Question

starting at sunrise, the temperature rose 2.5 degrees every hour. after 8 hours, the temperature was 67 degrees.
a) write an equation to model the temperature, y, after x hours after sunrise.
b) what does the slope & y - intercept mean in terms of the problem?
c) if sunrise was at 6:00 a.m., what is the temperature at noon?

Explanation:

Step1: Find the y - intercept

The temperature rises at a rate of \(m = 2.5\) degrees per hour. After \(x = 8\) hours, \(y=67\) degrees. Using the slope - intercept form \(y=mx + b\), substitute \(m = 2.5\), \(x = 8\), and \(y = 67\) into the equation:
\(67=2.5\times8 + b\)
\(67 = 20 + b\)
\(b=67 - 20=47\)

Step2: Write the equation

The slope - intercept form of a linear equation is \(y = mx + b\). Since \(m = 2.5\) and \(b = 47\), the equation is \(y=2.5x + 47\)

Step3: Interpret the slope and y - intercept

• Slope: The slope \(m = 2.5\) means that the temperature rises \(2.5\) degrees per hour after sunrise.
• y - intercept: The y - intercept \(b = 47\) means that the temperature at sunrise (\(x = 0\) hours after sunrise) was \(47\) degrees.

Step4: Calculate the temperature at noon

Sunrise is at 6:00 a.m. and noon is 6 hours after 6:00 a.m. So \(x = 6\). Substitute \(x = 6\) into the equation \(y=2.5x + 47\):
\(y=2.5\times6+47\)
\(y = 15+47\)
\(y = 62\)

Answer:

a) \(y = 2.5x+47\)
b) Slope: Temperature rises \(2.5\) degrees per hour. Y - intercept: Temperature at sunrise (\(47\) degrees).
c) \(62\) degrees.