Question

a hotel manager is adding a tile border around the hotels rectangular pool. let x represent the width of the pool, in feet. the length is 2 more than 2 times the width, as shown. write two expressions that represent the perimeter of the pool.
click the icon for a diagram of the pool.
which expressions will give the perimeter of the pool? select all that apply.
a. 2x + 2x + 2
b. 3x + 2
c. 4(x + 2x + 2)
d. 6x + 2
e. 6x + 4
f. 2(3x + 2)

Explanation:

Step1: Find the length of the pool

The width of the pool is \( x \) feet. The length is 2 more than 2 times the width, so length \( l = 2x + 2 \) feet.

Step2: Recall the perimeter formula for a rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2\times(\text{length} + \text{width}) \). Substituting the values of length and width, we get \( P = 2\times((2x + 2)+x) \).

Step3: Simplify the first expression

Simplify \( 2\times((2x + 2)+x) \):
First, combine like terms inside the parentheses: \( (2x + 2)+x=3x + 2 \). Then multiply by 2: \( 2\times(3x + 2)=6x + 4 \). Wait, no, wait: Wait, \( (2x + 2)+x=3x + 2 \)? Wait, \( 2x+x = 3x \), so \( 3x + 2 \), then \( 2\times(3x + 2)=6x + 4 \)? Wait, no, let's re - do the perimeter formula. Wait, perimeter of rectangle is \( 2l + 2w \), where \( l \) is length and \( w \) is width. So \( l = 2x+2 \), \( w = x \). So \( P = 2(2x + 2)+2x \).

Simplify \( 2(2x + 2)+2x \):
First, distribute the 2: \( 4x+4 + 2x \). Then combine like terms: \( 4x+2x+4=6x + 4 \).

Now let's check each option:

• Option A: \( 2x+2x + 2=4x + 2 \). Not equal to \( 6x + 4 \).
• Option B: \( 3x+2 \). Not equal to \( 6x + 4 \).
• Option C: \( 4(x + 2x+2) \). First, combine like terms inside the parentheses: \( x + 2x+2=3x + 2 \). Then multiply by 4: \( 4\times(3x + 2)=12x+8 \). Not equal to \( 6x + 4 \). Wait, I think I made a mistake in the length. Wait, the problem says "the length is 2 more than 2 times the width". So length \( l=2x + 2 \)? Wait, no, "2 more than 2 times the width" is \( 2x+2 \)? Wait, no, "2 times the width" is \( 2x \), "2 more than" that is \( 2x + 2 \). Then width is \( x \). Then perimeter \( P = 2(l + w)=2((2x + 2)+x)=2(3x + 2)=6x + 4 \).

Wait, let's re - evaluate the options:

Option A: \( 2x+2x + 2=4x + 2 \). No.

Option B: \( 3x + 2 \). No.

Option C: \( 4(x + 2x+2)=4(3x + 2)=12x + 8 \). No.

Option D: \( 6x+2 \). No.

Option E: \( 6x + 4 \). Yes, this matches our calculated perimeter.

Option F: \( 2(3x + 2) \). Distribute the 2: \( 6x+4 \). Yes, this is equal to \( 6x + 4 \).

Wait, let's re - derive the perimeter correctly.

Perimeter of rectangle: \( P=2l + 2w \), where \( l \) is length and \( w \) is width.

Given \( w=x \), \( l = 2x + 2 \).

So \( P=2(2x + 2)+2x=4x + 4+2x=6x + 4 \).

Now, let's check option F: \( 2(3x + 2)=6x + 4 \), which is equal to the perimeter.

Option E: \( 6x + 4 \), which is equal to the perimeter.

Wait, let's check option A again: \( 2x+2x + 2=4x + 2 \). No.

Option B: \( 3x + 2 \). No.

Option C: \( 4(x + 2x+2)=4(3x + 2)=12x + 8 \). No.

Option D: \( 6x+2 \). No.

So the correct options are E and F. Wait, but let's check the perimeter formula again. Wait, maybe the length is \( 2x+2 \)? Wait, no, "2 more than 2 times the width" is \( 2x + 2 \). Width is \( x \). Then perimeter \( P=2(l + w)=2((2x + 2)+x)=2(3x + 2)=6x + 4 \), which is option E (\( 6x + 4 \)) and option F (\( 2(3x + 2)=6x + 4 \)).

Answer:

E. \( 6x + 4 \), F. \( 2(3x + 2) \)