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John, Sally, and Natalie would all like to save some money. John decides that it
would be best to save money in a jar in his closet every single month. He decides
to start with $300, and then save $100 each month. Sally has $6000 and decides
to put her money in the bank in an account that has a 7% interest rate that is
compounded annually. Natalie has $5000 and decides to put her money in the
bank in an account that has a 10% interest rate that is compounded continuously.

Write the model equation for Sally’s situation

How much money will sally have after 2 years?

How much money will sally have after 10 years?

What type of exponential model is Natalie’s situation?

Write the model equation for Natalie’s situation

How much money will Natalie have after 2 years?

How much money will Natalie have after 10 years?

Respuesta :

calculista

Answer:

Part 1) [tex]A=6,000(1.07)^{t}[/tex]

Part 2)  [tex]\$6,869.40[/tex]

Part 3) [tex]\$11,802.91[/tex]

Part 4) Is a exponential growth function

Part 5) [tex]A=5,000(e)^{0.10t}[/tex]    or  [tex]A=5,000(1.1052)^{t}[/tex]

Part 6) [tex]\$6,107.01[/tex]

Part 7)  [tex]\$13,591.41[/tex]

Step-by-step explanation:

Part 1) Write the model equation for Sally’s situation

we know that    

The compound interest formula is equal to  

[tex]A=P(1+\frac{r}{n})^{nt}[/tex] 

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest  in decimal

t is Number of Time Periods  

n is the number of times interest is compounded per year

in this problem we have  

[tex]P=\$6,000\\ r=7\%=0.07\\n=1[/tex]

substitute in the formula above

[tex]A=6,000(1+\frac{0.07}{1})^{1*t}\\ A=6,000(1.07)^{t}[/tex]

Part 2) How much money will Sally have after 2 years?

For t=2 years

substitute  the value of t in the exponential growth function

[tex]A=6,000(1.07)^{2}=\$6,869.40[/tex]

Part 3) How much money will Sally have after 10 years?

For t=10 years

substitute  the value of t in the exponential growth function

[tex]A=6,000(1.07)^{10}=\$11,802.91[/tex] 

Part 4) What type of exponential model is Natalie’s situation?

we know that

The formula to calculate continuously compounded interest is equal to

[tex]A=P(e)^{rt}[/tex] 

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest in decimal  

t is Number of Time Periods  

e is the mathematical constant number

we have  

[tex]P=\$5,000\\r=10\%=0.10[/tex]

substitute in the formula above

[tex]A=5,000(e)^{0.10t}[/tex]

Applying property of exponents

[tex]A=5,000(1.1052)^{t}[/tex]

 therefore

Is a exponential growth function

Part 5) Write the model equation for Natalie’s situation

[tex]A=5,000(e)^{0.10t}[/tex]    or  [tex]A=5,000(1.1052)^{t}[/tex]

see Part 4)

Part 6) How much money will Natalie have after 2 years?

For t=2 years

substitute

[tex]A=5,000(e)^{0.10*2}=\$6,107.01[/tex]

Part 7) How much money will Natalie have after 10 years?

For t=10 years

substitute

[tex]A=5,000(e)^{0.10*10}=\$13,591.41[/tex]

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