Question

the amount of money (in trillions of dollars) that is invested in passively managed index funds can be approximated by the function ( a(x) = 0.1211(1.201)^x ), where ( x = 10 ) corresponds to the year 2010.\\( \text{(a)} \\) what was the amount of money in passively managed index funds in 2015 and 2019?\\( \text{(b)} \\) if the trend continues, what is the first full year when the amount of passively managed index funds exceeds $9.4 trillion?\\( \text{(a)} \\) which of the following describes how to find the amount of money in passively managed index funds in 2015 using the given information? select the correct choice and fill in the answer box to complete your choice.\\( \text{(type an integer or a decimal.)} \\)\\( \text{\\o a.} \\) to find the amount of money in passively managed index funds in 2015, substitute \\( \boldsymbol{15} \\) for ( x ) and evaluate to find ( f(x) ).\\( \text{\\o b.} \\) to find the amount of money in passively managed index funds in 2015, find the intersection point of the graphs ( y = 0.1211(1.201)^x ) and ( y = \\). the amount of money in passively managed index funds in 2015 is represented by the ( y )-coordinate.\
in 2015, the amount of money in passively managed index funds was about $\square$ trillion.\\( \text{(do not round until the final answer. then round to the nearest hundredth as needed.)} \\)

Explanation:

Step1: Identify x for 2015

Since $x=10$ corresponds to 2010, for 2015: $x = 2015 - 2010 + 10 = 15$

Step2: Calculate 2015 fund amount

Substitute $x=15$ into $A(x)=0.1211(1.201)^x$:
$A(15)=0.1211\times(1.201)^{15}$
First compute $(1.201)^{15}\approx16.4422$
Then $A(15)\approx0.1211\times16.4422\approx1.991$

Step3: Calculate 2019 fund amount

For 2019: $x = 2019 - 2010 + 10 = 19$
Substitute $x=19$ into $A(x)$:
$A(19)=0.1211\times(1.201)^{19}$
First compute $(1.201)^{19}\approx27.8935$
Then $A(19)\approx0.1211\times27.8935\approx3.378$

Step4: Solve for x when A(x)>9.4

Set up equation: $0.1211(1.201)^x > 9.4$
First rearrange: $(1.201)^x > \frac{9.4}{0.1211}\approx77.6218$
Take natural log of both sides: $x\times\ln(1.201) > \ln(77.6218)$
$\ln(1.201)\approx0.1824$, $\ln(77.6218)\approx4.352$
Solve for x: $x > \frac{4.352}{0.1824}\approx23.86$
Since x must be integer, $x=24$. The year is $2010 + (24-10)=2024$

Answer:

(a)

2015 amount:

$1.99$ trillion

2019 amount:

$3.38$ trillion

Correct choice for finding 2015 amount:

A. To find the amount of money in passively managed index funds in 2015, substitute 15 for x and evaluate to find f(x).

(b)

2024