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Calculus - Differentiation - Applied max/min questions.
Type 5: Rates - Test Yourself 2.


 

Time. 1. Tracey and Andrew are outback adventurers.

Tracey drives her car west along a road at a constant speed of 72 km/hour (which is 1.2 km/minute) while Andrew drives his car south along another road at a constant speed of 90 kph (1.5 km/minute). The two roads intersect at right angles (typical of outback roads).

Tracey starts 40 km from the intersection and Andrew starts 50 km from the intersection. Both drive towards the intersection which is at the well-known Birdsville Hotel so they plan to eat lunch together if they arrive at about the same time.

(i) Explain why Tracey is (40 - 1.2t) km from the Birdsville Hotel after t minutes.

(ii) Show that the distance between Tracey and Andrew after t minutes is given by
D2 = 4100 - 246t + 3.69t2.Hint.As noted on the video, we do not take the square root before differentiating because all terms would be under the root sign. Just differentiate normally.

(ii) Find the time at which the distance between Tracey and Andrew is the closest. Can they have lunch together?
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Costs. 5. When a ship is cruising at an average speed of v kph, its rate of consumption of fuel in tonnes per hour is given by 125 + 0.004v3.

(i) Show that, for a voyage of 5,000 km and a cruising speed of v km/hour, the formula for the total fuel used, T tonnes, is given by

(ii) Determine the speed for the ship if fuel economy has to be maximised and also the amount of fuel which is used at that speed.

 

 
 

 
 

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