a) In a right triangle ABC with ∠B equal to 90, find angle A and C so that sin(C) = cos(C). b) The lengths of side AB and side BC of a scalene triangle ABC are 10 cm and 8 cm respectively. The size of angle C is 60. Find the length of side AC.
a) Let a be the length of the side opposite angle A, b be the length of the side opposite angle B and c the length of the side opposite angle C. Since ∠B = 90 b^2 = a^2 + c^2 Approach I Since, sin(C) = cos(C) From trigonometric tables, this is possible only when C = 45 In a traingle A + B + C = 180 A + 90 + 45 = 180 A + 135 = 180 A = 45 Approach II sin(C) = c/b and cos(B) = a/b sin(C) = cos(C) means c/b = a/b which gives c = a The two sides are equal in length means that the triangle is isosceles and angles B and C are equal in size of 45. b) Let x be the length of side AC. Use the cosine law 10^2 = 8^2 + x^2 - 2*8*x*cos(60) Solve the quadratic equation for x: x = 11.21 and x = -3.21 x cannot be negative and therefore the solution is x = 11.2 (rounded to one decimal place).
a) Suppose that f(x) is a function with f(100)=54 and f'(100)=14 . Estimate f(101.5). b) Integrate the following: ∫ x^2*sin(x^3) dx
a) The question is an application of Tangent Line Approximation. L(x) = f(a) + f'(a)(x − a) = f(101.5) = f(100) + f'(100)*(100-101.5) = f(100) + f(100)*1.5 = 54 + 14*1.5 = 54 + 21 = 75 b) Let I = ∫ x^2*sin(x^3) dx and t = x^3, => dt = 3x^2*dx, => x^2 dx = dt/3 Substituting, we get ∫ x^2*sin(x^3) dx = ∫ (sin(x^2))*(x^2 dx) = ∫ sin(t)*dt/3 = -(1/3)*cost(t) + constant Substituting back = -(1/3)*cost(x^3) + constant
In the face of disappointing earnings results and increasingly assertive institutional stockholders, Eastman Kodak was considering the sale of its health division which had an EBIT of $560 million in its most recent year from revenues of $528.5 billion. The firm’s expected growth rate is expected to be 6% for the next 5 years and fall to 2% thereafter. Capital expenditure in the health division was $420 million last year and depreciation was $350 million. Both are expected to grow at 4% in the long run. Working capital requirements are 10% of sales. The average equity beta of companies competing with Eastman Kodak’s health division was 1.15. Eastman Kodak’s health division has a debt ratio of .20 and a debt cost of 7.5%. The company has a tax rate of .40 and the T-Bond rate is 7%. and Market risk premium is 5.5%. Estimate the division’s WACC.
We need to focus on what is asked in the question. Most of the information is given just to intimidate the student. Only a few parameters among this problem are relevant. WACC = Ke(E/E+D) + Kd(1 - t)*(D/E+D) Ke = cost of equity Kd = cost of debt D/E+D = debt as a % of total Value E/E+D = equity as a % of total Value t = tax rate (marginal) From the above question, sort out the variables mentioned in the question and variables which are missing, Ke =? Kd = 7.5% D/E+D = 0.2 E/E+D = 1 - D/E+D = 1- 0.2 = 0.8 t = 0.4 So our equation becomes: = Ke*0.8 + 7.5%*(1 - 0.4)*0.2 = Ke*0.8 + 7.5%*0.6*0.2 = Ke*0.8 + 0.9% We have to find Ke. For finding Ke First, we need to calculate expected a return on equity. Using CAPM model: E = Rf + β(Rm-Rf); where Rf = risk-free rate of return, (or the T-Bond rate) β = Beta (systematic risk measure) Rm = Expected return on market (Rm-Rf) = Market Risk Premium Rf = 7% β = 1.15 (Rm - Rf) = 5.5% plugging in E = 7% + 1.15*5.5% = 13.325% Use this value as cost of equity WACC = Ke*0.8 + 0.9% = 13.325*0.8 + 0.9% = 11.56 % Hence WACC = 11.56 %