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.
(a) what was the amount of money in passively managed index funds in 2015 and 2019?
(b) if the trend continues, what is the first full year when the amount of passively managed index funds exceeds $9.4 trillion?
(a) which of the following describes how to find 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.
(type an integer or a decimal.)
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. 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 $\boxed{1.98} trillion.
(do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: 2015 x-value calculation

$x = 2015 - 2000 = 15$

Step2: 2015 fund amount calculation

$A(15) = 0.1211(1.201)^{15}$
First calculate $(1.201)^{15} \approx 16.350$
Then $A(15) = 0.1211 \times 16.350 \approx 1.98$

Step3: 2019 x-value calculation

$x = 2019 - 2000 = 19$

Step4: 2019 fund amount calculation

$A(19) = 0.1211(1.201)^{19}$
First calculate $(1.201)^{19} \approx 29.972$
Then $A(19) = 0.1211 \times 29.972 \approx 3.63$

Step5: Set up inequality for $9.4T$

$0.1211(1.201)^x > 9.4$

Step6: Isolate exponential term

$(1.201)^x > \frac{9.4}{0.1211} \approx 77.622$

Step7: Solve for x via logarithm

$x > \frac{\ln(77.622)}{\ln(1.201)} \approx \frac{4.352}{0.183} \approx 23.78$

Step8: Find full year

$x=24$ corresponds to $2000+24=2024$

Step9: Confirm part (a) method

Substitute $x=15$ into $A(x)$

Answer:

(a)

• Correct choice: A. To find the amount of money in passively managed index funds in 2015, substitute 15 for x and evaluate to find f(x).
• Amount in 2015: 1.98 trillion dollars
• Amount in 2019: 3.63 trillion dollars

(b) 2024