Optimization Problems: Fire Safety

1.According to fire safety codes, a potential exit window needs to be at least 20 inches wide, at least 24 inches tall, and at least 5.7 square feet in area.

a)If a window is 20" wide X 24" tall, does it meet the requirements for safety?

b)How tall would a 20" wide window have to be to meet code? Give your answer in inches.

c)How wide would a 24" tall window have to be to meet code? Give your answer in inches.

2.You are designing low-cost housing units, and need to economize on the cost of the windows. Every bedroom window in each unit requires a legal exit window.

a.The glass costs $3.18 per square foot. The window framing material costs $1.42 per linear foot for the sides and top, and $2.38 per linear foot for the base. Assume that each window will have an area of exactly 5.7 square feet, and write a formula for C(w) = the total cost per window of frame and glass, where w represents the width of the window.

b.Find the dimensions of the window that will minimize the cost.

c.Use either the first derivative or the second derivative test to confirm your answer for b.

3.After completing your calculations, you learn that the vendor has raised prices on materials. The new costs per linear foot are $1.50 for the sides, $2.62 for the top, and $3.97 for the base.

a.Find the new cost function C(w).

b.Find the value of w that minimizes the cost function.

c.Sketch a graph of the cost function. Show the minimum value.

d.Find the domain and range of C(w).