Question

1. a wallpaper border (trim) is pasted on the wall.

a) what is the total length needed?
b) 12-foot rolls are available to purchase. how many rolls need to be purchased?
c) if each roll costs $12.49, how much will the bill be, before any taxes?

Explanation:

Part (a)

Step1: Recall Perimeter Formula

The wall is a rectangle with a gap. The perimeter of a rectangle is \( P = 2\times(\text{length}+\text{width}) \), then subtract the gap.
Length \( l = 12'9'' \), Width \( w = 8'1'' \), Gap \( g = 2'6'' \).

Step2: Convert to Inches (Optional, but easier)

1 foot = 12 inches.
\( 12'9'' = 12\times12 + 9 = 153 \) inches.
\( 8'1'' = 8\times12 + 1 = 97 \) inches.
\( 2'6'' = 2\times12 + 6 = 30 \) inches.

Step3: Calculate Perimeter of Rectangle

\( P_{\text{rectangle}} = 2\times(153 + 97) = 2\times250 = 500 \) inches.

Step4: Subtract the Gap

\( P_{\text{needed}} = 500 - 30 = 470 \) inches.

Step5: Convert Back to Feet and Inches

\( 470 \div 12 = 39 \) feet with a remainder of \( 2 \) inches. Wait, no—wait, maybe better to calculate in feet directly.

Alternative (Direct in Feet):
\( 12'9'' = 12 + \frac{9}{12} = 12.75 \) ft.
\( 8'1'' = 8 + \frac{1}{12} \approx 8.0833 \) ft.
\( 2'6'' = 2 + \frac{6}{12} = 2.5 \) ft.

Perimeter of rectangle: \( 2\times(12.75 + 8.0833) = 2\times20.8333 \approx 41.6666 \) ft.
Subtract gap: \( 41.6666 - 2.5 = 39.1666 \) ft, which is \( 39'2'' \) (since \( 0.1666\times12 \approx 2 \) inches).

Part (b)

Step1: Determine Rolls Needed

Each roll is 12 feet. Total length needed: \( 39.1666 \) ft.
Number of rolls \( n = \lceil \frac{39.1666}{12}
ceil \) (ceiling function, since we can't buy a fraction of a roll).

Step2: Calculate Division

\( 39.1666 \div 12 \approx 3.2639 \).

Step3: Round Up

Since we can't buy a fraction of a roll, we need \( 4 \) rolls (because 3 rolls would give \( 3\times12 = 36 \) ft, which is less than \( 39.1666 \) ft).

Part (c)

Step1: Cost per Roll

Each roll costs \( \$12.49 \), and we need 4 rolls.

Step2: Total Cost

Total cost \( = 4\times12.49 = 49.96 \) dollars.

Final Answers:

a) The total length needed is \( \boldsymbol{39'2''} \) (or \( 39.17 \) ft approximately).
b) The number of rolls needed is \( \boldsymbol{4} \).
c) The total bill is \( \boldsymbol{\$49.96} \).

Answer:

Step1: Cost per Roll

Each roll costs \( \$12.49 \), and we need 4 rolls.

Step2: Total Cost

Total cost \( = 4\times12.49 = 49.96 \) dollars.

Final Answers:

a) The total length needed is \( \boldsymbol{39'2''} \) (or \( 39.17 \) ft approximately).
b) The number of rolls needed is \( \boldsymbol{4} \).
c) The total bill is \( \boldsymbol{\$49.96} \).