The length of an open-top box is 4 cm longer than its width. The box was made from a 480-cm^2 rectangular sheet of material with 6cm by 6cm squares cut from each corner. The height of the box is 6cm. Find the dimensions of the box.
Please show me in detail how to set this up and solve.
area of sides = 2 L *6 + 2 W*6
= 12 L + 12 W
area of bottom = L W
total area = L W + 12 L + 12 W = 480-4*36
so
L W + 12 L + 12 W = 336
but L = W + 4
(W+4)W + 12(W+4) + 12 W - 336 = 0
W^2 + 28 W -288 = 0
(W -16)(W+18) = 0
W = 16
L = 16+4 = 20
l=4+w
w=l-4
h=6
v=480
v=lwh
480=12 x 8 x 6
To solve this problem, we can follow these steps:
Step 1: Visualize the problem
Imagine a rectangular sheet of material where squares measuring 6cm by 6cm are removed from each corner. When the remaining material is folded, it forms an open-top box with a height of 6cm.
Step 2: Understand the given information
From the problem, we know that the rectangular sheet has an area of 480 cm², and the squares cut from each corner measure 6cm by 6cm. The height of the box is given as 6cm.
Step 3: Define the variables
Let's assign variables to the length and width of the box to solve for them. Let's say:
- Length of the box = L cm
- Width of the box = W cm
Step 4: Set up an equation for the area
The area of the rectangular sheet can be computed by multiplying the length (L + 2 * 6) and the width (W + 2 * 6):
(L + 2 * 6) * (W + 2 * 6) = 480
Step 5: Simplify the equation
Simplify the equation by distributing and combining like terms:
(L + 12) * (W + 12) = 480
LW + 12L + 12W + 144 = 480
LW + 12L + 12W = 480 - 144
LW + 12L + 12W = 336
Step 6: Substitute the relationship between length and width
We know from the given information that the length of the box is 4 cm longer than its width. So, we can write the relationship as:
L = W + 4
Substitute this relationship into the equation we have:
(W + 4) * W + 12(W + 4) + 12W = 336
Step 7: Solve the equation
Expand and simplify the equation further:
W² + 4W + 12W + 48 + 12W = 336
W² + 28W + 48 = 336
W² + 28W - 288 = 0
Step 8: Solve the quadratic equation
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula:
W = (-b ± √(b² - 4ac))/(2a)
Where a = 1, b = 28, and c = -288:
W = (-28 ± √(28² - 4 * 1 * -288))/(2 * 1)
Simplify the equation:
W = (-28 ± √(784 + 1152))/2
W = (-28 ± √1936)/2
W = (-28 ± 44)/2
Now, we have two possible values for W:
1. W = (-28 + 44)/2 = 16/2 = 8
2. W = (-28 - 44)/2 = -72/2 = -36
Since width cannot be negative, we discard the second value.
Step 9: Calculate the length
Since we know L = W + 4, we can substitute W = 8 into this equation:
L = 8 + 4 = 12
Step 10: Determine the dimensions of the box
The dimensions of the box are:
- Length = L = 12 cm
- Width = W = 8 cm
Therefore, the length and width of the box are 12 cm and 8 cm, respectively.