To find the volume of the largest box that can be made from a square piece of cardboard with side lengths of 24 inches by cutting equal squares from each corner and turning up the sides, we can use the formula V = x(24-2x)(24-2x), where x represents the side length of the squares being cut.
The first step is to find the value of x that maximizes the volume. We can do this by finding the critical points of the function V = x(24-2x)(24-2x) and determining which one corresponds to a maximum.
To find the critical points, we can take the derivative of V with respect to x:
dV/dx = (24-2x)(24-2x) + x(-2)(24-2x) + x(24-2x)(-2)
= (24-2x)(24-2x) - 4x(24-2x) - 4x(24-2x)
= (24-2x)(24-2x) - 8x(24-2x)
Next, we set the derivative equal to 0 and solve for x:
(24-2x)(24-2x) - 8x(24-2x) = 0
Simplifying this equation, we get:
576 - 96x + 4x^2 - 192x + 32x^2 - 64x^2 = 0
- 24x^2 + 4x^2 - 96x - 192x - 64x^2 + 576 = 0
- 84x^2 - 288x + 576 = 0
Now, we can solve this quadratic equation for x. We can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For this equation, a = -84, b = -288, and c = 576. Plugging these values into the quadratic formula, we get:
x = (-(-288) ± sqrt((-288)^2 - 4(-84)(576))) / (2(-84))
x = (288 ± sqrt(82944 + 192768)) / 168
x = (288 ± sqrt(275712)) / 168
x = (288 ± 524.63) / 168
x ≈ 4.26 or x ≈ 4.81
Since the length of the side must be positive, we can ignore the negative solution. Therefore, the value of x that maximizes the volume is approximately 4.81 inches.
To find the volume of the largest box, we substitute this value of x into the formula V = x(24-2x)(24-2x):
V ≈ 4.81(24-2(4.81))(24-2(4.81))
V ≈ 4.81(24-9.62)(24-9.62)
V ≈ 4.81(14.38)(14.38)
V ≈ 978.075
Therefore, the volume of the largest box that can be made from the square piece of cardboard is approximately 978.075 cubic inches.
What is the volume of the largest box that can be made from a square piece of cardboard with side lengths of 24 inches by cutting equal squares from each corner and turning up the sides? use V = x (24-2x)(24-2x)
1 answer