Question

Question 3 Consider the triangle constructed from the points () = (0,0), Q = (2,0) and P = (L cos 0, L sin (). (a) Write a function for the area of the triangle in terms of 0 and L, namely A(0, L). 3 (b) Show that if the triangle has a perimeter of 6 then L = 2 - cose (c) Sketch the domain of A on a OL-axes to ensure that the following conditions hold • L represent the length of the vector OP. • The point P is in the first quadrant. • The perimeter of the triangle is at most 6. Hint: use a computer to draw the curve corresponding to the perimeter. (d) By increasing L or we will increase the area of the triangle A(0,L). If we are only able to make an arbitrarily small increase L or 0 (but not both) and our goal is to obtain a larger area for the triangle, is it more efficient to increase L or 6 for following triangles: (i) (0, 1) = (6.1). (ii) (0, L) = 4' 2 (e) Find a relationship between L and 0 describing the times when it is equally efficient to increase both L or 6 to increase A(0, L). Sketch this on your figure from part (c) and indicate all (0, L) coordinates where it is more efficient to increase each variable.

Answer 1

Solution d3 (a) The given Vertices are O= coo>, Q=42,0), P =< LESO LSul> The ares of ди a is given by A(0,2)= 09 0 - as (AOOP) = 2 2 ) Los Lsind I o 2 -o-o 0-0 +1(21.simo) 2 LEGO Lont I 2 ILsino or A(0, 2) = 7L sino x (6) Using distance formuller we obtain the sides of as, - Og = √(2-0)²+(0-0)² 122=2 deLcOS 0 -2) + (Lsme)? = 122col +9 -4 Lloso+ OP L2cos a + L²sull +4-4 Leosto L² +4-4 Lloso.

OP= N(Leos 0-0) + (Lsindo) ² 1210520 +L? Sinc - Se CC05204520) √2² こ EL Now Perimeter is given to be 6 so, OQ+QP top 6 or 2 + √22 +4 4Leosa + L = 6 L²+4 -4 Loose = 6-2-L J 22 +4 4 Llosa 4-L Squaring 6/8 we get L² +4-4 Lloso = (4+2) = 16+1²_8L 4+4-4lcoso 16+/ 8L OL-4L coso 116-4 4L(2-coso) =12 } } } 23 3 L 4(2-coso) 2- Cosa

(C) from (a) ACO, L) = L sind .: Arer ,A(0, L) > 0 L sind > 0 DE (OT) – I Quadrano => Sind 0 of Perimeter is at most 6 Then L at most is 2-osa L sama- 2 0 10 L to arer g (& for CO, L) We increase get the maxima Value for (Q, 2) = ( 17 ) es to get to get maxima aver Lsina, we onerese o so, dA Lloso [ A da & 이다 Sind So, noresse a Then onerase 니