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:

a=60 b=28 V=(a−2c)(b−2c)c V=(60−2c)(28−2c)c V=4c3−176c2+1680c V′=12c2−352c+1680 V′=0 12c2−352c+1680=0  12c2−352c+1680=0 12c2−352c+1680=0  p=12;q=−352;r=1680 D=q2−4pr=3522−4⋅12⋅1680=43264 D>0  c1,2=−q±D2p=352±4326424 c1,2=352±20824 c1,2=14.66666667±8.6666666666667 c1=23.333333333333 c2=6   Factored form of the equation:  12(c−23.333333333333)(c−6)=0 c=c1=23.3333=6 a1=a−2⋅ c1=60−2⋅ 23.3333≐403≐13.3333 b1=b−2⋅ c1=28−2⋅ 23.3333≐−563≐−18.6667 V1=a1⋅ b1⋅ c1=13.3333⋅ (−18.6667)⋅ 23.3333≐−5807.4074 a2=a−2⋅ c2=60−2⋅ 6=48 b2=b−2⋅ c2=28−2⋅ 6=16 V2=a2⋅ b2⋅ c2=48⋅ 16⋅ 6=4608 V2>V1 c=c2=6 cma=60 \ \\ b=28 \ \\ V=(a-2c)(b-2c)c \ \\ V=(60-2c)(28-2c)c \ \\ V=4c^3-176c^2+1680c \ \\ V'=12c^2- 352c +1680 \ \\ V'=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}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 352 \pm \sqrt{ 43264 } }{ 24 } \ \\ c_{1,2}=\dfrac{ 352 \pm 208 }{ 24 } \ \\ c_{1,2}=14.66666667 \pm 8.6666666666667 \ \\ c_{1}=23.333333333333 \ \\ c_{2}=6 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 12 (c -23.333333333333) (c -6)=0 \ \\ c=c_{1}=23.3333=6 \ \\ a_{1}=a-2 \cdot \ c_{1}=60-2 \cdot \ 23.3333 \doteq \dfrac{ 40 }{ 3 } \doteq 13.3333 \ \\ b_{1}=b-2 \cdot \ c_{1}=28-2 \cdot \ 23.3333 \doteq - \dfrac{ 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 \ \text{cm}

Checkout calculation with our calculator of quadratic equations.


Try another example


Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!


Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?
Tip: Our volume units converter will help you with the conversion of volume units.

You need to know the following knowledge to solve this word math problem:

Next similar math problems:

  • Swimming pool
    basen The pool shape of cuboid is 299 m3 full of water. Determine the dimensions of its bottom if water depth is 282 cm and one bottom dimension is 4.7 m greater than the second.
  • Cylindrical container
    valec2_6 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.
  • Cone
    diag22 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.
  • Inverse matrix
    matrix_3 Find how many times is the larger determinant is the matrix A, which equals 9 as the determinant of its inverse matrix.
  • The determinant
    matrix_13 The determinant of the unit matrix equals 7. Check how many rows the A matrix contains.
  • Discriminant
    Quadratic_equation_discriminant Determine the discriminant of the equation: ?
  • Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  • Quadratic equation
    kvadrat_2 Find the roots of the quadratic equation: 3x2-4x + (-4) = 0.
  • Roots
    parabola Determine the quadratic equation absolute coefficient q, that the equation has a real double root and the root x calculate: ?
  • Jar
    sklenice From the cylinder shaped jar after tilting spilled water so that the bottom of the jar reaches the water level accurately into half of the base. Height of jar h = 7 cm and a jar diameter D is 12 cm. How to calculate how much water remains in the jar?
  • Curve and line
    parabol The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3 (1) Find the x co-ordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?
  • Cubes - diff
    cube_2 Second cubes edge is 2 cm longer than the edge of the first cube. Volume difference blocks is 728 cm3. Calculate the sizes of the edges of the two dice.
  • Combinations
    trezor_1 How many elements can form six times more combinations fourth class than combination of the second class?
  • Combinations
    math_2 From how many elements we can create 990 combinations 2nd class without repeating?
  • Solve 3
    eq2_4 Solve quadratic equation: (6n+1) (4n-1) = 3n2
  • Quadratic function 2
    parabola1 Which of the points belong function f:y= 2x2- 3x + 1 : A(-2, 15) B (3,10) C (1,4)
  • Gasoline canisters
    fuel_4 35 liters of gasoline is to be divided into 4 canisters so that in the third canister will have 5 liters less than in the first canister, the fourth canister 10 liters more than the third canister and the second canister half of that in the first canis