An agricultural research station operates two types of automated irrigation pumps: Type Al… — GMAT

An agricultural research station operates two types of automated irrigation pumps: Type Alpha and Type Beta. Operating 4 Type Alpha pumps and 3 Type Beta pumps simultaneously for 5 hours consumes a total of $215\text{ kilowatt-hours (kWh)}$ of electricity. Operating 2 Type Alpha pumps and 5 Type Beta pumps simultaneously for 4 hours consumes a total of $156\text{ kWh}$ of electricity. Based on the information provided, match each given pump metric on the left with its correct hourly electricity consumption value on the right.

Hourly electricity consumption rate of one Type Alpha pump $7\text{ kWh per hour}$
Hourly electricity consumption rate of one Type Beta pump $5\text{ kWh per hour}$
Combined hourly electricity consumption rate of one Type Alpha pump and one Type Beta pump $12\text{ kWh per hour}$

Cevap

The hourly consumption rate of one Type Alpha pump is

7\text{ kWh per hour}

, one Type Beta pump is

5\text{ kWh per hour}

, and their combined hourly rate is

12\text{ kWh per hour}

Solving the system of simultaneous linear equations

4a + 3b = 43

and

2a + 5b = 39

yields

a = 7

for Type Alpha pumps and

b = 5

for Type Beta pumps. The sum

a + b

equals

12

Adım Adım Çözüm

Set up equations for the total hourly consumption from the given scenario data.

Let

a

be the hourly consumption of Type Alpha in kWh and

b

be the hourly consumption of Type Beta in kWh. From 5 hours of operation:

5(4a + 3b) = 215 \implies 4a + 3b = 43

. From 4 hours of operation:

4(2a + 5b) = 156 \implies 2a + 5b = 39

Dividing total energy by total hours converts total consumption into hourly rates for each operating group.

Solve the system of simultaneous linear equations for

a

and

b

Multiply

2a + 5b = 39

by 2 to get

4a + 10b = 78

. Subtract

4a + 3b = 43

from

4a + 10b = 78

(4a + 10b) - (4a + 3b) = 78 - 43 \implies 7b = 35 \implies b = 5

Eliminating variable

a

allows direct solution for variable

b

Substitute

b = 5

back into one of the linear equations to find

a

2a + 5(5) = 39 \implies 2a + 25 = 39 \implies 2a = 14 \implies a = 7

Determines the single unit rate for Type Alpha pumps.

Calculate the combined hourly rate for one Type Alpha and one Type Beta pump.

a + b = 7 + 5 = 12\text{ kWh per hour}

Calculates the sum of both individual unit rates.

Anahtar Kavram

Formulating and solving a system of two linear equations in two variables from rates and total work/consumption.

Tahmini Süre:2m 0s

Soru

Cevap

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