1. Total Area of the Side of the House [tex]\[ \approx 394.696 \, \text{sq ft} \][/tex]
2. Paint Comparison:
| Paint Type | Price per Gallon | Gallons Needed | Total Cost |
|---------------------|------------------|----------------|------------|
| Standard Paint | $25 | 2 | $50 |
| Paint with Primer | $35 | 1 | $35 |
| High-Quality Paint | $45 | 1 | $45 |
Best Option: Paint with Primer - Total Cost: $35
This option is the most cost-effective.
Start by calculating the area of the side of the house using the given measurements and trigonometry.
1. Divide the shape into simpler figures:
- The side of the house consists of a rectangle at the bottom and an isosceles triangle at the top.
2. Calculate the area of the rectangle:
- The rectangle has a width of 35 ft and a height of 10 ft.
[tex]\[ \text{Area of the rectangle} = \text{width} \times \text{height} = 35 \, \text{ft} \times 10 \, \text{ft} = 350 \, \text{ft}^2 \][/tex]
3. Calculate the area of the isosceles triangle:
- To find the area of the isosceles triangle, we need to determine the base and the height.
- The given angle at the peak of the triangle is 117.2°.
- The two base angles of the isosceles triangle are each (180° - 117.2°) / 2 = 31.4°.
Draw a perpendicular from the peak of the triangle to the base, dividing the isosceles triangle into two right triangles. Each right triangle will have:
- One angle of 31.4°
- An adjacent side (half of the base of the isosceles triangle)
- A hypotenuse of 10 ft (height of the triangle)
Using trigonometry (specifically the sine function):
[tex]\[ \sin(31.4^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{height}}{10 \, \text{ft}} \][/tex]
[tex]\[ \text{height} = 10 \, \text{ft} \times \sin(31.4^\circ) \][/tex]
Calculating the height:
[tex]\[ \text{height} \approx 10 \, \text{ft} \times 0.5214 \approx 5.214 \, \text{ft} \][/tex]
The base of the isosceles triangle is:
2 x (adjacent side) = 2 x (10 ft x cos(31.4°))
[tex]\[ \text{base} \approx 2 \times (10 \, \text{ft} \times 0.8572) \approx 2 \times 8.572 \, \text{ft} \approx 17.144 \, \text{ft} \][/tex]
The area of the isosceles triangle:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 17.144 \, \text{ft} \times 5.214 \, \text{ft} \][/tex]
[tex]\[ \text{Area} \approx \frac{1}{2} \times 89.392 \, \text{ft}^2 \approx 44.696 \, \text{ft}^2 \][/tex]
Calculate the total area of the side of the house:
Total area = Area of the rectangle + Area of the triangle = [tex]350 \, \text{ft}^2 + 44.696 \, \text{ft}^2 \approx 394.696 \, \text{ft}^2[/tex]
Create a table comparing the prices of different types of paint. For simplicity, assume the following types of paint:
| Paint Type | Price per Gallon | Coverage per Gallon (sq ft) | Contains Primer |
|---------------------|------------------|-----------------------------|-----------------|
| Standard Paint | $25 | 350 | No |
| Paint with Primer | $35 | 400 | Yes |
| High-Quality Paint | $45 | 450 | Yes |
Estimate the amount of paint needed based on the total area of 394.696 sq ft and compare the costs.
1. Standard Paint:
- Coverage per gallon: 350 sq ft
- Total area to be painted: 394.696 sq ft
- Number of gallons needed:
[tex]\[ \text{Number of gallons} = \frac{\text{Total area}}{\text{Coverage per gallon}} = \frac{394.696 \, \text{sq ft}}{350 \, \text{sq ft/gallon}} \approx 1.13 \, \text{gallons} \][/tex]
- Since you can't buy a fraction of a gallon, need 2 gallons.
- Cost:
Cost = 2 gallons x $25 per gallon = $50
2. Paint with Primer:
- Coverage per gallon: 400 sq ft
- Total area to be painted: 394.696 sq ft
- Number of gallons needed:
[tex]\[ \text{Number of gallons} = \frac{394.696 \, \text{sq ft}}{400 \, \text{sq ft/gallon}} \approx 0.99 \, \text{gallons} \][/tex]
- 1 gallon will be needed.
- Cost:
Cost = 1 gallon x $35 per gallon = $35
3. High-Quality Paint:
- Coverage per gallon: 450 sq ft
- Total area to be painted: 394.696 sq ft
- Number of gallons needed:
[tex]\[ \text{Number of gallons} = \frac{394.696 \, \text{sq ft}}{450 \, \text{sq ft/gallon}} \approx 0.88 \, \text{gallons} \][/tex]
- 1 gallon will be needed.
- Cost:
Cost = 1 gallon x $45 per gallon = $45
Comparison Table:
| Paint Type | Price per Gallon | Coverage per Gallon (sq ft) | Contains Primer | Gallons Needed | Total Cost |
|---------------------|------------------|-----------------------------|-----------------|----------------|------------|
| Standard Paint | $25 | 350 | No | 2 | $50 |
| Paint with Primer | $35 | 400 | Yes | 1 | $35 |
| High-Quality Paint | $45 | 450 | Yes | 1 | $45 |
Based on this comparison, Paint with Primer is the most cost-effective option since it only requires 1 gallon and has a total cost of $35.
