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Nitrogen is the "active ingredient" in many quick-acting fertilizers. You are operating a farm of 1500 acres to produce soybeans. Which of the following fertilizers will you choose as the most economical for your farm?

A. [tex]$NH_4NO_3$[/tex] at [tex]$\$[/tex] 625[tex]$ for 25 kg

B. $[/tex]\left( NH_4\right)_2HPO_4[tex]$ at $[/tex]\[tex]$ 55$[/tex] for 1 kg
C. ammonia, [tex]$NH_3$[/tex], at [tex]$\$[/tex] 128[tex]$ for 50 kg

D. urea, $[/tex]CO \left( NH_2\right)_2[tex]$, at $[/tex]\[tex]$ 60$[/tex] for 5 kg

Answer :

Sure! Let's determine the most economical fertilizer for your farm by comparing the cost per kilogram for each type of fertilizer. Here are the fertilizers and their prices:

1. Ammonium nitrate ([tex]$NH_4NO_3$[/tex]): [tex]$625 for 25 kg
2. Diammonium hydrogen phosphate ($[/tex]\left(NH_4\right)_2HPO_4[tex]$): $[/tex]55 for 1 kg
3. Ammonia ([tex]$NH_3$[/tex]): [tex]$128 for 50 kg
4. Urea ($[/tex]CO\left(NH_2\right)_2[tex]$): $[/tex]60 for 5 kg

We'll calculate the cost per kilogram for each fertilizer to find out which one is the cheapest. Here are the calculations:

1. Ammonium nitrate ([tex]$NH_4NO_3$[/tex]):
- Total cost: [tex]$625
- Total weight: 25 kg
- Cost per kilogram: $[/tex]625 / 25 kg = [tex]$25.0 per kg

2. Diammonium hydrogen phosphate ($[/tex]\left(NH_4\right)_2HPO_4[tex]$):
- Total cost: $[/tex]55
- Total weight: 1 kg
- Cost per kilogram: [tex]$55 / 1 kg = $[/tex]55.0 per kg

3. Ammonia ([tex]$NH_3$[/tex]):
- Total cost: [tex]$128
- Total weight: 50 kg
- Cost per kilogram: $[/tex]128 / 50 kg = [tex]$2.56 per kg

4. Urea ($[/tex]CO\left(NH_2\right)_2[tex]$):
- Total cost: $[/tex]60
- Total weight: 5 kg
- Cost per kilogram: [tex]$60 / 5 kg = $[/tex]12.0 per kg

Comparing the costs per kilogram for these fertilizers:
- Ammonium nitrate ([tex]$NH_4NO_3$[/tex]): [tex]$25.0 per kg
- Diammonium hydrogen phosphate ($[/tex]\left(NH_4\right)_2HPO_4[tex]$): $[/tex]55.0 per kg
- Ammonia ([tex]$NH_3$[/tex]): [tex]$2.56 per kg
- Urea ($[/tex]CO\left(NH_2\right)_2[tex]$): $[/tex]12.0 per kg

The most economical fertilizer for your farm is Ammonia ([tex]$NH_3$[/tex]) at a cost of $2.56 per kg. This makes it the best choice in terms of cost-efficiency compared to the other options.

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Rewritten by : Barada