Differentiation - max min Type 3 TYS 1

The questions on this page address the following situations:

Boxes	1. A block of wood is in the shape of a square-based prism. The sum of the height of the block and the perimeter of the base is 20 cm. (i) Show that the volume of the block can be expressed as V = 20x² - 4x³. (ii) What is the volume of the largest block (nearest cm³)? Answer.Max. volume = 593 cm³.
	2. A rectangular storage unit, closed on all sides except for the front, is to be constructed with a shelf as shown. Its dimensions are 2x m wide, y m deep and x m high. It is to be built with 24 m² of pineboard (ignore the thickness of the pineboard for the purposes of this question). (i) Show that 2x² + 8xy = 24. (ii) Develop an equation expressing the volume of the box in terms of x. (iii) Determine the maximum volume of the storage unit. Answer.Max. volume = 21.33m³.
	3. An open tank is to be constructed with a square base of side x metres and with four rectangular sides. The tank is to have a capacity of 32 m³. (i) Show that the height of the tank can be expressed as m. (ii) Find the least area of sheet metal from which the tank can be constructed (answer to nearest m²). Answer.Least area = 23 m².
	4. A trough of depth h metres and length 1 metre was constructed out of stainless steel sheeting. The cross-section of the trough is an isosceles trapezium with the acute angles each being 45^o. The width of the base of the trough is a metres. The area of the cross-section measures 60 m². (i) Show that . (ii) Show that the amount of stainless steel, A m², required to construct the trough is expressed by . (iii) Find the depth of the trough, to the nearest cm, if the amount of stainless steel used was a minimum. Answer.Height must be 5.73 m.
	5. A piece of paper in the shape of an equilateral triangle with an edge length of 20 cm is to be used to make an open box with no cover on the top. Quadrilateral shapes are to be cut out of the corners in the manner shown. The height of the box is h cm. (i) Show that the side of the equilateral triangular base is . (ii) Prove that the volume of the box is . (iii) Find the height of the box that will produce the maximum volume. Answer.Height = m.
Cones.	6. The sum of the radius and the slant height of a cone is 20 cm. Find the length of the radius if the volume of the cone is a maximum. Answer.For max. volume, r = 1.6 cm.
	7. A cone has a base with radius r cm and a height h cm. Its slant height is 6 cm. (i) Show that its volume can be written as . (ii) Hence find the exact value for the height h if the cone is to have maximum volume. Answer.For max. volume, h = 2√3 cm.
	8. A right circular cone of radius r and height h has a total surface area S and volume V. We know that and that . (i) Show that 9V² = r²(S² - 2π r²S). (ii) Given this cone has a fixed surface area S, find . (iii) Hence find the semi-vertical angle θ that gives the maximum volume of the cone for a fixed surface area. Express your angle to the nearest minute.
	9.
Cylinders.	10. The sum of the diameter and the height of a closed cylinder is 40 cm. Find the length of the radius and the height if the volume of the cylinder is a maximum. Answer. r = h = 40/3 cm.
	11. Find the dimensions of a closed cylindrical can of greatest volume that can be made from 150π cm² of sheet aluminium. Answer. r = 5 cm, height = 10 cm.
	12. The sum of the height (h cm) of a cylinder and the circumference of its base with radius (r cm) is 10 cm. (i) Show that h = 10 - 2π r. (ii) Show that the volume of the cylinder can be expressed as V = π r² (10 - 2πr). (iii) Find the exact value of r at which the volume of the cylinder is a maximum. (iv) Hence find the maximum volume of the cylinder (to the nearest cm²). Answer.Max vol = 148 cm³.
	13. A closed cylindrical can is to have a volume of 16π cm³. What should be the radius and height of the can if the surface area of the can is to be a minimum? Answer. r = 2 and h = 4 cm.
	14. A cylindrical container, closed at both ends, is made from thin sheet metal. The container is to have a radius of r cm and a height of h cm. The volume of the container is to be 1,000 cm³. (i) Show that the area of sheet metal required to make the container can be expressed as cm³. (ii) Hence find the minimum area of sheet metal required to make the container (to the nearest cm²). Answer.Min. area = 1,187 cm².