3. suppose $20,000 is invested for 7 years. find the accumulated amount under the following plans, rounded…

3. suppose $20,000 is invested for 7 years. find the accumulated amount under the following plans, rounded to the nearest cent. circle the plan that yields the greatest return. a. 3.6% interest compounded monthly b. 3.8% interest compounded continuously 4. the population of florida panthers in 1980 was about 24 panthers, and in the year 2010 about 117 panthers. at this rate, the population of florida panthers can be modelled by the function p(t)=24(1.0542)^t where t is the number of years since 1980. a. fill in the blanks. the input variable of the function is _, and it represents the number of _ since _. the output is given symbolically by _ which represents the _ of _. b. use the function to approximate the projected population of florida panthers for the year 2032. label your numerical answer with the appropriate unit. (hints: how do we determine the value of t for which we should evaluate the function? if were counting the number of animals, what place value should we round to?)

3. suppose $20,000 is invested for 7 years. find the accumulated amount under the following plans, rounded to the nearest cent. circle the plan that yields the greatest return. a. 3.6% interest compounded monthly b. 3.8% interest compounded continuously 4. the population of florida panthers in 1980 was about 24 panthers, and in the year 2010 about 117 panthers. at this rate, the population of florida panthers can be modelled by the function p(t)=24(1.0542)^t where t is the number of years since 1980. a. fill in the blanks. the input variable of the function is _, and it represents the number of _ since _. the output is given symbolically by _ which represents the _ of _. b. use the function to approximate the projected population of florida panthers for the year 2032. label your numerical answer with the appropriate unit. (hints: how do we determine the value of t for which we should evaluate the function? if were counting the number of animals, what place value should we round to?)

Answer

3.

Explanation:

Step1: Recall compound - interest formula for monthly compounding

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times compounded per year, and $t$ is the number of years. Given $P=$20000$, $t = 7$ years, $r_a=0.036$, and $n = 12$ (monthly compounding). [A_a=20000(1 +\frac{0.036}{12})^{12\times7}=20000(1 + 0.003)^{84}=20000\times(1.003)^{84}] Using a calculator, $(1.003)^{84}\approx1.28377$, so $A_a=20000\times1.28377=$25675.40$.

Step2: Recall continuous - compounding formula

The continuous - compounding formula is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the number of years. Given $P = 20000$, $t=7$ years, and $r_b = 0.038$. [A_b=20000e^{0.038\times7}=20000e^{0.266}] Using a calculator, $e^{0.266}\approx1.30416$, so $A_b=20000\times1.30416=$26083.20$.

Since $A_b>A_a$, the plan with 3.8% interest compounded continuously yields the greatest return.

4.

Explanation:

a.

  • The input variable of the function is $t$, and it represents the number of years since 1980.
  • The output is given symbolically by $p(t)$ which represents the number of Florida panthers.

b.

First, find the value of $t$ for the year 2032. Since $t$ is the number of years since 1980, $t=2032 - 1980=52$. Then, use the function $p(t)=24(1.0542)^t$. Substitute $t = 52$ into the function: [p(52)=24(1.0542)^{52}] Using a calculator, $(1.0542)^{52}\approx17.797$, so $p(52)=24\times17.797 = 427.128$. Since we are counting the number of animals, we round to the nearest whole number. So the projected population of Florida panthers in 2032 is 427 panthers.

Answer:

  1. The plan with 3.8% interest compounded continuously yields the greatest return.
  2. a. $t$, years, 1980, $p(t)$, number of Florida panthers b. 427 panthers