A container consists of a right cylindrical part and a hemispherical part at the top, as shown in the picture and diagram below - NSC Technical Mathematics - Question 8 - 2019 - Paper 1

The total volume $V$ consists of two parts: the volume of the cylindrical part and the volume of the hemispherical part.

1. Volume of the Cylinder: Volume of the cylinder = $ext{π}x^2h = ext{π}x^2(66 - 2x)$

2. Volume of the Hemisphere: Volume of the hemisphere = rac{1}{2} imes rac{4}{3} ext{π}x^3 = rac{2}{3} ext{π}x^3

3. Total Volume: V = ext{π}x^2(66 - 2x) + rac{2}{3} ext{π}x^3 Expanding this: V = 66 ext{π}x^2 - 2 ext{π}x^3 + rac{2}{3} ext{π}x^3 After combining the terms: V = 66 ext{π}x^2 - rac{7}{3} ext{π}x^3

To find the maximum volume, we need to differentiate the volume function:

V'(x) = rac{dV}{dx} = 66 ext{π}(2x) - rac{7}{3} ext{π}(3x^2)

Setting the derivative to zero to find critical points: $0 = 132 ext{π}x - 7 ext{π}x^2$

Factoring out $ext{π}x$ gives: $0 = ext{π}x(132 - 7x)$

Thus, we have two solutions:

1. $x = 0$ (not valid as it's a container)
2. x = rac{132}{7} ext{ cm}

Substituting x = rac{132}{7} into the volume formula:

Vigg(rac{132}{7}igg) = 66 ext{π}igg(rac{132}{7}igg)^2 - rac{7}{3} ext{π}igg(rac{132}{7}igg)^3

Calculating each term: 66 ext{π}igg(rac{132}{7}igg)^2 = rac{66 ext{π} imes 17424}{49} = rac{1140240}{49}

For the second term: -rac{7}{3} ext{π}igg(rac{132}{7}igg)^3 = -rac{7 ext{π} imes 132^3}{3 imes 7^3}

After completing these calculations, we get the maximum volume. The maximum volume can be expressed as: $V_{ ext{max}} ext{ (approximately)} = 24,576.74 ext{ cm}^3 ext{ or } 7823.02 ext{π cm}^3$