Question

1. a rectangular backyard is x feet long and y feet wide. the new design of the backyard increases the length by 35% and the width by 20%. additionally, a 8 - foot wide pathway is added around the backyard. enter the simplest expression for the perimeter, in feet, of the redesigned backyard, including the pathway.

Explanation:

Step1: Find new length

Original length is \( x \) feet. A 35% increase means the new length is \( x + 0.35x = 1.35x \) feet. Then, a 8 - foot pathway is added around, so we need to consider the perimeter contribution from the length. Wait, actually, the 8 - foot wide pathway around the backyard: when we add a pathway around a rectangle, the length and width each increase by \( 2\times8 \) (since the pathway is on both sides) ? Wait, no, the problem says "the new design of the backyard increases the length by 35% and the width by 20%. Additionally, a 8 - foot wide pathway is added around the backyard." Wait, maybe the 35% and 20% increase is before adding the pathway? Or maybe the pathway is an additional 8 feet on each side? Wait, let's re - read: "A rectangular backyard is \( x \) feet long and \( y \) feet wide. The new design of the backyard increases the length by 35% and the width by 20%. Additionally, a 8 - foot wide pathway is added around the backyard. Enter the simplest expression for the perimeter, in feet, of the redesigned backyard, including the pathway."

So first, increase length by 35%: new length (before pathway) is \( x(1 + 0.35)=1.35x \). Then, add an 8 - foot wide pathway around. So the length will increase by \( 2\times8 \) (because the pathway is on both the left and right sides of the length) ? Wait, no, the width of the pathway is 8 feet. So for the length of the backyard including the pathway: the original length (after 35% increase) is \( 1.35x \), and then we add 8 feet on the left and 8 feet on the right, so total length \( L = 1.35x+8 + 8=1.35x + 16 \).

For the width: original width is \( y \), increase by 20%: \( y(1 + 0.20)=1.2y \). Then add 8 feet on the top and 8 feet on the bottom, so total width \( W=1.2y + 16 \).

Now, the perimeter of a rectangle is \( P = 2(L + W) \).

So \( P=2[(1.35x + 16)+(1.2y + 16)] \)

Step2: Simplify the expression inside the brackets

First, simplify \( (1.35x + 16)+(1.2y + 16)=1.35x+1.2y + 32 \)

Step3: Multiply by 2 for the perimeter

\( P = 2(1.35x+1.2y + 32)=2.7x + 2.4y+64 \)

Wait, maybe I misinterpreted the pathway. Maybe the 8 - foot wide pathway: the "width" of the pathway is 8 feet, so when we add a pathway of width \( w \) around a rectangle, the length becomes \( L_{original}+2w \) and the width becomes \( W_{original}+2w \), where \( w \) is the width of the pathway. But in this problem, the 35% and 20% increase is applied first, then the pathway.

Wait, another interpretation: the 35% increase and 20% increase already include the pathway? No, the problem says "additionally, a 8 - foot wide pathway is added around the backyard". So the 35% and 20% are for the backyard itself, and then the pathway is an extra.

Wait, maybe the pathway is 8 feet wide, so the length of the entire area (backyard + pathway) is the length of the backyard (after 35% increase) plus \( 2\times8 \) (since the pathway is on both sides of the length), and the width is the width of the backyard (after 20% increase) plus \( 2\times8 \).

So length after 35% increase: \( 1.35x \), then add \( 2\times8 \) for the pathway: \( L = 1.35x+16 \)

Width after 20% increase: \( 1.2y \), then add \( 2\times8 \) for the pathway: \( W = 1.2y + 16 \)

Perimeter \( P = 2(L + W)=2[(1.35x + 16)+(1.2y + 16)]=2(1.35x+1.2y + 32)=2.7x+2.4y + 64 \)

Wait, but let's check the percentage increase again. 35% increase of \( x \) is \( 0.35x \), so new length is \( x+0.35x = 1.35x \). 20% increase of \( y \) is \( 0.2y \), so new width is \( y + 0.2y=1.2y \). Then, the pathway is 8 feet wide around, so the length becomes \( 1.35x+8 + 8=1.35x + 16 \) (because the pathway is on both sides of the length), and the width becomes \( 1.2y+8 + 8=1.2y + 16 \) (pathway on both sides of the width). Then the perimeter of a rectangle is \( 2\times( length + width) \), so \( P = 2[(1.35x + 16)+(1.2y + 16)]=2(1.35x+1.2y + 32)=2.7x + 2.4y+64 \)

Alternatively, maybe the 8 - foot wide pathway is such that the length increases by 8 feet (not 16) and the width increases by 8 feet? But that doesn't make sense because a pathway around the rectangle would add to both sides. For example, if you have a rectangle and you put a border around it, the length of the bordered rectangle is original length + 2*border width, and the width is original width + 2*border width.

So let's re - derive:

1. Calculate the length after 35% increase: \( L_1=x(1 + 0.35)=1.35x \)
2. Calculate the width after 20% increase: \( W_1=y(1 + 0.20)=1.2y \)
3. Calculate the length including the pathway: The pathway has a width of 8 feet, so we add 8 feet to the left and 8 feet to the right of \( L_1 \), so \( L = L_1+8 + 8=1.35x + 16 \)
4. Calculate the width including the pathway: We add 8 feet to the top and 8 feet to the bottom of \( W_1 \), so \( W=W_1 + 8+8 = 1.2y+16 \)
5. Calculate the perimeter: The perimeter of a rectangle is \( P = 2(L + W) \)
- Substitute \( L \) and \( W \) into the formula: \( P = 2[(1.35x + 16)+(1.2y + 16)] \)
- First, simplify the expression inside the brackets: \( (1.35x + 16)+(1.2y + 16)=1.35x+1.2y + 32 \)
- Then multiply by 2: \( P=2\times(1.35x + 1.2y+32)=2.7x + 2.4y+64 \)

Answer:

\( 2.7x + 2.4y + 64 \)