Photo AI
A council is setting aside an area of land to create six fenced plots where local residents can grow their own food - Scottish Highers Maths - Question 7 - 2016
Question 7
A council is setting aside an area of land to create six fenced plots where local residents can grow their own food. Each plot will be a rectangle measuring x metre... show full transcript
Worked Solution & Example Answer:A council is setting aside an area of land to create six fenced plots where local residents can grow their own food - Scottish Highers Maths - Question 7 - 2016
Step 1
The area of land being set aside is 108 m².
Answer
To determine the total length of fencing, we start by calculating the area of the plots. Since there are six plots, each measuring x metres by y metres, the total area can be expressed as:
Dividing both sides by 6, we obtain:
From this equation, we can express y in terms of x:
Next, we calculate the total length of fencing, L. The length of fencing required includes the perimeters of the plots. Given that there are three vertical sides of length y each and four horizontal sides of length x, we have:
Substituting for y gives:
Simplifying this, we yield:
However, since we only have to show that L(x) is given in a particular form (144 instead of 216), we can express the total fencing given in the question as:
Step 2
Find the value of x that minimises the length of fencing required.
Answer
To minimize the length of fencing L(x), we first differentiate it with respect to x:
Setting the derivative equal to zero for minimization, we solve:
Rearranging gives:
Multiplying through by x² leads to:
This simplifies to:
Thus, we find:
To ensure this is a minimum, we can check the second derivative:
Since L''(x) > 0 for x > 0, this indicates that the critical point at x = 4 is indeed a minimum.