Let’s break down the steps to solve each part of the problem systematically.

Problem Description Recap

The task is to calculate the wall area that will be wallpapered, determine the number of wallpaper rolls required, and calculate the total cost with and without sales tax. The wall is 18 feet wide and 14 feet high, with a window (4 ft × 4 ft) and a door (3 ft × 7 ft) excluded from the wallpaper area.

Step-by-Step Solutions

Part a: Find the area of the surface to be wallpapered

1. Calculate the total wall area: $\text{Wall Area} = \text{Width} \times \text{Height} = 18 \, \text{ft} \times 14 \, \text{ft} = 252 \, \text{ft}^2$

2. Calculate the area of the window: $\text{Window Area} = 4 \, \text{ft} \times 4 \, \text{ft} = 16 \, \text{ft}^2$

3. Calculate the area of the door: $\text{Door Area} = 3 \, \text{ft} \times 7 \, \text{ft} = 21 \, \text{ft}^2$

4. Subtract the areas of the window and door from the total wall area: $\text{Wallpaper Area} = \text{Wall Area} - (\text{Window Area} + \text{Door Area})$ $\text{Wallpaper Area} = 252 \, \text{ft}^2 - (16 \, \text{ft}^2 + 21 \, \text{ft}^2) = 252 \, \text{ft}^2 - 37 \, \text{ft}^2 = 215 \, \text{ft}^2$

Part b: Determine the number of wallpaper rolls required

1. Find the area covered by one roll: Each roll covers $35 \, \text{ft}^2$.

2. Divide the total wallpaper area by the area covered per roll: $\text{Number of Rolls} = \frac{\text{Wallpaper Area}}{\text{Area Covered by One Roll}}$ $\text{Number of Rolls} = \frac{215}{35} \approx 6.14$

3. Round up to the nearest whole number (since partial rolls cannot be purchased): $\text{Number of Rolls Needed} = 7$

Part c: Calculate the total cost before sales tax

1. Find the cost per roll: Each roll costs $38.

2. Multiply the number of rolls by the cost per roll: $\text{Total Cost (Before Tax)} = \text{Number of Rolls Needed} \times \text{Cost per Roll}$ $\text{Total Cost (Before Tax)} = 7 \times 38 = 266$

Part d: Calculate the total cost including sales tax

1. Determine the sales tax rate: $7\% = 0.07$.

2. Calculate the tax amount: $\text{Tax Amount} = \text{Total Cost (Before Tax)} \times \text{Sales Tax Rate}$ $\text{Tax Amount} = 266 \times 0.07 = 18.62$

3. Add the tax to the total cost before tax: $\text{Total Cost (Including Tax)} = \text{Total Cost (Before Tax)} + \text{Tax Amount}$ $\text{Total Cost (Including Tax)} = 266 + 18.62 = 284.62$

Summary of Answers

• a. Wallpaper Area: $215 \, \text{ft}^2$
• b. Number of Rolls Needed: 7
• c. Total Cost Before Tax: $266
• d. Total Cost Including Tax: $284.62

Would you like me to provide further details or help with any specific part? Here are 5 related questions for further practice:

1. How would the calculations change if the wall had two windows of the same size instead of one?
2. What if the rolls of wallpaper each covered $50 \, \text{ft}^2$? How many rolls would you need then?
3. How would the total cost change if the price of each roll increased to $45?
4. What percentage of the wall area is occupied by the window and door combined?
5. How can you calculate the total cost for multiple walls of different dimensions?

Tip:

Always round up when calculating quantities like wallpaper rolls, tiles, or other materials since partial units are usually not sold!