Arlan needs to create a box from a piece of cardboard. The dimensions of his cardboard are 10 inches by 8 inches. He must cut a square from each corner of the cardboard, in order to form a box. What size square should he cut from each corner, in order to create a box with the largest possible volume?
A.0.5 inches
B. 1 inch
C. 1.5 inches
D. 2 inches
I think its C ?
L = 10 - 2x
w = 8 - 2 x
v = L * w * x
v = (10-2x)(8-2x)x
v = (80 - 36 x + 4 x^2)x
v = 4 x^3 - 36 x^2 + 80 x
do you do calculus?
if so the
min or max when dv/dx = 0
0 = 12 x^2 -72 x + 80
0 = 3 x^2 - 18 x + 20
x = 3 +/- (1/3)sqrt 21
= 3 +/- 1.52
the plus sign is too big a width
so
3 - 1.52
is 1.5 close enough
no , So its D ? 2 inches
Nope, C 1.5 inches
To find the size of the square Arlan needs to cut from each corner in order to create a box with the largest possible volume, let's break down the problem step-by-step:
1. Start with a piece of cardboard that measures 10 inches by 8 inches.
2. Cut squares from each corner of the cardboard. Let's assume the size of the square Arlan cuts is 'x' inches.
3. After cutting the squares, the dimensions of the remaining cardboard will be (10 - 2x) inches by (8 - 2x) inches.
4. To create a box, the next step is to fold up the remaining flaps along the sides and secure them together.
5. The volume of the resulting box can be calculated by multiplying the lengths of the three sides: length, width, and height.
So, the volume of the box can be expressed as:
V = (10 - 2x) * (8 - 2x) * x
Now, we can simplify this equation and find the value of 'x' that maximizes the volume.
To do this, we can expand the equation:
V = (80 - 16x - 20x + 4x^2) * x
V = 4x^3 - 36x^2 + 80x
To find the maximum volume, we need to find the value of 'x' that maximizes this equation. This can be achieved by finding the derivative of the equation with respect to 'x' and setting it equal to zero:
dV/dx = 12x^2 - 72x + 80
Now, set dV/dx = 0 and solve for 'x':
12x^2 - 72x + 80 = 0
Using the quadratic formula, we find two possible values for 'x':
x = (72 ± √(72^2 - 4*12*80)) / (2*12)
x = (72 ± √(5184 - 3840)) / 24
x = (72 ± √1344) / 24
x = (72 ± 36.7) / 24
Simplifying further, we find:
x ≈ 4.2 or x ≈ 1.3
Since the dimension of the cardboard is given in inches, we can only have whole number values for 'x'. Therefore, the closest option from the given choices is:
D. 2 inches
So, Arlan should cut 2 inch squares from each corner to create a box with the largest possible volume.