Differentiation - max min Type 2 TYS 2

The questions on this page focus on situations:

2. addressing areas including paddocks, etc

4. measuring the distance between two curves.

Creating 2 shapes.	1. Two circles with radii x and y are such that the sum of their radii has a constant value R. (i) Show that the sum of the areas of the two circles may be expressed as S = πx² + π(R - x)². (ii) Show that the sum of their areas will be least when the radii are equal. (iii) Find the least sum of their areas. Answer.Min area = (πR²)/2.
	2. A wire of length 24 cm is cut into two pieces. One piece is then bent to form a square with sides a cm. The other piece is bent to form an equilateral triangle with each side of length b cm. Let A_S be the area of the square and A_T be the area of the triangle. (i) Show that . (ii) Show that the total area A = A_S + A_T = . (iii) Hence show that the total area A is at its minimum when .
Areas and paddocks.	3. In the diagram below, P is a point on the curve y = x(6 - x) and Q is a point on the curve y = x(x - 4). PQ cuts the x axis at right angles at R. (i) Show that the length of PQ is given by 10x - 2x². (ii) Find an expression for the area of PQST as a function of x. (iii) Find the value of x which gives the maximum area for PQST (0 ≤ x ≤ 5). Answer.Max area = 37.04 u².
	4. A flag has two vertical green stripes at the ends and a vertical yellow stripe in the centre. The design and measurements are shown below: The perimeter of the flag is 376 cm. The total green area is 6567 cm². (i) Show that a + b = 188 - x - y. (ii) Show that the width of the yellow band is y = 188 - x - 6567x^-1. (iii) Find the dimensions of the flag if the vertical width is a maximum. Answer.Dimensions are 81.04 × 106.96.
	5. Robin has designed a garden bed which consists of a rectangle and a semicircle as shown in the diagram. The perimeter of Robin's garden is 20 m. (i) Find an expression for h in terms of r. (ii) Show that the area of the garden bed can be given by (iii) Find the value of r that gives the maximum area of the garden and hence calculate this area to the nearest square metre. Answer.(i) h = 10 - r - πr/2 (iii) r = 20/(4 + π) = 2.8. Area = 28 m².
	6. The diagram shows a triangular piece of land ABC with dimensions BC = a metres, AC = b metres and AB = c metres (where a ≤ b ≤ c). The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let S and T be points on AB and AC respectively so that ST divides the land as intended. Let AS = x metres, AT = y metres and ST = z metres. (i) Show that . (ii) Use the cosine rule in triangle AST to show that (iii) Show that the value of z² in the equation in part (ii) is a minimum when . (iv) Show that the minimum length of the fence ST in metres can be expressed as where P = a + b + c (assume that the value of x determined in part (iii) is possible).
	7. In the diagram below, the point is a variable point on the curve . The points S (-5, 2) and T (3, -4) lie on the straight line 3x + 4y + 7 = 0. (i) Show that the area of the triangle RST is . (ii) Find the value of a which will produce the triangle on minimum area. Express your answer correct to one decimal place. (iii) Hence find the minimum area of triangle RST.
	8. In the diagram below, the curve (where h and k are constants) has a minimum turning point at (0, -6) and it passes through the point (5, -1). A rectangle PQRS is inscribed within the curve as shown with its axis of symmetry at x = 0. (i) Find the values of h and k. (ii) If Q has coordinates (β, 0), find the coordinates of R in terms of β. (iii) Show that the area A of the rectangle PQRS is . (iv) Hence show that the maximum area of the rectangle PQRS is units².
Geometric shapes.	9. The diagram above represents the dimensions (in metres) of a small garden. (i) Show that . (ii) Develop an expression in terms of x for the perimeter of the garden P. (iii) Find the minimum perimeter P of the garden (nearest centimetre). Answer.The minimum perimeter is 14.47 m.
	10. In the diagram below, AB and CD are parallel railings which are 8 m apart. The points C and D are 10 m apart on the lower railing. Two crossbars AD and BC intersect at T as shown. The line through T perpendicular to AB intersects AB at G and CD at H. The length of TH is y metres. (i) By using similar triangles or otherwise, show that (ii) Show that (iii) Hence show that the total area of ATB and CTD is given by (iv) Find the value of y that minimises the total area. Justify your answer. Answer.For min area, y = 8 cm.
	11. The parabola y = 4 - x² cuts the x-axis at R and S. The point P (x, y) lies on the parabola in the first quadrant. Q also lies on the parabola such that PQ is parallel to the x-axis. (i) Write down the coordinates of R and S. (ii) Show that the area of the trapezium PQRS can be expressed as A = 8 + 4x - 2x² - x³ (iii) Hence find the value of x which gives a maximum value of A. Justify your answer.
	12. A circular window of radius√5 m requires three metal strips AB, DC and FG for reinforcement as shown in the diagram below. O is the centre of the window. OF = OG = y metres and FB = FA = CG = GD = x metres. (i) Given that L is the total length of the metal strips (so L = AB + CD + FG), show that (ii) The window will have maximum strength when the total length L of the window is a maximum. Find the value of x for which the window has maximum strength. Answer.Max L when x = 2 m.
	13. An isosceles trapezium is inscribed in a semi-circle of radius R. The longer parallel side of the trapezium forms the diameter of the semi-circle. Show that the length of the shorter parallel side of the trapezium equals the radius of the semi-circle when the area of the trapezium is a maximum. (You can assume the concavity is negative).
	14. The cross-section of a metal cam is shown in the diagram below. The cross-section consists of a semi-circle AXC centered at O with radius r cm and a sector ABC of radius 2r cm centered at A with angle θ. (i) What is the perimeter of the metal cam in terms of r and θ? (ii) If the area of the cross-section is fixed at 4 cm², show that the perimeter in part (i) can be expressed as (iii) Show that the perimeter in part (i) is smallest when . Answer.P = 2r + 2rθ + πr.
Distance between 2 curves.	15. P is a point on the curve y = x(6 - x) and Q is a point on the curve y = x(x - 4). (i) Sketch these two curves and mark the points P and Q which have the same x value. (ii) Show that the length PQ can be expressed as PQ = L = 10x - 2x² (iii) Hence find the minimum length of PQ for 0 ≤ x ≤ 5. Answer.Min length is 12.5 u.
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