Question

a rectangular wooden walkway in a city park has a length of 50 ft and a width of 10 ft. the city wants a bigger walkway, so it hires a landscaper to expand the current one. the current walkway has an area of 500 ft². the new walkway will be a larger rectangle. the landscaper plans on making the new length 4 times the current length and the new width 4 times the current width.
(a) fill in the blanks to describe the size of the new walkway.
new length × new width = new area
□ ft × □ ft = □ ft²
(b) how will the expansion change the area of the walkway?
the area of the new walkway will be □ times the area of the current walkway.
(c) the citys art museum also wants to expand a rectangular walkway. it wants the new walkway to be a rectangle that has 12 times the area of the current walkway. the landscaper proposes a plan for the expansion. fill in the blank so that the landscapers plan is correct.
make the new length □ times the current length, and make the new width 2 times the current width.

Explanation:

Step1: Calculate new length and width

Current length \( l = 50\) ft, new length \(L=4\times50 = 200\) ft.
Current width \(w = 10\) ft, new width \(W = 4\times10=40\) ft.

Step2: Calculate new area

Using the formula for the area of a rectangle \(A = L\times W\), we have \(A=200\times40=8000\) \(ft^{2}\).

Step3: Find the ratio of new - to - current area

Current area \(A_{1}=500\) \(ft^{2}\), new area \(A_{2} = 8000\) \(ft^{2}\). The ratio \(\frac{A_{2}}{A_{1}}=\frac{8000}{500}=16\).

Step4: Solve for the length factor in part (c)

Let the current length be \(l\), current width be \(w\), current area \(A_{current}=l\times w\). New width \(w_{new}=2w\), let new length \(l_{new}=x\times l\). New area \(A_{new}=12A_{current}\).
Using the area formula \(A_{new}=l_{new}\times w_{new}\), we substitute \(A_{new}=12(l\times w)\) and \(l_{new}=x\times l\), \(w_{new}=2w\). So \(12(l\times w)=(x\times l)\times(2w)\).
Cancel out \(l\) and \(w\) (since \(l
eq0\) and \(w
eq0\)), we get \(12 = 2x\), then \(x = 6\).

Answer:

(a) \(200\) ft \(\times40\) ft \(=8000\) \(ft^{2}\)
(b) \(16\)
(c) \(6\)