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]
