Question

Рядом с прямым шоссе строят земляную насыпь для защиты от шума. Перпендикулярное сечение насыпи на всем ее протяжении имеет вид параболы, ветви которой направлены вниз. Высота насыпи равна 3 м , ширина на поверхности земли - 6 м и длина - 50 м . Сколько кубических метров грунта требуется для постройки этой насыпи ?

Answer 1

Решение в приложении.

Answer 2

Calculation of the Volume of Earth Required for the Construction of the Embankment

To calculate the volume of earth required for the construction of the embankment, we need to determine the shape of the embankment and its dimensions.

According to the given information, the cross-section of the embankment along its entire length has the shape of a downward-facing parabola. The height of the embankment is 3 meters, the width on the surface of the ground is 6 meters, and the length is 50 meters.

To calculate the volume of the embankment, we can use the formula for the volume of a solid of revolution obtained by rotating a curve around an axis. In this case, the curve is the parabolic cross-section of the embankment, and the axis is the length of the embankment.

The formula for the volume of a solid of revolution is:

V = π∫[a,b] y^2 dx

Where: - V is the volume of the solid - π is a mathematical constant approximately equal to 3.14159 - ∫[a,b] represents the definite integral over the interval [a, b] - y is the function that describes the shape of the curve - dx represents an infinitesimally small change in x

In this case, the parabolic cross-section of the embankment can be described by the equation of a parabola in vertex form:

y = a(x - h)^2 + k

Where: - a is a constant that determines the shape of the parabola - (h, k) are the coordinates of the vertex of the parabola

Since the vertex of the parabola is at the highest point of the embankment, the equation becomes:

y = a(x - 25)^2 + 3

To determine the value of the constant a, we can use the fact that the width of the embankment on the surface of the ground is 6 meters. Plugging in the values (25, 3) and (31, 0) into the equation, we can solve for a:

6 = a(31 - 25)^2 + 3

Simplifying the equation, we get:

6 = 36a + 3

36a = 3

a = 3/36 = 1/12

Now that we have the equation of the parabola, we can calculate the volume of the embankment by integrating the function y^2 with respect to x over the interval [0, 50]:

V = π∫[0,50] (a(x - 25)^2 + 3)^2 dx

Evaluating this integral will give us the volume of the embankment in cubic meters.

Please note that due to the complexity of the integral, the exact calculation may require numerical methods or software.