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Answer :
To solve this problem, we are given a function that describes the number of brake pads remaining after installing them on a certain number of cars, denoted as [tex]\( f(x) = 480 - 4x \)[/tex]. Additionally, there is a seemingly conflicting expression [tex]\( 148 + 4x \)[/tex]. Our task is to reconcile these expressions and find the number of cars, [tex]\( x \)[/tex], for which the function holds true.
Here's a step-by-step explanation:
1. Understanding the problem:
- Each car requires 4 brake pads.
- The manufacturing plant starts with 480 brake pads.
- After installing brake pads on [tex]\( x \)[/tex] cars, the remaining number of brake pads is modeled by the function [tex]\( f(x) = 480 - 4x \)[/tex].
2. Analyzing the conflicting information:
- We also have an expression [tex]\( 148 + 4x \)[/tex].
- It seems this expression needs to equal the remaining brake pads as described by [tex]\( f(x) \)[/tex]. Thus, we set up the equation:
[tex]\[
480 - 4x = 148 + 4x
\][/tex]
3. Solving the equation:
- Combine like terms:
[tex]\[
480 - 4x = 148 + 4x
\][/tex]
- Add [tex]\( 4x \)[/tex] to both sides to eliminate [tex]\( x \)[/tex] from one side:
[tex]\[
480 = 148 + 8x
\][/tex]
- Subtract 148 from both sides:
[tex]\[
332 = 8x
\][/tex]
- Divide both sides by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{332}{8} = 41.5
\][/tex]
4. Interpretation:
- The resulting [tex]\( x = 41.5 \)[/tex] suggests that brake pads can be installed on 41.5 cars, which in practical terms implies the number of full cars that can be completed is 41. Since we typically consider whole cars, the plant can fully service 41 cars with the given brake pads.
Therefore, the solution indicates that after determining the point where both expressions were equal, the plant can fully outfit 41 cars with brake pads based on the initial stock of 480 brake pads.
Here's a step-by-step explanation:
1. Understanding the problem:
- Each car requires 4 brake pads.
- The manufacturing plant starts with 480 brake pads.
- After installing brake pads on [tex]\( x \)[/tex] cars, the remaining number of brake pads is modeled by the function [tex]\( f(x) = 480 - 4x \)[/tex].
2. Analyzing the conflicting information:
- We also have an expression [tex]\( 148 + 4x \)[/tex].
- It seems this expression needs to equal the remaining brake pads as described by [tex]\( f(x) \)[/tex]. Thus, we set up the equation:
[tex]\[
480 - 4x = 148 + 4x
\][/tex]
3. Solving the equation:
- Combine like terms:
[tex]\[
480 - 4x = 148 + 4x
\][/tex]
- Add [tex]\( 4x \)[/tex] to both sides to eliminate [tex]\( x \)[/tex] from one side:
[tex]\[
480 = 148 + 8x
\][/tex]
- Subtract 148 from both sides:
[tex]\[
332 = 8x
\][/tex]
- Divide both sides by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{332}{8} = 41.5
\][/tex]
4. Interpretation:
- The resulting [tex]\( x = 41.5 \)[/tex] suggests that brake pads can be installed on 41.5 cars, which in practical terms implies the number of full cars that can be completed is 41. Since we typically consider whole cars, the plant can fully service 41 cars with the given brake pads.
Therefore, the solution indicates that after determining the point where both expressions were equal, the plant can fully outfit 41 cars with brake pads based on the initial stock of 480 brake pads.
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