# Largest parabolic section from right circular cone

**Situation**

A right circular cone has a base diameter of 24 cm. The maximum area of parabolic segment that can be cut from this cone is 207.8 cm^{2}.

Part 1: Determine the base width of the parabola.

A. 22.32 cm

B. 18.54 cm

C. 15.63 cm

D. 20.78 cm

Part 2: Determine the altitude of the parabola.

A. 14 cm

B. 18 cm

C. 15 cm

D. 16 cm

Part 2: Determine the altitude of the cone.

A. 20 cm

B. 14 cm

C. 16 cm

D. 18 cm

**Solution**

$A_{max} = \frac{2}{3}(2b)h$

$207.8 = \frac{4}{3}bh$

$h = \dfrac{155.85}{b}$ → Equation (1)

Triangle DEC is similar to triangle ABC

$\dfrac{h}{24 - x} = \dfrac{L}{24}$

$h = \frac{1}{24}L(24 - x)$ → Equation (2)

For the right triangle at the base of the cone

$b^2 + (12 - x)^2 = 12^2$

$b^2 = 12^2 - (12 - x)^2$

$b^2 = 12^2 - (12^2 - 24x + x^2)$

$b = \sqrt{24x - x^2}$ → Equation (3)

Area of parabolic section at any distance x

$A = \frac{2}{3}(2b)h$

$A = \frac{4}{3}\sqrt{24x - x^2}[ \, \frac{1}{24}L(24 - x) \, ]$

$A = \frac{1}{18}L(24 - x)\sqrt{24x - x^2}$

$\dfrac{\partial A}{\partial x} = \frac{1}{18}L \left[ (24 - x) \dfrac{24 - 2x}{2\sqrt{24x - x^2}} + \sqrt{24x - x^2} ~ (-1) \right] = 0$

$\dfrac{2(12 - x)(24 - x)}{2\sqrt{24x - x^2}} - \sqrt{24x - x^2} = 0$

$\dfrac{(12 - x)(24 - x)}{\sqrt{24x - x^2}} = \sqrt{24x - x^2}$

$(12 - x)(24 - x) = 24x - x^2$

$288 - 36x + x^2 = 24x - x^2$

$2x^2 - 60x + 288 = 0$

$x^2 - 30x + 144 = 0$

$x = 24 \, \text{ and } \, 6$

Use

$x = 6 \, \text{ cm}$

Thus,

$b = \sqrt{24x - x^2}$ → From Equation (3)

$b = \sqrt{24(6) - 6^2}$

$b = 6\sqrt{3} \, \text{ cm}$

Base-width of parabolic section

$2b = 12\sqrt{3}$

$2b = 20.78 \, \text{ cm}$ Part 1: [ D ]

Altitude of parabolic section

$h = \dfrac{155.85}{b}$ → From Equation (1)

$h = \dfrac{155.85}{6\sqrt{3}}$

$h = 15 \, \text{ cm}$ Part 2: [ C ]

From

$h = \frac{1}{24}L(24 - x)$ → From Equation (2)

$15 = \frac{1}{24}L(24 - 6)$

$L = 20 \, \text{ cm}$

By Pythagorean theorem

$H^2 + 12^2 = L^2$

$H^2 + 12^2 = 20^2$

$H = 16 \, \text{ cm}$ Part 3: [ C ]

**Note:**

For maximum area of parabola that can be cut from a right circular cone:

$h = 0.75L$

$x = 0.25D$

**Solving the problem using the above formulas**

$b = 24 \sin 60^\circ$

$b = 20.78 \, \text{ cm}$ *answer*

Altitude of parabola

$A_{max} = \frac{2}{3}bh$

$207.8 = \frac{2}{3}(20.78)h$

$h = 15 \, \text{ cm}$ *answer*

Lateral height of cone

$h = 0.75L$

$15 = 0.75L$

$L = 20 \, \text{ cm}$

Altitude of cone

$H^2 + 12^2 = L^2$

$H^2 + 12^2 = 20^2$

$H = 16 \, \text{ cm}$ *answer*