Question

The monthly charge for a waste collection service is 1448 dollars for 100 kg of waste and 2498 dollars for 175 kg of waste.A) Find a linear model for the cost, C. of waste collection as a function of the number of kilograms, w.C= B) What is the slope of the line found in part (a)?Slope =Think about the interpretation of the slope: are the units of the slopeA dollarsB. kilogramsC. kilograms per dollarD. dollars per kilogram(c) What is the value of the vertical intercept of the line found in part (a)?Vertical intercept=Think about the interpretation of the intercept: are the units of the intercept-A. kilograms per dollarB. kilogramsC. dollars per kilogramD. dollars

Answer 1

a) C = 14w + 48

b) slope =

Step-by-step explanation:

A) Let C represent the cost of the waste

w represent the number of kilograms of waste

when cost = $1448, the amount of waste = 100kg

(w, C) = (100, 1448)

when cost = $2498, the amount of waste = 175kg

(w, C) = (175, 2498)

we need to find the slope of the points we were given

Slope formula is given as:

`\begin{gathered} m\text{ = }(y_2-y_1)/(x_2-x_1) \\ In\text{ this case: }m\text{ = }(C_2-C_1)/(w_2-w_1) \\ \end{gathered}`
`\begin{gathered} w_1=100,C_1=1448,w_2=175,C_2\text{ = 2}498 \\ m\text{ = }(2498-1448)/(175-100)\text{ = }(1050)/(75) \\ m\text{ = slope = 14} \end{gathered}`
`\begin{gathered} Lin\text{ear equation is given as:} \\ y\text{ =mx + b};\text{ }m\text{ = slope, b = y-intercept} \\ C\text{ = mw + b} \\ m\text{ = slope, b = y-intercept} \\ To\text{ get b, we will use the slope and any of the points:} \\ point\colon(100,\text{ 1448) = (w, C)} \\ 1448\text{ = 14(100) + b} \\ b\text{ = 1448 -1400} \\ b\text{ = 48} \end{gathered}`
`\begin{gathered} \text{The linear function becomes:} \\ C\text{ = 14w + 48} \end{gathered}`

B) slope = 14

Slope = cost/amount of waste

In units, we are dividing the dollar by the kilogram = $/kg

The units of the slope is dollars per kilogram (option D)

C) Vertical intercpet is the y -intercept

From our calculation, y-intercept is = b = 48

Hence, the vertical intercept is 48

The y-intercept is the intial value or initial cost of the waste. This is the cost when the kilogram of waste is zero

C = 14w + 48

w = 0

C = 14(0) + 48

C =48

Since the unit of C is dollars. Then the unit of the intercept will also be in dollars (option D).