Question

a rectangular wooden walkway in a city park has a length of 25 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 250 ft². the new walkway will be a larger rectangle. the landscaper plans on making the new length 2 times the current length and the new width 2 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 city’s art museum also wants to expand a rectangular walkway. it wants the new walkway to be a rectangle that has 6 times the area of the current walkway. the landscaper proposes a plan for the expansion. fill in the blank so that the landscaper’s plan is correct.
make the new length 3 times the current length, and make the new width □ times the current width.

Explanation:

Part (a)

Step1: Find new length

Current length is 25 ft, new length is 2 times current. So new length = \(2 \times 25 = 50\) ft.

Step2: Find new width

Current width is 10 ft, new width is 2 times current. So new width = \(2 \times 10 = 20\) ft.

Step3: Find new area

Area of rectangle is length × width. So new area = \(50 \times 20 = 1000\) \(ft^2\).

Step1: Recall current area

Current area is \(250\) \(ft^2\).

Step2: Find ratio of new to current area

New area is \(1000\) \(ft^2\), so ratio = \(\frac{1000}{250} = 4\).

Step1: Let current length be \(l\), width be \(w\), area \(A = lw\).

New length is \(3l\), let new width be \(kw\) (where \(k\) is the multiple). New area should be \(6A\).

Step2: Set up equation for new area

New area = \(3l \times kw = 3k \times lw\). But new area = \(6lw\), so \(3k \times lw = 6lw\).

Step3: Solve for \(k\)

Divide both sides by \(3lw\) (assuming \(l,w
eq 0\)), we get \(k = \frac{6}{3} = 2\).

Answer:

New length: \(50\) ft, New width: \(20\) ft, New area: \(1000\) \(ft^2\)

Part (b)