Solve the following problems involving direct and inverse proportions. Round answer to two decimal places as necessary.
Two gears, A and B, are in mesh. Gear A has 22 teeth and revolves at 262 rpm. Determine the number of revolutions per minute of gear B which has 41 teeth.
Answer (2)
The problem involves inverse proportion between the number of teeth and revolutions per minute of two gears. We set up the equation T A R A = T B R B and substitute the given values T A = 22 , R A = 262 , and T B = 41 . Solving for R B , we get:
Establish the inverse proportion equation: T A R A = T B R B .
Substitute given values: 22 × 262 = 41 × R B .
Solve for R B : R B = 41 22 × 262 = 41 5764 ≈ 140.58536585 .
Round to two decimal places: 140.59 .
Explanation
Problem Setup We are given that two gears, A and B, are in mesh. Gear A has 22 teeth and revolves at 262 rpm. Gear B has 41 teeth. We need to find the number of revolutions per minute of gear B.
Inverse Proportionality The number of teeth and revolutions per minute are inversely proportional. This means that the product of the number of teeth and the number of revolutions per minute is constant for both gears. Let T A be the number of teeth of gear A, and R A be the number of revolutions per minute of gear A. Let T B be the number of teeth of gear B, and R B be the number of revolutions per minute of gear B. Then, we have the relationship:
T A R A = T B R B
Substitution We are given T A = 22 , R A = 262 , and T B = 41 . We want to find R B . Substituting the given values into the equation, we get:
22 × 262 = 41 × R B
Solving for Revolutions of Gear B Now, we solve for R B :
R B = 41 22 × 262
R B = 41 5764
R B ≈ 140.58536585
Final Answer We are asked to round the answer to two decimal places. Therefore, the number of revolutions per minute of gear B is approximately 140.59 rpm.
Examples
Understanding inverse proportions is crucial in many real-world scenarios, such as designing gear systems. For instance, in a bicycle, the ratio of teeth on the front and rear gears determines the effort required to pedal. If you increase the number of teeth on the rear gear, you'll need to pedal more to cover the same distance, but you'll gain mechanical advantage, making it easier to climb hills. Similarly, in engines and machinery, adjusting gear ratios allows engineers to optimize speed and torque output, balancing power and efficiency.
The number of revolutions per minute of gear B is approximately 140.59 rpm, calculated using the relationship of inverse proportion between the number of teeth and the revolutions per minute of the gears. By applying the formula T A × R A = T B × R B and substituting the known values, we can derive the value for R B .
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