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Zorluk: Çok zorRatios, Rates, and Proportions

A chemical synthesis plant uses two reaction vessels, Vessel 1 and Vessel 2, to produce a liquid solution composed entirely of Compound X, Compound Y, and water.

- Vessel 1 produces the solution at a constant rate of 6060 liters per hour, with Compound X, Compound Y, and water in the volume ratio 2:1:32 : 1 : 3, respectively.
- Vessel 2 produces the solution at a constant rate of 9090 liters per hour, with Compound X, Compound Y, and water in the volume ratio 1:3:21 : 3 : 2, respectively.

If both vessels operate simultaneously to fill an initially empty storage tank, which of the following statements must be true regarding the mixture in the storage tank after operating for any duration HH hours (H>0H > 0)? Select all such statements.

  1. Compound Y accounts for exactly 1130\frac{11}{30} of the total volume of solution in the storage tank.Cevap
  2. The ratio of the volume of Compound X to the volume of water in the storage tank is 7:127 : 12.Cevap
  3. C
    The volume of Compound X in the storage tank is greater than the volume of Compound Y in the storage tank.
  4. D
    Water constitutes strictly more than 40%40\% of the total volume of solution in the storage tank.
  5. E
    Compound X represents exactly 718\frac{7}{18} of the total volume of solution in the storage tank.

Cevap

The statements confirming that Compound Y accounts for exactly 11/30 of the total solution volume and that the ratio of Compound X to water is 7:12 are both correct.
The total volume rate of solution produced by both vessels combined is 150 liters per hour. Within this total, Compound X is produced at 35 liters per hour, Compound Y at 55 liters per hour, and water at 60 liters per hour. Consequently, Compound Y accounts for 55 / 150 = 11/30 of the total volume, and the ratio of Compound X to water is 35 : 60 = 7 : 12. Both of these statements must be true.

Adım Adım Çözüm

1
Calculate the hourly output rates for each component from Vessel 1.
Total rate = 60 L/hr. Ratio X:Y:Water = 2:1:3 (sum of parts = 6). Compound X = 60 * (2/6) = 20 L/hr; Compound Y = 60 * (1/6) = 10 L/hr; Water = 60 * (3/6) = 30 L/hr.
Decompose the total production rate into component rates using part-to-whole fractions.
2
Calculate the hourly output rates for each component from Vessel 2.
Total rate = 90 L/hr. Ratio X:Y:Water = 1:3:2 (sum of parts = 6). Compound X = 90 * (1/6) = 15 L/hr; Compound Y = 90 * (3/6) = 45 L/hr; Water = 90 * (2/6) = 30 L/hr.
Decompose the second vessel's production rate into component rates.
3
Combine the component rates from both vessels.
Total solution rate = 60 + 90 = 150 L/hr. Compound X = 20 + 15 = 35 L/hr; Compound Y = 10 + 45 = 55 L/hr; Water = 30 + 30 = 60 L/hr.
Find the overall composition delivered to the storage tank per hour.
4
Evaluate each candidate statement against the combined component rates.
Fraction of Y = 55 / 150 = 11/30 (True). Ratio X to Water = 35 : 60 = 7 : 12 (True). X (35 L/hr) > Y (55 L/hr) (False). Water percentage = 60 / 150 = 40% (Not strictly greater than 40%, so False). Fraction of X = 35 / 150 = 7/30 != 7/18 (False).
Determine which statements are mathematically true for any operating duration H > 0.

Anahtar Kavram

Weighted rate aggregation and multi-component ratio analysis
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Konular

Ratios, Rates, and Proportions

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