# 15 - 17 Box open at the top in maxima and minima

**Problem 15**

A box is to be made of a piece of cardboard 9 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made in this way.

**Solution:**

$V = (9 - 2x)^2 \, x$

$V = 81x - 36x^2 + 4x^3$

$\dfrac{dV}{dx} = 81 - 72x + 12x^2 = 0$

$4x^2 - 24x + 27 = 0$

Using quadratic formula

$a = 4; \,\, b = -24; \,\, c = 27$

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \dfrac{-(-24) \pm \sqrt{(-24)^2 - 4(4)(27)}}{2(4)}$

$x = \dfrac{24 \pm 12}{8}$

$x = 4.5 \, \text{ and } \, 1.5$

Use x = 1.5 inches

Maximum volume:

$V = [ \, 9 - 2(1.5) \, ]^2 \, (1.5)$

$V = 54 \, \text{ in}^2$ *answer*

**Problem 16**

Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides.

**Solution:**

$V = (24 - 2x)(15 - 2x) x$

$V = 360x - 78x^2 + 4x^3$

$\dfrac{dV}{dx} = 360 - 156x + 12x^2 = 0$

$x^2 - 13x + 30 = 0$

$(x - 10)(x - 3) = 0$

$x = 10 \, \text{ (meaningless) and } 3$

$V_{max} = [ \, 24 - 2(3) \, ] \, [ \, 15 - 2(3) \, ] \, 3$

$V_{max} = 486 \, \text{ in}^3$ *answer*

**Problem 17**

Find the depth of the largest box that can be made by cutting equal squares of side x out of the corners of a piece of cardboard of dimensions 6a, 6b, (b ≤ a), and then turning up the sides. To select that value of x which yields a maximum volume, show that

**Solution:**

$V = (6a - 2x)(6b - 2x)x$

$V = 36abx - 12(a + b)x^2 + 4x^3$

$dV / dx = 36ab - 24(a + b)x + 12x^2 = 0$

$x^2 - 2(a + b)x + 3ab = 0$

$A = 1; \,\, B = -2(a + b); \,\, C = 3ab$

$x = \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

$x = \dfrac{2(a + b) \pm \sqrt{4(a + b)^2 - 4(1)(3ab)}}{2(1)}$

$x = \dfrac{2(a + b) \pm 2\sqrt{(a^2 + 2ab + b^2) - 3ab}}{2}$

$x = (a + b) + \sqrt{a^2 - ab + b^2} \,\, $ and

$x = (a + b) - \sqrt{a^2 - ab + b^2}$

If a = b:

$x = (a + b) + \sqrt{a^2 - ab + b^2}$

$x = (b + b) + \sqrt{b^2 - b^2 + b^2}$

$x = 3b$ (x is equal to ½ of 6b - meaningless)

From

$x = (a + b) - \sqrt{a^2 - ab + b^2}$

$x = (b + b) - \sqrt{b^2 - b^2 + b^2}$

$x = b$ *okay*

Use $x = a + b - \sqrt{a^2 - ab + b^2}$ *answer*