The answer is, Fertilizer A = 20, Fertilizer B = 56.
What is fertilizer and example?
- Among the organic fertilizers that occur naturally are manure, slurry, worm castings, peat, seaweed, and guano. Crops made of green manure are also raised to enrich the soil with nutrients.
- Organic fertilizers also include naturally occurring minerals including mine rock phosphate, sulfate of potash, and limestone.
Set up the equations:
- Nitrogen: 8x + 5y ≥ 440.
- Phosphorous: 2x + 5y ≥ 260.
- Potassium: 4x + 5y ≥ 360.
Find the vertices:
- It is easiest to graph the equations to find the vertices. (see attachment).
- You can also solve each system of equations to find the intersected points.
The following satisfy the "greater than or equal to" requirement:
- (0, 88) = y-intercept of Nitrogen equation.
- (20, 56) = intersection of Nitrogen and Potassium equations.
- (50, 32) = intersection of Phosphorous and Potassium.
- (130, 0) = x-intercept of Potassium.
Use vertices in cost function C(x) to find the minimum:
- C(x) = $30x + $20y.
- (0, 88): $30(0) + $20(88) = $1760.
- (20, 56): $30(20) + $20(56) = $1720 = This is the minimum!
- (50, 32): $30(50) + $20(32) = $2140.
- (130, 0): $30(130) + $20(0) = $3900.
The minimum cost occurs when 20 bags of Fertilizer A and 56 bags of Fertilizer B are purchased.
Learn more about Fertilizer here:
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