A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions W inches by L inches by cutting out equal squares of side x at each corner and then folding up the sides. (W = 12 in. and L = 20 in). Find the values of x for which the volume is greater than 200 in3. (Give each answer correct to three decimal places
v = x(W-2x)(L-2x)
so,
x(12-2x)(30-2x) > 200
8-3√6 < x < 5
To find the values of x for which the volume is greater than 200 in³, we need to solve the following equation for x:
Volume = (Length - 2x) * (Width - 2x) * x > 200
Given that W = 12 in. and L = 20 in., we can substitute the values into the equation:
(20 - 2x) * (12 - 2x) * x > 200
Simplifying, we get:
240x - 64x² - 4x³ > 200
Rearranging the terms, we have a cubic equation:
4x³ - 64x² + 240x - 200 > 0
To solve the inequality, follow these steps:
Step 1: Factor out common factor (x - 5) from equation to get:
(x - 5)(4x² - 24x + 40) > 0
Step 2: Solve the quadratic equation 4x² - 24x + 40 = 0 using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
x = (-(-24) ± √((-24)² - 4(4)(40))) / (2(4))
x = (24 ± √(576 - 640)) / 8
x = (24 ± √(-64)) / 8
Since we cannot take the square root of a negative number, there are no real solutions for the quadratic equation. Therefore, we only need to consider x - 5 > 0.
Step 3: Solve x - 5 > 0 to find the range of values for x:
x > 5
So, the values of x for which the volume is greater than 200 in³ are x > 5.
To find the values of x for which the volume is greater than 200 in³, we need to follow these steps:
1. Start with the given dimensions of the rectangle: W = 12 in. and L = 20 in.
2. Cut out squares of side x from each corner. This will reduce the dimensions of the rectangle.
The new length of the rectangle is L - 2x, and the new width is W - 2x.
3. Fold up the sides to form the open-top box. The height of the box is x.
4. Calculate the volume of the box using the formula: Volume = Length × Width × Height.
The volume is given as greater than 200 in³, so we have the inequality:
(L - 2x) × (W - 2x) × x > 200.
5. Substitute the given values into the inequality: (20 - 2x) × (12 - 2x) × x > 200.
6. Simplify the equation and solve for x:
(20 - 2x) × (12 - 2x) × x > 200
(240 - 52x + 4x²) × x > 200
4x³ - 52x² + 240x > 200
Rearrange the inequality to the form:
4x³ - 52x² + 240x - 200 > 0
7. This is a cubic inequality that we can solve graphically or algebraically. However, in this case, let's solve it algebraically.
To find the values of x where the inequality is true, we need to find the solutions to the equation:
4x³ - 52x² + 240x - 200 = 0
Use numerical methods or factoring to solve this equation. The solutions will be the values of x that satisfy the original inequality. Round the answers to three decimal places.
8. Once you find the values of x that satisfy the equation and the original inequality, those will be the values that make the volume greater than 200 in³.