Question

Прямоугольная трапеция с основаниями 2 и 5 см и меньшей боковой стороной 4 см вращается вокруг большего основания. Найдите полную поверхность тела вращения

Answer 1

См  рисунок
Дано :     
ABCD трапеция 
AD | | BC ;
AB ⊥ AD  ;
AD = a =5 см ;
BC  =b =2 см ;
AB = r = 4  см .
------
Sпол  -? 

Проведем   СH ⊥ AD ⇒ CH = AB = r
Sпол = π*AB² +2π*AB*BC +π* CH*CD   || обозначаем  CD =L ||    
* * * (цилиндрический  и конический  поверхности)
Из ΔСHD по теореме Пифагора :
L=√ (CH² +HD²) =√ (AB² +(AD -BC)²)  = √ (r² +(a -b)²) =√ (4² +(5 -2)² )  =5 (см) .
Sпол = π*r² +2π*r*b + πr*L =r(r +2b + L)π = 4(4 +2*2 +5) π =52π  (см²).
 
ответ : 52π см².

Answer 2

Problem Analysis

We are given a rectangular trapezoid with base lengths of 2 cm and 5 cm, and a smaller side length of 4 cm. The trapezoid is rotated around the larger base. We need to find the total surface area of the resulting solid.

Solution

To find the total surface area of the solid, we need to consider the lateral surface area and the two circular bases.

1. Lateral Surface Area: - The lateral surface area of the solid is the curved surface formed by rotating the trapezoid around the larger base. - The lateral surface area can be calculated by finding the circumference of the circle formed by the larger base and multiplying it by the height of the trapezoid. - The height of the trapezoid can be found using the Pythagorean theorem, as the smaller side length (4 cm) forms the height of a right triangle with the height of the trapezoid and the difference between the base lengths. - Let's calculate the height of the trapezoid: - The difference between the base lengths is 5 cm - 2 cm = 3 cm. - Using the Pythagorean theorem, we have: height^2 + 3^2 = 4^2. - Solving for height, we get: height^2 + 9 = 16. - height^2 = 16 - 9 = 7. - Taking the square root of both sides, we get: height = √7 cm. - Now, let's calculate the lateral surface area: - The circumference of the circle formed by the larger base can be calculated using the formula: circumference = 2πr, where r is the radius of the circle. - The radius of the circle is half of the larger base length, so r = 5 cm / 2 = 2.5 cm. - The lateral surface area is given by: lateral surface area = circumference * height. - Substituting the values, we get: lateral surface area = 2π * 2.5 cm * √7 cm.

2. Circular Bases: - The circular bases of the solid are formed by the two ends of the trapezoid when it is rotated. - The area of each circular base can be calculated using the formula: area = πr^2, where r is the radius of the circle. - The radius of each circular base is half of the corresponding base length, so the radius of the larger base is 5 cm / 2 = 2.5 cm, and the radius of the smaller base is 2 cm / 2 = 1 cm. - The area of each circular base is given by: area = π * radius^2. - Substituting the values, we get: area of larger base = π * (2.5 cm)^2 and area of smaller base = π * (1 cm)^2.

3. Total Surface Area: - The total surface area of the solid is the sum of the lateral surface area and the areas of the two circular bases. - Let's calculate the total surface area: - Total surface area = lateral surface area + area of larger base + area of smaller base. - Substituting the values, we get: total surface area = 2π * 2.5 cm * √7 cm + π * (2.5 cm)^2 + π * (1 cm)^2.

Calculation

Let's calculate the total surface area of the solid using the given measurements:

- Lateral surface area = 2π * 2.5 cm * √7 cm - Area of larger base = π * (2.5 cm)^2 - Area of smaller base = π * (1 cm)^2 - Total surface area = 2π * 2.5 cm * √7 cm + π * (2.5 cm)^2 + π * (1 cm)^2

Now, let's substitute the values and calculate the total surface area.