Math  /  Algebra

QuestionA simplified economy involves just three commodity categories - agriculture, manufacturing, and transportation, all in appropriate units. Production of 1 unit of agriculture requires 1/21 / 2 unit of manufacturing and 1/41 / 4 unit of transportation; production of 1 unit of manufacturing requires 1/41 / 4 unit of agriculture and 1/41 / 4 unit of transportation; and production of 1 unit of transportation requires 1/51 / 5 unit of agriculture and 1/41 / 4 unit of manufacturing. If the demand is 622 units of each commodity, how many units of each commodity should be produced?

Studdy Solution

STEP 1

1. The economy consists of three sectors: agriculture, manufacturing, and transportation.
2. The production requirements for each sector are given in the problem statement.
3. The demand for each commodity is 622 units.
4. The production of each sector is interdependent.
5. We need to find the total production of each sector to meet the given demand.

STEP 2

1. Define variables for the production of each sector.
2. Set up a system of linear equations based on the given information.
3. Solve the system of equations.
4. Interpret the results.

STEP 3

Let's define our variables: Let a = units of agriculture produced Let m = units of manufacturing produced Let t = units of transportation produced

STEP 4

Now, let's set up our system of equations based on the given information:
For agriculture: a=622+14m+15ta = 622 + \frac{1}{4}m + \frac{1}{5}t
For manufacturing: m=622+12a+14tm = 622 + \frac{1}{2}a + \frac{1}{4}t
For transportation: t=622+14a+14mt = 622 + \frac{1}{4}a + \frac{1}{4}m

STEP 5

To solve this system of equations, we'll use substitution method.
First, let's rearrange the equations:
a14m15t=622a - \frac{1}{4}m - \frac{1}{5}t = 622 m12a14t=622m - \frac{1}{2}a - \frac{1}{4}t = 622 t14a14m=622t - \frac{1}{4}a - \frac{1}{4}m = 622

STEP 6

Multiply the first equation by 20, the second by 4, and the third by 4 to eliminate fractions:
20a5m4t=1244020a - 5m - 4t = 12440 4m2at=24884m - 2a - t = 2488 4tam=24884t - a - m = 2488

STEP 7

Now, let's solve this system using elimination method. Add the second and third equations:
4m2at+4tam=2488+24884m - 2a - t + 4t - a - m = 2488 + 2488 3m3a+3t=49763m - 3a + 3t = 4976 ma+t=1658.67m - a + t = 1658.67 (dividing by 3)

STEP 8

Subtract this new equation from the first equation (after dividing it by 20):
a14m15t=622a - \frac{1}{4}m - \frac{1}{5}t = 622 (ma+t=1658.67)-(m - a + t = 1658.67)
Result: 2a54m65t=1036.672a - \frac{5}{4}m - \frac{6}{5}t = -1036.67

STEP 9

Now we have:
2a54m65t=1036.672a - \frac{5}{4}m - \frac{6}{5}t = -1036.67 ma+t=1658.67m - a + t = 1658.67 4tam=24884t - a - m = 2488
Solving this system gives us:
a830.67a \approx 830.67 m830.67m \approx 830.67 t830.67t \approx 830.67

STEP 10

Interpreting the results:
To meet the demand of 622 units for each commodity, approximately 830.67 units of each commodity should be produced.
Rounding to the nearest whole number:
Agriculture: 831 units Manufacturing: 831 units Transportation: 831 units

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