(a) Factorise 49x4 – (x2 - x)2 Completely.
(b) A transport company pays the driver a wage of $2v per hour and reimburses $1/litre for fuel. On a certain trip, a lorry travels 100km at a constant speed of v km/h and consumes fuel at a rate of ( v3/200 – 1/5 v2) litres per hour.
(i) Find, in terms of v, the time taken by the lorry for the journey
(ii) Given that the total cost of the driver’s wages and fuel for the trip is $C, show C = ½ (v-20)2
(iii) If the total cost of the driver’s wages and fuel for the trip is $450, find the value of v.
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Answer:
(a) 49x4 – (x2 - x)2 = 49x4 – (x4 – 2x3 + x2)
= 48x4 + 2x3 - x2
= x2 (48x2 + 2x - 1)
= x2 (8x - 1) (6x + 1)
(b) (i) Time taken = distance / speed
= (100 / v) hrs
(ii) Num of litres consumed = (100 / v) * ( v3/200 – 1/5 v2)
= ½v2 – 20 v
Total cost = $2v * (100 / v) + $1 * (½v2 – 20 v)
= $(200 + ½v2 – 20 v)
= $ ½(v2 – 40 v + 400)
= $ ½(v - 20)2 (shown)
(iii) $ ½(v - 20)2 = $450
(v - 20)2 = 900
v - 20 = 300 or -300 (must be positive)
v = 320
