Question

Участок земли, выделенный под зону отдыха, имеет форму треугольника. Для составления плана участка измерили две его стороны и угол между ними. Найдите периметр участка, если его стороны равны 4 км и 6 км, а градусная мера угла между ними — 120 градусов. Найдите площадь участка с точностью до 0,1 га. Выполните вычисления на микрокалькуляторе. Запишите обоснованное решение и ответ.

Answer 1

Ответ:

↓↓↓

Объяснение:

Третья сторона  АВ  в ΔАВС по т. косинусов

АВ²=4²+6²-2*4*6*cos 120

AB²=16+36-2*4*6*(-0,5)

AB²=52+24

AB=√76≈8,7 км

Р=18,7 км

S=1\2*4*6* sin120=12*√3\2=6√3≈10,4 км²

Answer 2

Problem Analysis

We are given a triangular plot of land for a recreational area, and we need to find the perimeter and area of the plot. We are given the lengths of two sides of the triangle (4 km and 6 km) and the angle between them (120 degrees). We need to calculate the perimeter and area of the plot with a precision of 0.1 hectares.

Solution

To find the perimeter of the triangular plot, we need to sum the lengths of all three sides. Since we are given the lengths of two sides, we can use the Law of Cosines to find the length of the third side.

The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our case, we know the lengths of sides a and b (4 km and 6 km) and the angle C (120 degrees). We can substitute these values into the equation to find the length of side c.

Let's calculate the length of side c:

c^2 = 4^2 + 6^2 - 2 * 4 * 6 * cos(120) c^2 = 16 + 36 - 48 * cos(120) c^2 = 52 - 48 * (-0.5) c^2 = 52 + 24 c^2 = 76 c = sqrt(76) c ≈ 8.7178 km

Now that we have the lengths of all three sides, we can calculate the perimeter of the triangular plot by summing the lengths of the sides:

Perimeter = 4 + 6 + 8.7178 Perimeter ≈ 18.7178 km

To find the area of the triangular plot, we can use Heron's formula, which states that the area of a triangle with sides a, b, and c is given by:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, given by:

s = (a + b + c) / 2

Let's calculate the area of the triangular plot:

s = (4 + 6 + 8.7178) / 2 s ≈ 9.3589 km

Area = sqrt(9.3589 * (9.3589 - 4) * (9.3589 - 6) * (9.3589 - 8.7178)) Area ≈ 19.5193 km^2

To convert the area from square kilometers to hectares, we can use the conversion factor 1 hectare = 10,000 square meters. Since 1 square kilometer = 100 hectares, we have:

Area in hectares = 19.5193 * 100 Area in hectares ≈ 1951.93 hectares

Therefore, the perimeter of the triangular plot is approximately 18.7178 km, and the area is approximately 1951.93 hectares.

Answer: The perimeter of the triangular plot is approximately 18.7178 km, and the area is approximately 1951.93 hectares.