Question

ivanna’s vegetable garden has an area of 400 ft². it is the shape of a rectangle with a length of 40 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 2 times the current length and the new width 2 times the current width.
(a) fill in the blanks to describe the size of ivanna’s 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) ivanna’s friend elsa also wants to expand her rectangular garden. elsa wants her new garden to be a rectangle that has 14 times the area of her current garden. she goes to ivanna for advice. fill in the blank so that ivanna’s advice to elsa is correct.
make the new length □ times the current length, and make the new width 2 times the current width.

Explanation:

Step1: Analyze Ivanna's Garden Expansion (a)

Current length \( l = 40 \) ft, current width \( w = 10 \) ft. New length is \( 2 \times 40 = 80 \) ft, new width is \( 2 \times 10 = 20 \) ft. New area \( = 80 \times 20 = 1600 \) \( \text{ft}^2 \).

Step2: Determine Area Change (b)

Original area \( A = 40 \times 10 = 400 \) \( \text{ft}^2 \). New area is 1600 \( \text{ft}^2 \). \( \frac{1600}{400} = 4 \), so new area is 4 times original.

Step3: Solve for Elsa's Length Multiplier (c)

Let length multiplier be \( k \), width multiplier is 2. Area multiplier: \( k \times 2 = 14 \). So \( k = \frac{14}{2} = 7 \).

Answer:

s:
(a) New length: \( 80 \) ft, New width: \( 20 \) ft, New area: \( 1600 \) \( \text{ft}^2 \)
(b) \( 4 \)
(c) \( 7 \)