2 ﷐2−﷐𝑡﷮3﷯﷯ at t = 4 ﷐﷐𝑥﷯﷮𝑡=4﷯ = ﷐﷐4﷯﷮2﷯﷐2−﷐4﷮3﷯﷯ = 16 ﷐﷐6 − 4﷮3﷯﷯ = 16 ﷐﷐2﷮3﷯﷯ = ﷐32﷮3﷯ 𝑚. Hence, Distance PQ = ﷐𝟑𝟐﷮𝟑﷯ 𝒎.

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