Math  /  Data & Statistics

QuestionThe graph shows the relationship between the temperature of a cup of coffee and the time it is sitting on a table. AA trend line has been drawn to approximate the linear relationship of the data.
5. Describe the association and what it means for the situation.
6. According to the trend line, what would the temperature of the coffee be after 30 minutes?

Studdy Solution

STEP 1

1. The trend line represents a linear relationship between temperature and time.
2. The slope of the trend line is negative, indicating a decrease in temperature over time.
3. The trend line can be used to estimate the temperature at any given time within the range of the data.

STEP 2

1. Describe the association between temperature and time.
2. Use the trend line to estimate the temperature after 30 minutes.

STEP 3

The association between temperature and time is negative. As time increases, the temperature of the coffee decreases. This means that the longer the coffee sits on the table, the cooler it becomes.

STEP 4

Identify the equation of the trend line. The trend line can be approximated as a linear equation in the form y=mx+b y = mx + b , where y y is the temperature, x x is the time, m m is the slope, and b b is the y-intercept.
Given the description, approximate the slope m m as:
m=40180500=14050=2.8 m = \frac{40 - 180}{50 - 0} = \frac{-140}{50} = -2.8

STEP 5

Determine the y-intercept b b . From the graph description, the temperature at time x=0 x = 0 is approximately 180°F. Therefore, b=180 b = 180 .
The equation of the trend line is:
y=2.8x+180 y = -2.8x + 180

STEP 6

Substitute x=30 x = 30 into the trend line equation to find the temperature after 30 minutes:
y=2.8(30)+180 y = -2.8(30) + 180
y=84+180 y = -84 + 180
y=96 y = 96
According to the trend line, the temperature of the coffee after 30 minutes would be:
96F \boxed{96^\circ \text{F}}

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