Question

Из металлического листа ,имеющего форму прямоугольника ,длина которого в 1.5 раза больше ширины ,сделан открытый сверху ящик .Для этого по углам листа вырезаны квадраты со стороной 3дм и получившиеся боковые грани загнуты.Найдите размеры листа ,если объем получившегося ящика оказался равным 216 дм3. + Объяснение как вы это решили.

Answer 1

х - ширина листа, (х-3-3)=(x-6) - ширина ящика

у=1.5*х - длина листа, (1.5*х-6) - длина ящика

Так как вырезали квадраты со стороной 3дм, то загнутые грани( они

же высоты ящика) имеют сторону 3дм

h=3дм

обьем ящика: (х-6)*(1.5*х-6)*h=216

1.5*х*х-15*х+36=72

1.5*х*х-15*х-36=0

х*х-10*х-24=0

D=196

x1=12

x2=-2<0

Итого

х=12 - ширина

у=1.5х=18 - длина 

Answer 2

Problem Analysis

We are given a rectangular metal sheet that is 1.5 times longer than its width. The sheet is used to create an open-top box by cutting squares with a side length of 3 dm from the corners and bending the resulting flaps. We need to find the dimensions of the sheet if the volume of the box is 216 dm³.

Solution

Let's assume that the width of the metal sheet is x dm. According to the problem statement, the length of the sheet is 1.5 times the width, so the length is 1.5x dm.

To create the box, squares with a side length of 3 dm are cut from the corners of the sheet. This reduces the length and width of the sheet by 6 dm (3 dm on each side). After cutting the squares, the length of the sheet becomes 1.5x - 6 dm, and the width becomes x - 6 dm.

The height of the box is equal to the side length of the cut squares, which is 3 dm.

To find the volume of the box, we multiply the length, width, and height: Volume = Length × Width × Height

Substituting the values we have: 216 dm³ = (1.5x - 6) dm × (x - 6) dm × 3 dm

Now we can solve this equation to find the value of x and then calculate the dimensions of the metal sheet.

Calculation

Let's solve the equation step by step:

216 dm³ = (1.5x - 6) dm × (x - 6) dm × 3 dm

Expanding the equation: 216 dm³ = (4.5x - 18) dm² × (x - 6) dm

Multiplying the terms: 216 dm³ = 4.5x² - 27x - 18x + 108 dm³

Combining like terms: 216 dm³ = 4.5x² - 45x + 108 dm³

Subtracting 216 dm³ from both sides: 0 = 4.5x² - 45x - 108 dm³

Dividing the equation by 4.5 to simplify: 0 = x² - 10x - 24 dm³

Now we have a quadratic equation. We can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula to find the value of x:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, a = 1, b = -10, and c = -24.

Substituting the values into the quadratic formula: x = (-(-10) ± √((-10)² - 4(1)(-24))) / (2(1))

Simplifying: x = (10 ± √(100 + 96)) / 2 x = (10 ± √196) / 2 x = (10 ± 14) / 2

Simplifying further: x₁ = (10 + 14) / 2 = 24 / 2 = 12 x₂ = (10 - 14) / 2 = -4 / 2 = -2

Since the width cannot be negative, we discard the negative value of x.

Therefore, the width of the metal sheet is 12 dm.

Using this value, we can calculate the length of the sheet: Length = 1.5x = 1.5 × 12 = 18 dm

So, the dimensions of the metal sheet are 12 dm × 18 dm.

Conclusion

The dimensions of the metal sheet are 12 dm × 18 dm. This was determined by solving the equation derived from the given information and using the quadratic formula to find the width of the sheet.