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Answer :
To find the equation for the proportional relationship between the volume of paint [tex]\( x \)[/tex] (in gallons) and the area of fence [tex]\( y \)[/tex] (in square feet) that Zeke can paint, we need to determine the constant of proportionality.
Here's a step-by-step guide on how to solve this:
1. Identify the Given Information:
- Zeke uses 4.3 gallons of paint to cover an area of 597.7 square feet.
2. Set Up the Proportion:
- Since [tex]\( x \)[/tex] and [tex]\( y \)[/tex] have a proportional relationship, we can express it using the equation [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
3. Calculate the Constant of Proportionality ([tex]\( k \)[/tex]):
- To find [tex]\( k \)[/tex], divide the area of the fence by the volume of paint:
[tex]\[
k = \frac{597.7}{4.3}
\][/tex]
4. Substitute to Find [tex]\( k \)[/tex]:
- Solving the above will give us:
[tex]\[
k \approx 139
\][/tex]
5. Write the Equation:
- Now, using the value of [tex]\( k \)[/tex], you can write the equation representing the proportional relationship:
[tex]\[
y = 139x
\][/tex]
So, the equation for the relationship between the volume of paint [tex]\( x \)[/tex] and the area of fence [tex]\( y \)[/tex] is [tex]\( y = 139x \)[/tex]. This means for each gallon of paint, Zeke can paint approximately 139 square feet of the fence.
Here's a step-by-step guide on how to solve this:
1. Identify the Given Information:
- Zeke uses 4.3 gallons of paint to cover an area of 597.7 square feet.
2. Set Up the Proportion:
- Since [tex]\( x \)[/tex] and [tex]\( y \)[/tex] have a proportional relationship, we can express it using the equation [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
3. Calculate the Constant of Proportionality ([tex]\( k \)[/tex]):
- To find [tex]\( k \)[/tex], divide the area of the fence by the volume of paint:
[tex]\[
k = \frac{597.7}{4.3}
\][/tex]
4. Substitute to Find [tex]\( k \)[/tex]:
- Solving the above will give us:
[tex]\[
k \approx 139
\][/tex]
5. Write the Equation:
- Now, using the value of [tex]\( k \)[/tex], you can write the equation representing the proportional relationship:
[tex]\[
y = 139x
\][/tex]
So, the equation for the relationship between the volume of paint [tex]\( x \)[/tex] and the area of fence [tex]\( y \)[/tex] is [tex]\( y = 139x \)[/tex]. This means for each gallon of paint, Zeke can paint approximately 139 square feet of the fence.
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