A drilling machine is to have six speeds ranging from [tex]$50 rev/min$[/tex] to [tex]$750 rev/min$[/tex]. The speeds form a geometric progression.
Answer (1)
Identify the first term a 1 = 50 and the last term a 6 = 750 of the geometric progression.
Use the formula a 6 = a 1 5 to find the common ratio r = 5 15 ≈ 1.7188 .
Calculate the intermediate terms using a n = a 1 n − 1 for n = 2 , 3 , 4 , 5 .
The six speeds are approximately 50 , 85.94 , 147.71 , 253.88 , 436.36 , 750 rev/min. 50 , 85.94 , 147.71 , 253.88 , 436.36 , 750
Explanation
Understanding the Problem We are given that a drilling machine has six speeds that range from 50 rev/min to 750 rev/min. These speeds form a geometric progression. Our goal is to find these six speeds.
Setting up the Geometric Progression Let the six speeds be a 1 , a 2 , a 3 , a 4 , a 5 , a 6 . We know that a 1 = 50 and a 6 = 750 . Since the speeds form a geometric progression, we can express each term as a n = a 1 ⋅ r n − 1 , where r is the common ratio.
Finding the Common Ratio We can use the information about a 1 and a 6 to find the common ratio r . We have a 6 = a 1 ⋅ r 5 , so 750 = 50 ⋅ r 5 . Dividing both sides by 50, we get r 5 = 50 750 = 15 . Taking the fifth root of both sides, we find r = 5 15 .
Calculating the Speeds Now that we have the common ratio, we can find the other speeds. We have:
a 1 = 50 a 2 = 50 ⋅ r = 50 ⋅ 5 15 ≈ 50 ⋅ 1.7188 ≈ 85.94 a 3 = 50 ⋅ r 2 = 50 ⋅ ( 5 15 ) 2 ≈ 50 ⋅ ( 1.7188 ) 2 ≈ 147.71 a 4 = 50 ⋅ r 3 = 50 ⋅ ( 5 15 ) 3 ≈ 50 ⋅ ( 1.7188 ) 3 ≈ 253.88 a 5 = 50 ⋅ r 4 = 50 ⋅ ( 5 15 ) 4 ≈ 50 ⋅ ( 1.7188 ) 4 ≈ 436.36 a 6 = 50 ⋅ r 5 = 50 ⋅ 15 = 750
Final Answer Therefore, the six speeds are approximately 50, 85.94, 147.71, 253.88, 436.36, and 750 rev/min.
Examples
Geometric progressions are not just abstract math; they appear in many real-world scenarios. For example, consider the depreciation of a car's value over time. If a car loses 20% of its value each year, the car's value each year forms a geometric progression. Similarly, compound interest on a savings account grows geometrically. Understanding geometric progressions helps us predict future values in these situations, whether it's the resale value of an asset or the growth of an investment.
