Question

Стороны основания прямого параллелепипеда равны 8 и 12 см,угол между ними -60 градусов. Диагональ большей боковой грани равна 15 см. Вычислите площадь полной поверхности параллелепипеда. Помоги пожалуйста Подробно

Answer 1

Есть формула площади параллелограмма : S=abSin альфа. S=8×12×Sin60°==48√3. Площади оснований равны 2×48=96√3. Находим высоту из прямоугольного треугольника по теореме Пифагора .Н=√15^2-12^2 =9. Площадь боковой поверхности равна S=P×H=2a×g×H=2×8×12×9 =1728. Общая площадь равна 1728+96√3.

Answer 2

Given Information:

We are given the following information about a parallelepiped: - The sides of the base are 8 cm and 12 cm. - The angle between the sides is -60 degrees. - The diagonal of the larger lateral face is 15 cm.

Approach:

To find the surface area of the parallelepiped, we need to calculate the areas of all its faces and then sum them up.

Solution:

Let's break down the solution into steps:

Step 1: Find the length of the third side of the base using the given angle and sides. To find the length of the third side, we can use the Law of Cosines. The formula is as follows:

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

Where: - c is the length of the third side - a and b are the lengths of the other two sides - C is the angle between the two sides

In this case, a = 8 cm, b = 12 cm, and C = -60 degrees.

Step 2: Calculate the area of the base. The area of the base of the parallelepiped is given by the formula:

Area = length * width

In this case, the length is 8 cm and the width is 12 cm.

Step 3: Calculate the area of the larger lateral face. To calculate the area of the larger lateral face, we need to find the length of the diagonal of the base. We can use the Pythagorean theorem to find the length of the diagonal. The formula is as follows:

diagonal^2 = length^2 + width^2

In this case, the length is 8 cm and the width is 12 cm.

Step 4: Calculate the area of the smaller lateral face. The area of the smaller lateral face is the same as the area of the larger lateral face.

Step 5: Calculate the area of the top face. The area of the top face is the same as the area of the base.

Step 6: Calculate the area of the bottom face. The area of the bottom face is the same as the area of the base.

Step 7: Calculate the total surface area. To find the total surface area, we need to sum up the areas of all the faces.

Let's calculate the values step by step:

Step 1: Find the length of the third side of the base. Using the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C) c^2 = 8^2 + 12^2 - 2 * 8 * 12 * cos(-60) c^2 = 64 + 144 + 192 * cos(60) c^2 = 208 + 192 * 0.5 c^2 = 208 + 96 c^2 = 304 c = sqrt(304) c ≈ 17.46 cm

Step 2: Calculate the area of the base. Area = length * width Area = 8 cm * 12 cm Area = 96 cm^2

Step 3: Calculate the area of the larger lateral face. Using the Pythagorean theorem: diagonal^2 = length^2 + width^2 diagonal^2 = 8^2 + 12^2 diagonal^2 = 64 + 144 diagonal^2 = 208 diagonal = sqrt(208) diagonal ≈ 14.42 cm

Step 4: Calculate the area of the smaller lateral face. The area of the smaller lateral face is the same as the area of the larger lateral face. Area = diagonal * width Area = 14.42 cm * 12 cm Area ≈ 173.04 cm^2

Step 5: Calculate the area of the top face. The area of the top face is the same as the area of the base. Area = 96 cm^2

Step 6: Calculate the area of the bottom face. The area of the bottom face is the same as the area of the base. Area = 96 cm^2

Step 7: Calculate the total surface area. Total Surface Area = 2 * (base area) + 2 * (lateral face area) Total Surface Area = 2 * 96 cm^2 + 2 * 173.04 cm^2 Total Surface Area = 192 cm^2 + 346.08 cm^2 Total Surface Area ≈ 538.08 cm^2

Therefore, the approximate total surface area of the parallelepiped is 538.08 cm^2.

Note: The calculations are approximate due to rounding of decimal values during intermediate steps.