Question

kiras vegetable garden has an area of 300 ft². it is the shape of a rectangle with a length of 30 ft and a width of 10 ft. she has decided to expand the garden to make room for several new varieties of lettuce. the new garden will be a larger rectangle. she plans on making the new length 4 times the current length and the new width 5 times the current width.
(a) fill in the blanks to describe the size of kiras new garden.
new length × new width = new area
□ ft × □ ft = □ ft²
(b) how will the expansion change the area of the garden?
the area of the new garden will be □ times the area of the current garden.
(c) kiras friend alonzo also wants to expand his rectangular garden. alonzo wants his new garden to be a rectangle that has 27 times the area of his current garden. he goes to kira for advice. fill in the blank so that kiras advice to alonzo is correct.
make the new length □ times the current length, and make the new width 3 times the current width.

Explanation:

Part (a)

Step 1: Find new length

Current length is 30 ft, new length is 4 times current. So new length = \( 4 \times 30 = 120 \) ft.

Step 2: Find new width

Current width is 10 ft, new width is 5 times current. So new width = \( 5 \times 10 = 50 \) ft.

Step 3: Find new area

Area of rectangle is length × width. So new area = \( 120 \times 50 = 6000 \) \( \text{ft}^2 \).

Step 1: Recall area scale factor

When length is scaled by \( a \) and width by \( b \), area scales by \( a \times b \). Here, \( a = 4 \), \( b = 5 \).

Step 2: Calculate scale factor

Scale factor = \( 4 \times 5 = 20 \).

Step 1: Let current length be \( l \), width \( w \), area \( A = l \times w \). New length \( = x \times l \), new width \( = 3 \times w \), new area \( = 27A \).

Step 2: Set up equation for new area

\( (x \times l) \times (3 \times w) = 27 \times (l \times w) \)

Step 3: Simplify and solve for \( x \)

\( 3x \times l \times w = 27 \times l \times w \). Divide both sides by \( 3 \times l \times w \): \( x = \frac{27}{3} = 9 \).

Answer:

\( 120 \) ft \( \times \) \( 50 \) ft \( = \) \( 6000 \) \( \text{ft}^2 \)

Part (b)