Question

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 sheets thickness (in meters) after x weeks. complete the equation for the relationship between the thickness and number of weeks.

Explanation:

Step1: Identify the linear - equation form

The general form of a linear equation is $y = mx + b$, where $m$ is the slope (rate of change) and $b$ is the y - intercept. Here, the rate of decrease of ice - sheet thickness is $m=- 0.2$ (negative because the thickness is decreasing).

Step2: Find the y - intercept

We know that when $x = 7$, $y = 2.4$. Substitute these values into $y=mx + b$: $2.4=-0.2\times7 + b$.

Step3: Solve for b

First, calculate $-0.2\times7=-1.4$. Then, the equation becomes $2.4=-1.4 + b$. Add 1.4 to both sides: $b=2.4 + 1.4=3.8$.

Step4: Write the final equation

Substitute $m=-0.2$ and $b = 3.8$ into $y=mx + b$. We get $y=-0.2x+3.8$.

Answer:

$y=-0.2x + 3.8$