Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applied max/min questions.
Type 5: Rates - Test Yourself 1.


 

Questions on this page focus on:
1. Times to complete tasks.
2. Costs to complete tasks.
 

 

Time.

1.

The above diagram shows a 1 km wide river (I guess after the Queensland floods). Rory is standing at A and needs to get to point D. The point B is across the river and directly opposite A.

The point D is 3 km down the river from B.

Rory intends to swim to point C (which is x km from B) and then run to point D. He can swim at 4 kph and and he can run at 10 kph (very fit is our friend Rory!!).

(i) Show that the time (t hours) it takes for Rory to reach point D can be expressed as

(ii) Find the distance BC representing the distance down river at which Rory must finish his swim for him to minimise his time to reach D.

Answer.Distance BC = 436 m.
   
  3. Alyssa intends to enter a triathlon competition and is practising on a course consisting of three parts:
  • a straight line swim from S to X. She can swim at 5 kph.
    She can stop her swim at any point before X which is a km from O.
  • a bike ride from X to Y. She can ride her bike at 10 kph.
  • a run around a quarter of a circular track from Y to F.
    She can run at 8 kph.

The perpendicular distance from S to O is 2 km. Let the distance from O to where she lands be a km. The distance from O to Y is 4 km.

(i) Show that the time Alyssa takes to complete the three legs of the triathlon course SXYF is

(ii) Find the value of a which will enable Alyssa to minimise the time she takes to complete the three legs of the course (3 dp - so to the nearest metre).

Answer. a = (2√3)/3 kms = 1.155 km.
  4.
  5.
   
Costs. 6. The hourly cost (C) of running a car at an average speed of v kph can be expressed as

(i) We want to drive from Carnavon in WA to Ningaloo (one of the great National Parks in the world) - a distance of say 280 km. Calculate the average speed for this journey (nearest kph) for which the total cost of running the car is a minimum.

(ii) Calculate the minimum cost.

(iii) What caution must be made about the time taken for this journey?
  7. A factory can produce 350 widgets every week. Their production cost can be expressed by the equation
where x is the number of widgets produced each week.

The widgets are sold for $3 each (really a bargain if you are into widgets).

Find the maximum weekly profit which can be made if the factory mst produce at least 100 widgets to supply curent orders.

Answer.Max profit is $827.62
when x = 330 widgets.
  8. When a ship is travelling at a speed of v km/hr, its rate of consumption of fuel per hour is given by 125 + 0.004v3.

(i) Show that on a voyage of 5,000 km at a speed of v km/hour, the total for the fuel tonnes/hour is given by .

(ii) Hence find the speed for the greatest speed economy and the amount of fuel used at this speed. Justify your answer.