CHAPTER 7. DIFFERENTIAL CALCULUS 7.6
- (a) After doing some research, a transport company has determinedthat the rate at
which petrol is consumed by one of its large carriers, travelling at anaverage
speed of x km per hour, is given by:
P (x) =
55 �h−^1
2 x
+
x
200 km^2 �−^1 h−^1
i. Assume that the petrol costs R 4 , 00 per litre and the driverearns R 18 , 00 per
hour (travelling time). Now deduce that the total cost, C, in Rands, for a
2 000 km trip is given by:
C(x) =
256000 kmh−^1 R
x
+ 40x
ii. Hence determine theaverage speed to be maintained to effect a minimum
cost for a 2 000 km trip.
(b) During an experiment the temperature T (in degrees Celsius), varies with time t
(in hours), according tothe formula:
T (t) = 30 + 4t−
1
2
t^2 t∈ [1; 10]
i. Determine an expression for the rate of change of temperature with time.
ii. During which time interval was the temperature dropping?
- The depth, d, of water in a kettle t minutes after it starts to boil, is given by d =
86 −^18 t−^14 t^3 , where d is measured in millimetres.
(a) How many millimetres of water are there in the kettle just before it starts to boil?
(b) As the water boils, the level in the kettle drops. Find the rate at which the water
level is decreasing when t = 2 minutes.
(c) How many minutesafter the kettle starts boiling will the water levelbe dropping
at a rate of 1218 mm/minute?
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(1.) 01g7 (2.) 01g8 (3.) 01g9 (4.) 01ga (5.) 01gb (6.) 01gc
(7.) 01gd (8.) 01ge (9.) 01gf (10.) 01gg (11.) 01gh (12.) 01gi
(13.) 01gj (14.) 01gk (15.) 01gm (16.) 021h
