A open-topped box can be created by cutting congruent squares from each of the four corners of a piece of cardboard that has dimensions of 20cm by 30cm and folding up the sides. Derermine the dimensions of the square that must be cut to create a box with a volume of 100cm3. Widht= 20-2x, length =30-2x, height=x.
just plug the values into the equation
100 = x(20-2x)(30-2x)
4x^3 - 100x^2 + 600x - 100 = 0
x^3 - 25x^2 + 150x - 25 = 0
There are 3 real solutions, approximately
0.17, 9.52, 15.31
15.31 is out, since the sides aren't long enough.
Not an easy problem, if you have no tools for solving cubics.
To determine the dimensions of the square that must be cut to create a box with a volume of 100 cm³, we can use the formula for the volume of a rectangular box:
Volume = Length × Width × Height
Given that the width is 20 cm, the length is 30 cm, and the height is x cm, we can substitute these values into the formula:
100 cm³ = (30 - 2x) cm × (20 - 2x) cm × x cm
Simplifying the equation, we have:
100 cm³ = (600 - 60x - 40x + 4x²) × x
Expanding the brackets:
100 cm³ = 600x - 100x² + 4x³
Rearranging the equation to form a cubic equation:
4x³ - 100x² + 600x - 100 = 0
Now, we can solve this cubic equation to find the value(s) of x that satisfies the equation using mathematical methods such as factoring, synthetic division, or numerical approximation methods.
To determine the dimensions of the square that must be cut to create a box with a volume of 100 cm³, we need to use the formula for the volume of a rectangular prism:
Volume = Length × Width × Height
Given that the length is (30 - 2x), the width is (20 - 2x), and the height is x, we can plug these values into the formula:
100 = (30 - 2x) × (20 - 2x) × x
Simplifying the equation, we get:
100 = (600 - 60x - 40x + 4x²) × x
100 = (4x² - 100x + 600) × x
100 = 4x³ - 100x² + 600x
Rearranging the equation, we have a cubic equation:
4x³ - 100x² + 600x - 100 = 0
We can solve this equation to find the value of x using numerical methods like the Newton-Raphson method or by using software/tools like calculators or math software.
Once we find the value of x, we can substitute it back into the expressions for the length and width to determine the dimensions of the square that must be cut.