In a certain country, a letter up to 1 ounce requires postage in the amount of [tex][tex]$\$[/tex]0.42[/tex], and each additional ounce (or fraction of an ounce) has an additional cost. The function below represents the cost of mailing one letter with a weight of [tex][tex]$x$[/tex][/tex] ounces in this country.

[tex]
f(x) = 0.42 + 0.20\lceil x - 1 \rceil
[/tex]

Josiah has two letters. The first weighs 2.9 ounces, and the second weighs 4.1 ounces. What is the total amount Josiah can expect to pay for postage?

A. [tex][tex]$\$[/tex]1.62[tex]$[/tex]

B. [tex]$[/tex]\[tex]$1.64$[/tex][/tex]

C. [tex][tex]$\$[/tex]1.84[tex]$[/tex]

D. [tex]$[/tex]\[tex]$2.04$[/tex][/tex]



Answer :

To determine the total amount Josiah needs to pay for postage for his letters, let's break down the cost calculation step-by-step according to the given function:

[tex]\[ f(x) = 0.42 + 0.20 \lceil x - 1 \rceil \][/tex]

where [tex]\(x\)[/tex] is the weight of the letter in ounces, [tex]\(0.42\)[/tex] is the base cost for the first ounce, and each additional ounce or fraction of an ounce adds [tex]\(0.20\)[/tex].

Let's start by calculating the postage for each letter.

### Letter 1: Weighs 2.9 ounces

1. Compute [tex]\( x - 1 \)[/tex]:
[tex]\[ 2.9 - 1 = 1.9 \][/tex]
2. Take the ceiling of this value:
[tex]\[ \lceil 1.9 \rceil = 2 \][/tex]
3. Calculate the total cost:
[tex]\[ 0.42 + 0.20 \times 2 = 0.42 + 0.40 = 0.82 \][/tex]

Thus, the cost to mail the first letter weighing 2.9 ounces is [tex]\( \$0.82 \)[/tex].

### Letter 2: Weighs 4.1 ounces

1. Compute [tex]\( x - 1 \)[/tex]:
[tex]\[ 4.1 - 1 = 3.1 \][/tex]
2. Take the ceiling of this value:
[tex]\[ \lceil 3.1 \rceil = 4 \][/tex]
3. Calculate the total cost:
[tex]\[ 0.42 + 0.20 \times 4 = 0.42 + 0.80 = 1.22 \][/tex]

Thus, the cost to mail the second letter weighing 4.1 ounces is [tex]\( \$1.22 \)[/tex].

### Total Amount to Pay

Now we add the costs for both letters:

[tex]\[ 0.82 + 1.22 = 2.04 \][/tex]

Therefore, the total amount Josiah can expect to pay for postage for both letters is [tex]\(\$2.04\)[/tex].

The correct answer is:
[tex]\(\$2.04\)[/tex]