Question

A delivery​ company's charge for an overnight package weighing in excess of one pound is given by the formula c equals 2.01 w plus 14.29 comma where w is the weight of package in pounds. Find the following.  
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Part 1
A 5​-pound package costs​ $
  
24.34 to send overnight.
​(Round to the nearest​ cent.)
Part 2
A 12​-pound package costs​ $
  
38.41 to send overnight.
​(Round to the nearest​ cent.)
Part 3
A 13​-pound package costs​ $
  
40.42 to send overnight.
​(Round to the nearest​ cent.)
Part 4
The​ figure, shown to the​ right, is the graph of the formula for the​ company's charge, with w on the horizontal​ axis, in​ pounds, and c on the vertical axis in dollars.
Part 5
Choose the approximate cost of sending a 13one half
​-pound package overnight.
A.
​$27
B.
​$41
Your answer is correct.C.
​$58
Part 6
A package that costs ​$160.31 to mail weighs about
  
enter your response here pounds.

Answer 1

To find the costs of sending different weights of packages using the given formula \( c = 2.01w + 14.29 \), we can plug in the respective weights for \( w \) and calculate \( c \).

### Part 1: Cost of a 5-pound package
- \( w = 5 \)
\[ c = 2.01(5) + 14.29 \]
\[ c = 10.05 + 14.29 \]
\[ c = 24.34 \]

### Part 2: Cost of a 12-pound package
- \( w = 12 \)
\[ c = 2.01(12) + 14.29 \]
\[ c = 24.12 + 14.29 \]
\[ c = 38.41 \]

### Part 3: Cost of a 13-pound package
- \( w = 13 \)
\[ c = 2.01(13) + 14.29 \]
\[ c = 26.13 + 14.29 \]
\[ c = 40.42 \]

### Part 5: Cost of a 13.5-pound package
- \( w = 13.5 \)
\[ c = 2.01(13.5) + 14.29 \]
\[ c = 27.135 + 14.29 \]
\[ c \approx 41.43 \]

The closest option to \( 41.43 \) is **B. $41**.

### Part 6: Finding weight of a package that costs $160.31
To find \( w \) when \( c = 160.31 \):
\[
160.31 = 2.01w + 14.29
\]
Subtract \( 14.29 \) from both sides:
\[
160.31 - 14.29 = 2.01w
\]
\[
146.02 = 2.01w
\]
Now divide both sides by \( 2.01 \):
\[
w = \frac{146.02}{2.01} \approx 72.7
\]

The weight of the package is approximately **72.7 pounds**.

### Final Answers
1. Cost of a 5-pound package: **$24.34**
2. Cost of a 12-pound package: **$38.41**
3. Cost of a 13-pound package: **$40.42**
4. Cost of a 13.5-pound package: **$41**
5. Weight of a package costing $160.31: **72.7 pounds**