Question

You are building a rectangular doorway. The doorway must be at least 1 meter high
and at least 0.5 meters wide. It must also enclose a door with area 2.5 m2
. The
material for the sides of the doorway cost $30 per meter and the material for the top
of the doorway costs $50 per meter. The bottom of the doorway requires no material.
What is the cheapest you can build the doorway?

Answer 1

first try to analyse the question

Answer 2

wow.. nice drawing!

Answer 3

Using Mathematica's constrained optimization function, Minimize:\[\text{Minimize}\left[\left\{50w+30(2h),h\geq 1,w\geq \frac{1}{2},h*w==\frac{25}{10}\right\},\{w,h\}\right]\to \]\[\left\{100 \sqrt{3},\left\{w\to \sqrt{3},h\to \frac{5}{2 \sqrt{3}}\right\}\right\} \]\[\{173.205,\{w\to 1.73205,h\to 1.44338\}\} \]

Answer 4

$173.21 rounding up to the nearest cent.