Paper box

Hard rectangular paper has dimensions of 60 cm and 28 cm. The corners are cut off equal squares and the residue was bent to form an open box. How long must be side of the squares to be the largest volume of the box?

Correct result:

c =  6 cm

Solution:

$$\begin{gathered} a=60 \\ b=28 \\ V=(a-2c)(b-2c)c \\ V=(60-2c)(28-2c)c \\ V=4c^3-176c^2+1680c \\ {V}^{\mathrm{\prime }}=12c^2-352c+1680 \\ {V}^{\mathrm{\prime }}=0 \\ 12c^2-352c+1680=0 \\ 12c^2-352c+1680=0 \\ 12c^2-352c+1680=0 \\ p=12;q=-352;r=1680 \\ D=q^2-4pr=352^2-4\cdot 12\cdot 1680=43264 \\ D>0 \\ {c}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{352\pm \sqrt{43264}}{24}} \\ {c}_{1,2}={\displaystyle \frac{352\pm 208}{24}} \\ {c}_{1,2}=14.66666667\pm 8.66666666667 \\ {c}_1=23.3333333333 \\ {c}_2=6 \\ \text{ Factored form of the equation: } \\ 12(c-23.3333333333)(c-6)=0 \\ c=c_1=23.3333=6 \\ {a}_1=a-2\cdot c_1=60-2\cdot 23.3333={\displaystyle \frac{40}{3}}\doteq 13.3333 \\ {b}_1=b-2\cdot c_1=28-2\cdot 23.3333=-{\displaystyle \frac{56}{3}}\doteq -18.6667 \\ {V}_1=a_1\cdot b_1\cdot c_1=13.3333\cdot (-18.6667)\cdot 23.3333\doteq -5807.4074 \\ {a}_2=a-2\cdot c_2=60-2\cdot 6=48 \\ {b}_2=b-2\cdot c_2=28-2\cdot 6=16 \\ {V}_2=a_2\cdot b_2\cdot c_2=48\cdot 16\cdot 6=4608 \\ {V}_2>V_1 \\ c=c_2=6=6\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

Try another example

Showing 0 comments:

Next similar math problems:

• Cylindrical container
  An open-topped cylindrical container has a volume of V = 3140 cm³. Find the cylinder dimensions (radius of base r, height v) so that the least material is needed to form the container.
• Sphere and cone
  Within the sphere of radius G = 33 cm inscribe cone with largest volume. What is that volume and what are the dimensions of the cone?
• Cone
  Into rotating cone with dimensions r = 8 cm and h = 8 cm incribe cylinder with maximum volume so that the cylinder axis is perpendicular to the axis of the cone. Determine the dimensions of the cylinder.
• Cardboard box
  Peter had square cardboard. The length of the pages was an integer in decimetres. He cut four squares with a side of 3 dm from the corners and made a box out of it, which fit exactly 108 cubes with an edge 1 dm long. Julia cut four squares with a side of
• Rectangle pool
  Determine dimensions of open pool with a square bottom with a capacity 32 m³ to have painted/bricked walls with least amount of material.
• Secret treasure
  Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
• Sphere in cone
  A sphere of radius 3 cm desribe cone with minimum volume. Determine cone dimensions.
• Max - cone
  From the iron bar (shape = prism) with dimensions 6.2 cm, 10 cm, 6.2 cm must be produced the greatest cone. a) Calculate cone volume. b) Calculate the waste.
• Alien ship
  The alien ship has the shape of a sphere with a radius of r = 3000m, and its crew needs the ship to carry the collected research material in a cuboid box with a square base. Determine the length of the base and (and height h) so that the box has the large
• Box
  Cardboard box shaped quadrangular prism with a rhombic base. Rhombus has a side 5 cm and one diagonal 8 cm long and height of the box is 12 cm. The box will open at the top. How many cm² of cardboard we need to cover overlap and joints that are 5% of are
• Cross-sections of a cone
  Cone with base radius 16 cm and height 11 cm divide by parallel planes to base into three bodies. The planes divide the height of the cone into three equal parts. Determine the volume ratio of the maximum and minimum of the resulting body.
• Ratio of edges
  The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid.
• Carpet
  The room is 10 x 5 meters. You have the role of carpet width of 1 meter. Make rectangular cut of roll that piece of carpet will be longest possible and it fit into the room. How long is a piece of carpet? Note .: carpet will not be parallel with the diago
• Aquarium
  The box-shaped aquarium is 40 cm high; the bottom has dimensions of 70 cm and 50 cm. Simon wanted to create an exciting environment for the fish, so he fixed three pillars to the bottom. They all have the shape of a cuboid with a square base. The base edg
• Minimum surface
  Find the length, breadth, and height of the cuboid shaped box with a minimum surface area, into which 50 cuboid shaped blocks, each with length, breadth and height equal to 4 cm, 3 cm and 2 cm respectively can be packed.
• Cardboard box
  We want to make a cardboard box shaped quadrangular prism with rhombic base. Rhombus has a side of 5 cm and 8 cm one diagonal long. The height of the box to be 12 cm. The box will be open at the top. How many square centimeters cardboard we need, if we ca
• Two rectangular boxes
  Two rectangular boxes with dimensions of 5 cm, 8 cm, 10 cm, and 5 cm, 12 cm, 1 dm are to be replaced by a single cube box of the same cubic volume. Calculate its surface.