Question

The short-term

(no more than $24$hours) parking fee F (in dollars) for

parking x hours at O’Hare International Airport’s main

parking garage can be modeled by the function

role="math" localid="1645828501222" $F\left(x\right)=\left\{\begin{array}{ll}2& if0<x\le 1\\ 4& if1<x\le 3\\ 10& if3<x\le 4\\ 5int(x+1)+2& if4<x<9\\ 51& if9\le x\le 24\end{array}\right.$

Determine the fee for parking in the short-term parking

garage for

(a) $2$hours

(b) $7$hours

(c) $15$hours

(d) $8$hours and $24$minutes

Short answer

(a) The parking fee for$2$ hours is $$4$

(b) The parking fee for $7$hours is $$42$

(c) The parking fee for $15$hours is $$51$

(c) The parking fee for $8$hours and$24$ minutes is $$47$

Step-by-step solution

Step 1. given data

function for the parking fee for x hours is $F\left(x\right)=\left\{\begin{array}{ll}2& if0<x\le 1\\ 4& if1<x\le 3\\ 10& if3<x\le 4\\ 5int(x+1)+2& if4<x<9\\ 51& if9\le x\le 24\end{array}\right.$

parking time is

(a) $2$hours

(b) $7$hours

(c) $15$hours

(d) $8$hours and $24$minutes

Step 2. Part (a)

Parking time is $2$hours so $x=2$

substitute $2$ for x in the function

role="math" localid="1645829332024" $$\begin{gathered} F\left(x\right)=4if1<x\le 3 \\ F\left(2\right)=4 \end{gathered}$$

so the parking fee for$2$hoursis $$4$

Step 3. Part (b)

Parking time is $7$hours so$x=7$

substitute $7$for x in the function

role="math" localid="1645829508088" $$\begin{gathered} F\left(x\right)=5int(x+1)+2if4<x<9 \\ F\left(7\right)=5int(7+1)+2 \\ F\left(7\right)=5\left(8\right)+2 \\ F\left(7\right)=42 \end{gathered}$$

so the parking fee for$7$ hours is $$42$

Step 4. Part (c)

Parking time is$15$ hours so$x=15$

substitute $15$for x in the function

role="math" localid="1645829258895" $$\begin{gathered} F\left(x\right)=51if9\le x\le 24 \\ F\left(15\right)=51 \end{gathered}$$

so the parking fee for $15$hours is $$51$

Step 5. Part (d)

Parking time is$8$hours and $24$minutes

so$$\begin{gathered} x=8+\frac{24}{60} \\ x=8+0.4 \\ x=8.4 \end{gathered}$$

substitute $8.4$for x in the function

$$\begin{gathered} F\left(x\right)=5int(x+1)+2if4<x<9 \\ F(8.4)=5int(8.4+1)+2 \\ F(8.4)=5int(9.4)+2 \\ F(8.4)=5\left(9\right)+2 \\ F(8.4)=45+2 \\ F(8.4)=47 \end{gathered}$$

so the parking fee for $8$hours and $24$minutes is $$47$