A logistics company uses two types of automated cargo drones, Drone X and Drone Y, to comp… — GMAT

A logistics company uses two types of automated cargo drones, Drone X and Drone Y, to complete long-distance freight deliveries.

- Drone X travels at a constant speed of $40\text{ km/h}$ and consumes energy at a constant rate of $15\text{ kWh/h}$ .
- Drone Y travels at a constant speed of $60\text{ km/h}$ and consumes energy at a constant rate of $25\text{ kWh/h}$ .

On a completed joint mission, the two drones traveled a combined total distance of $480\text{ km}$ and consumed a combined total energy of $190\text{ kWh}$ .

In the table below, select the number of hours that Drone X operated and the number of hours that Drone Y operated during this mission.

Drone X Operating Time	Drone Y Operating Time	Hours
[ ]	[ ]	3
[ ]	[ ]	4
[ ]	[ ]	5
[ ]	[ ]	6
[ ]	[ ]	8

Which of the following correctly identifies the operating time, in hours, for Drone X and Drone Y, respectively?

A
Drone X: 4 hours; Drone Y: 6 hours
Drone X: 6 hours; Drone Y: 4 hoursCevap
C
Drone X: 5 hours; Drone Y: 5 hours
D
Drone X: 3 hours; Drone Y: 6 hours
E
Drone X: 8 hours; Drone Y: 3 hours

Cevap

Drone X operated for 6 hours and Drone Y operated for 4 hours.

The system of equations

40 t_X + 60 t_Y = 480

and

15 t_X + 25 t_Y = 190

uniquely yields

t_X = 6

and

t_Y = 4

. Substituting Drone X = 6 hours and Drone Y = 4 hours gives a total distance of

40(6) + 60(4) = 480\text{ km}

and a total energy consumption of

15(6) + 25(4) = 190\text{ kWh}

, perfectly matching both given conditions.

Adım Adım Çözüm

Set up a system of two linear equations representing total distance and total energy consumption.

Let

t_X

be the operating time of Drone X in hours, and

t_Y

be the operating time of Drone Y in hours.
Distance equation:

40 t_X + 60 t_Y = 480

Energy equation:

15 t_X + 25 t_Y = 190

Distance is calculated as

\text{speed} \times \text{time}

and energy as

\text{rate} \times \text{time}

Simplify both linear equations by dividing by their greatest common divisors.

Dividing the distance equation by 20 gives:

2 t_X + 3 t_Y = 24

.
Dividing the energy equation by 5 gives:

3 t_X + 5 t_Y = 38

Simplifying coefficients reduces computational complexity during elimination.

Solve the system using elimination.

Multiply

2 t_X + 3 t_Y = 24

by 3:

6 t_X + 9 t_Y = 72

.
Multiply

3 t_X + 5 t_Y = 38

by 2:

6 t_X + 10 t_Y = 76

.
Subtract the first multiplied equation from the second:

(6 t_X + 10 t_Y) - (6 t_X + 9 t_Y) = 76 - 72 \implies t_Y = 4\text{ hours}

Eliminating

t_X

allows direct solving for

t_Y

Substitute

t_Y = 4

back into

2 t_X + 3 t_Y = 24

to solve for

t_X

2 t_X + 3(4) = 24 \implies 2 t_X + 12 = 24 \implies 2 t_X = 12 \implies t_X = 6\text{ hours}

Determines the exact value for Drone X's operating duration.

Anahtar Kavram

Simultaneous Linear Equations with Work and Rate Constraints

Tahmini Süre:2m 30s

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