Jane grows several varieties of plants in a rectangular-shaped garden. She uses fencing to divide the garden into 16 squares that are each 1 m by 1 m. She also puts fencing around the perimeter of the garden.

What should the dimensions of the garden be so that Jane uses the least amount of fencing?

Can you turn this into a calculus max/min problem?

To determine the dimensions of the garden that require the least amount of fencing, we need to consider how much fencing Jane will need for each possible dimension.

Let's start by assuming the length of the garden is L meters. If the width of the garden is W meters, then we can calculate the total amount of fencing needed. We have the length of the garden (L) and the width of the garden (W).

Since we're dividing the garden into 16 squares, and each square is 1m by 1m, the length of the garden (L) should be divisible by 4 since there are 4 squares in each row. Similarly, the width of the garden (W) should be divisible by 4 since there are 4 squares in each column.

Let's consider the possible dimensions of the garden that meet this requirement:

Possible values for L: 4m, 8m, 12m, 16m, 20m, 24m, ...

Possible values for W: 4m, 8m, 12m, 16m, 20m, 24m, ...

For each possible value of L and W, we can calculate the perimeter of the garden. The perimeter is calculated by adding the lengths of all four sides.

Perimeter = 2L + 2W

Now, let's calculate the perimeter for each possible dimension and find the one that requires the least amount of fencing:

For L = 4m and W = 4m, the perimeter = 2(4) + 2(4) = 8 + 8 = 16m
For L = 8m and W = 8m, the perimeter = 2(8) + 2(8) = 16 + 16 = 32m
For L = 12m and W = 12m, the perimeter = 2(12) + 2(12) = 24 + 24 = 48m
For L = 16m and W = 16m, the perimeter = 2(16) + 2(16) = 32 + 32 = 64m
For L = 20m and W = 20m, the perimeter = 2(20) + 2(20) = 40 + 40 = 80m
For L = 24m and W = 24m, the perimeter = 2(24) + 2(24) = 48 + 48 = 96m

Based on the calculations, it is evident that the dimensions of the garden that require the least amount of fencing are L = 4m and W = 4m. The perimeter of this garden is 16m.

Therefore, Jane should make the rectangular-shaped garden with dimensions 4m by 4m to use the least amount of fencing.