Problem-Solving and Data Analysis

A hybrid vehicle uses $2$ gallons of fuel to travel $80$ miles in the city, and $2.5$ gallons of fuel to travel $120$ miles on the highway. A driver takes a trip that includes $40$ miles of city driving and $180$ miles of highway driving. What is the total number of gallons of fuel the vehicle is expected to use for the entire trip?

The number of active participants in an online educational forum has been growing exponentially since the forum was launched. At the launch of the forum ($t = 0$), there were $5,000$ active participants. Two months after the launch ($t = 2$), there were $7,200$ active participants. If the number of active participants continues to grow exponentially at this constant monthly percent rate, what is the number of active participants in the forum four months after the launch ($t = 4$)?

A scientist measures the concentration of a chemical solution across five different trials. The concentrations recorded for the first four trials are $12.1$ grams per liter ($\text{g/L}$), $12.2\text{ g/L}$, $12.4\text{ g/L}$, and $12.5\text{ g/L}$. The concentration for the fifth trial is $x\text{ g/L}$, where $x < 12.0$. If the mean of the concentrations from all five trials is equal to their median, what is the value of $x$?

A local library recorded the number of books checked out by 9 patrons on a certain day. The numbers of books checked out were:

If a 10th patron who checked out $x$ books is included, the mean of the numbers of books checked out by the 10 patrons is equal to the median of the numbers of books checked out by the 10 patrons. If $x$ is a positive integer, what is the value of $x$?

A software developer recorded the number of software bugs resolved each day for 20 days, as shown in the table below.

On the 21st day, the developer resolved 10 bugs. Which of the following statements best describes the effect of this new data point on the mean and the median of the dataset?

An investment account and a savings account are opened at the same time. The value of the investment account, in dollars, is modeled by the function $I(t) = 5,000(1.08)^t$, where $t$ is the number of years since the account was opened. The value of the savings account increases linearly by $\$400$ each year, starting with an initial deposit of $\$5,000$. What is the difference, in dollars, between the value of the investment account and the value of the savings account $2$ years after they are opened?

A small design firm employs $10$ people. The mean of their annual salaries is $\$50,000$, and the median is $\$48,000$. The employee with the highest salary of $\$95,000$ receives a $\$10,000$ raise, while the salaries of all other employees remain unchanged. What are the mean and the median of the salaries after the raise?

To create a specific shade of green paint, a painter mixes yellow paint and blue paint in a ratio of $3:5$ by volume. To create a specific shade of teal paint, the painter mixes yellow paint and blue paint in a ratio of $1:3$ by volume. If the painter mixes equal volumes of the green paint and the teal paint to create a new mixture, what is the ratio of yellow paint to blue paint in the new mixture?

At the start of $2018$, City A and City B each recycled $12,000$ tons of plastic. For the next several years, the annual amount of plastic recycled in City A increased by $800$ tons each year. During the same period, the annual amount of plastic recycled in City B increased by $5\%$ each year. Which of the following is closest to the difference, in tons, between the annual amount of plastic recycled in City B and the annual amount of plastic recycled in City A at the start of $2023$?

An environmental science class recorded the daily PM2.5 air quality index (AQI) values for a city over a 7-day period. The recorded values for 6 of the days were $38$, $42$, $45$, $49$, $52$, and $58$. The AQI value for the 7th day, $x$, is unknown. If the mean AQI value for the 7 days is $48$, what is the median AQI value for the 7 days?

A scientist is tracking the growth of a cell culture. The cell culture's area, in square millimeters, can be modeled by either a linear function or an exponential function. The table below shows the area of the cell culture at the end of Day 1 and Day 2.

If the area of the cell culture grows exponentially, the area on Day 4 would be $E$ square millimeters. If the area grows linearly, the area on Day 4 would be $L$ square millimeters. What is the value of $E - L$?

An agricultural drone sprays liquid fertilizer at a constant rate. The drone covers $3$ acres of cropland every $20$ minutes. If the drone operates continuously at this rate, how many hours will it take the drone to spray $54$ acres of cropland?

A researcher wants to estimate the proportion of registered voters in a county who support a proposed sales tax increase. The county has $50,000$ registered voters. The researcher randomly selects $400$ registered voters from a list of those who voted in the most recent school board election. Of the selected voters, $60\%$ support the proposed tax increase. Which of the following is the most appropriate conclusion?

A municipal library system models the growth of its e-book collection at two different branches. At the start of the year ($t = 0$ months), Branch A and Branch B each have $1,000$ e-books. The number of e-books at Branch A increases by $150$ e-books each month. The number of e-books at Branch B increases by $20\%$ each month. What is the difference between the number of e-books at Branch B and the number of e-books at Branch A at $t = 3$ months?

A laboratory technician prepares a liquid mixture by combining compound $X$, compound $Y$, and water. The ratio of the volume of compound $X$ to the volume of compound $Y$ is $3$ to $5$, and the ratio of the volume of compound $Y$ to the volume of water is $2$ to $3$. If the technician prepares a total of $1,240$ milliliters of the mixture, how many milliliters of compound $Y$ are in the mixture?

A municipal utility department wants to estimate the number of residential water service lines that contain lead in a community with $3,600$ homes. The department selects a random sample of $120$ homes and inspects their water service lines. The inspection reveals that $9$ of the sampled homes have lead service lines. Based on the results of this sample, what is the estimated number of homes in the entire community that have lead service lines?

A construction project requires a specific concrete mixture made by combining cement, sand, and gravel in a ratio of $1 : 2 : 4$ by weight. A builder needs to prepare exactly $350$ pounds of this concrete mixture. If the builder currently has $80$ pounds of sand and an unlimited supply of cement and gravel, how many additional pounds of sand must the builder purchase to make the concrete mixture?

An agricultural department monitored the population of honeybee colonies in two adjacent conservation areas starting in 2018 ($t = 0$). In Area 1, the number of colonies increases by a constant 230 colonies each year. In Area 2, the number of colonies increases by a constant 10% each year. At $t = 0$, both areas had 2,000 colonies. If the model for Area 1 predicts $L$ colonies at $t = 3$ and the model for Area 2 predicts $E$ colonies at $t = 3$, what is the value of $L - E$?

A transportation planner wants to estimate the proportion of commuters in a city who regularly use public transit to travel to work. The city has $60,000$ commuters. The planner conducts a survey by randomly selecting $300$ commuters from a database of registered transit pass holders in the city and finds that $240$ of them regularly use public transit to travel to work. Which of the following is the most appropriate conclusion based on the design of the survey?

A research group conducted a survey using a random sample of $250$ residents of a town with a total population of $18,000$ residents. Of the residents surveyed, $64\%$ reported that they recycle regularly. The survey has a margin of error of $3\%$. Based on the survey results, what is the difference between the maximum and minimum estimated number of residents in the town who recycle regularly?