Question

for each model described, explain what the slope and intercept mean (in context)
a) \\( \hat { c } = 25 + 2 w \\), where c is the cost of shipping a package (in dollars and w is the weight of the package in pounds.
b) \\( \hat { f } = 40 + \frac { 1 } { 4 } c \\) where f is temperature in degrees fahrenheit and c is the number of chirps a cricket makes in 1 minute. (believe it or not, this generally is true for crickets in temperatures between \\( 55 ^ { \circ } f \\) and \\( 100 ^ { \circ } f \\).)
c) \\( \hat { p } = 15 + 0.1 m \\), where p is your cell phone plans monthly charge (in dollars) and m is the number of minutes you used.
d) \\( \hat { c } = 11 - 0.5 h \\), where c is how tall a candle is (in inches) after it has been burning for h hours.

Explanation:

Step1: Analyze the linear equation form

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y -\)intercept.

Step2: For part (a)

• Slope:

In the equation \(\hat{c}=25 + 2w\), the slope \(m = 2\). In the context of shipping cost, it means that for each additional pound of the package's weight, the cost of shipping increases by \(2\) dollars.

• Intercept:

The \(y -\)intercept \(b = 25\). It represents the base cost of shipping a package (the cost when the weight \(w = 0\) pounds).

Step3: For part (b)

• Slope:

In the equation \(\hat{F}=40+\frac{1}{4}c\), the slope \(m=\frac{1}{4}\). In the context of temperature and cricket chirps, it means that for each additional chirp a cricket makes in one minute, the temperature (in Fahrenheit) increases by \(\frac{1}{4}\) degree.

• Intercept:

The \(y -\)intercept \(b = 40\). It represents the temperature (in Fahrenheit) when the number of cricket chirps \(c = 0\) in one minute.

Step4: For part (c)

• Slope:

In the equation \(\hat{P}=15+0.1m\), the slope \(m = 0.1\). In the context of a cell - phone plan, it means that for each additional minute of usage, the monthly charge of the cell - phone plan increases by \(0.1\) dollars.

• Intercept:

The \(y -\)intercept \(b = 15\). It represents the base monthly charge of the cell - phone plan (the charge when the number of minutes used \(m = 0\)).

Step5: For part (d)

• Slope:

In the equation \(\hat{C}=11-0.5h\), the slope \(m=- 0.5\). In the context of a burning candle, it means that for each additional hour the candle burns, the height of the candle decreases by \(0.5\) inches.

• Intercept:

The \(y -\)intercept \(b = 11\). It represents the initial height of the candle (the height when the candle has been burning for \(h = 0\) hours).

Answer:

a) Slope: Cost increases by $2 per pound. Intercept: Base cost is $25.
b) Slope: Temperature increases by \(\frac{1}{4}^{\circ}F\) per chirp. Intercept: Temperature is \(40^{\circ}F\) when \(c = 0\) chirps.
c) Slope: Monthly charge increases by $0.1 per minute. Intercept: Base monthly charge is $15.
d) Slope: Candle height decreases by \(0.5\) inches per hour. Intercept: Initial candle height is \(11\) inches.