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Zorluk: ZorRatios, Rates, and Proportions

An agricultural drone is programmed to spray fertilizer on two adjacent fields, Field X and Field Y. The ratio of the area of Field X to the area of Field Y is 3:53:5. The drone sprays Field X at a constant rate of 44 acres per hour, and it sprays Field Y at a constant rate of 66 acres per hour. If it takes the drone a total of 7676 hours of continuous operation to spray both fields completely, what is the area, in acres, of Field Y?

  1. A
    20
  2. B
    48
  3. C
    144
  4. 240Cevap
  5. E
    475

Cevap

240
The area of Field Y is 240240 acres. Since the ratio of the area of Field X to Field Y is 3:53:5, we can define their areas as 3a3a and 5a5a. Using the relation Time=AreaRate\text{Time} = \frac{\text{Area}}{\text{Rate}}, the total time is 3a4+5a6=76\frac{3a}{4} + \frac{5a}{6} = 76. Finding a common denominator of 1212 gives 9a12+10a12=76\frac{9a}{12} + \frac{10a}{12} = 76, which simplifies to 19a12=76\frac{19a}{12} = 76. Solving for aa yields a=48a = 48. The area of Field Y is then 5×48=2405 \times 48 = 240 acres.

Adım Adım Çözüm

1
Define variables for the areas of the fields based on the given ratio.
Let the area of Field X be 3a3a acres and the area of Field Y be 5a5a acres, where aa is a positive constant.
Since the ratio of the area of Field X to Field Y is 3:53:5, expressing them in terms of a single variable aa allows us to set up a solvable equation.
2
Express the time spent spraying each field in terms of the variable aa.
Time for Field X = 3a4\frac{3a}{4} hours; Time for Field Y = 5a6\frac{5a}{6} hours.
Using the relation Time=AreaRate\text{Time} = \frac{\text{Area}}{\text{Rate}}, we substitute the respective areas and constant rates to get the time expressions.
3
Set up an equation for the total time spent spraying both fields.
3a4+5a6=76\frac{3a}{4} + \frac{5a}{6} = 76
The sum of the time spent on each field must equal the total operational time of 7676 hours.
4
Solve the equation for the constant aa.
Find the common denominator of 1212: 9a12+10a12=76  ⟹  19a12=76  ⟹  19a=912  ⟹  a=48\frac{9a}{12} + \frac{10a}{12} = 76 \implies \frac{19a}{12} = 76 \implies 19a = 912 \implies a = 48.
Finding the least common multiple of the denominators allows us to clear fractions and isolate the variable aa.
5
Calculate the area of Field Y.
Area of Field Y = 5×48=2405 \times 48 = 240 acres.
The area of Field Y was defined as 5a5a, so substituting a=48a = 48 gives the final area.

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Setting up and solving equations involving ratios, rates, and proportions.
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Ratios, Rates, and Proportions

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