Question

A prize winning animal is going to be shown at the State Fair. A pen must be constructed to hold the animal.

The rectangular pen will have three wooden sides, which cost 30 dollars per foot to construct. The front of the pen will be made of a special reinforced glass, which costs 80 dollars per foot to construct.
The animal requires an area of 40 square feet in order to be comfortable.
a.If the pen is to be built as cheaply as possible, what are the dimensions of the pen that should be built?
Length of glass front of pen : feet
Depth of pen : feet
b.How much will the pen cost to build?
Cost of pen : 3dollars

Answer 1

To build the pen as cheaply as possible, the dimensions of the pen should be 4 feet for the length of the glass front and 10 feet for the depth. The cost of the pen would be $422.

Step-by-step explanation:

Dimensions of the Pen

To build the pen as cheaply as possible, we need to minimize the cost of constructing the pen. Since the three wooden sides cost $30 per foot to construct and the front made of reinforced glass costs $80 per foot to construct, we should construct the three wooden sides to minimize the cost.

Since the area required by the animal is 40 square feet, we can calculate the dimensions of the pen using the formula for the area of a rectangle: length × width = area. Let's assume the length of the pen is L and the depth is D. Therefore, we have L × D = 40. Rearranging the equation, we get D = 40 / L.

To minimize the cost, we can minimize the length of the wooden sides. Let's substitute D in terms of L into the formula for the cost of the wooden sides, which is 3 × (2L + D). Substituting D = 40 / L, we get the equation for the cost of the wooden sides as C = 3 × (2L + 40 / L). To find the minimum cost, we can differentiate C with respect to L and set it equal to zero. After solving, we find L = 4 and D = 10. Therefore, the dimensions of the pen that should be built are 4 feet for the length of the glass front and 10 feet for the depth.

Cost of the Pen

To calculate the cost of the pen, we need to calculate the cost of the wooden sides and the glass front. The cost of the wooden sides is given by 3 × (2L + D), where L = 4 and D = 10. Therefore, the cost of the wooden sides is 3 × (2 × 4 + 10) = $102. The cost of the glass front is given by 80 × L, where L = 4. Therefore, the cost of the glass front is 80 × 4 = $320. The total cost of the pen is the sum of the cost of the wooden sides and the cost of the glass front, which is $102 + $320 = $422.

Answer 2

To build the pen as cheaply as possible, choose the dimensions of 5 feet by 8 feet. The cost of the pen would be 1195 dollars.

Step-by-step explanation:

To build the pen as cheaply as possible, we can minimize the cost of the wooden sides by using the minimum amount of wood necessary. Since the animal requires an area of 40 square feet, we can first find the dimensions of the rectangle that has an area of 40 square feet. The dimensions can be found by finding the factors of 40. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. Using these factors, the possible dimensions for the pen are:

Length: 1 foot, Width: 40 feet (perimeter: 81 feet)

Length: 2 feet, Width: 20 feet (perimeter: 44 feet)

Length: 4 feet, Width: 10 feet (perimeter: 28 feet)

Length: 5 feet, Width: 8 feet (perimeter: 26 feet)

The cheapest option would be to choose the dimensions of 5 feet by 8 feet since it has the smallest perimeter of 26 feet. To calculate the cost of the pen, we can multiply the cost per foot for the wooden sides by the perimeter of the pen and add it to the cost per foot for the glass front multiplied by the length of the front glass. The cost of the pen would be:

[3(5) + 30(26)] + 80(5) = 15 + 780 + 400 = 1195 dollars