7. OR. If you prefer, calculate Proj_CS(A)b using the approach of Example 8 on page 266 instead. You should get an infinite number of solutions for x to the normal equation. Any solution for x will work, and then calculate Ax to get Proj_CS(A)
   • For either approach, briefly explain why Proj_CS(A)b is the closest vector in your plane to the vector (See Theorem 5.15 if needed.)
8. Choose 5 arbitrary unique vectors in R² that are not on the same line or scalar multiples of each other. What are your 5 vectors in R²?
   • Use normal equations (see page 265, if needed) to determine the equation of the least square regression line for this set of 5 points.
9. Choose any 2 of the 5 vectors in the above problem that will form a nonstandard basis for R², which we will call B’. (12 points for entire problem)
   • What are your two vectors in B’?
   • If B is the standard basis for R² (I.e. B ={(1.0). (0.1)}, determine the transition matrix (called P^-1 in our text) from B to B’.
   • Use P^-1 to calculate the coordinates for (1,5) with respect to the basis B’.
10. Investigate the following linear differential equation: y’’ + 4y = 0; Solutions {sin(2x), cos(2x)}

• Verify that each solution satisfies the differential equation.
• Use the Wronskian to verify that the solution set is linearly independent.
• Write the general solution of the differential equation.

Suppose you have an electric hot water heater for your house which is an aluminum cylinder which has a 0.56 m radius and is 2 m high. The walls are 1.0 cm thick. The thermal conductivity of Aluminum is 217 W/(m K). Assume that the temperature of the hot water inside the hot water heater is kept at a constant 90 C, and the external temperature is 27 C.

Part A:

What is the surface area of the cylinder?

Part B:

How much energy is lost through the walls of the hot water heater in one week? (Assume thinner surface of the heater is 90 C and the outer surface is 27 C.)

Part C:

Assume you pay $0.10 per kW-hour for electricity. How much would it cost just to keep the hot water inside the heater for one week?

Part D:

Suppose that you wrap the hot water heater on all sides with a 10 cm thick blanket of fiberglass insulation which has a thermal conductivity of 0.04 W/(m K). Assume the inner surface of the fiberglass insulation is at 90 C and the outer surface is at 27 C, and the total surface area is still what you calculated in part A.

Part E:

Assume you pay $0.10 per kW-hour for electricity. How much would it cost just to keep the hot water inside the fiberglass-wrapped heater for one week?

A wing-body model is tested in a wind tunnel with a flow of 100 m/s at standard sea-level conditions.

The wing area is 1.5 m^2 and the mean aerodynamic chord length is 0.45 m. Measurements of lift force, L, and moment about the center of gravity, MCG are made using the wind tunnel force balance.

The lead mass is removed and a horizontal tail without an elevator is added to the model. The distance from the airplane center of gravity to the aerodynamic center of the tail is 1.0 m.

The area of the tail is 0.4 m2 and the tail-setting angle is -2.0°. The tail has a lift-curve slope of 0.12 per degree. Experimental measurements give the downwash in terms of angle of attack, ε = α 0.42 , i.e. the downwash is zero for zero angle of attack, 0
ε = 0

For the case where the angle of attack, α = 5.0°, and the lift, L= 4134 N:

(i) Calculate MCG
(ii) Determine whether the model possesses longitudinal static stability. Calculate Mo
(iii) Calculate the location of the neutral point and the static margin if h = 0.26.
(iv) An elevator is added to the horizontal tail with the property / 0.04 L E T
∂ ∂ = C δ per degree.

In two ways calculate the elevator deflection angle required to trim the aircraft at an angle of attack of 8.0°. Answer should be the same.

Target Corporation prepares its financial statements according to U.S. GAAP. Target’s financial statements and disclosure notes for the year ended February 3, 2018, are available here (https://corporate.target.com/annual-reports/2017/10-K/10-k-cover). This material is also available under the Investor Relations link at the company’s website (www.target.com).

Required:

1. By what name does Target label its balance sheet?

2. What amounts did Target report for the following items on February 3, 2018?

3. What was Target’s largest current asset? What was its largest current liability?

4. Compute Target’s current ratio and debt to equity ratio in 2018? (Round your answers to 2 decimal places. Enter your answers in millions, not in dollars (i.e., 10,00,000 should be entered as 10).)

5. Assuming Target’s industry had an average current ratio of 1.0 and an average debt to equity ratio of 2.5, comment on Target’s liquidity and long-term solvency.

1. A new car is tested on a 290-m-diameter track. If the car speeds up at a steady 1.2 m/s2 , how long after starting is the magnitude of its centripetal acceleration equal to the tangential acceleration? Express your answer to two significant figures and include the appropriate units.

2. A concrete highway curve of radius 80.0 m is banked at a 15.0 ∘ angle.

Part A: What is the maximum speed with which a 1300 kg rubber-tired car can take this curve without sliding? (Take the static coefficient of friction of rubber on concrete to be 1.0.) Express your answer with the appropriate units.

3. A 4 kg ball swings in a vertical circle on the end of an 80-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is θ=30∘.

Part A: What is the ball's speed when θ=30∘? Express your answer to two significant figures and include the appropriate units.

Part B: What is the magnitude of the ball's acceleration when θ=30∘? Express your answer to two significant figures and include the appropriate units.

Part C: What is the direction of the ball's acceleration when θ=30∘? Give the direction as an angle from the r-axis. Express your answer to two significant figures and include the appropriate units.

WACC and Optimal Capital Structure

F. Pierce Products Inc. is considering changing its capital structure. F. Pierce currently has no debt and no preferred stock, but it would like to add some debt to take advantage of low interest rates and the tax shield. Its investment banker has indicated that the pre-tax cost of debt under various possible capital structures would be as follows:

* Use the exact value of 2/3 in your calculations.

F. Pierce uses the CAPM to estimate its cost of common equity, r_s and at the time of the analaysis the risk-free rate is 7%, the market risk premium is 8%, and the company's tax rate is 35%. F. Pierce estimates that its beta now (which is "unlevered" because it currently has no debt) is 1.2. Based on this information, what is the firm's optimal capital structure, and what would be the weighted average cost of capital at the optimal capital structure? Do not round intermediate calculations. Round your answers to two decimal places.

Debt:   %
Equity:   %
WACC:   %

Given here are the data from a dependent variable and two independent variables. The second independent variable is an indicator variable with several categories. Hence, this variable is represented by x₂, x₃, and x₄. How many categories are there for this independent
variable? Use a computer to perform a multiple regression analysis on this data to predict y from the x values. Discuss the output and pay particular attention to the dummy variables.

Appendix A Statistical Tables

Number of categories =

.

The regression equation is: y− =

+

x1 +

x2 -

x3 -

x4 (Round answers to 3 decimals.)