Question

1. the value of a stock increases by 8% each year.

a. explain why the stock value does not increase by 80% each decade.
b. does the value increase by more or less than 80% each decade?

1. decide if each statement is true or false. (lesson 4-3)

a. $50^{\frac{1}{2}}=25$
b. $sqrt{30}$ is a solution to $y^{2}=30$
c. $243^{\frac{1}{3}}$ is equivalent to $sqrt3{243}$.
d. $sqrt{20}$ is a solution to $m^{4}=20$

1. lin is saving $300 per year in an account that pays 4.5% interest per year, compounded annually. about how much money will she have 20 years after she started? (lesson 2-26)

a. $545.45
b. $3,748.78
c. $9,411.43
d. $1,124,634.54

Explanation:

For Question 5:

Step1: Define compound growth formula

Let initial value be $P$, growth rate $r=0.08$, time $t=10$.
Compound value: $P(1+r)^t$

Step2: Calculate total growth factor

$\text{Factor}=(1+0.08)^{10}=1.08^{10}\approx2.1589$

Step3: Calculate total growth percentage

$\text{Growth \%}=(2.1589-1)\times100\approx115.89\%$

Step4: Compare to 80% (part a/b)

Simple 8%/year for 10 years gives $10\times8\%=80\%$, but compound growth earns interest on accumulated value, so it is not simple addition. The total growth is more than 80%.

Step1: Evaluate $50^\frac{1}{2}$

$50^\frac{1}{2}=\sqrt{50}=5\sqrt{2}\approx7.07
eq25$, so false.

Step2: Test $\sqrt{30}$ for $y^2=30$

$(\sqrt{30})^2=30$, which matches the equation, so true.

Step3: Relate $243^\frac{1}{5}$ to roots

$a^\frac{1}{n}=\sqrt[n]{a}$, so $243^\frac{1}{5}=\sqrt[5]{243}$, so true.

Step4: Test $\sqrt{20}$ for $m^4=20$

$(\sqrt{20})^4=(20^\frac{1}{2})^4=20^2=400
eq20$, so false.

Step1: Use annuity future value formula

The formula for future value of an ordinary annuity is:
$$FV = P\times\frac{(1+r)^t - 1}{r}$$
where $P=300$, $r=0.045$, $t=20$.

Step2: Calculate $(1+r)^t$

$(1+0.045)^{20}=1.045^{20}\approx2.4117$

Step3: Compute numerator and fraction

$(2.4117-1)=1.4117$
$\frac{1.4117}{0.045}\approx31.3711$

Step4: Calculate final future value

$FV=300\times31.3711\approx9411.33$

Answer:

a. The stock grows via compound growth, not simple growth—each year's increase is applied to the already increased value, not the original principal, so the total growth is not just $10\times8\%$.
b. More than 80%

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For Question 6: