The management of the UNICO department store has decided to enclose a 917 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 $7/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

wood side ft
steel side ft

The management of the UNICO department store has decided to enclose a 855 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 $2/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

wood side ft
steel side ft

Let's assume the length of the wood side is x ft, and the width of the wood side is y ft.

So, the area of the wood side can be calculated as:
AreaWood = x * y

The steel side is straight, so we need to subtract the length of the wood side from the total length of the enclosure to get the length of the steel side. The width of the steel side will be the same as the width of the wood side, y ft.

So, the area of the steel side can be calculated as:
AreaSteel = (917 - x) * y

To calculate the cost, we need to find the perimeter of each side and multiply it by the respective cost per running foot.

PerimeterWood = 2x + 2y
CostWood = 7 * PerimeterWood

PerimeterSteel = (917 - x) + 2y
CostSteel = 5 * PerimeterSteel

The total cost can be calculated as:
TotalCost = CostWood + CostSteel

To minimize the cost, we need to find the minimum value of TotalCost by differentiating it with respect to x and setting it equal to 0.

d(TotalCost)/dx = 0

Simplifying the equation, we get:

7(2) + 5(-1) = 10x - 3y + 15

10x - 3y = -19

To solve this equation and find the values of x and y, we need another equation. Let's use the fact that the area of the wood side is 917 ft^2.

AreaWood = x * y = 917 ft^2

Now we have a system of two equations:

10x - 3y = -19
x * y = 917

We can solve this system of equations using any method (substitution, elimination, etc.). Let's use substitution.

First, solve the second equation for x:

x = 917 / y

Now substitute this expression of x into the first equation:

10(917/y) - 3y = -19

Simplifying this equation, we get:

9170/y - 3y = -19

Multiply both sides of the equation by y:

9170 - 3y^2 = -19y

Rearrange the equation:

3y^2 - 19y + 9170 = 0

Now we can use the quadratic formula to solve for y:

y = (-b ± √(b^2 - 4ac)) / 2a

Using values a = 3, b = -19, c = 9170, we get:

y ≈ 28.4 ft or y ≈ 34.0 ft

Now we can substitute the value of y back into the second equation to solve for x:

x = 917 / y

Substituting y = 28.4 ft:

x ≈ 32.3 ft

Substituting y = 34.0 ft:

x ≈ 27.0 ft

Therefore, the dimensions of the enclosure that can be erected at minimum cost are approximately:

Wood side: 27.0 ft x 28.4 ft
Steel side: 890.0 ft x 28.4 ft

To determine the dimensions of the enclosure that can be erected at minimum cost, we need to find the dimensions that minimize the cost of the fencing.

Let's assume the length of the pine board fencing is L ft. Since there are two sides made of pine boards, the total length of the pine board fencing is 2L ft.

The length of the steel fencing is equal to the remaining length after subtracting the length of the pine board fencing from the total perimeter of the enclosure. The total perimeter of the enclosure is the sum of the lengths of all four sides.

The given information tells us that one side is formed by the external wall of the store, so we need to find the lengths of the other three sides.

Let's assume the width of the enclosure is W ft. The length of the external wall of the store will then be W ft.

The total perimeter of the enclosure is the sum of the four sides:

Total Perimeter = Length of external wall + 2L + W + Length of steel fencing

The area of the enclosure is given as 917 ft^2:

Area = Length (W) * Width (L) = 917 ft^2

Now we can substitute the value of W from the area equation into the perimeter equation:

Total Perimeter = W + 2L + W + Length of steel fencing

Since the length of the steel fencing is equal to the remaining length after subtracting the length of the pine board fencing from the total perimeter, it can be written as:

Length of steel fencing = Total Perimeter - 2L - W - W

Next, we need to determine the cost of the fencing.

The cost of the pine board fencing is $7 per running foot, so the cost for 2L ft of pine board fencing is:

Cost of pine board fencing = 7 * 2L = 14L

The cost of the steel fencing is $5 per running foot, so the cost for the length of steel fencing is:

Cost of steel fencing = 5 * (Total Perimeter - 2L - W - W) = 5 * (Total Perimeter - 4L - 2W)

The total cost of the fencing is the sum of the cost of the pine board fencing and the cost of the steel fencing:

Total Cost = Cost of pine board fencing + Cost of steel fencing = 14L + 5 * (Total Perimeter - 4L - 2W)

To find the dimensions that minimize the total cost, we can take the partial derivatives of the cost equation with respect to L and W, and set them equal to zero:

∂(Total Cost) / ∂L = 0
∂(Total Cost) / ∂W = 0

Solving these partial derivative equations will give us the values of L and W that minimize the cost.

assuming the fenced area is rectangular,

Let steel side be x, wood side be y

xy = 917, so
y = 917/x

The cost of the fence is thus

c(x) = 5x + 2*7(917/x)

Then find where c'(x) = 0