Help with a minimum cost pre-calc problem?

Question

two gasolines, type A and typeB, have octane ratings of 87 and 93, respectively. Type A costs$1.13 per gallon and type B costs $1.28 per galllon. Determine the blend of minimum cost with an octane rating of at least 87. WHat is the minimum cost? is the cost cheaper than the advertized cost of $1.20 per gallon for gasoline with an octanr rating of 89?(Hint: let x be the fraction of each gallon that is type A and let y be the fraction that is type B.)

Answer 1

I assume that you want to determine the blend of octane rating 89? Not 87.

x = fraction of type A
y = 1-x = fraction of type B.

Cost per gallon of blend = 1.28*(1-x) + 1.13*x
= 1.28 - 0.15x

This cost is minimum when x = 1.
Now if to get an 89 octane rating, you require
87x + 93*(1-x) = 89
we can determine x = 2/3
In that case cost of blend = $1.18 per gallon

Yes, those bastards are ripping us off.

Answer 2

The octane number of A itself is 87 so if you mix even a little quanty of B the octane number of the mixture will be more than 87.That means the cost of octane number 87 gasoline is $1.13.
To get the cost at octane rating 89.
For 87---------------->1.13
For 93----------------->1.28
use inerpolation 1.13+(1.28-1.13)/(93-87)x(89-87)
=1.18

Answer 3

x + y = 1
87x + 93y = 89
87x + 87y = 87
6y = 2
y = 1/3
x = 2/3

C = 1.13(2/3) + 1.28(1/3)
C = $1.18