the perimeter of a piece of cardboard is 34 inches. squares measuring 2 inches on a side are cut from each corner so that when the sides are folded up, the diagonal of the resulting box has the length 7 inches.what are the original dimensions of the card board?
let the width of the cardboard be x
then it's length is 17-x
2 inches cut off each dimensions leaves the base of the box to be
x-4 and 17-x-4
x-4 and 13-x
(x-4)^2 + (13-x)^2 = 49
x^2 - 8x + 16 + 169 - 26x + x^2 = 49
2x^2 - 34x + 136 = 0
x^2 - 17x + 68 = 0
x = (17 ± √17)/2
x = appr 10.56 inches or x = appr 6.438 inches
mmmhh, was expecting "nicer" numbers
check:
perimeter = 2(10.56+6.44) = 34 , check
new base is 6.56 by 2.438
hypotenuse??
5.56^2 + 2.438^2
= 6.998 , close enough to 7 based on round-off
Let's assume the original dimensions of the cardboard are length (L) and width (W).
1. From the given information, we know that the perimeter of the cardboard is 34 inches.
Perimeter of a rectangle = 2 * (Length + Width)
So we have: 2L + 2W = 34 ----(Equation 1)
2. We also know that squares measuring 2 inches on a side are cut from each corner.
If we cut 2 inches from each corner of the length and width, then the resulting dimensions would be:
(L - 2) for the length and (W - 2) for the width.
And the height of the resulting box will be 2 inches.
3. We are also given that when the sides are folded up, the diagonal of the resulting box has a length of 7 inches.
Using the Pythagorean theorem, the diagonal (D) of a rectangular box can be calculated as:
D = √(L^2 + W^2 + H^2)
Substituting the values, we have: 7 = √((L-2)^2 + (W-2)^2 + 2^2)
Simplifying, we get: 49 = (L-2)^2 + (W-2)^2 + 4 ----(Equation 2)
To find the original dimensions of the cardboard (L and W), we need to solve equations 1 and 2 simultaneously.
Let's solve these equations step by step:
Step 1: Rearrange Equation 1 to get one variable in terms of the other.
2L + 2W = 34
L + W = 17
L = 17 - W
Step 2: Substitute the value of L from Step 1 into Equation 2.
49 = (L-2)^2 + (W-2)^2 + 4
49 = [(17-W)-2]^2 + (W-2)^2 + 4
49 = (15-W)^2 + (W-2)^2 + 4
49 = (W^2 - 30W + 225) + (W^2 - 4W + 4) + 4
49 = 2W^2 - 34W + 233
Step 3: Simplify the equation obtained in Step 2 and rewrite it in standard quadratic form (Ax^2 + Bx + C = 0).
2W^2 - 34W + 233 - 49 = 0
2W^2 - 34W + 184 = 0
Step 4: Solve the quadratic equation obtained in Step 3 using the quadratic formula.
The quadratic formula is: W = (-B ± √(B^2 - 4AC)) / (2A)
In our case, A = 2, B = -34, and C = 184.
W = (-(-34) ± √((-34)^2 - 4 * 2 * 184)) / (2 * 2)
W = (34 ± √(1156 - 1472)) / 4
W = (34 ± √(-316)) / 4
Since we cannot take the square root of a negative number, it means that there are no real solutions for W.
Therefore, there are no valid original dimensions for the cardboard that satisfy the given conditions.
To find the original dimensions of the cardboard, we need to go through the problem step by step.
Let's assume the length of the cardboard is "L" inches and the width is "W" inches.
1. Cut squares from each corner: Squares measuring 2 inches on a side are cut from each corner. Therefore, the length and width of the resulting box will be reduced by 4 inches (2 inches cut from each corner).
2. Calculate the new dimensions after cutting: The new dimensions of the cardboard will be (L - 4) inches and (W - 4) inches.
3. Folding up the sides: When the sides are folded up, the resulting box will have a height equal to the 2-inch squares that were cut out.
4. Finding the diagonal of the resulting box: According to the problem, the diagonal of the resulting box measures 7 inches.
Using the Pythagorean theorem, we can find the new dimensions of the box:
(New length)^2 + (New width)^2 = (Diagonal)^2
((L - 4)^2 + (W - 4)^2 = 7^2
Solving this equation will give us the new dimensions of the box.
5. Calculating the perimeter: The perimeter of the cardboard can be calculated by adding the original length, original width, and twice the reduction caused by cutting the squares. In this case, it is given that the perimeter is 34 inches.
So, we have the following equation:
2(L - 4) + 2(W - 4) = 34
Now, you can solve these two equations simultaneously to find the original dimensions of the cardboard.