Question

a monkey is swinging from a tree. On the first swing, she passed through an arc of 20m. With each swing , she passes through an arc 4/5 the length of the previous swing. What is the total distance the monkey has traveled when she completes her 10th swing

Answer 1

Answer:

the answer is 89

Step-by-step explanation:

Answer 2

Answer:

89.26 m

Step-by-step explanation:

We are given that

Distance covered by Monkey in first swig=20 m

Distance covered by monkey in second swing=$\frac{4}{5}\times 20=16 m$

Distance covered by monkey in third swing=$\frac{4}{5}\times \frac{4}{5}\times 20=12.8 m$

Distance covered by monkey in fourth swing=$\frac{4}{5}\times (\frac{4}{5})^2\times 20=(\frac{4}{5})^3(20) m$

From above pattern

We can write

$a_n=(\frac{4}{5})^{n-1}(20)$

$a_{10}=(\frac{4}{5})^9(20)$

Total distance covered by monkey when she completed hr 10th swing

$D=a_1+a_2+...+a_{10}$

$D=20+\frac{4}{5}(20)+(\frac{4}{5})^2(20)+...+(\frac{4}{5})^9(20)$

$D=20(1+\frac{4}{5}+(\frac{4}{5})^2+....+(\frac{4}{5})^9)$

$\frac{a_2}{a_1}=\frac{\frac{4}{5})^2}{\frac{4}{5}}=\frac{4}{5}$

$\frac{a_3}{a_2}=\frac{(\frac{4}{5})^3}{(\frac{4}{5})^2}=\frac{4}{5}$

It is in G.P form

Sum of n terms in G.P

$S_n=\frac{a(1-r^n)}{1-r}$

When $r<1$

a=1 and r=$\frac{4}{5}$,n=10

Using the formula

$D=20(\frac{1(1-(\frac{4}{5})^{10})}{1-\frac{4}{5}})$

$D=20\times 5(1-(0.8)^{10})$

D=89.26 m