In an industrial robotics laboratory, three distinct robotic arms perform repetitive cycli… — State PSC Exam

In an industrial robotics laboratory, three distinct robotic arms perform repetitive cyclic tasks. Arm X completes one full operational cycle every $\frac{16}{15}$ seconds, Arm Y completes a cycle every $\frac{24}{25}$ seconds, and Arm Z completes a cycle every $\frac{32}{35}$ seconds. If all three robotic arms begin their cycles simultaneously from a synchronized home position, what is the minimum time required for all three arms to return to the home position at the exact same instant?

A
$\frac{8}{525}$ seconds
B
$\frac{32}{175}$ seconds
$\frac{96}{5}$ secondsAnswer
D
$\frac{8}{5}$ seconds

Answer

The minimum time required for all three robotic arms to synchronize again is

\frac{96}{5}

seconds.

To determine the next simultaneous occurrence of repeating events, we compute the Lowest Common Multiple (LCM) of their cycle durations. For fractions, the LCM is determined by dividing the LCM of the numerators by the Highest Common Factor (HCF) of the denominators. Thus,

\frac{\text{LCM}(16, 24, 32)}{\text{HCF}(15, 25, 35)} = \frac{96}{5}

Step-by-Step Solution

Identify the mathematical operation required.

Calculate the Lowest Common Multiple (LCM) of the three cycle times.

To find the first instance when multiple independent repeating events synchronize, we must find the LCM of their individual intervals.

State the formula for finding the LCM of fractions.

\text{LCM of fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}

This is the standard algebraic rule for determining the lowest common multiple of rational numbers.

Calculate the LCM of the numerators: 16, 24, and 32.

The LCM of 16, 24, and 32 is 96.

The prime factorizations are

16 = 2^4

24 = 2^3 \times 3

, and

32 = 2^5

. The highest powers of prime factors are

2^5

and

3^1

, so

2^5 \times 3 = 32 \times 3 = 96

Calculate the HCF of the denominators: 15, 25, and 35.

The HCF of 15, 25, and 35 is 5.

The prime factorizations are

15 = 3 \times 5

25 = 5^2

, and

35 = 5 \times 7

. The only common prime factor is 5.

Apply the values to the fraction LCM formula.

\frac{96}{5}

Dividing the computed numerator LCM (96) by the computed denominator HCF (5) yields the final answer.

Key Concept

Calculating the Lowest Common Multiple (LCM) of fractional values to synchronize repeating events.

Estimated Time:1m 30s

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