Question

100 баллов!Если основанием пирамиды есть треугольник со сторонами 4, 5 и 7см, а высота пирамиды равна наибольшей высоте основания, то чему равен объем пирамиды? Решение обязательно с рисунком

Answer 1

Если основанием пирамиды есть треугольник со сторонами 4, 5 и 7см, а высота пирамиды равна наибольшей высоте основания, то чему равен объем пирамиды?

РЕШЕНИЕ:

• Рассмотрим тр. АВС:
По формуле Герона найдём площадь треугольника АВС:

$s = \sqrt{p(p - a)(p - b)(p - c)}$
где р = ( а + b + c ) / 2 - полупериметр, а, b и с - стороны треугольника

$s = \sqrt{8 \times (8 - 4)(8 - 5)(8 - 7)} = \sqrt{8 \times 4 \times 3 \times 1} = \\ = \sqrt{4 \times4 \times 6} = 4 \sqrt{6}$
• Площадь треугольника АВС равна:
S abc = ( 1/2 ) • AC • h1
4V6 = ( 1/2 ) • 4 • h1
h1 = 2V6

• S abc = ( 1/2 ) • AB • h2
4V6 = ( 1/2 ) • 5 • h2
h2 = 8V6 / 5

• S abc = ( 1/2 ) • BC • h3
4V6 = ( 1/2 ) • 7 • h3
h3 = 8V6 / 7

Наибольшая высота треугольника АВС равна 2V6
Значит, ED = 2V6

• Обьём пирамиды ЕАВС равен:
V = ( 1/3 ) • S abc • ED = ( 1/3 ) • 4V6 • 2V6 = 2 • 8 = 16


ОТВЕТ: 16

Answer 2

Calculation of the Volume of a Pyramid

To find the volume of a pyramid, we can use the formula:

Volume = (1/3) * Base Area * Height

In this case, the base of the pyramid is a triangle with sides measuring 4 cm, 5 cm, and 7 cm. The height of the pyramid is equal to the maximum height of the base.

Let's calculate the volume step by step:

1. Find the area of the base triangle using Heron's formula. Heron's formula 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, calculated as:

s = (a + b + c) / 2

In this case, the sides of the triangle are 4 cm, 5 cm, and 7 cm. Let's calculate the area:

s = (4 + 5 + 7) / 2 = 8

Area = sqrt(8 * (8 - 4) * (8 - 5) * (8 - 7)) = sqrt(8 * 4 * 3 * 1) = sqrt(96) ≈ 9.8 cm².

2. Since the height of the pyramid is equal to the maximum height of the base, we can use the Pythagorean theorem to find the height. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the sides of the right triangle are 4 cm, 5 cm, and the height (h) of the pyramid. Let's calculate the height:

h² = 5² - 4² = 25 - 16 = 9

h = sqrt(9) = 3 cm.

3. Now that we have the base area (9.8 cm²) and the height (3 cm), we can calculate the volume of the pyramid:

Volume = (1/3) * Base Area * Height

Volume = (1/3) * 9.8 cm² * 3 cm = 9.8 cm² * 1 cm = 9.8 cm³.

Therefore, the volume of the pyramid is 9.8 cm³.

Diagram:

Here is a diagram to help visualize the pyramid:

       /\
      /  \
     /    \
    /______\
      4 cm
     /      \
    /        \
   /__________\
     5 cm
    /          \
   /            \
  /______________\
    7 cm

|-----|
      3 cm

Please note that the diagram is not to scale and is only for illustrative purposes.

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