Question: The district health officers are planning to construct a rectangular fence across the quarantine centre with one gate measuring $12m$ in length. There is $1000m$ of fencing available. The construction team wants to know the dimensions of the field that will yield a maximum area that can be fenced off before starting the installation. However, the community is constrained with funds and they can manage to fence a total of $64020m^2$ only. The community members are not sure if the fences can be constructed given the available funds or they collect funds from each family.
1. **State the problem:**
We need to find the dimensions of a rectangular field with one side as a gate of length $12m$ and the rest fenced with $1000m$ of fencing material. We want to maximize the area of this field.
2. **Define variables:**
Let the length of the rectangle be $L$ and the width be $W$. The gate is on one length side, so fencing is needed for the other three sides plus the other length side.
3. **Write the fencing constraint:**
The total fencing used is for two widths and one length minus the gate length (since gate is not fenced):
$$ 2W + (L - 12) = 1000 $$
4. **Express $L$ in terms of $W$:**
$$ L - 12 = 1000 - 2W $$
$$ L = 1012 - 2W $$
5. **Write the area formula:**
$$ A = L \times W = (1012 - 2W)W = 1012W - 2W^2 $$
6. **Maximize the area:**
Take derivative of $A$ with respect to $W$ and set to zero:
$$ \frac{dA}{dW} = 1012 - 4W = 0 $$
$$ 4W = 1012 $$
$$ W = \frac{1012}{4} = 253 $$
7. **Find corresponding $L$:**
$$ L = 1012 - 2(253) = 1012 - 506 = 506 $$
8. **Calculate maximum area:**
$$ A_{max} = L \times W = 506 \times 253 = 127,918 $$
9. **Compare with community fund constraint:**
The community can only manage $64020m^2$, which is less than the maximum area $127,918m^2$.
10. **Conclusion:**
The maximum area possible with $1000m$ fencing and a $12m$ gate is $127,918m^2$, which exceeds the community's fund constraint of $64020m^2$. Therefore, they cannot fence the maximum area with the available funds and may need to collect more funds or reduce the fenced area.