Question

Найдите площадь полной поверхности тела полученного при вращении прямоугольного треугольника с катетом 6 см и гипотенузой 10см вокург большего катета

Answer 1

При вращении Δ   вокруг большего катета образуется конус, в основании будет круг с радиусом r = 6 см, высота h=10 см , найдем L - образующую конус по теореме Пифагора: L =√(r²+h²)
L = √(10²+6²) = √(100+36) = √136=√(2²·34)=2√31 =≈ 11,66
S полная  = S бок.поверх +S основания
S бок.поверх. =π·r·L
S основания = π·r²
π = ≈3,14
S полная = π·r·L +πr² = πr(r+L) =π·6·(6+11.66) =6·π·17,66 =105,96·π =≈ 332.71

Answer 2

Calculating the Surface Area of a Revolved Right-Angled Triangle

To calculate the surface area of the solid obtained by revolving a right-angled triangle with a leg of 6 cm and a hypotenuse of 10 cm around the larger leg, we can use the formula for the surface area of a solid of revolution.

The surface area of a solid of revolution formed by revolving a curve \(y = f(x)\) from \(x = a\) to \(x = b\) around the x-axis is given by the formula:

\[A = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\]

Where: - \(f(x)\) is the function that defines the curve - \(\frac{dy}{dx}\) is the derivative of the function with respect to \(x\)

Calculating the Surface Area

Given the right-angled triangle with a leg of 6 cm and a hypotenuse of 10 cm, we can use the Pythagorean theorem to find the length of the other leg. The formula for the surface area of the solid of revolution can then be applied to find the surface area.

Using the Pythagorean theorem, we find that the length of the other leg is 8 cm.

Now, we can use the formula for the surface area of a solid of revolution to calculate the surface area of the obtained solid.

\[A = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\]

Conclusion

The surface area of the solid obtained by revolving the right-angled triangle with a leg of 6 cm and a hypotenuse of 10 cm around the larger leg is 20 cm^2.