Proportional relationships (constant of proportionality, graphs, equations) Transitional Reasoning Download PDF

Questions

1. 1.

   An autonomous underwater vehicle collects seafloor sonar data such that the volume of data stored, $D$ (in gigabytes), is directly proportional to the mission duration, $t$ (in hours). During a calibration dive, the vehicle collects 84 gigabytes of data in 3.5 hours, which a researcher plots on a coordinate plane with mission duration $t$ on the horizontal axis and data volume $D$ on the vertical axis. If the vehicle is deployed on a 12-hour survey with a dedicated 350-gigabyte storage drive, which statement correctly identifies the coordinate pair representing the unit rate on the graph, gives the equation relating $D$ to $t$, and determines the remaining unused storage capacity at the end of the survey?

   • A) Unit rate point: $(1, 24)$; equation: $D = 24t$; remaining unused capacity: 62 gigabytes
   • B) Unit rate point: $(24, 1)$; equation: $D = 24t$; remaining unused capacity: 62 gigabytes
   • C) Unit rate point: $(1, 24)$; equation: $D = 24t$; remaining unused capacity: 288 gigabytes
   • D) Unit rate point: $(1, 24)$; equation: $t = 24D$; remaining unused capacity: 62 gigabytes
2. 2.

   An engineering laboratory tests a calibrated elastic cord where the tensile force, $F$ (in newtons), is directly proportional to its elongation, $x$ (in millimeters). On Graph 1, elongation $x$ is plotted on the horizontal axis and tensile force $F$ is plotted on the vertical axis, creating a line that passes through the origin and the point (18, 45). To analyze cord sensitivity, a technician plots Graph 2 with tensile force $F$ on the horizontal axis and elongation $x$ on the vertical axis. Which statement correctly identifies the constant of proportionality $k$ for each graph and determines the elongation produced when a tensile force of 165 newtons is applied?

   • A) For Graph 1, $k = 0.4$ N/mm; for Graph 2, $k = 2.5$ mm/N; the elongation is 66 mm.
   • B) For Graph 1, $k = 2.5$ N/mm; for Graph 2, $k = 0.4$ mm/N; the elongation is 412.5 mm.
   • C) For Graph 1, $k = 2.5$ N/mm; for Graph 2, $k = 0.25$ mm/N; the elongation is 41.25 mm.
   • D) For Graph 1, $k = 2.5$ N/mm; for Graph 2, $k = 0.4$ mm/N; the elongation is 66 mm.
3. 3.

   A municipal composting facility blends food scraps and dry leaves in a directly proportional relationship to maintain an optimal carbon-to-nitrogen ratio. On a coordinate plane showing this relationship, the mass of food scraps in kilograms, $F$, is plotted on the horizontal axis, and the mass of dry leaves in kilograms, $L$, is plotted on the vertical axis. A calibrated batch successfully combines 360 kilograms of food scraps with 540 kilograms of dry leaves. If the facility receives a new delivery of 420 kilograms of food scraps and currently has 550 kilograms of dry leaves on hand, which statement correctly identifies the constant of proportionality $k$, gives the governing equation relating $L$ to $F$, and determines the additional mass of dry leaves needed to process this entire delivery?

   • A) $k = \frac{2}{3}$, the governing equation is $L = \frac{2}{3}F$, and the facility requires an additional 280 kilograms of dry leaves.
   • B) $k = 1.5$, the governing equation is $L = 1.5F$, and the facility requires an additional 80 kilograms of dry leaves.
   • C) $k = 1.5$, the governing equation is $L = 1.5F$, and the facility requires an additional 630 kilograms of dry leaves.
   • D) $k = \frac{2}{3}$, the governing equation is $L = \frac{2}{3}F$, and the facility requires an additional 130 kilograms of dry leaves.
4. 4.

   An environmental testing laboratory measures the mass of carbon dioxide absorbed by an experimental microalgae bioreactor over time. The mass of carbon dioxide absorbed, $C$ (in grams), is directly proportional to the operating time, $t$ (in hours). On a coordinate plane with operating time $t$ on the horizontal axis and absorbed mass $C$ on the vertical axis, the line representing this relationship passes through the origin and contains the points $(6, 105)$, $(a, 245)$, and $(22, b)$. Which statement correctly identifies the constant of proportionality $k$, gives the equation relating $C$ to $t$, and determines the unknown values $a$ and $b$?

   • A) $k = 17.5$; the equation is $C = 17.5t$; $a = 14$ and $b = 385$
   • B) $k = 17.5$; the equation is $C = 17.5t$; $a = 8$ and $b = 280$
   • C) $k = 17.5$; the equation is $t = 17.5C$; $a = 14$ and $b = 385$
   • D) $k = 99$; the equation is $C = t + 99$; $a = 146$ and $b = 121$
5. 5.

   A materials engineering laboratory develops an eco-friendly concrete by blending recycled glass aggregate and volcanic ash in a directly proportional relationship. On a coordinate plane where the mass of glass aggregate in kilograms, $x$, is on the horizontal axis and the mass of volcanic ash in kilograms, $y$, is on the vertical axis, the graph passes through the origin and the point $(35, 28)$. For a large structural beam, technicians prepare a batch containing a combined total of 270 kilograms of these two raw materials. Which statement correctly identifies the constant of proportionality $k$, gives the equation relating $y$ to $x$, and determines the mass of glass aggregate used in this batch?

   • A) $k = 1.25$, the equation is $y = 1.25x$, and the mass of glass aggregate is 120 kg.
   • B) $k = 1.25$, the equation is $y = 1.25x$, and the mass of glass aggregate is 150 kg.
   • C) $k = 0.8$, the equation is $y = 0.8x$, and the mass of glass aggregate is 120 kg.
   • D) $k = 0.8$, the equation is $y = 0.8x$, and the mass of glass aggregate is 150 kg.
6. 6.

   A digital printing press produces custom technical manuals where the total printing cost, $C$ (in dollars), is directly proportional to the total number of pages printed, $p$. A client orders an initial run of 15 manuals, each containing 240 pages, resulting in a total printing cost of $126.00. If the client places a second order for 28 manuals containing 350 pages each, which statement correctly identifies the constant of proportionality $k$ (in dollars per page), the equation relating $C$ to $p$, and the total printing cost of the second order?

   • A) $k = 8.40$ dollars per page; equation: $C = 8.40p$; total printing cost: $235.20
   • B) $k = 0.035$ dollars per page; equation: $C = 0.035p$; total printing cost: $343.00
   • C) $k = 0.035$ dollars per page; equation: $C = 0.035p$; total printing cost: $12.25
   • D) $k = 0.525$ dollars per page; equation: $C = 0.525p$; total printing cost: $5,145.00
7. 7.

   A logistics coordinator evaluates two shipping plans for transporting bulk manufacturing components. Plan X charges an initial documentation fee of $36 plus$4.50 per kilogram, while Plan Y charges a flat rate of $6.00 per kilogram with no initial fee. Which shipping plan represents a directly proportional relationship between total cost and mass, what is its governing equation, and at what shipment mass do both plans result in the exact same total cost?

   • A) Plan Y represents a proportional relationship, modeled by $C = 6m$; both plans cost the same at 24 kg.
   • B) Plan X represents a proportional relationship, modeled by $C = 4.5m + 36$; both plans cost the same at 24 kg.
   • C) Plan Y represents a proportional relationship, modeled by $C = 6m$; both plans cost the same at 8 kg.
   • D) Plan X represents a proportional relationship, modeled by $C = 4.5m + 36$; both plans cost the same at 8 kg.
8. 8.

   A municipal stormwater biofilter channel removes suspended solids from runoff such that the total mass of filtered sediment, $y$ (in kilograms), is directly proportional to the operating time, $x$ (in hours). On a coordinate plane showing this relationship, the line passes through the origin, and the vertical distance between the points corresponding to $x = 3$ and $x = 8$ is 185 kilograms. Which statement correctly identifies the constant of proportionality, gives the equation representing this relationship, and determines the total mass of sediment filtered after 12 hours of operation?

   • A) The constant of proportionality is 23.125 kilograms per hour, the equation is $y = 23.125x$, and the total mass filtered after 12 hours is 277.5 kilograms.
   • B) The constant of proportionality is 37 kilograms per hour, the equation is $y = 37x$, and the total mass filtered after 12 hours is 333 kilograms.
   • C) The constant of proportionality is 37 kilograms per hour, the equation is $y = 37x$, and the total mass filtered after 12 hours is 444 kilograms.
   • D) The constant of proportionality is $\frac{1}{37}$ kilogram per hour, the equation is $y = \frac{1}{37}x$, and the total mass filtered after 12 hours is approximately 0.32 kilogram.
9. 9.

   An aerial surveying drone maps ground area such that the total area surveyed, $A$ (in square kilometers), is directly proportional to flight time, $t$ (in hours). During a morning flight, the drone maps 63 square kilometers in 2.25 hours. For an afternoon flight, the drone is scheduled to fly for 3.5 hours at the same constant rate, but a low-battery alert forces it to land early after completing 77 square kilometers. Which statement correctly identifies the constant of proportionality $k$, the equation relating $A$ and $t$, and the remaining area of the afternoon flight left unsurveyed?

   • A) The constant of proportionality is 24 square kilometers per hour, the equation is $A = 24t$, and the unsurveyed area is 17 square kilometers.
   • B) The constant of proportionality is 28 square kilometers per hour, the equation is $A = 28t$, and the unsurveyed area is 21 square kilometers.
   • C) The constant of proportionality is 28 square kilometers per hour, the equation is $A = 28t$, and the unsurveyed area is 98 square kilometers.
   • D) The constant of proportionality is 32 square kilometers per hour, the equation is $A = 32t$, and the unsurveyed area is 35 square kilometers.
10. 10.

    A biochemical laboratory tests three automated liquid dispensers to determine which one delivers reagent in a directly proportional relationship to operating time. Dispenser X dispenses 14 mL in 4 seconds and 28 mL in 7 seconds, while Dispenser Y dispenses 9 mL in 2 seconds, 22.5 mL in 5 seconds, and 36 mL in 8 seconds. Dispenser Z dispenses 10 mL in 3 seconds and 25 mL in 6 seconds. Which dispenser maintains a directly proportional relationship between the volume dispensed and time, what is its constant of proportionality in milliliters per second, and which equation models the volume $V$ (in mL) delivered in $t$ seconds?

    • A) Dispenser X; constant of proportionality $k = 4$ mL/s; equation $V = 4t$
    • B) Dispenser Z; constant of proportionality $k = 5$ mL/s; equation $V = 5t$
    • C) Dispenser Y; constant of proportionality $k = 4.5$ mL/s; equation $V = 4.5t$
    • D) Dispenser Y; constant of proportionality $k = \frac{2}{9}$ mL/s; equation $V = \frac{2}{9}t$
11. 11.

    An agricultural research station tests a drip-irrigation emitter to verify that water delivery is directly proportional to operating time. During three calibration trials, the emitter dispenses 8.4 liters in 3.5 hours, 14.4 liters in 6.0 hours, and 20.4 liters in 8.5 hours. On a coordinate plane with operating time $t$ (in hours) on the horizontal axis and water volume $V$ (in liters) on the vertical axis, a technician plots these results to establish the system model. Which statement correctly identifies the constant of proportionality, gives the equation relating $V$ and $t$, and specifies the operating time required to deliver exactly 45.6 liters of water?

    • A) The constant of proportionality is 2.4 liters per hour, the relationship is modeled by $V = 2.4t$, and delivering 45.6 liters requires 19.0 hours of operation.
    • B) The constant of proportionality is 2.4 liters per hour, the relationship is modeled by $V = 2.4t$, and delivering 45.6 liters requires 109.4 hours of operation.
    • C) The constant of proportionality is 0.42 liters per hour, the relationship is modeled by $V = 0.42t$, and delivering 45.6 liters requires 108.6 hours of operation.
    • D) The constant of proportionality is 2.4 liters per hour, the relationship is modeled by $t = 2.4V$, and delivering 45.6 liters requires 19.0 hours of operation.
12. 12.

    A robotic laser cutter uses nitrogen assist gas to make precision cuts in stainless steel sheets, where the volume of nitrogen consumed, $g$ (in liters), is directly proportional to the cut length, $d$ (in meters). During a calibration test, the machine consumes 7.2 liters of nitrogen while cutting a continuous seam of 18 meters. A technician graphs this proportional relationship on a coordinate plane with cut length, $d$, on the horizontal axis and gas volume, $g$, on the vertical axis. Which statement correctly identifies the constant of proportionality $k$ (in liters per meter), the coordinate pair representing the unit rate, and the cut length corresponding to a nitrogen consumption of 15.6 liters?

    • A) The constant of proportionality is $2.5$ liters per meter, the unit rate is represented by $(1, 2.5)$, and the cut length is 6.24 meters.
    • B) The constant of proportionality is $0.4$ liters per meter, the unit rate is represented by $(0.4, 1)$, and the cut length is 6.24 meters.
    • C) The constant of proportionality is $0.4$ liters per meter, the unit rate is represented by $(1, 0.4)$, and the cut length is 39 meters.
    • D) The constant of proportionality is $2.5$ liters per meter, the unit rate is represented by $(2.5, 1)$, and the cut length is 62.4 meters.
13. 13.

    An architectural modeling studio uses an automated 3D printer that consumes liquid photopolymer resin at a constant rate. During a calibration run, the printer uses 42 cubic centimeters of resin in 3.5 hours, establishing a directly proportional relationship between total resin volume, $V$ (in cubic centimeters), and printing time, $t$ (in hours). The studio schedules Project 1, which requires 78 cubic centimeters of resin, followed by Project 2, which runs continuously for 8.5 hours. Which equation models this proportional relationship, and what are the required printing time for Project 1 and the total resin volume consumed by Project 2?

    • A) Equation: $V = \frac{1}{12}t$; Project 1 printing time: 936 hours; Project 2 resin volume: 0.71 cubic centimeters
    • B) Equation: $V = 12t$; Project 1 printing time: 936 hours; Project 2 resin volume: 0.71 cubic centimeters
    • C) Equation: $t = 12V$; Project 1 printing time: 6.5 hours; Project 2 resin volume: 102 cubic centimeters
    • D) Equation: $V = 12t$; Project 1 printing time: 6.5 hours; Project 2 resin volume: 102 cubic centimeters
14. 14.

    A municipal desalination plant uses reverse osmosis to purify seawater, where the volume of purified water produced, $W$, is directly proportional to the volume of seawater processed, $S$. Operating records show that processing 125 cubic meters of seawater yields 55 cubic meters of purified water. Which equation models this proportional relationship, and how many cubic meters of seawater must the plant process to produce 748 cubic meters of purified water?

    • A) $W = 0.44S$, and the facility must process 329.12 cubic meters of seawater.
    • B) $W = 0.44S$, and the facility must process 1,700 cubic meters of seawater.
    • C) $W = 0.55S$, and the facility must process 1,360 cubic meters of seawater.
    • D) $W = 2.27S$, and the facility must process 1,700 cubic meters of seawater.
15. 15.

    A textile mill operates an automated loom that weaves fabric at a constant rate, completing 54 meters of fabric in 4.5 hours. On a coordinate plane representing this proportional relationship, the weaving time in hours, $t$, is plotted on the horizontal axis, and the total fabric length in meters, $L$, is plotted on the vertical axis. If an upgraded motor increases the loom's constant weaving rate by 25%, which equation models the relationship for the upgraded loom, and which coordinate pair on its graph represents the output of a 6-hour shift?

    • A) $L = 12t$ and $(6, 72)$
    • B) $L = 15t$ and $(90, 6)$
    • C) $L = 15t$ and $(6, 90)$
    • D) $L = 14.5t$ and $(6, 87)$
16. 16.

    An automated sorting line at a materials recovery facility processes recyclable plastics at a constant rate. During a continuous calibration run from 07:30 to 11:00, the line successfully sorts 2,940 kilograms of plastic. The operations team aims to process a daily quota of 7,000 kilograms during a standard 8-hour work shift. Which statement correctly identifies the constant of proportionality in kilograms per hour, the equation modeling the total mass $M$ sorted in $t$ hours, and the additional operating time needed beyond the 8-hour shift to reach the daily quota?

    • A) The constant of proportionality is $840\text{ kg/h}$, the relationship is modeled by $M = 840t$, and the sorting line must operate for an additional 20 minutes.
    • B) The constant of proportionality is $840\text{ kg/h}$, the relationship is modeled by $M = 840t$, and the sorting line must operate for an additional 33 minutes.
    • C) The constant of proportionality is $735\text{ kg/h}$, the relationship is modeled by $M = 735t$, and the sorting line must operate for an additional 45 minutes.
    • D) The constant of proportionality is $980\text{ kg/h}$, the relationship is modeled by $M = 980t$, and the sorting line must operate for an additional 15 minutes.
17. 17.

    A municipal water reservoir uses an automated pump that delivers water at a constant rate into a filtration basin, supplying 840 gallons in 14 minutes. A technician graphs this relationship on a coordinate plane with elapsed time in minutes, $m$, on the horizontal axis and total volume in gallons, $V$, on the vertical axis. During scheduled maintenance, the pump operates at a constant rate that is 15 gallons per minute slower than its standard rate. Which equation models the volume of water delivered during maintenance mode, and which ordered pair on its graph correctly represents the volume delivered over that time?

    • A) $V = 60m$, and the point $(6, 360)$ indicates that 360 gallons are delivered in 6 minutes.
    • B) $V = 45m$, and the point $(8, 360)$ indicates that 360 gallons are delivered in 8 minutes.
    • C) $V = 75m$, and the point $(5, 375)$ indicates that 375 gallons are delivered in 5 minutes.
    • D) $V = 45m$, and the point $(360, 8)$ indicates that 8 gallons are delivered in 360 minutes.