The tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of cylindrical part are 9 m and 30 m respectively and height of conical part is 8 m with same diameter as that of the cylindrical part, then find
(1) the area of the canvas used in making the tent;
(2) the cost of the canvas bought for the tent at the rate 200 per sq m. if 30 sq m canvas was wasted during stitching.
(1) Assuming that the tent has no floor, then the area is just the lateral area of the cylinder, plus that of the cone.
A = 2πrh + 1/3 πr^2 h = 2π*15*9 + 1/3 π * 15^2 * 8 = 870π
(2) 200(870π + 30)
To find the area of the canvas used in making the tent, we need to calculate the lateral surface area of both the cylindrical and conical parts and add them together.
(1) The lateral surface area of the cylindrical part can be found using the formula:
Lateral Surface Area of Cylinder = 2πrh
where r is the radius and h is the height.
Given that the height (h) of the cylindrical part is 9 m and the diameter is 30 m, we can find the radius (r) by dividing the diameter by 2:
r = 30 m / 2 = 15 m
Substituting r = 15 m and h = 9 m into the formula, we have:
Lateral Surface Area of Cylinder = 2π * 15 m * 9 m
(2) The lateral surface area of the conical part can be found using the formula:
Lateral Surface Area of Cone = πrl
where r is the radius and l is the slant height.
Given that the height (h) of the conical part is 8 m and the diameter is 30 m (same as the cylindrical part), we can find the radius (r) again as:
r = 30 m / 2 = 15 m
The slant height (l) can be found using the Pythagorean theorem, because we have a right triangle formed by the radius (15 m), slant height (l), and height (8 m) of the conical part:
l^2 = r^2 + h^2
l^2 = 15 m^2 + 8 m^2
l^2 = 225 m^2 + 64 m^2
l^2 = 289 m^2
l = √(289 m^2)
l = 17 m
Now, substituting r = 15 m and l = 17 m into the formula, we have:
Lateral Surface Area of Cone = π * 15 m * 17 m
Finally, we can calculate the total area of the canvas used in making the tent by adding the lateral surface area of the cylindrical part and the lateral surface area of the conical part.
Total Area of Canvas = Lateral Surface Area of Cylinder + Lateral Surface Area of Cone
To find the cost of the canvas bought for the tent, we need to subtract the wasted area during stitching and multiply by the cost per square meter.
Let's proceed with the calculations.
To find the area of the canvas used in making the tent, we need to calculate the curved surface area of the cylindrical part and the curved surface area of the conical part separately, and then add them together.
(1) Curved Surface Area of the Cylindrical Part:
The formula for the curved surface area of a cylinder is given by: CSA = 2πrh, where r is the radius of the base and h is the height.
Given that the diameter of the cylindrical part is 30 m, the radius (r) can be calculated as half of the diameter: r = 30/2 = 15 m.
Using the provided height (h) for the cylindrical part, which is 9 m, we can calculate the curved surface area (CSA) as follows:
CSA of the cylindrical part = 2πrh = 2π(15)(9) = 270π sq m.
(2) Curved Surface Area of the Conical Part:
The formula for the curved surface area of a cone is given by: CSA = πrl, where r is the radius of the base and l is the slant height of the cone.
Since the diameter of the conical part is the same as the cylindrical part, the radius of the conical part is also 15 m.
To calculate the slant height (l) of the cone, we can use the Pythagorean theorem. The height (h) of the conical part and the radius (r) form a right-angled triangle, where the hypotenuse is the slant height (l). Using the height of the conical part, which is 8 m, we can calculate the slant height as follows:
l = √(r^2 + h^2) = √(15^2 + 8^2) = √(225 + 64) = √289 = 17 m.
Using the calculated radius (r) and slant height (l), we can find the curved surface area (CSA) of the conical part as follows:
CSA of the conical part = πrl = π(15)(17) = 255π sq m.
Now, we can find the total area of the canvas used in making the tent by adding the curved surface areas of the cylindrical and conical parts together:
Total Area = CSA of cylindrical part + CSA of conical part
= 270π + 255π
= 525π sq m.
Therefore, the area of the canvas used in making the tent is 525π square meters.
To find the cost of the canvas bought for the tent, we need to subtract the wasted canvas and then multiply by the cost per square meter.
(3) Wasted Canvas:
Given that 30 square meters of canvas was wasted during stitching.
Total Canvas Bought = Total Area + Wasted Canvas
= 525π + 30
≈ 525π + 94.25 (approximating π as 3.14)
Now, we can find the cost of the canvas bought at the rate of $200 per square meter as follows:
Cost = Total Canvas Bought * Rate
= (525π + 94.25) * 200
Therefore, the cost of the canvas bought for the tent at the rate of $200 per square meter, with 30 square meters wasted, is approximately (525π + 94.25) * 200.