Question

1. two cleaning services charge money at an hourly rate, plus an initial one - time fee that is the same no matter how long the job takes.

sparkle team cleaners charges according to this table:

time (hours)	cost (dollars)
2	72
4	104
5	120
8	168

Explanation:

Since the problem is not fully presented (it likely has a follow - up question about the cost - time relationship, maybe finding the hourly rate and the one - time fee or comparing with another service), we can start by analyzing the cost - time data for Sparkle Team Cleaners. Let's assume we want to find the linear equation for the cost \(C\) in terms of time \(t\) (in hours).

Step 1: Find the hourly rate (slope of the linear equation)

The general form of a linear equation is \(C = mt + b\), where \(m\) is the hourly rate (slope) and \(b\) is the one - time fee (y - intercept).

We can use two points \((t_1,C_1)\) and \((t_2,C_2)\) from the table. Let's take \((t_1 = 1,C_1 = 56)\) and \((t_2=2,C_2 = 72)\).

The slope \(m=\frac{C_2 - C_1}{t_2 - t_1}=\frac{72 - 56}{2 - 1}=\frac{16}{1}=16\). Let's verify with another pair of points. Take \((t_1 = 2,C_1 = 72)\) and \((t_2 = 4,C_2=104)\).

\(m=\frac{104 - 72}{4 - 2}=\frac{32}{2}=16\). So the hourly rate \(m = 16\) dollars per hour.

Step 2: Find the one - time fee (\(b\))

Using the point \((t = 1,C = 56)\) and the equation \(C=mt + b\) with \(m = 16\).

Substitute into the equation: \(56=16\times1 + b\)

\(56=16 + b\)

Subtract 16 from both sides: \(b=56 - 16=40\).

We can check with another point. Let's take \(t = 5\), \(C=16\times5+40=80 + 40=120\), which matches the table. And for \(t = 8\), \(C=16\times8 + 40=128+40 = 168\), which also matches.

So the cost equation for Sparkle Team Cleaners is \(C = 16t+40\), where the hourly rate is 16 dollars per hour and the one - time fee is 40 dollars.

If you have a specific question (like comparing with another service, finding cost for a certain time, etc.), please provide more details.

Answer:

Since the problem is not fully presented (it likely has a follow - up question about the cost - time relationship, maybe finding the hourly rate and the one - time fee or comparing with another service), we can start by analyzing the cost - time data for Sparkle Team Cleaners. Let's assume we want to find the linear equation for the cost \(C\) in terms of time \(t\) (in hours).

Step 1: Find the hourly rate (slope of the linear equation)

The general form of a linear equation is \(C = mt + b\), where \(m\) is the hourly rate (slope) and \(b\) is the one - time fee (y - intercept).

We can use two points \((t_1,C_1)\) and \((t_2,C_2)\) from the table. Let's take \((t_1 = 1,C_1 = 56)\) and \((t_2=2,C_2 = 72)\).

The slope \(m=\frac{C_2 - C_1}{t_2 - t_1}=\frac{72 - 56}{2 - 1}=\frac{16}{1}=16\). Let's verify with another pair of points. Take \((t_1 = 2,C_1 = 72)\) and \((t_2 = 4,C_2=104)\).

\(m=\frac{104 - 72}{4 - 2}=\frac{32}{2}=16\). So the hourly rate \(m = 16\) dollars per hour.

Step 2: Find the one - time fee (\(b\))

Using the point \((t = 1,C = 56)\) and the equation \(C=mt + b\) with \(m = 16\).

Substitute into the equation: \(56=16\times1 + b\)

\(56=16 + b\)

Subtract 16 from both sides: \(b=56 - 16=40\).

We can check with another point. Let's take \(t = 5\), \(C=16\times5+40=80 + 40=120\), which matches the table. And for \(t = 8\), \(C=16\times8 + 40=128+40 = 168\), which also matches.

So the cost equation for Sparkle Team Cleaners is \(C = 16t+40\), where the hourly rate is 16 dollars per hour and the one - time fee is 40 dollars.

If you have a specific question (like comparing with another service, finding cost for a certain time, etc.), please provide more details.