The problem you're solving involves finding the dimensions of five identical pens arranged side by side, where the total area is $1000 \, m^2$, and we need to minimize the amount of fencing used. Here's how we approach it:

1. Area Equation:

The total area of the five pens is given by: $1000 = 5xy$ where $x$ is the length of the pens and $y$ is the width of each pen. This equation can be rewritten to solve for $x$: $x = \frac{1000}{5y} = \frac{200}{y}$

2. Fence Equation:

The total length of fencing required (perimeter) is given by: $F = 6x + 10y$ This includes the lengths of the horizontal sections (6 $x$, as there are six vertical boundaries) and the vertical sections (10 $y$, as there are five pens requiring five boundaries and two external ones).

3. Substituting $x$ into the Fence Equation:

Now substitute $x = \frac{200}{y}$ into the equation for the fence: $F = 6 \left( \frac{200}{y} \right) + 10y = \frac{1200}{y} + 10y$

4. Minimizing the Fence Equation:

To minimize the amount of fence used, take the derivative of $F$ with respect to $y$ and set it equal to zero: $\frac{dF}{dy} = -\frac{1200}{y^2} + 10$ Setting this equal to zero: $-\frac{1200}{y^2} + 10 = 0$ $\frac{1200}{y^2} = 10$ $y^2 = \frac{1200}{10} = 120$ $y = \sqrt{120} \approx 10.95 \, m$

5. Finding $x$:

Now, substitute $y \approx 10.95$ back into the equation for $x$: $x = \frac{200}{y} = \frac{200}{10.95} \approx 18.27 \, m$

Final Dimensions:

• Width $y \approx 10.95 \, m$
• Length $x \approx 18.27 \, m$

These dimensions will minimize the amount of fencing used while keeping the total area of the pens at $1000 \, m^2$.

Would you like more details or have any questions?

Related Questions:

1. How do we confirm that the calculated dimensions give the correct total area of $1000 \, m^2$?
2. Can we use Lagrange multipliers to solve this optimization problem?
3. What happens to the fence equation if the number of pens changes from 5 to another value?
4. What if the area constraint changes to something other than $1000 \, m^2$?
5. How does the shape of the pens (square vs rectangular) affect the fencing requirements?

Tip:

When optimizing a quantity, always check if the second derivative confirms a minimum or maximum!