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

Questions

1. 1.

   A digital sensor monitors the total mass of metal brackets produced, $y$ (in kilograms), which has a proportional relationship with the machine's operating time, $x$ (in minutes). The sensor records that 28 kilograms had been produced at an operating time of $t$ minutes, and exactly 6 minutes later, at time $t + 6$ minutes, the total production reached 52 kilograms. Which equation models this proportional relationship, and what was the machine's initial operating time, $t$?

   • A) $y = 0.25x$ and $t = 112$ minutes
   • B) $y = 4x$ and $t = 13$ minutes
   • C) $y = 4.67x$ and $t = 6$ minutes
   • D) $y = 4x$ and $t = 7$ minutes
2. 2.

   A marine research team deploys an underwater drone that descends at a constant speed, establishing a proportional relationship between its depth, $y$ (in meters), and the elapsed time, $x$ (in seconds). On a coordinate plane graphing this relationship with time on the horizontal axis and depth on the vertical axis, the line passes through the points $(15, 24)$ and $(m, 56)$. What is the drone's unit rate of descent represented by the point $(1, k)$ on this graph, and what is the value of $m$?

   • A) Unit rate: 1.6 meters per second; $m = 35$
   • B) Unit rate: 0.625 meters per second; $m = 35$
   • C) Unit rate: 1.6 meters per second; $m = 89.6$
   • D) Unit rate: 0.625 meters per second; $m = 89.6$
3. 3.

   An automated irrigation system in an eco-greenhouse dispenses water at a constant rate such that the total volume of water delivered, $V$ (in liters), is proportional to the operating time, $t$ (in hours). A digital sensor registers that the system delivered exactly 84 liters of water during an uninterrupted 3.5-hour cycle. Based on this proportional relationship, which equation correctly expresses $V$ in terms of $t$, and how many liters of water will be dispensed over an 8-hour period?

   • A) $V = 28t$ and 224 liters
   • B) $t = 24V$ and 192 liters
   • C) $V = 24t$ and 192 liters
   • D) $V = 21t$ and 168 liters

Answers and step-by-step solutions appear after you finish the test.