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A lake near the Arctic Circle is covered by a 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 rate of 0.2 meters per week. After 7 weeks, the sheet is only 2.4 meters thick.
Let
y represent the ice sheet's thickness (in meters) after
x weeks.
Complete the equation for the relationship between the thickness and number of weeks.

y=◻

A lake near the Arctic Circle is covered by a 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 rate of $0.2$ meters per week. After $7$ weeks, the sheet is only $2.4$ meters thick. $\newline$ Let $y$ represent the ice sheet's thickness (in meters) after $x$ weeks. $\newline$ Complete the equation for the relationship between the thickness and number of weeks. $\newline$ $y=\square$

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Write exponential functions: word problems

Full solution

Q. A lake near the Arctic Circle is covered by a 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 rate of

0.2

meters per week. After

7

weeks, the sheet is only

2.4

meters thick.

\newline

Let

y

represent the ice sheet's thickness (in meters) after

x

weeks.

\newline

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

\newline

y=\square

Rephrasing the Equation: Let's first rephrase the ＂What is the equation that models the relationship between the thickness of the ice sheet and the number of weeks?＂
Calculating Initial Thickness: To find the initial thickness of the ice sheet before it started melting, we need to work backwards from the given information. We know that after $7$ weeks, the ice is $2.4$ meters thick and it decreases by $0.2$ meters each week. So, we can calculate the initial thickness by adding $7$ weeks' worth of melting to the final thickness. $\newline$ Initial thickness $=$ Final thickness $+$ (Rate of melting per week $*$ Number of weeks) $\newline$ Initial thickness $= 2.4$ meters $+$ ( $0.2$ meters/week $*$ $7$ weeks)
Performing Calculation: Performing the calculation for the initial thickness: $\newline$ Initial thickness = $2.4\,\text{meters} + 1.4\,\text{meters}$ $\newline$ Initial thickness = $3.8\,\text{meters}$ $\newline$ This is the value of ' $a$ ' in the equation $y = a - bx$ , where ' $b$ ' is the rate of melting per week, and ' $x$ ' is the number of weeks.
Writing the Equation: Now we can write the equation using the initial thickness $a$ as $3.8$ meters and the rate of melting $b$ as $0.2$ meters per week. The equation will model the thickness $y$ of the ice sheet after $x$ weeks. $\newline$ $y = a - bx$ $\newline$ $y = 3.8 - 0.2x$

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