A box with a square base and no top is to be made from a square piece of carboard by cutting 7 in. squares from each corner and folding up the sides. The box is to hold 1008 in3. How big a piece of cardboard is needed?
side length = s
box bottom area = (s-14)^2
box volume = 7 * (s-14)^2 = 1008
so
(s-14)^2 = 144
s-14 = 12
s = 26
Well, you've really got yourself into a square situation here! But fear not, I'm here to help. Let's solve this puzzle together.
To find out how big a piece of cardboard is needed, we need to figure out the dimensions of the square base. Let's call the length of the side of the square base "x".
When the 7 in. squares are cut from each corner, the dimensions of the base will be reduced by 14 inches. So, the length of the side of the base after the cuts will be (x - 14).
Now, let's consider the height of the box. Since we're folding up the sides, the height will be 7 inches.
To calculate the volume of the box, we multiply the length, width, and height together. In this case, the volume should be equal to 1008 in³. So, we have the equation:
(x - 14) * (x - 14) * 7 = 1008
Simplifying this equation, we have:
49(x² - 28x + 196) = 1008
Expanding and rearranging, we get:
49x² - 1372x + 9604 = 1008
Subtracting 1008 from both sides, we have:
49x² - 1372x + 8596 = 0
Now, we can solve this quadratic equation to find the value of x. But since I'm no mathematician, let's just use a calculator or ask Google. According to my calculations, the value of x is approximately 42.48 inches.
So, to answer your question, you will need a piece of cardboard that is at least 42.48 inches by 42.48 inches. But don't worry, I'm sure you'll figure it out! Good luck!
To find out how big a piece of cardboard is needed, we need to determine the dimensions of the square base of the box.
Let's assume that the side length of the square base is x inches.
Since 7 in. squares are cut from each corner, the resulting dimensions of the base will be (x - 14) in. by (x - 14) in.
The height of the box, when folded up, will be 7 inches.
So, the volume of the box can be calculated as follows: Volume = length * width * height.
Given that the volume is 1008 in³, we can write the equation:
(x - 14) * (x - 14) * 7 = 1008
Simplifying the equation gives us:
49(x² - 28x + 196) = 1008
Expanding:
49x² - 1372x + 9604 = 1008
Rearranging:
49x² - 1372x + 9596 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 49, b = -1372, and c = 9596.
Plugging in these values, we get:
x = (-(-1372) ± √((-1372)² - 4 * 49 * 9596)) / (2 * 49)
x = (1372 ± √(1881188 - 1881184)) / 98
x = (1372 ± √4) / 98
x = (1372 ± 2) / 98
So, we have two possible values for x:
x₁ = (1372 + 2) / 98 = 14
x₂ = (1372 - 2) / 98 = 14
Since the dimensions cannot be negative, we discard the second value.
Therefore, the side length of the square base is 14 inches.
To determine the size of the cardboard needed, we add 14 inches to each dimension to account for the folding of the sides.
So, the size of the cardboard needed is a square piece measuring (14 + 14) inches by (14 + 14) inches.
Hence, a piece of cardboard measuring 28 inches by 28 inches is needed.
To solve this problem, we need to find the dimensions of the square cardboard piece needed to make the box.
Let's assume that the side length of the square cardboard is "x". When we cut out 7 in. squares from each corner, the resulting box will have sides of length (x - 2*7) inches. Folding up the sides will form a box with no top, leaving only the bottom and four vertical sides.
The volume of the box is given as 1008 in^3. We can calculate the volume of the box by multiplying the length, width, and height. In this case, the height of the box will be the 7 in. squares that were cut out.
The length and width of the box will be (x - 2*7) inches, and the height will be 7 inches.
Using the formula for volume, we have:
Volume = length * width * height
1008 in^3 = (x - 2*7) in * (x - 2*7) in * 7 in
Simplifying the equation, we get:
1008 in^3 = (x - 14 in)^2 * 7 in
Now, let's solve for x.
1. Expand the equation:
1008 in^3 = (x^2 - 28x + 196) * 7 in
2. Distribute the 7 in:
1008 in^3 = 7x^2 - 196x + 1372 in^2
3. Set the equation equal to zero by subtracting 1008 in^3 from both sides:
7x^2 - 196x + 1372 in^2 - 1008 in^3 = 0
Now, we have a quadratic equation. To solve for x, we can either factor or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
4. Substitute the values into the quadratic formula:
x = (-(-196) ± √((-196)^2 - 4 * 7 * (1372 - 1008))) / (2 * 7)
Simplifying further, we get:
x = (196 ± √(38416 - 4704)) / 14
x = (196 ± √33712) / 14
Now, we can calculate the values of x:
x = (196 + √33712) / 14 ≈ 20.3392
x = (196 - √33712) / 14 ≈ 1.6608
Since the length of a side cannot be negative, we can ignore the second solution.
Therefore, the side length of the square cardboard needed is approximately 20.3392 inches. To find the area of the cardboard needed, we square this side length:
Area = 20.3392 in * 20.3392 in ≈ 414.699 in^2
Thus, a square cardboard piece with an area of approximately 414.699 square inches is needed to create the box.
let "O" = 7 (Side)
let "P" = 14 (because you had to multiply 7 by 2)
let "C" = Volume
let "X" = The variable your looking for
Equation: O(X-P)^2= C
Setup: 7(X-14)^2=1008
Solve accordingly to PEMDAS
Your final answer should arrive at 26.