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

so fresh & so clean charges according to this graph:
graph with time (hours) on x - axis from 0 to 7 and cost (dollars) on y - axis from 0 to 140, with a line starting at (0,30) (approx) and increasing

a. a customer would like a company to visit their home and only give a quote for their services, without doing further work. which company is the most cost effective for this customer? explain or show your reasoning.

b. another customer would like the company to perform 7 hours of work. which company should this customer choose?

c. what is the smallest number of hours of work for which sparkle team cleaners is cheaper than so fresh & so clean?

Explanation:

Part a

Step1: Find initial fee for Sparkle Team Cleaners

For Sparkle Team Cleaners, when time \( t = 1 \) hour, cost \( C = 56 \) dollars; when \( t = 2 \) hours, \( C = 72 \) dollars. The hourly rate \( r \) is \( 72 - 56 = 16 \) dollars per hour. Using \( t = 1 \), initial fee \( f = 56 - 16\times1 = 40 \) dollars.

Step2: Find initial fee for So Fresh & So Clean

From the graph, when \( t = 0 \), the cost is the initial fee. The y - intercept is 30 dollars (since at \( t = 0 \), \( C = 30 \)).

Step3: Compare initial fees

Sparkle Team Cleaners has an initial fee of 40 dollars, So Fresh & So Clean has an initial fee of 30 dollars. For a quote (0 hours of work), the cost is the initial fee. So So Fresh & So Clean is more cost - effective.

Step1: Find the cost function for Sparkle Team Cleaners

We know the cost function is of the form \( C(t)=f + rt \), where \( f = 40 \) (initial fee) and \( r = 16 \) (hourly rate). So \( C_{Sparkle}(t)=40 + 16t \). For \( t = 7 \), \( C_{Sparkle}(7)=40+16\times7=40 + 112 = 152 \) dollars.

Step2: Find the cost function for So Fresh & So Clean

From the graph, the slope (hourly rate) \( r=\frac{y_2 - y_1}{t_2 - t_1}\). Using points (0, 30) and (2, 70) (from the graph, at \( t = 2 \), \( C = 70 \)), \( r=\frac{70 - 30}{2-0}=20 \) dollars per hour. The cost function is \( C_{Fresh}(t)=30+20t \). For \( t = 7 \), \( C_{Fresh}(7)=30 + 20\times7=30+140 = 170 \) dollars.

Step3: Compare the costs

Since \( 152<170 \), Sparkle Team Cleaners is cheaper for 7 hours of work.

Step1: Set up the inequality

We want to find \( t \) such that \( C_{Sparkle}(t)<C_{Fresh}(t) \). Substitute the cost functions: \( 40 + 16t<30+20t \).

Step2: Solve the inequality

Subtract \( 16t \) from both sides: \( 40<30 + 4t \). Subtract 30 from both sides: \( 10<4t \). Divide both sides by 4: \( t>\frac{10}{4}=2.5 \). Since \( t \) represents the number of hours and we want the smallest integer (or real - valued) number of hours, the smallest number of hours is 3 (since for \( t = 2.5 \), they cost the same, and for \( t>2.5 \), Sparkle is cheaper. If we consider non - integer hours, the smallest \( t \) is just greater than 2.5, but if we consider whole number of hours, the smallest whole number is 3).

Answer:

So Fresh & So Clean. Because the initial (one - time) fee for Sparkle Team Cleaners is 40 dollars (calculated as \( 56-16\times1 = 40 \), where 16 is the hourly rate \( 72 - 56\)) and for So Fresh & So Clean, from the graph, the initial fee (when time = 0) is 30 dollars. For a quote (0 hours of work), the cost is the initial fee, so So Fresh & So Clean is cheaper.

Part b