Question

1. a vacation home in clearwater, florida, rents for $105 per day. the home pays a $105 gas bill each time it is rented for x days. what is the domain of this function?

a. ( x geq 0 )
b. ( {0, 1, 2, 3, dots} )
c. ( {0, 105, 210, 315, dots} )
d. all real numbers

1. the scatter plot below shows the average traffic volume and the average traffic speed on interstate 75 for 50 days in 2020.

scatter plot image with average traffic volume on x - axis and average traffic speed on y - axis, showing a negative trend
which statement best describes the relationship between average traffic volume and average traffic speed shown on the scatter plot?
a. as traffic volume increases, vehicle speed increases
b. as traffic volume increases, vehicle speed decreases
c. as traffic volume increases, vehicle speed increases at first, then decreases
d. as traffic volume decreases, vehicle speed increases

1. the scatter plot shows the monthly heating bill cost for the williams family compared to the average temperature per month. which statement best describes the trend for the scatter plot?

scatter plot image with average temperature (°f) on x - axis and monthly heating bill (in dollars) on y - axis, showing a negative trend
a. the lower the average monthly temperature, the lower the monthly heating bill
b. the lower the average monthly temperature, the higher the monthly heating bill
c. there is a low correlation between the average monthly temperature and the monthly heating bill
d. there is no correlation between the average monthly temperature and the monthly heating bill

Explanation:

Question 17 (Assuming the vacation home problem, let's solve it if we can infer the function)

Step1: Identify the function

The vacation home costs $105 per day, and there's a $100 cleaning fee. So the total cost \( f(x) \) for \( x \) days is \( f(x)=105x + 100 \), where \( x\geq0 \) and \( x \) is a non - negative integer (since you can't rent for a fraction of a day in a practical sense, but also \( x = 0 \) would mean just the cleaning fee? Wait, maybe \( x \) is the number of days, so \( x\in\{0,1,2,\dots\} \), and the range: when \( x = 0 \), \( f(0)=100 \); \( x = 1 \), \( f(1)=105 + 100=205 \); \( x = 2 \), \( f(2)=210+100 = 310 \)? Wait, maybe the options are about the domain or range. Let's assume the function is \( f(x)=105x + 100 \), \( x\geq0 \) (days, so \( x \) is a non - negative integer, but also \( x = 0 \) could be a possible input (just paying cleaning fee). The range would be \( f(x)=100,205,310,\dots \) which is \( \{100 + 105x|x\geq0,x\in\mathbb{Z}\} \). Looking at the options, option c: \( \{0,105,210,315,\dots\} \) doesn't match, wait maybe I misread. Wait the original problem: "A vacation home in Tarpon Springs, Florida, rents for $105 per day. The homeowner pays a $100 g... (maybe cleaning) fee in addition to costing the home for \( x \) days. What is the domain of this function?" Wait domain is the possible values of \( x \). \( x \) is the number of days, so \( x\geq0 \) and \( x \) is a non - negative integer (0,1,2,3,...), so option b: \( \{0,1,2,3,\dots\} \) or option a: \( x\geq0 \) (real numbers) but in reality, you can't rent for a fraction of a day, so domain is non - negative integers. But maybe the question is about range. Wait the options: a. \( x\geq0 \) (domain), b. \( \{0,1,2,3,\dots\} \) (domain, non - negative integers), c. \( \{0,105,210,315,\dots\} \) (range, but 0 is not in range if \( x\geq0 \) and \( f(x)=105x + 100 \), when \( x = 0 \), \( f(x)=100 \)), d. all real numbers. So if it's domain, \( x \) is the number of days, so \( x\geq0 \) and \( x \) is a non - negative integer, so option b. But maybe I made a mistake. Alternatively, if the function is total cost, \( f(x)=105x+100 \), \( x\geq0 \) (days, so \( x \) is a non - negative real number? No, days are discrete. So domain is \( x\geq0 \) (non - negative integers), so option b.

Step2: Analyze the options

• Option a: \( x\geq0 \) (real numbers) is incorrect because you can't rent for a fraction of a day in a practical rental situation (domain should be non - negative integers).
• Option b: \( \{0,1,2,3,\dots\} \) is the set of non - negative integers, which is the domain (number of days can be 0,1,2,...).
• Option c: This looks like a range (105x) but with 0, which is not correct as the minimum cost is 100 (when \( x = 0 \)).
• Option d: All real numbers is incorrect as days can't be negative or a fraction in a practical sense.

Step1: Analyze the scatter plot trend

A scatter plot with average traffic volume on the x - axis and average traffic speed on the y - axis. From the description, the points show a downward trend (as x (traffic volume) increases, y (traffic speed) decreases).

• Option a: Says speed increases with volume, which is opposite of the trend.
• Option b: As traffic volume increases, vehicle speed decreases. This matches the downward trend in the scatter plot.
• Option c: Says speed increases then decreases, but the plot shows a consistent downward trend.
• Option d: Says as traffic volume decreases, speed increases, which is the same as option b (since if volume increases, speed decreases, then volume decreases, speed increases), but the question is about the relationship as volume increases. The main trend is that with increasing volume, speed decreases.

Step1: Analyze the scatter plot trend

The scatter plot has average temperature (x - axis) and monthly heating bill (y - axis). As the average temperature (x) increases, the monthly heating bill (y) decreases. So when the average monthly temperature is lower (smaller x), the monthly heating bill is higher (larger y).

• Option a: Says lower temperature leads to lower heating bill, which is incorrect.
• Option b: The lower the average monthly temperature, the higher the monthly heating bill. This matches the trend (as temperature increases, heating bill decreases, so inverse: lower temperature, higher bill).
• Option c: The correlation is strong (negative correlation), not low.
• Option d: There is a negative correlation, so there is a relationship.

Answer:

b. \( \{0,1,2,3,\dots\} \)

Question 18