Question

a lake near the arctic circle is covered by a 2 - meter - thick sheet of ice during the cold winter months. when spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. after 3 weeks, the sheet is only 1.25 meters thick. let y represent the ice sheets thickness (in meters) after x weeks. complete the equation for the relationship between the thickness and number of weeks. y =

Explanation:

Step1: Calculate the rate of ice - thickness decrease

The initial thickness of the ice is $2$ meters and after $3$ weeks, it is $1.25$ meters. The change in thickness $\Delta y=2 - 1.25=0.75$ meters. The change in time $\Delta x = 3$ weeks. The rate of change (slope $m$) of the linear relationship is $m=\frac{\Delta y}{\Delta x}=\frac{2 - 1.25}{3}=\frac{0.75}{3}=0.25$ meters per week. Since the thickness is decreasing, the slope is negative, $m=- 0.25$.

Step2: Use the slope - intercept form of a linear equation

The slope - intercept form of a linear equation is $y=mx + b$, where $m$ is the slope and $b$ is the $y$ - intercept. The initial thickness of the ice (when $x = 0$) is $b = 2$ meters. Substituting $m=-0.25$ and $b = 2$ into the equation, we get $y=-0.25x+2$.

Answer:

$y=-0.25x + 2$