Answer:
[tex](a)\ Total = \$67.89[/tex]
[tex](b)[/tex]
[tex]c(r) = 24.50(1.095)[/tex] ----- [tex]r \le 2[/tex]
[tex]c(r) = [24.50 + (r - 2)*12.50] * [1.095][/tex] ---- [tex]r > 2[/tex]
Explanation:
Given
[tex]Copies = 550[/tex]
[tex]First\ 250 = \$24.50[/tex]
[tex]Every\ extra\ 100 = \$12.50[/tex]
Solving (a): Cost of 550 copies
We have:
[tex]First\ 250 = \$24.50[/tex]
This means that, there are 300 copies left (i.e. 550 - 250)
[tex]Every\ extra\ 100 = \$12.50[/tex]
There are 3 hundreds in 300
So, the cost of the 300 copies is:
[tex]300\ copies = 3 * \$12.50 =\$37.50[/tex]
[tex]Total_{(Before\ Tax)} = First\ 250 + 300\ copies[/tex]
[tex]Total_{(Before\ Tax)}= \$24.50 + \$37.50[/tex]
[tex]Total_{(Before\ Tax)} = \$62.00[/tex]
Apply sales tax of 9.5%
[tex]Total = Total_{(Before\ Tax)} *(1 + Sales\ Tax)[/tex]
[tex]Total = \$62.00 *(1 + 9.5\%)[/tex]
Express percentage as decimal
[tex]Total = \$62.00 *(1 + 0.095)[/tex]
[tex]Total = \$62.00 *(1.095)[/tex]
[tex]Total = \$67.89[/tex]
Solving (b): The piece wise function
From the question, we understand that the first 250 cost $24.50
First, we calculate the number of 100s in 250
[tex]r = \frac{250}{100}[/tex]
[tex]r =2.5[/tex]
r must be an integer; So, we round down
[tex]r =2[/tex]
This means that there are 2 whole hundreds in 150.
So, the first function (before tax) is:
[tex]c(r) =24.50[/tex] ---- [tex]r \le 2[/tex]
For every other 100 after the first 250
The charge is:
Charge = First 250 + Number of 100s * 12.50
r has a maximum value of 2 for the first 250, this means that the next copies of 100s will have a factor of r - 2
So, the next function (before tax) is:
[tex]c(r) = 24.50 + (r - 2) * 12.50[/tex] ----- [tex]r > 2[/tex]
At this point, we have:
[tex]c(r) =24.50[/tex] ---- [tex]r \le 2[/tex]
[tex]c(r) = 24.50 + (r - 2) * 12.50[/tex] ----- [tex]r > 2[/tex]
Apply sales tax of 9.5%
[tex]c(r) = 24.50(1.095)[/tex] ----- [tex]r \le 2[/tex]
[tex]c(r) = [24.50 + (r - 2)*12.50] * [1.095][/tex] ---- [tex]r > 2[/tex]
