Claire and Chad want to design a rectangular pen for their new puppy. They want the pen to have

an area of 48 square feet. Fencing costs $0.85 per foot.
b. What are the dimensions and the cost of the most expensive pen Claire and Chad could
build, assuming the side lengths are whole numbers? Explain.

a rectangle with equal sides (a square)

... encloses the most area with the least fencing

the most fencing (with integer side lengths)
... would be a rectangle 48 feet long and one foot wide

To find the dimensions and cost of the most expensive pen they can build, we need to consider that the area of the pen should be 48 square feet.

Let's start by listing all possible pairs of factors for 48:

1 x 48 = 48
2 x 24 = 48
3 x 16 = 48
4 x 12 = 48
6 x 8 = 48

Out of these pairs, the dimensions of the pen would be 6 feet by 8 feet, as this is the larger pair.

Now, to find the cost, we need to calculate the perimeter of the pen, which is the sum of all the sides:

Perimeter = 2(length + width)

Perimeter = 2(6 + 8)
Perimeter = 2(14)
Perimeter = 28 feet

Now we can find the cost by multiplying the perimeter by the cost per foot:

Cost = Perimeter x Cost per foot
Cost = 28 x $0.85
Cost = $23.80

Therefore, the dimensions of the most expensive pen they could build are 6 feet by 8 feet, and it would cost $23.80.

To find the dimensions of the most expensive pen that Claire and Chad could build, we need to determine the possible side lengths by factoring the area of the pen.

The area of the rectangular pen is given as 48 square feet. We want to find two whole numbers whose product is 48.

To factor 48, we can start by finding the smallest whole number greater than or equal to the square root of 48. The square root of 48 is approximately 6.93. So we start by checking whole numbers from 7 downwards.

Starting from 7, we see that 48 divided by 7 is not a whole number. Next, we try 6. The division 48 รท 6 gives us 8, which is a whole number. This means that the dimensions of the pen could be 6 feet by 8 feet.

Now that we have the dimensions of the pen, we can calculate the perimeter and then multiply it by the cost of fencing per foot to find the cost of the pen.

The perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, the perimeter is (2 * length) + (2 * width).

For the pen with dimensions 6 feet by 8 feet, the perimeter is (2 * 6) + (2 * 8) = 12 + 16 = 28 feet.

Now, to find the cost of the pen, we multiply the perimeter by the cost per foot. The cost of fencing is $0.85 per foot, so the cost of the pen is 28 * $0.85 = $23.80.

Therefore, the dimensions of the most expensive pen Claire and Chad could build, assuming the side lengths are whole numbers, are 6 feet by 8 feet. The cost of this pen would be $23.80.