# 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:

$a=60 \ \\ b=28 \ \\ V=(a-2c)(b-2c)c \ \\ V=(60-2c)(28-2c)c \ \\ V=4c^3-176c^2+1680c \ \\ V'=12c^2- 352c +1680 \ \\ V'=0 \ \\ 12c^2- 352c +1680=0 \ \\ \ \\ 12c^2- 352c +1680=0 \ \\ 12c^2 -352c +1680=0 \ \\ \ \\ p=12; q=-352; r=1680 \ \\ D=q^2 - 4pr=352^2 - 4\cdot 12 \cdot 1680=43264 \ \\ D>0 \ \\ \ \\ c_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 352 \pm \sqrt{ 43264 } }{ 24 } \ \\ c_{1,2}=\dfrac{ 352 \pm 208 }{ 24 } \ \\ c_{1,2}=14.66666667 \pm 8.66666666667 \ \\ c_{1}=23.3333333333 \ \\ c_{2}=6 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 12 (c -23.3333333333) (c -6)=0 \ \\ c=c_{1}=23.3333=6 \ \\ a_{1}=a-2 \cdot \ c_{1}=60-2 \cdot \ 23.3333 \doteq \dfrac{ 40 }{ 3 } \doteq 13.3333 \ \\ b_{1}=b-2 \cdot \ c_{1}=28-2 \cdot \ 23.3333 \doteq - \dfrac{ 56 }{ 3 } \doteq -18.6667 \ \\ V_{1}=a_{1} \cdot \ b_{1} \cdot \ c_{1}=13.3333 \cdot \ (-18.6667) \cdot \ 23.3333 \doteq -5807.4074 \ \\ a_{2}=a-2 \cdot \ c_{2}=60-2 \cdot \ 6=48 \ \\ b_{2}=b-2 \cdot \ c_{2}=28-2 \cdot \ 6=16 \ \\ V_{2}=a_{2} \cdot \ b_{2} \cdot \ c_{2}=48 \cdot \ 16 \cdot \ 6=4608 \ \\ V_{2} > V_{1} \ \\ c=c_{2}=6 \ \text{cm}$

Checkout calculation with our calculator of quadratic equations.

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...):

Be the first to comment!

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.

## Next similar math problems:

• Swimming pool
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
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
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.
• The determinant
The determinant of the unit matrix equals 7. Check how many rows the A matrix contains.
• Roots
Determine the quadratic equation absolute coefficient q, that the equation has a real double root and the root x calculate: ?
• Discriminant
Determine the discriminant of the equation: ?
Find the roots of the quadratic equation: 3x2-4x + (-4) = 0.
• Equation
Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
• Jar
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
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?
• Combinations
From how many elements we can create 990 combinations 2nd class without repeating?