Question

area and volume
investigating the effects on the area for non - proportional and proportion...
ssica has a patio that is in the shape of a rectangle with an area of ( 600 \text{ft}^2 ). she has decided to expand the patio so that it can hold more furniture. the new patio will be a larger rectangle. the current patio has a length of 30 ft and a width of 20 ft. she plans on making the new length 3 times the current length and the new width 3 times the current width.
(a) fill in the blanks to describe the size of jessicas new patio.
\\(\underbrace{\text{new length}}_{\square \text{ft}}\times\underbrace{\text{new width}}_{\square \text{ft}} = \underbrace{\text{new area}}_{\square \text{ft}}\\)
(b) how will the expansion change the area of the patio?
the area of the new patio will be \\(\square\\) times the area of the current patio.
(c) jessicas friend maria also wants to expand her rectangular patio. maria wants her new patio to be a rectangle that has 28 times the area of her current patio. she goes to jessica for advice. fill in the blank so that jessicas advice to maria is correct.
make the new length \\(\square\\) times the current length, and make the new width 7 times the current width.

Explanation:

Part (a)

Step1: Calculate new length

Current length is 30 ft, new length is 3 times current length. So new length = \( 3 \times 30 = 90 \) ft.

Step2: Calculate new width

Current width is 20 ft, new width is 3 times current width. So new width = \( 3 \times 20 = 60 \) ft.

Step3: Calculate new area

Area of rectangle is length × width. So new area = \( 90 \times 60 = 5400 \) \( \text{ft}^2 \).

Step1: Recall current area

Current area is 600 \( \text{ft}^2 \).

Step2: Find ratio of new to current area

New area is 5400 \( \text{ft}^2 \), so ratio = \( \frac{5400}{600} = 9 \).

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

Step2: Let new length be \( k \times l \), new width be \( 7 \times w \), new area \( A' = k \times l \times 7 \times w = 7k \times l \times w \).

Step3: Set new area to 28 times current area

We know \( A' = 28A \), and \( A = l \times w \), \( A' = 7k \times A \). So \( 7k \times A = 28A \). Divide both sides by \( A \) (since \( A
eq 0 \)): \( 7k = 28 \), so \( k = \frac{28}{7} = 4 \).

Answer:

\( 90 \) ft \( \times \) \( 60 \) ft \( = \) \( 5400 \) \( \text{ft}^2 \)

Part (b)