A oceanographic research institute uses three autonomous submersibles��Submersible A, Sub… — GMAT

A oceanographic research institute uses three autonomous submersibles��Submersible A, Submersible B, and Submersible C—to perform high-resolution seabed mapping. Working together at their respective constant rates, Submersible A and Submersible B can complete a full mapping mission in $15$ hours. Working together at their respective constant rates, Submersible B and Submersible C can complete the same mapping mission in $24$ hours. During a specialized operation, Submersible A works alone for $6$ hours, after which it is recalled. Submersible B then works alone for $10$ hours, after which Submersible C joins Submersible B, and both work together for an additional $12$ hours to complete the remaining portion of the mission. How many hours would it take Submersible A to complete the entire seabed mapping mission working alone at its constant rate?

A
18
B
20
24Answer
D
30
E
36

Answer

24 hours

The option stating 24 is correct because setting up the system of work rates

r_A + r_B = \frac{1}{15}

and

r_B + r_C = \frac{1}{24}

, along with the multi-stage work equation

6r_A + 10r_B + 12(r_B + r_C) = 1

, allows us to eliminate

r_C

and solve for

r_A = \frac{1}{24}

. Taking the reciprocal gives 24 hours.

Step-by-Step Solution

Define rate variables and set up combined rate equations.

Let

r_A

r_B

, and

r_C

be the hourly rates of Submersibles A, B, and C respectively (in fraction of mission per hour). We are given: (1)

r_A + r_B = \frac{1}{15}

and (2)

r_B + r_C = \frac{1}{24}

Work rates are additive reciprocals of completion times for constant-rate work problems.

Express total mission work completed in the multi-stage operation.

6r_A + 10r_B + 12(r_B + r_C) = 1

, which simplifies to

6r_A + 22r_B + 12r_C = 1

Total work equals the sum of work done across all sequential operational phases.

Substitute equation (2) into the multi-stage work equation.

Since

r_C = \frac{1}{24} - r_B

, we get

6r_A + 22r_B + 12\left(\frac{1}{24} - r_B\right) = 1 \implies 6r_A + 10r_B + \frac{1}{2} = 1 \implies 6r_A + 10r_B = \frac{1}{2}

Substituting

r_C

eliminates one variable, reducing the system to two equations in

r_A

and

r_B

Solve the two-variable system for

r_A

Multiply equation (1) by

10

to get

10r_A + 10r_B = \frac{10}{15} = \frac{2}{3}

. Subtract

6r_A + 10r_B = \frac{1}{2}

from this to get

4r_A = \frac{2}{3} - \frac{1}{2} = \frac{1}{6} \implies r_A = \frac{1}{24}

Eliminating

r_B

isolates Submersible A's rate.

Calculate the time taken by Submersible A working alone.

Time

= \frac{1}{r_A} = \frac{1}{1/24} = 24

hours.

Completion time is the reciprocal of the hourly work rate.

Key Concept

System of Work Rate Equations for Multi-Stage Combined Work

Estimated Time:2m 30s

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Step-by-Step Solution

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