Question

2. (18 pts) Consider an isosceles triangle that has its base on the x-axis. The apex of the triangle is at the point (O.y), where y > 0, and the remaining two vertices are at the points (x, 0) and (-x,0), where x > 0 (see figure). The area of the triangle is 120 in? This triangle lies inside a bigger isosceles triangle that also has its base on the x-axis. The apex of the bigger triangle is at the point (0.y + 3), and its two other vertices are at the points (x + 1.0) and (-x-1,0). Find the value of x which minimizes the area of the bigger triangle. Show your work and justify your answer. NOTE: Picture is not to scale. Solve the problem by performing the steps 1-4. STEP 1: Label the picture. Make sure to include all the relevant numbers and variables. STEP 2: Let A be the area of the bigger triangle. Find an expression for A, in terms of x and y.

Answer 1

€ 10,4+3) We are given two brangles, IsoSCELES A ABC and ADEF au 4+3 TRIANGLE y A_^ * Ama SABC = (Base) theight), (-x-1,0) (-xo) M 120 - $(2494) 120 = ay (given) (Domain of A) Ayera 4 Spef = {(EM) CDF) = £ 14+3)(22+27 Alany) = 14+3)(4+1) We need to Minimize Altry), Almry) = (x+1)(y+3) ; 4= Domain 44) Alx) = (x+1)(12+3) Alx) = 120+34 +130 +3 = 34+10+123 * A'in) = 3-622 tou CRITICAL POINTS ; A'(x)=0 3–120 = 0 => 3x2_120=0 => x²=40 => x== /40 > lc 2 10 ; x can't be negative 6'1 x) = -240 3 A(2.J10) - - 240 8(10) To Jio 2 [x=2170] is point of Local Minima. Since, it is only point , ... It is point of ABSOLUTE Minina