Interpreting Linear Relationships in Context

A commercial agricultural irrigation system distributes liquid fertilizer from a storage tank. The volume of fertilizer remaining in the tank, $V$, in liters, $m$ minutes after the system is turned on is modeled by the equation:

where $c$ is the number of times the system's nozzles are cleaned during the irrigation process. During each cleaning cycle, the flow of fertilizer is completely paused, and no fertilizer is distributed. Based on the model, what is the duration, in minutes, of a single cleaning cycle?

A tutor charges a total fee, $C$, in dollars, for a tutoring session of $h$ hours according to the equation $C = 25h + 40$. What is the best interpretation of $40$ in this context?

A shipping company offers two types of delivery services: standard and express. The total shipping cost, in dollars, for a package sent via standard service is modeled by the function $C(w) = 1.25w + b_s$, where $w$ is the weight of the package, in pounds, and $b_s$ is a constant representing the flat handling fee. The total shipping cost, in dollars, for a package sent via express service is modeled by the function $E(k) = m_e k + b_e$, where $k$ is the weight of the package, in kilograms, and $b_e$ is a constant representing the flat handling fee.

The rate of change of the express shipping cost with respect to the package's weight, in dollars per kilogram, is $2.4$ times the rate of change of the standard shipping cost with respect to the package's weight, in dollars per pound. The flat handling fee for the express service is $4.50$ dollars more than the flat handling fee for the standard service. If it costs $58.50$ dollars to ship a package weighing $26.4$ pounds using the express service, what is the cost, in dollars, to ship a package weighing $20$ pounds using the standard service? (Assume $1\text{ kilogram} = 2.2\text{ pounds}$.)

A worker at a distribution center packages boxes at a constant rate. The total number of boxes, $B$, the worker has packaged $h$ hours after starting their shift can be modeled by the equation $B = 12h + 15$. According to the model, how many boxes were already packaged at the start of the worker's shift?

A hiker begins a climb at an elevation of $1,200$ feet above sea level and climbs at a constant rate of $350$ feet per hour. The hiker's elevation, $E$, in feet, $t$ hours after beginning the climb is given by the equation $E = 350t + 1,200$. Which of the following is the best interpretation of $350$ in this context?

A logistics company operates two distribution centers, Center A and Center B. The daily operating cost, $C_A$, in dollars, at Center A when processing $p$ packages is given by the equation $C_A = 1.75p + 3,200$. The daily operating cost, $C_B$, in dollars, at Center B when processing $p$ packages is given by the equation $C_B = 2.25(p - 400) + 2,800$, where $p \geq 400$. The difference in daily operating costs, $D$, in dollars, between Center B and Center A is defined as $D = C_B - C_A$. For $p \geq 400$, this difference is modeled by the equation $D = 0.50p - 1,300$. Which of the following is the best interpretation of the number $0.50$ in this context?

An electric vehicle's battery is being charged. The charge of the battery, $C$, as a percentage of its full capacity, $t$ minutes after the charging begins is modeled by the equation $C = 1.2t + 18$, where $t \le 60$. According to the model, what was the battery's charge percentage when the charging began?

A custom t-shirt printing company charges a one-time design setup fee plus a fixed price for each t-shirt printed. The total cost, $C$, in dollars, for an order of $n$ t-shirts is given by the equation $C = 15n + 50$. What does the number $50$ represent in this equation?

An oceanographic research vessel measures hydrostatic pressure during deep-sea dives. During a dive in temperate waters, the pressure $P$, in atmospheres (atm), at a depth of $d$ meters below the surface is modeled by the linear equation $P = \rho d + 1.0$, where $\rho$ is a constant representing the rate of change of pressure with respect to depth. At a depth of $150$ meters, the pressure is $16.0$ atm. During a second dive in a high-salinity region, the rate of change of pressure with respect to depth is $25\%$ greater than the rate in the temperate waters. According to the model for the second dive, at what depth, in meters, is the pressure $26.0$ atm?

A scientist is monitoring the volume of liquid nitrogen in a storage tank. The volume $V$, in liters, of liquid nitrogen remaining in the tank $t$ hours after a cooling system malfunction is modeled by the equation $V = 240 - 1.5t$. After how many hours of malfunction will there be exactly $180$ liters of liquid nitrogen remaining in the tank?

A manufacturing company uses the equation below to model the total daily cost, $K$, in thousands of dollars, of operating a factory when $x$ units are produced:

where $150 \le x \le 800$. The company plans to transition to a new production setup that will increase the daily fixed operating cost by $\$3,000$ but will decrease the cost to produce each unit by $\$20$. Which of the following equations best models the new total daily cost, $K_{\text{new}}$, in thousands of dollars, to produce $x$ units under the new setup?

An architect is tracking the height of a new skyscraper under construction. The total height $H$, in feet, of the building $w$ weeks after construction of the main frames began is modeled by the equation $H = 120 + 15w$. According to the model, by how many feet does the height of the building increase each week?

A botanist tracks the growth of a bamboo plant. The height $h$, in inches, of the plant $d$ days after the tracking began can be modeled by the equation $h = 4.5d + 12$. Which of the following is the best interpretation of $12$ in this context?

A deep space communications satellite transmits a telemetry data file to a ground station on Earth. The remaining size of the file to be received at the ground station, $S$, in megabytes (MB), can be modeled by the equation $S = 850 - 4.5(t - 18)$, where $t$ is the number of seconds since the satellite initiated its transmission sequence, and $t \ge 18$. Which of the following is the best interpretation of the number $18$ in this context?

A landscaping service uses a water tank to irrigate lawns. The volume of water, $V$, in gallons, remaining in the tank after $n$ lawns have been irrigated is modeled by the equation $V = 850 - 25n$. According to the model, by how many gallons does the volume of water in the tank decrease for each lawn that is irrigated?

An environmental cleanup crew is removing a contaminant from a soil site. The remaining mass of the contaminant, $C$, in kilograms, after $d$ days of treatment is modeled by the equation $C = 400 - p d$, where $p$ is the daily removal rate, in kilograms per day, under the original protocol. Under a new treatment protocol, the daily removal rate is increased by $25\%$, and the treatment time required to completely remove the contaminant is reduced by $8$ days. What was the daily removal rate, in kilograms per day, under the original protocol?

An industrial oven is used in a bakery. The temperature of the oven chamber, $C$, in degrees Fahrenheit ($^\circ\text{F}$), $t$ minutes after the heating element is turned on is modeled by the linear equation:

After a system upgrade, the starting temperature of the oven is $8^\circ\text{F}$ warmer, and the rate at which the oven heats up is $20\%$ faster. During a test of the upgraded oven, the heating element is turned on for $15$ minutes, after which the oven is turned off and cools down at a constant rate of $12^\circ\text{F}$ per minute. If the cooling process is also linear, which of the following functions models the temperature of the upgraded oven, $U$, in degrees Fahrenheit, $m$ minutes after it is turned off?

To rent an electric scooter, a rider pays a flat unlocking fee plus a fee for each minute of the ride. The total cost $C$, in dollars, for a ride of $m$ minutes is given by the equation $C = 0.15m + 1.20$. Which of the following is the best interpretation of the number $1.20$ in this context?