Question

Найти сторону основания правильной четырёхугольной призмы,если её высота равна 8,а площадь полной поверхности 264

Answer 1

 Площадь полной поверхности равна:
Sполн=2Sосн+Sбок=2а²+4аh=264, где а - сторона основания.
2а²+32а=264,
а²+16а-132=0,
а₁=-22, а<0, значение не подходит.
а₂=6.
Ответ: в основании призмы сторона равна 6.

Answer 2

Finding the Side Length of the Base of a Regular Prism

To find the side length of the base of a regular quadrilateral prism, we need to use the given information of the prism's height and the total surface area.

Let's denote the side length of the base of the prism as s.

The total surface area of a regular quadrilateral prism can be calculated using the formula:

Total Surface Area = 2 * Base Area + Lateral Surface Area

The base area of a regular quadrilateral prism is given by:

Base Area = s^2

The lateral surface area of a regular quadrilateral prism is given by:

Lateral Surface Area = Perimeter of Base * Height

Since the base of a regular quadrilateral prism is a square, the perimeter of the base is given by:

Perimeter of Base = 4 * s

Substituting these values into the formula for the total surface area, we get:

264 = 2 * s^2 + 4 * s * 8

Simplifying the equation, we have:

264 = 2s^2 + 32s

Rearranging the equation to a quadratic form, we get:

2s^2 + 32s - 264 = 0

To solve this quadratic equation, we can use the quadratic formula:

s = (-b ± √(b^2 - 4ac)) / (2a)

In this case, the coefficients of the quadratic equation are:

a = 2, b = 32, c = -264

Substituting these values into the quadratic formula, we can calculate the possible values for the side length of the base of the prism.

Let's calculate the values using the quadratic formula:

s = (-32 ± √(32^2 - 4 * 2 * -264)) / (2 * 2)

Simplifying the equation further, we have:

s = (-32 ± √(1024 + 2112)) / 4

s = (-32 ± √3136) / 4

s = (-32 ± 56) / 4

This gives us two possible solutions:

s1 = (-32 + 56) / 4 = 6

s2 = (-32 - 56) / 4 = -22

Since the side length of a geometric shape cannot be negative, we discard the negative solution.

Therefore, the side length of the base of the regular quadrilateral prism is 6.

Please note that the solution is based on the given information and calculations.