Question

lesson 2c: modeling real-world situations with linear equations
learning target 2c: i can find and use linear equations to model data, interpret slope, y - intercept, average rate of change, and the solutions from those models.
what does it mean to model?
the price of a kit kat chocolate bar has been increasing over time.
let p represent the price (in dollars) of a kit kat bar, and t represent the number of years since 2000.
$p = 0.05t + 0.75$
what was the price of a kit kat bar in 2000?________
how much has the price increased each year?________
when emily returned from vacation, she turned the heat back on in her home. she set the temperature as low as it could go.
q represents the temperature in emily’s home (in degrees celsius) after t minutes.
$q = 15 - 0.4t$
what was the temperature when emily returned from vacation?________
how much does the temperature change every minute?________
a paintball court charges an initial entrance fee plus a fixed price per ball.
p represents the total price (in dollars) as a function of the number of balls used n.
$p = 0.80n + 5.50$
what is the entrance fee?________
what is the price for 10 balls, not including the entrance fee?________

Explanation:

Step1: Analyze the linear equation \(P = 0.05t+0.75\)

For the price of Kit - Kat bar in 2000, \(t = 0\) (since \(t\) is the number of years since 2000). Substitute \(t = 0\) into \(P=0.05t + 0.75\).
\(P=0.05\times0+0.75=0.75\)

The slope of the linear equation \(y = mx + b\) is \(m\). In the equation \(P = 0.05t+0.75\), the slope \(m = 0.05\), which represents the rate of change of the price per year.

Step2: Analyze the linear equation \(Q = 15-0.4t\)

When Emily returned from vacation, \(t = 0\). Substitute \(t = 0\) into \(Q = 15-0.4t\).
\(Q=15-0.4\times0 = 15\)

The slope of the equation \(Q = 15-0.4t\) (or \(Q=- 0.4t + 15\)) is \(m=-0.4\), which represents the rate of change of temperature per minute.

Step3: Analyze the linear equation \(P = 0.80n+5.50\)

The entrance fee is the value when \(n = 0\). Substitute \(n = 0\) into \(P = 0.80n+5.50\).
\(P=0.80\times0+5.50 = 5.50\)

The price for \(n\) balls (not including the entrance fee) is given by the term with \(n\). For \(n = 10\) balls (not including the entrance fee), use the formula \(P_{balls}=0.80n\). Substitute \(n = 10\) into \(P_{balls}=0.80n\).
\(P_{balls}=0.80\times10=8\)

Answer:

• The price of a Kit Kat bar in 2000: \(\$0.75\)
• The price increase each year for Kit Kat bar: \(\$0.05\) per year
• The temperature when Emily returned from vacation: \(15^{\circ}C\)
• The temperature change every minute: \(- 0.4^{\circ}C\) per minute (the negative sign indicates a decrease)
• The entrance fee for the paint - ball court: \(\$5.50\)
• The price for 10 balls (not including entrance fee): \(\$8\)