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?
no you cannot,turn this into a calculus max/min problem.
share as many sides as possible. Make it square :)
There are two logical answers.
1) A square that's four meters on each side. That will give you 16 meters of outside fence and 24 meters of inside fence.
2) A rectangle that's 2 by 8 meters. That will give you 20 meters of outside fence and 22 meters of inside fence.
To find the dimensions of the garden that would require the least amount of fencing, we can start by visualizing the rectangular garden.
Since the garden is divided into 16 squares of 1m by 1m, let's assume it has x squares along the length and y squares along the width.
The total perimeter of the garden would be formed by three horizontal sides and one vertical side. The horizontal sides would have lengths 2x (since there are two horizontal sides) and y (the left vertical side doesn't contribute to the perimeter). The vertical side would have a length y as well.
Therefore, the total perimeter, P, can be calculated as P = 2x + y + y = 2x + 2y.
Now, let's consider the constraint that the garden is divided into 16 squares. Since the area of a rectangle is equal to its length multiplied by its width, we know that the area of the garden (A) is equal to xy.
According to the problem, Jane wants to find the dimensions of the garden that use the least amount of fencing. In other words, she wants to minimize the total perimeter, P, while maintaining the constraint that the area, A, is equal to 16 (since the 16 squares form the entire garden).
To find the dimensions that minimize the perimeter, we can use algebraic methods by substituting A = 16 into the perimeter equation:
P = 2x + 2y (Equation 1)
A = xy (Equation 2)
Substituting A = 16 into Equation 2, we have:
16 = xy.
Solving Equation 2 for y, we get:
y = 16/x.
Now, substituting y = 16/x into Equation 1, we have:
P = 2x + 2(16/x).
Simplifying, we get:
P = 2x + 32/x.
To minimize the perimeter, we can take the derivative of P with respect to x and set it equal to zero. However, as per your request, we won't convert this into a calculus problem.
Instead, let's assume some reasonable values for x and calculate the corresponding values for y and P to find the values that minimize P and fit the constraint of A = 16.
Let's start with x = 1:
y = 16/1 = 16.
P = 2(1) + 2(16/1) = 2 + 32 = 34.
Next, x = 2:
y = 16/2 = 8.
P = 2(2) + 2(16/2) = 4 + 16 = 20.
Continuing this process, we can calculate the perimeters for different values of x and choose the one that minimizes P while ensuring A = 16.
Based on this method, you can try different values of x and calculate the corresponding values of y and P to find the dimensions of the garden that require the least amount of fencing while fitting the given constraint.