Question

An open-top bin is to be made from a 15-centimeter by 40-centimeter piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What size square should be cut out of each corner to get a bin with the maximum volume?

Answer 1

Answer:

The size of the square is  $\frac{10}{3}$ cm by  

Step-by-step explanation:

To find the side of the square which removing from each corner of the plastic and makes the volume of the bin maximum assume that the side of the square is x and write the formula of the volume in terms of x, then differentiate it and equate the differentiation by 0 to find the value of x

∵ The dimensions of the piece of plastic are 15 cm and 50 cm

∵ The side of the square which removed from each corner is x cm

- That means each dimension will be less by 2x (x from right and

    x from left)

∴ The dimensions of the base of the bin are (15 - 2x) , (40 - 2x)

∴ The height of the bin is x

The formula of the volume of the bin is V = l × w × h

∵ The volume of the bin V = (15 - 2x)(40 - 2x)(x)

- Multiply the two brackets at first

∵ (15 - 2x)(40 - 2x) = (15)(40) + (15)(-2x) + (-2x)(40) + (-2x)(-2x)

∴ (15 - 2x)(40 - 2x) = 600 + (-30x) + (-80x) + 4x²

- Add the like terms

∴ (15 - 2x)(40 - 2x) = 600 - 110x + 4x²

- Substitute it in the formula of the volume

∴ V = (x)(600 - 110x + 4x²)

- Multiply the bracket by x

∴ V = 600x - 110x² + 4x³

Now differentiate V with respect to x

∵ V' = 600 - 220x + 12x²

- Equate the differentiation by 0

∴ 600 - 220x + 12x² = 0

- Re-arrange the terms of the equation from greatest power of x

∴ 12x² - 220x + 600 = 0

- Use your calculator to solve the quadratic equation and give

    you the values of x

∴ x = 15  OR x = $\frac{10}{3}$

One of them will give the maximum volume and the other will give the minimum volume to find that fin V" and substitute the values of x on it if the answer is negative then the volume is maximum if the answer is positive, then the volume is minimum

∵ V' = 600 - 220x + 12x²

∴ V" = -220 + 24x

- Substitute x by 15

∴ V" = -220 + 24(15) = 140 (positive value)

∴ x = 15 gives the minimum volume

- Substitute x by $\frac{10}{3}$

∴ V" = -220 + 24( $\frac{10}{3}$ ) = -140 (negative value)

∴ x =  $\frac{10}{3}$ gives the maximum volume

∴ The side of the square is $\frac{10}{3}$ cm

The size of the square is  $\frac{10}{3}$ cm by  $\frac{10}{3}$ cm