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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. Complete the equation for the relationship between the thickness and number of weeks.

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Write and solve equations for proportional relationships

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Q. 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. Complete the equation for the relationship between the thickness and number of weeks.

Given Information: We know: $\newline$ Initial thickness of the ice: $2$ meters $\newline$ Thickness after $3$ weeks: $1.25$ meters $\newline$ Duration: $3$ weeks $\newline$ We need to find the rate at which the ice thickness decreases per week.
Calculate Rate of Decrease: To find the rate of decrease per week, we subtract the final thickness from the initial thickness and then divide by the number of weeks. $\newline$ Rate of decrease per week = $(\text{Initial thickness} - \text{Final thickness}) / \text{Duration}$ $\newline$ Rate of decrease per week = $(2 \, \text{meters} - 1.25 \, \text{meters}) / 3 \, \text{weeks}$
Calculate Rate per Week: Now we calculate the rate of decrease per week. $\newline$ Rate of decrease per week = $(0.75 \, \text{meters}) / 3 \, \text{weeks}$ $\newline$ Rate of decrease per week = $0.25 \, \text{meters per week}$
Equation for Relationship: The equation for the relationship between the thickness $T$ of the ice and the number of weeks $w$ is: $\newline$ $T = \text{Initial thickness} - (\text{Rate of decrease per week} \times w)$ $\newline$ $T = 2 \, \text{meters} - (0.25 \, \text{meters/week} \times w)$

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