A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides. The box is to hold&nbsp;100&nbsp;∈3. What size cardboard piece is required?

Question

A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides. The box is to hold&amp;nbsp;100&amp;nbsp;∈3. What size cardboard piece is required?

abonirali59 · Accepted Answer

Step 1  Let x be the length of one side of the cardboard, so our initial piece of cardboard is x by x.  When 4 inches are removed from eac side, the base of thebox is x−8 by x−8  Since the volume is 100 ∈3  Step 2  4(x−8)2=100  x2−16x+64=25  x2−16x+39=0  (x−3)(x−13)=0  So  x=3 or x=13  x=3 is not possible, since then the length of the base would be 3−8=−5  Thus x=13, and the piece of cardboard is 13 inches by 13 inches.

Lakisha Archer · Accepted Answer

Step 1  To find the length of each side of the square piece of cardboard  The Volume will be V=L×W×H  We are cutting 4 inches from each corner, so the length and width will be x−8  The height will be 4.  100=4×(x−8)×(x−8)  100=4×(x−8)2  (x−8)2=25  x−8=5  x=13 inches  Hence, the piece of cardboard needed will be 13 by 13 inches.

user_27qwe · Accepted Answer

To solve the problem, let&#039;s denote the length of the side of the square base as x. We are cutting out 4-inch squares from each corner of the square cardboard, which reduces the dimensions of the base and height of the box.The base of the resulting box will have side length x−2×4 (since we remove two 4-inch squares from each side). The height of the box will be 4 inches.The volume of a box is given by the formula:Volume=base&amp;nbsp;area×heightFor our box, the base area is (x−2×4)2 and the height is 4 inches. We know that the volume should be 100 cubic inches, so we can set up the equation:100=(x−2×4)2×4Simplifying the equation, we have:100=(x−8)2×4Now we can solve for x by isolating it. Dividing both sides of the equation by 4, we get:1004=(x−8)2Simplifying further:25=(x−8)2Taking the square root of both sides, we have:25=(x−8)2Which simplifies to:5=|x−8|To solve for x, we consider two cases:Case 1: x−8=5Solving for x in this case gives us x=13.Case 2: −(x−8)=5Solving for x in this case gives us x=3.Since we are cutting squares from each corner, the side length x should be larger than 8 inches. Therefore, the correct solution is x=13.Thus, a cardboard piece with side length 13 inches is required to make the box.

karton · Accepted Answer

Result:13 inchesSolution:When we cut out a square with a side length of 4 inches from each corner of the cardboard, the dimensions of the resulting box will be (x−8) by (x−8) by 4 (height).To find the volume of the box, we multiply the three dimensions together:V=(x−8)×(x−8)×4.According to the problem, the box is required to hold 100 cubic inches, so we can set up the equation:(x−8)×(x−8)×4=100.To determine the size of the cardboard piece required, we need to solve this equation for x. Let&#039;s proceed with the solution.First, let&#039;s simplify the equation:4(x−8)2=100.Now, divide both sides of the equation by 4:(x−8)2=1004.Simplifying the right side:(x−8)2=25.Taking the square root of both sides:x−8=±25.Simplifying further:x−8=±5.Now, solve for x:For x−8=5, we have x=5+8=13.For x−8=−5, we have x=−5+8=3.Since the length cannot be negative, we discard the solution x=3.Therefore, the size of the cardboard piece required is 13 inches.

alenahelenash · Accepted Answer

Step 1:The height of the box will be 4 inches, as we are folding up the sides. Thus, the volume of the box can be calculated as the product of the base area and the height:V=base&amp;nbsp;area×heightSince the base is a square, the base area is given by:base&amp;nbsp;area=(side&amp;nbsp;length)2Substituting the given values, we have:100in3=(x−8)2×4Step 2:Now, let&#039;s solve the equation to find the value of x:100in34=(x−8)225in3=(x−8)2To remove the square, we can take the square root of both sides:25in3=(x−8)25in=x−8Adding 8 to both sides, we get:5in+8in=xThus, the length of each side of the original square piece of cardboard is x=13inches. Therefore, a square piece of cardboard with dimensions 13 inches by 13 inches is required.

A box with a square base and no top is

Answered question

Answer & Explanation

New Questions in Precalculus