Question

1. complete the circles with multiplication and dividing expression.
2. a car is bought for $20,000. it value decreases by 15% each year.

(a) what will be the value after 2 years?
(b) after how many years will it first be worth less than $10,000?

1. create your own two - step function machines.

(a) write your function
(b) create a table showing at 5 pairs of values.
(c) draw a mapping diagram.

Explanation:

2.

Let's assume we want to find expressions for multiplication and division with $\frac{4}{9}x^{12}$.

• For multiplication: If we multiply $\frac{4}{9}x^{12}$ by $3$, we get $b = \frac{4}{9}x^{12}\times3=\frac{4}{3}x^{12}$. If we multiply it by $\frac{1}{2}x^{3}$, we get $c=\frac{4}{9}x^{12}\times\frac{1}{2}x^{3}=\frac{2}{9}x^{15}$
• For division: If we divide $\frac{4}{9}x^{12}$ by $x^{4}$, we get $d = \frac{\frac{4}{9}x^{12}}{x^{4}}=\frac{4}{9}x^{8}$. If we divide it by $\frac{2}{3}$, we get $e=\frac{\frac{4}{9}x^{12}}{\frac{2}{3}}=\frac{2}{3}x^{12}$

3.

(a)

Step1: Identify the formula for depreciation

The formula for exponential - decay is $A = P(1 - r)^t$, where $P$ is the initial value, $r$ is the rate of decay as a decimal, and $t$ is the number of years. Here, $P=\$20000$, $r = 0.15$, and $t = 2$.

Step2: Substitute the values into the formula

$A=20000\times(1 - 0.15)^2=20000\times0.85^2=20000\times0.7225=\$14450$

Step1: Set up the inequality

We want to find $t$ such that $A=20000\times(0.85)^t<10000$. So, $(0.85)^t<\frac{10000}{20000}=0.5$

Step2: Take the natural - logarithm of both sides

$\ln(0.85)^t<\ln(0.5)$. Using the property of logarithms $\ln a^b = b\ln a$, we get $t\ln(0.85)<\ln(0.5)$

Step3: Solve for $t$

Since $\ln(0.85)<0$, when we divide both sides of the inequality by $\ln(0.85)$ the inequality sign flips. So, $t>\frac{\ln(0.5)}{\ln(0.85)}\approx\frac{- 0.6931}{-0.1625}\approx4.27$
Since $t$ represents the number of years and it must be an integer, $t = 5$

Answer:

$\$14450$

(b)