Question

calculator active use the piece - wise function to answer the following.

1. $f(x)=\

$$\begin{cases}9x - x^{2},&-2\\leq x\\leq3\\2x - 17,&3 < x < 10\\5,&10\\leq x\\leq12\\end{cases}$$

$
a. find $f(1)$.
b. find $f(10)$.

1. a rental car costs $20 to rent plus 70 cents every mile driven for the first 100 miles. miles driven over 100 only cost 40 cents.

$c(m)=\

$$\begin{cases}20 + 0.7m,&0\\leq m\\leq100\\0.4(m - 100)+90,&m > 100\\end{cases}$$

$
a. find $c(128)$.
b. how many miles did a customer drive that spent 24 dollars on their rental car?
multiple choice - calculator active

1. the table below shows the average price of a movie ticket during certain years.

year	2012	2015	2018	2019	2020	2022
price	7.96	8.17	8.97	9.11	9.16	10.12

a linear regression is used to construct a function model $p$ that models the price over the given years. if $t = 1$ corresponds to 2012, $t = 4$ corresponds to 2015, and this pattern continues, which of the following defines function $p$?
(a) $p(t)=0.206x + 7.541$
(b) $p(t)=0.397x + 7.524$
(c) $p(t)=0.206x + 7.747$
(d) $p(t)=0.206x - 407.07$

1. the weight of an object is inversely proportional to the square of the distance between an object and the center of the earth. this relationship is modeled by the function $w$, where $w(d)=\frac{2.944\times10^{9}}{d^{2}}$ for distance, $d$, measured in feet, and weight where $w(d)$ measured in pounds. what is the average rate of change, in pounds per foot, if the distance between an object and the center of the earth is increased from 8500 feet to 9500 feet?

(a) 2944
(b) 8.127

Explanation:

9.

Step1: Determine the function - part for $x = 1$

Since $- 2\leq1\leq3$, use $f(x)=9x - x^{2}$.

Step2: Substitute $x = 1$ into the function

$f(1)=9\times1-1^{2}=9 - 1=8$.

Step3: Determine the function - part for $x = 10$

Since $10\leq10\leq12$, use $f(x)=5$.

Step1: Determine the function - part for $m = 128$

Since $128>100$, use $C(m)=0.4(m - 100)+90$.

Step2: Substitute $m = 128$ into the function

$C(128)=0.4(128 - 100)+90=0.4\times28 + 90=11.2+90=101.2$.

Step3: Solve for $m$ when $C(m)=24$

Since $24<20 + 0.7\times100=90$, use $C(m)=20 + 0.7m$. Set $20+0.7m=24$. Then $0.7m=24 - 20=4$, so $m=\frac{4}{0.7}=\frac{40}{7}\approx5.71$ (but this is wrong as we should first check the domain. Since $24<90$, we use $20 + 0.7m=24$. Solving for $m$ gives $m=\frac{24 - 20}{0.7}=\frac{4}{0.7}=\frac{40}{7}\approx5.71$ miles).

Answer:

a. $8$
b. $5$

10.