Question

Молодій родині необхідно перекрити дах будинку. Дах має конічну форму : діаметр 8 м і висота 3 м. Один лист металочерепиці має розміри 0,6 м × 1,2 м і коштує 40 грн. Один лист бітумної черепиці має розміри 0,8 м Х 1,25м і коштує 45 грн. На шви та обрізку витрачається 10 % від площі даху. Визначити площу даху та яке покриття даху фінансово вигідніше родині.

Answer 1

Calculation of Roof Area

To determine the roof area, we need to calculate the surface area of the conical roof. The formula for the surface area of a cone is given by:

Surface Area = π * r * (r + l)

where: - π is a mathematical constant approximately equal to 3.14159 - r is the radius of the base of the cone (half of the diameter) - l is the slant height of the cone

Given that the diameter of the roof is 8 meters, the radius would be 4 meters. The height of the cone is 3 meters. To calculate the slant height, we can use the Pythagorean theorem:

l = √(r^2 + h^2)

where: - h is the height of the cone

Substituting the values, we get:

l = √(4^2 + 3^2) = √(16 + 9) = √25 = 5

Now, we can calculate the surface area of the cone:

Surface Area = π * 4 * (4 + 5) = 36π square meters

Therefore, the roof area is approximately 36π square meters.

Cost of Metal Roofing

The dimensions of a single metal roofing sheet are 0.6 meters by 1.2 meters, and it costs 40 UAH. To calculate the number of sheets required, we divide the roof area by the area covered by a single sheet:

Number of metal roofing sheets = Roof area / Area covered by a single sheet

Area covered by a single sheet = 0.6 meters * 1.2 meters = 0.72 square meters

Substituting the values, we get:

Number of metal roofing sheets = 36π square meters / 0.72 square meters = 50π sheets

The cost of metal roofing for the entire roof would be:

Total cost of metal roofing = Number of metal roofing sheets * Cost per sheet

Total cost of metal roofing = 50π sheets * 40 UAH/sheet = 2000π UAH

Cost of Bituminous Roofing

The dimensions of a single bituminous roofing sheet are 0.8 meters by 1.25 meters, and it costs 45 UAH. To calculate the number of sheets required, we use the same formula as before:

Number of bituminous roofing sheets = Roof area / Area covered by a single sheet

Area covered by a single sheet = 0.8 meters * 1.25 meters = 1 square meter

Substituting the values, we get:

Number of bituminous roofing sheets = 36π square meters / 1 square meter = 36π sheets

The cost of bituminous roofing for the entire roof would be:

Total cost of bituminous roofing = Number of bituminous roofing sheets * Cost per sheet

Total cost of bituminous roofing = 36π sheets * 45 UAH/sheet = 1620π UAH

Calculation of Waste

To account for waste due to cutting and trimming, we need to calculate 10% of the roof area:

Waste = 10% of Roof area

Waste = 0.1 * 36π square meters = 3.6π square meters

Financial Comparison

To determine which roofing option is more cost-effective for the family, we compare the total costs of metal roofing and bituminous roofing, including the waste:

Total cost of metal roofing with waste = Total cost of metal roofing + Cost of waste

Total cost of bituminous roofing with waste = Total cost of bituminous roofing + Cost of waste

Substituting the values, we get:

Total cost of metal roofing with waste = 2000π UAH + 3.6π UAH = 2003.6π UAH

Total cost of bituminous roofing with waste = 1620π UAH + 3.6π UAH = 1623.6π UAH

Therefore, the family would find the metal roofing option more financially advantageous.

Please note that the calculations provided are based on the given dimensions and costs. It is always recommended to consult with a professional roofing contractor for accurate measurements and cost estimates.