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A rectangular bird feeder costs [tex]\[tex]$18.00[/tex]. A cylindrical bird feeder costs [tex]\$[/tex]24.00[/tex]. The expected cost to keep the rectangular bird feeder filled is [tex]\[tex]$3.00[/tex] per week. The expected cost to keep the cylindrical bird feeder filled is [tex]\$[/tex]2.00[/tex] per week. The equation below models the break-even point.

[tex]
18 + 3x = 24 + 2x
[/tex]

What does [tex]x[/tex] represent?

A. The total cost to fill the rectangular bird feeder after any number of weeks

B. The total cost to fill the cylindrical bird feeder after any number of weeks

C. The number of weeks the bird feeders are filled

D. The number of bird feeders purchased each week

Sagot :

GinnyAnswer
To solve the problem and determine what [tex]\( x \)[/tex] represents in the given equation [tex]\( 18 + 3x = 24 + 2x \)[/tex], we'll follow these steps:

1. Understand the Equation:
- [tex]\( 18 + 3x \)[/tex]: The total cost of the rectangular bird feeder, which is the initial cost of [tex]$\$[/tex]18.00[tex]$ plus \(\$[/tex]3.00\) per week.
- [tex]\( 24 + 2x \)[/tex]: The total cost of the cylindrical bird feeder, which is the initial cost of [tex]$\$[/tex]24.00[tex]$ plus \(\$[/tex]2.00\) per week.

2. Set the Totals Equal:
- The equation [tex]\( 18 + 3x = 24 + 2x \)[/tex] represents the point where the total costs of both bird feeders are equal.

3. Isolate [tex]\( x \)[/tex] to Find the Break-Even Point:
- Start by subtracting [tex]\( 2x \)[/tex] from both sides of the equation to simplify:
[tex]\[ 18 + 3x - 2x = 24 + 2x - 2x \][/tex]
which simplifies to:
[tex]\[ 18 + x = 24 \][/tex]

- Next, subtract 18 from both sides to further isolate [tex]\( x \)[/tex]:
[tex]\[ 18 + x - 18 = 24 - 18 \][/tex]
which simplifies to:
[tex]\[ x = 6 \][/tex]

4. Interpret [tex]\( x \)[/tex]:
- The solution [tex]\( x = 6 \)[/tex] means that it will take 6 weeks for the total costs of filling both bird feeders to be equal.

Therefore, [tex]\( x \)[/tex] represents the number of weeks the bird feeders are filled before their total costs equalize. Thus, the correct interpretation is:

- The number of weeks the bird feeders are filled.
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