A monkey is swinging from a tree. On the first swing, she passes through an arc of 20 m. With each swing, she passes through an arc [tex]\frac{4}{5}[/tex] the length of the previous swing.
What is the total distance the monkey has traveled when she completes her [tex]10^{\text {th }}[/tex] swing?
Round your final answer to the nearest meter.
[ ] m
Answer (2)
Identify the first term a = 20 and common ratio r = 5 4 of the geometric series.
Apply the formula for the sum of the first n terms of a geometric series: S n = 1 − r a ( 1 − r n ) .
Substitute a = 20 , r = 5 4 , and n = 10 into the formula: S 10 = 1 − 5 4 20 ( 1 − ( 5 4 ) 10 ) .
Calculate the sum and round to the nearest meter: S 10 ≈ 89 .
89
Explanation
Understanding the Problem We are given that the monkey swings through an arc of 20 m on the first swing. With each swing, the monkey travels 5 4 the length of the previous swing. We want to find the total distance the monkey has traveled when she completes her 1 0 th swing. This is a geometric series.
Identifying the Parameters The first term of the geometric series is a = 20 . The common ratio is r = 5 4 = 0.8 . The number of terms is n = 10 .
Stating the Formula The sum of the first n terms of a geometric series is given by the formula: S n = 1 − r a ( 1 − r n ) where S n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Calculating the Sum Substituting the values a = 20 , r = 5 4 , and n = 10 into the formula, we get: S 10 = 1 − 5 4 20 ( 1 − ( 5 4 ) 10 ) S 10 = 0.2 20 ( 1 − ( 0.8 ) 10 ) S 10 = 0.2 20 ( 1 − 0.1073741824 ) S 10 = 0.2 20 ( 0.8926258176 ) S 10 = 0.2 17.852516352 S 10 = 89.26258176
Rounding the Result We are asked to round the final answer to the nearest meter. Therefore, the total distance the monkey has traveled when she completes her 1 0 th swing is approximately 89 meters.
Final Answer Therefore, the total distance the monkey has traveled when she completes her 1 0 th swing, rounded to the nearest meter, is 89 meters.
Examples
Geometric series can be used to model scenarios where quantities decrease or increase by a constant factor. For example, the depreciation of a car's value each year can be modeled as a geometric series, where the initial value is the first term and the depreciation rate is related to the common ratio. Similarly, compound interest calculations involve geometric series, where the initial investment is the first term and the interest rate determines the common ratio. Understanding geometric series helps in predicting long-term financial outcomes and making informed decisions.
The total distance the monkey has traveled after 10 swings can be calculated using the sum of a geometric series. With the first swing of 20 meters and a common ratio of 5 4 , the total distance comes out to approximately 89 meters. Finally, after rounding to the nearest meter, the answer is 89 meters.
;
