Question

В основании прямой призмы лежит ромб со стороной 3см и углом между ними 30 градусов. Найдите объем призмы, если высота равна 5см.

Answer 1

Призма АВСДА1В1С1Д1, основание ромб АВСД, АВ=ВС=СД=АС=3, уголА=30, АА1=5, объем ризмы=площадь основания*высота=АВ в квадрате*sinA*АА1=3*3*1/2*5=22,5

Answer 2

Calculation of the Prism Volume

To find the volume of the prism, we need to multiply the area of the base by the height of the prism. In this case, the base of the prism is a rhombus with a side length of 3 cm and an angle of 30 degrees between the sides.

To calculate the area of the rhombus, we can use the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals of the rhombus.

Since the diagonals of a rhombus are perpendicular bisectors of each other, we can find the lengths of the diagonals using the side length and the angle between the sides.

Let's calculate the lengths of the diagonals:

1. The longer diagonal (d1): - We can use the formula: d1 = 2 * a * sin(angle/2), where a is the side length and angle is the angle between the sides. - Plugging in the values, we get: d1 = 2 * 3 cm * sin(30 degrees/2).

2. The shorter diagonal (d2): - We can use the formula: d2 = 2 * a * cos(angle/2), where a is the side length and angle is the angle between the sides. - Plugging in the values, we get: d2 = 2 * 3 cm * cos(30 degrees/2).

Now that we have the lengths of the diagonals, we can calculate the area of the rhombus using the formula mentioned earlier.

Finally, we can calculate the volume of the prism by multiplying the area of the base by the height of the prism.

Let's perform the calculations step by step:

1. Calculation of the longer diagonal (d1): - d1 = 2 * 3 cm * sin(30 degrees/2). - Using a calculator, we find that d1 ≈ 3.00 cm.

2. Calculation of the shorter diagonal (d2): - d2 = 2 * 3 cm * cos(30 degrees/2). - Using a calculator, we find that d2 ≈ 2.60 cm.

3. Calculation of the area of the rhombus: - Area = (d1 * d2) / 2. - Plugging in the values, we get: Area = (3.00 cm * 2.60 cm) / 2. - Using a calculator, we find that Area ≈ 3.90 cm^2.

4. Calculation of the volume of the prism: - Volume = Area * height. - Plugging in the values, we get: Volume = 3.90 cm^2 * 5 cm. - Using a calculator, we find that **Volume ≈ 19.50 cm^