Neilsen Cookie Company sells its assorted butter cookies in containers that have a net content of 1 lb. The estimated demand for the cookies is 700,000 1 lb containers. The setup cost for each production run is $546, and the manufacturing cost is $0.47 for each container of cookies. The cost of storing each container of cookies over the year is $0.35. Assuming uniformity of demand throughout the year and instantaneous production, how many containers of cookies should Neilsen produce per production run in order to minimize the production cost

46,734 units per run

Explanation:

total estimated demand = 700,000 containers

setup costs per production run = $546

manufacturing cost = $0.47 per container

holding cost = $0.35 per container

r = 700,000 / x

total setup costs = 546r = 546 (700,000/x) = 382,200,000/x

production costs = 0.47 x 700,000 = 329,000

storage cost per unit= 1/2r x 0.35 = 0.35/2(700,000/x) = 0.35x/1,400,000

total storage costs = 700,000 x 0.35x/1,400,000 = 0.175x

C(x) = 382,200,000/x + 0.175 x + 329,000

now we find the derivative:

C'(x) = -382,200,000/x² + 0.175

382,200,000/x² = 0.175

382,200,000 = 0.175x²

x² = 382,200,000 / 0.175 = 2,184,000,000

x = √2,184,000,000 = 46,733.28 ≈ 46,734 units per run

this answer is based on a continuous production process, there are 14.98 runs per year

bd = 9 cm

step-by-step explanation:

in a 30°-60°-90° triangle, the ratio of side lengths is 1 : √3 : 2. that is, the longest side (hypotenuse) is twice the length of the shortest side.

all of the triangles in your geometry are 30°-60°-90° triangles. ac is the hypotenuse of δacd, and the short side of δabc.

the short side ad of δacd will be half the length of ac, so 3 cm. the hypotenuse ab of δabc will be twice the length of ac, so 12 cm. segment bd is the difference of the lengths ab and ad, so is

bd = ab -ad

bd = 12 cm - 3 cm = 9 cm

comment on side length ratios

you can figure the ratios of side lengths in a 30°-60°-90° triangle by considering the trig ratios of the angles. or you can figure the length of the altitude of an equilateral triangle of side length 2 using the pythagorean theorem.