- Identify the perimeter \(P\) of the rectangular garden as \(P = 2(l + w)\).
- Determine the total distance \(D\) for four laps by multiplying the perimeter by 4, resulting in \(D = 4 \times 2(l + w)\).
- Simplify the expression to find the total distance: \(\boxed{8(l+w)}\).
Explanation
- Calculate total distance for four laps around a rectangle.
The problem asks for the total distance Jane runs around a rectangular garden in four laps, given its length \(l\) and width \(w\).
The perimeter \(P\) of a rectangle is the total distance around its boundary, which is calculated as the sum of all its sides: \(P = l + l + w + w = 2l + 2w = 2(l + w)\).
Since Jane completes four laps, the total distance \(D\) she runs is four times the perimeter of the garden.
Therefore, the total distance is:
\(D = 4 \times P\)
\(D = 4 \times [2(l + w)]\)
Simplifying this expression gives:
\(D = 8(l + w)\)
Thus, the expression that correctly represents the total distance Jane runs is \(8(l+w)\).
Examples
If a rectangular track measures \(50 \text{ meters}\) in length and \(30 \text{ meters}\) in width, the perimeter is \(P = 2(50 + 30) = 160 \text{ meters}\). Running four laps means Jane would cover \(4 \times 160 = 640 \text{ meters}\). Using our derived formula: \(D = 8(50 + 30) = 8(80) = 640 \text{ meters}\). This formula is a quick and efficient way to calculate total distance without needing to compute the perimeter separately first.