Respuesta :
Answer:
To find the width of the walkway (denoted as
�
x), we need to subtract the area of the pool from the total area of the pool and the walkway.
Given:
Length of the pool = 15 feet
Width of the pool = 6 feet
Total area of the pool and walkway = 252 square feet
Step 1: Calculate the area of the pool:
Area of the pool
=
Length
×
Width
=
15
×
6
=
90
square feet
Area of the pool=Length×Width=15×6=90square feet
Step 2: Subtract the area of the pool from the total area of the pool and walkway to find the area of the walkway:
Area of the walkway
=
Total area
−
Area of the pool
=
252
−
90
=
162
square feet
Area of the walkway=Total area−Area of the pool=252−90=162square feet
Step 3: Since the walkway surrounds the pool on all sides, the width of the walkway (
�
x) is added to both the length and width of the pool to get the new dimensions of the pool plus the walkway:
New length
=
15
+
2
�
feet
New length=15+2xfeet
New width
=
6
+
2
�
feet
New width=6+2xfeet
Step 4: Calculate the area of the pool plus the walkway using the new dimensions:
Area of pool plus walkway
=
(
New length
)
×
(
New width
)
Area of pool plus walkway=(New length)×(New width)
Step 5: Set up an equation using the area of the pool plus the walkway:
252
=
(
New length
)
×
(
New width
)
252=(New length)×(New width)
Now, plug in the expressions for the new length and width:
252
=
(
15
+
2
�
)
(
6
+
2
�
)
252=(15+2x)(6+2x)
Step 6: Expand and simplify the equation:
252
=
90
+
30
�
+
12
�
+
4
�
2
252=90+30x+12x+4x
2
0
=
4
�
2
+
42
�
−
162
0=4x
2
+42x−162
Step 7: Solve the quadratic equation. We can divide all terms by 2 to simplify it:
0
=
2
�
2
+
21
�
−
81
0=2x
2
+21x−81
We can then factor or use the quadratic formula to solve for
�
x.
�
=
−
�
±
�
2
−
4
�
�
2
�
x=
2a
−b±
b
2
−4ac
Where:
�
=
2
,
�
=
21
,
�
=
−
81
a=2,b=21,c=−81
�
=
−
21
±
2
1
2
−
4
(
2
)
(
−
81
)
2
(
2
)
x=
2(2)
−21±
21
2
−4(2)(−81)
�
=
−
21
±
441
+
648
4
x=
4
−21±
441+648
�
=
−
21
±
1089
4
x=
4
−21±
1089
�
=
−
21
±
33
4
x=
4
−21±33
This gives us two possible solutions:
�
1
=
−
21
+
33
4
=
12
4
=
3
x
1
=
4
−21+33
=
4
12
=3
�
2
=
−
21
−
33
4
=
−
54
4
=
−
27
2
=
−
13.5
x
2
=
4
−21−33
=
4
−54
=−
2
27
=−13.5
Since the width cannot be negative, the only valid solution is
�
=
3
x=3 feet.
So, Bella should make the walkway 3 feet wide.
Step-by-step explanation:
