To find Leonardo's manufacturing rate, we can set up a rational equation based on the times it takes for Sumaya and Leonardo to work individually and together.
Let:
- \( S \) be the rate at which Sumaya manufactures spark plugs (in spark plugs per minute)
- \( L \) be the rate at which Leonardo manufactures spark plugs (in spark plugs per minute)
Sumaya can manufacture a spark plug in 13 minutes, so her rate is: \[ S = \frac{1 \text{ spark plug}}{13 \text{ minutes}} = \frac{1}{13} \text{ spark plugs per minute} \]
Let’s assume Leonardo takes \( x \) minutes to make one spark plug, so his rate is: \[ L = \frac{1 \text{ spark plug}}{x \text{ minutes}} = \frac{1}{x} \text{ spark plugs per minute} \]
When they work together, they can manufacture a plug in 5.32 minutes. This gives us their combined rate: \[ \text{Combined Rate} = \frac{1 \text{ spark plug}}{5.32 \text{ minutes}} = \frac{1}{5.32} \text{ spark plugs per minute} \]
According to the formula for combined rates: \[ S + L = \text{Combined Rate} \] Substituting the values for \( S \) and \( L \): \[ \frac{1}{13} + \frac{1}{x} = \frac{1}{5.32} \]
Now, let's solve for \( x \):
-
First, we can find a common denominator for the left side: \[ \frac{x + 13}{13x} = \frac{1}{5.32} \]
-
Cross-multiplying gives us: \[ 5.32(x + 13) = 13x \]
-
Distributing the 5.32: \[ 5.32x + 69.16 = 13x \]
-
Rearranging the equation: \[ 69.16 = 13x - 5.32x \] \[ 69.16 = 7.68x \]
-
Solving for \( x \): \[ x = \frac{69.16}{7.68} \approx 9.01 \]
Therefore, Leonardo's time to manufacture one spark plug is approximately \( 9.01 \) minutes.
The answer is: 9.01 minutes
3 answers