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Use the situation below to answer all parts of question 3.

A lake near the Arctic Circle is covered by a 3-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 by 0.75 meters each week.

Part 1

Identify the equation that represents this situation.

Responses

y = 0.75x + 3
y = 0.75x + 3

y = -0.75x + 3
y = -0.75x + 3

y = 3x + 0.75
y = 3x + 0.75

y = -3x + 0.75
y = -3x + 0.75
Question 2
Part 2
The slope in this situation is
-0.75
and represents the
thickness of the ice in winter
.

The y-intercept in this situation is
3
and represents the
rate at which the ice melts
.

Question 3
Part 3

The relationship in this situation is proportional.

Responses

True
True

False
False
Question 4
Part 4

Graph the linear relationship described in this situation.

To graph: click one point and then a second point. The line will be created.

Keyboard Instructions
Initial graph state
The horizontal axis goes from -6.8 to 6.8 with ticks spaced every 1 unit(s).
The vertical axis goes from -6.8 to 6.8 with ticks spaced every 1 unit(s).

1 answer

Let's break down the situation and answer each part step by step.

Part 1: Identify the Equation

The ice thickness starts at 3 meters and decreases by 0.75 meters each week. Therefore, this can be modeled by a linear equation where:

  • y (the dependent variable) represents the thickness of the ice.
  • x (the independent variable) represents the number of weeks since spring began.
  • The initial thickness (y-intercept) is 3 meters.
  • Since the ice is melting, the slope will be negative, indicating the decrease in thickness.

Thus, the equation that represents this situation is: \[ y = -0.75x + 3 \]

Response:
y = -0.75x + 3

Part 2: Identify Slope and Y-intercept

  • The slope in this situation is -0.75. This represents the rate at which the ice thickness decreases each week (0.75 meters per week).
  • The y-intercept is 3. This represents the initial thickness of the ice at the beginning of spring.

Summarizing:

  • Slope: -0.75 (rate of ice melting)
  • y-intercept: 3 (initial ice thickness)

Responses:

  • The slope is -0.75 and represents the rate at which the ice melts.
  • The y-intercept is 3 and represents the thickness of the ice in winter.

Part 3: Determine if the Relationship is Proportional

For a relationship to be proportional, it must go through the origin (0,0). In this case, the ice thickness starts at 3 meters when x (weeks) is 0, indicating that the relationship does not pass through the origin.

Therefore, the relationship is not proportional.

Response:
False

Part 4: Graphing the Linear Relationship

To graph the linear relationship \( y = -0.75x + 3 \):

  1. Identify two points:

    • When \( x = 0 \): \( y = 3 \) → Point (0, 3)
    • When \( x = 4 \): \( y = -0.75(4) + 3 = -3 + 3 = 0 \) → Point (4, 0)

  2. Plot these points on the graph:

    • The first point (0, 3) indicates the thickness of ice at the start of spring.
    • The second point (4, 0) indicates that after 4 weeks, the ice thickness has reached 0.

  3. Draw a line through these points.

By following these steps, you will create the graph of the relationship described in this situation.