Answer:
the capacity of the tent and the height of the tent can be determined using the steps outlined above
Step-by-step explanation:
To find the capacity and height of the conical tent, we need to use the formula for the volume of a cone and relate it to the given information.
1. Volume of a cone formula: V = (1/3) * π * r^2 * h, where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cone.
2. The cloth used to make the tent has an area of 15 m². This area represents the curved surface area of the cone, which is given by the formula: A = π * r * l, where A is the area, r is the radius of the base, and l is the slant height of the cone.
3. We are given that the slant height of the cone is 5 m. Since the slant height, l, is related to the radius, r, and height, h, by the Pythagorean theorem, we can express it as l = √(r^2 + h^2).
Now, let's solve for the radius and height of the cone:
From the given information, we know that the area of the cloth is 15 m² and the slant height is 5 m.
4. Substitute the given values into the formula for the area of the cone: 15 = π * r * 5
5. Simplify the equation: 15 = 5πr
6. Solve for the radius: r = 15 / (5π)
7. Substitute the values of r and l into the equation for the slant height: 5 = √((15 / (5π))^2 + h^2)
8. Simplify the equation: 25 = (225 / (25π)) + h^2
9. Solve for the height: h^2 = 25 - (225 / (25π))
10. Simplify the equation: h^2 = 25 - (9 / π)
11. Calculate the height: h = √(25 - (9 / π))
Now, to find the capacity of the tent, substitute the radius and height into the formula for the volume of a cone:
12. Volume = (1/3) * π * r^2 * h
13. Volume = (1/3) * π * ((15 / (5π))^2) * √(25 - (9 / π))
14. Simplify the equation to find the capacity of the tent.
Therefore, the capacity of the tent and the height of the tent can be determined using the steps outlined above
