a system of inequalities can be used to determine the depth of a toy, in meters, in a pool depending on the…

a system of inequalities can be used to determine the depth of a toy, in meters, in a pool depending on the time, in seconds, since it was dropped. which constraint could be part of the scenario? the pool is 1 meter deep. the pool is 2 meters deep. the toy falls at a rate of at least a \\( \\frac { 1 } { 2 } \\) meter per second. the toy sinks at a rate of no more than a \\( \\frac { 1 } { 2 } \\) meter per second.

a system of inequalities can be used to determine the depth of a toy, in meters, in a pool depending on the time, in seconds, since it was dropped. which constraint could be part of the scenario? the pool is 1 meter deep. the pool is 2 meters deep. the toy falls at a rate of at least a \\( \\frac { 1 } { 2 } \\) meter per second. the toy sinks at a rate of no more than a \\( \\frac { 1 } { 2 } \\) meter per second.

Answer

Explanation:

Step1: Find the slope of the boundary line

The boundary line passes through ((-2,0)) and ((0, - 1)). The slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1-0}{0 - (-2)}=-\frac{1}{2}). The inequality representing the shaded region (since it is a solid line and we consider the rate) is (y\leqslant-\frac{1}{2}x - 1) (re - arranging for the context of depth (y) and time (x), the rate of change of (y) with respect to (x) is (\frac{\Delta y}{\Delta x}=-\frac{1}{2}). The magnitude of the slope gives the rate of descent.

Step2: Analyze each option

  • Option 1: The pool is 1 meter deep If we consider the (y) - values (depth), the maximum (y) - value (in the negative direction for depth, assuming (y = 0) is the surface) is not limited to (y=-1) from the graph.
  • Option 2: The pool is 2 meters deep There is no indication from the graph that the maximum depth (in terms of the lower bound of the inequality relevant to depth) is (y =- 2).
  • Option 3: The toy falls at a rate of at least a (\frac{1}{2}) meter per second The slope of the boundary line of the inequality (representing the rate of change of depth (y) with respect to time (x)) is (m =-\frac{1}{2}). The rate of descent (speed) is (\vert m\vert=\frac{1}{2}). Since the shaded region is below the line (y =-\frac{1}{2}x-1) (assuming (y) is depth and (x) is time), the toy falls at a rate of at least (\frac{1}{2}) meter per second.
  • Option 4: The toy sinks at a rate of no more than a (\frac{1}{2}) meter per second The slope of the boundary line implies a rate of (\frac{1}{2}) meter per second descent, and the inequality represents a rate of at least (\frac{1}{2}) meter per second (not no more than).

Answer:

The toy falls at a rate of at least a (\frac{1}{2}) meter per second.