We are tasked with finding the rate of change, the initial value, and the equation of the function describing the mass of ice lost over time. Given that the relationship is linear, we can use the standard linear equation:

$y = mx + b$

where:

• $y$ represents the total mass lost,
• $x$ represents the number of minutes,
• $m$ is the rate of change (slope),
• $b$ is the initial value (y-intercept).

Step 1: Identify the rate of change

The rate of change is how much mass the ice loses every minute. The problem tells us that the ice loses 5 grams of mass every minute. Therefore, the rate of change $m$ is:

$m = 5 \text{ grams per minute}.$

Step 2: Find the initial value (y-intercept)

The initial value $b$ represents the total mass lost at the start, when $x = 0$. We are given that after 11 minutes, the ice has lost 70 grams. This means when $x = 11$, $y = 70$.

We can use this point to find the initial value $b$. The equation of the line is:

$y = mx + b$

Substitute $m = 5$ and the point $(x, y) = (11, 70)$:

$70 = 5(11) + b$

Simplify the equation:

$70 = 55 + b$

Solve for $b$:

$b = 70 - 55 = 15$

Thus, the initial value $b$ is 15 grams. This means that at $x = 0$ (at the start), the ice had already lost 15 grams (perhaps because the melting process had already begun at that point).

Step 3: Write the equation

Now that we know the slope $m = 5$ and the initial value $b = 15$, we can write the equation of the line as:

$y = 5x + 15$

This equation describes the total mass of ice lost ($y$) after $x$ minutes.

Interpretation

• Rate of change: The ice loses mass at a constant rate of 5 grams per minute.
• Initial value: At the start (when $x = 0$), the ice had already lost 15 grams of mass.

Summary

The equation representing the total mass lost by the ice is:

$y = 5x + 15$

where $y$ is the mass lost in grams, and $x$ is the number of minutes that have passed.

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Do you need further clarification or have any other questions? Here are some related questions for you to consider:

1. How would the equation change if the ice lost mass at a different rate (e.g., 3 grams per minute)?
2. How could you use this equation to predict the total mass lost after 20 minutes?
3. What if the initial mass loss was different (e.g., 30 grams instead of 15 grams)? How would that affect the equation?
4. Can you think of a real-world situation where the ice melting process might not be linear?
5. How can we graph the equation $y = 5x + 15$ on a coordinate plane?

Tip: Always ensure to interpret the meaning of your slope and y-intercept in the context of the problem to better understand the relationship between variables.