Question

Развертка боковой поверхности цилиндра является прямоугольником диогональ которого = 8см а угол между диогоналями 60 градусов. найдите площадь боковой поверхности цилиндра

Answer 1

Т.к. угол между диагоналями прямоугольника равен 60 град, то значит и угол между основанием и диагональю прямоугольника равен 60 град, тогда

примем основание прямоугольника = а, высота прямоугольника (вторая сторона) = в, диагональ прямоугольника = с= 8см

тогда

cos60=а/с

а=с*cos 60=8*1/2=4 см

тогда

с^2=a^2+в^2

в^2=c^2-a^2=8^2-4^2=64-16=48 см

в=6,93 см

S=а*в=4*6,93=27,7 см^2

Answer 2

Finding the Area of the Lateral Surface of a Cylinder

To find the area of the lateral surface of a cylinder, we need to know the diagonal of the rectangle formed by the unwrapped lateral surface and the angle between the diagonals.

Given that the diagonal of the rectangle is 8 cm and the angle between the diagonals is 60 degrees, we can calculate the area of the lateral surface of the cylinder.

Let's break down the solution step by step:

1. The diagonal of the rectangle formed by the unwrapped lateral surface of the cylinder is given as 8 cm. 2. The angle between the diagonals is given as 60 degrees.

To find the length and width of the rectangle, we can use the properties of a rectangle and trigonometry:

3. Let's assume the length of the rectangle is l and the width is w. 4. Using trigonometry, we can find the relationship between l, w, and the angle between the diagonals. 5. In a rectangle, the diagonals are equal in length, so we have two congruent right triangles. 6. The angle between the diagonals divides each right triangle into two 30-60-90 triangles. 7. In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. 8. Since the angle between the diagonals is 60 degrees, the ratio of the sides in each right triangle is l:w:2l. 9. From this, we can set up the equation l:w = 1:√3. 10. Solving this equation, we find that l = √3w.

Now, let's calculate the area of the lateral surface of the cylinder:

11. The area of the lateral surface of a cylinder is given by the formula A = 2πrh, where r is the radius of the base and h is the height of the cylinder. 12. In our case, the height of the cylinder is equal to the width of the rectangle, which is w. 13. The radius of the base of the cylinder is equal to half the length of the rectangle, which is √3w/2. 14. Substituting the values into the formula, we have A = 2π(√3w/2)(w). 15. Simplifying, we get A = π√3w^2.

Therefore, the area of the lateral surface of the cylinder is π√3w^2.

Please note that the exact numerical value of the area cannot be determined without knowing the value of w. However, the formula π√3w^2 represents the general formula for the lateral surface area of the cylinder in terms of the width of the rectangle.

I hope this helps! Let me know if you have any further questions.