Answer:
The size of the square is [tex]\frac{10}{3}[/tex] 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 = [tex]\frac{10}{3}[/tex]
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 [tex]\frac{10}{3}[/tex]
∴ V" = -220 + 24( [tex]\frac{10}{3}[/tex] ) = -140 (negative value)
∴ x = [tex]\frac{10}{3}[/tex] gives the maximum volume
∴ The side of the square is [tex]\frac{10}{3}[/tex] cm
The size of the square is [tex]\frac{10}{3}[/tex] cm by [tex]\frac{10}{3}[/tex] cm
