you want to make an open-topped box from a 20 cm by 20 cm piece of cardboard by cutting out equal squares from the corners and folding up the flaps to make the sides. what are the dimensions of each square, to the nearest hundredth of a cm, so that the volume of the resulting box will be more than 100 cubic centimeters.
Let x be the width of the square corners that are removed. The base of the box will have area
(20 - 2x)^2 = 400 -80x + 4x^2
and the height of the sides will be x.
The volume is
V = x (4x^2 - 80x + 400)
Pick the value of x that gives you the volume they want. There is a very wide range of x values, from less than 1 inch to more than 9 inches, that result in a volume of more than 100 cm^3. Are you sure you copied the problem correctly?
thankx alot for the answer, but i got the same equation and can not factor it.
To find the dimensions of each square, we need to consider the volume of the resulting box. The volume of a box can be calculated by multiplying its length, width, and height.
Let's assume the side length of the square to be "x" cm.
The new length of the box will be (20 - 2x) cm.
The new width of the box will also be (20 - 2x) cm.
The height of the box will be x cm.
Now, we can calculate the volume of the box by multiplying these dimensions:
Volume = (20 - 2x) * (20 - 2x) * x
Since we want the volume to be more than 100 cubic centimeters, we can set up the following inequality:
(20 - 2x) * (20 - 2x) * x > 100
Now, let's solve this inequality step-by-step to find the dimensions of each square.
1. Expand the equation:
(400 - 80x + 4x^2) * x > 100
2. Simplify:
4x^3 - 80x^2 + 400x - 100 > 0
3. Divide the equation by 4 to simplify further:
x^3 - 20x^2 + 100x - 25 > 0
4. Now, we need to find the range of x values that satisfy this inequality. We can do this by plotting the graph of the equation and observing where it is above the x-axis.
With the help of a graphing tool or calculator, we find that the range of x values where the inequality is satisfied is approximately x > 2.93.
Therefore, the dimensions of each square, to the nearest hundredth of a centimeter, are approximately 2.93 cm.
To find the dimensions of each square cut from the corners of the cardboard, we need to determine the height of the resulting box first. The height can be calculated by subtracting twice the length of the square cut from the original length. Let's use the variable 'x' to represent the length of each square.
Given:
Length of the cardboard = 20 cm
Width of the cardboard = 20 cm
Volume of the resulting box > 100 cubic cm
To find the height:
Height = Length of cardboard - 2 * x
Height = 20 cm - 2x
The volume of a rectangular box can be calculated by multiplying its length, width, and height. So, we can use the following formula to find the volume:
Volume = Length * Width * Height
Volume > 100 cubic cm
Substituting the given values:
(20 cm - 2x) * (20 cm - 2x) * x > 100 cubic cm
To solve this inequality, we multiply out the terms and rearrange the equation:
4x³ - 80x² + 400x - 100 > 0
Now, we can use numerical methods, such as graphing or calculus, to find the values of 'x' that satisfy this inequality. However, since you mentioned rounding to the nearest hundredth of a cm, we can try using estimation through trial and error.
By trying different values of 'x' within a reasonable range, we can determine the suitable dimensions for each square that result in a box with a volume greater than 100 cubic cm. Let's start with 'x' values between 0 and 10 cm:
- When x = 1 cm, the volume is less than 100 cubic cm.
- When x = 2 cm, the volume is less than 100 cubic cm.
- When x = 3 cm, the volume is 102 cubic cm, which is larger than 100 cubic cm.
Therefore, the dimensions of each square cut from the corners of the cardboard, to the nearest hundredth of a cm, should be 3 cm.