MATH SOLVE

4 months ago

Q:
# The management of the unico department store has decided to enclose an 834 ft2 area outside the building for displaying potted plants and flowers. one side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. if the pine board fencing costs $6/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (round your answers to one decimal place.)

Accepted Solution

A:

The area is given by:

A = x * y = 834

The cost equation is given by:

C = 6 * (2x) + 3 * (y)

We express the equation in terms of a variable:

C (x) = 6 * (2x) + 3 * (834 / x)

Rewriting:

C (x) = 12x + 2502 / x

We derive the equation:

C '(x) = 12 - 2502 / x ^ 2

We match zero:

0 = 12 - 2502 / x ^ 2

We clear x:

x = root ((2502) / (12))

x = 14.4 feet

We look for the other dimension:

y = 834 / x

y = 834 / 14.4

y = 57.9 feet

Answer:

The dimensions of the enclosure that can be erected at minimum cost are:

x = 14.4 feet

y = 57.9 feet

A = x * y = 834

The cost equation is given by:

C = 6 * (2x) + 3 * (y)

We express the equation in terms of a variable:

C (x) = 6 * (2x) + 3 * (834 / x)

Rewriting:

C (x) = 12x + 2502 / x

We derive the equation:

C '(x) = 12 - 2502 / x ^ 2

We match zero:

0 = 12 - 2502 / x ^ 2

We clear x:

x = root ((2502) / (12))

x = 14.4 feet

We look for the other dimension:

y = 834 / x

y = 834 / 14.4

y = 57.9 feet

Answer:

The dimensions of the enclosure that can be erected at minimum cost are:

x = 14.4 feet

y = 57.9 feet