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Answer :
Each week, the factory receives [tex]$400$[/tex] kg of tomatoes. Out of these, [tex]$72\%$[/tex] are used for the soup, which amounts to
[tex]$$0.72 \times 400 = 288 \text{ kg.}$$[/tex]
This means that the remaining tomatoes left at the end of the week equal
[tex]$$400 - 288 = 112 \text{ kg.}$$[/tex]
The leftover tomatoes accumulate week by week. We need to find after how many weeks the accumulated leftover is at least [tex]$400$[/tex] kg so that the delivery is no longer necessary.
Let the number of weeks be [tex]$n$[/tex]. Since each week leaves [tex]$112$[/tex] kg unused, the total leftover after [tex]$n$[/tex] weeks is
[tex]$$112 \times n.$$[/tex]
We set up the inequality to determine [tex]$n$[/tex]:
[tex]$$112 \times n \ge 400.$$[/tex]
Dividing both sides by [tex]$112$[/tex] we get:
[tex]$$n \ge \frac{400}{112} \approx 3.57.$$[/tex]
Since [tex]$n$[/tex] must be a whole number (you can’t have a fraction of a week in this context), we round up to the next whole number. Thus, [tex]$n = 4$[/tex].
To confirm, let’s look at the cumulative totals:
- After Week 1: [tex]$112$[/tex] kg
- After Week 2: [tex]$112 \times 2 = 224$[/tex] kg
- After Week 3: [tex]$112 \times 3 = 336$[/tex] kg
- After Week 4: [tex]$112 \times 4 = 448$[/tex] kg
At Week 4, the cumulative leftover is [tex]$448$[/tex] kg, which is enough to meet the [tex]$400$[/tex] kg threshold.
Therefore, the factory does not need to take delivery of any more tomatoes after [tex]$\boxed{4}$[/tex] weeks.
[tex]$$0.72 \times 400 = 288 \text{ kg.}$$[/tex]
This means that the remaining tomatoes left at the end of the week equal
[tex]$$400 - 288 = 112 \text{ kg.}$$[/tex]
The leftover tomatoes accumulate week by week. We need to find after how many weeks the accumulated leftover is at least [tex]$400$[/tex] kg so that the delivery is no longer necessary.
Let the number of weeks be [tex]$n$[/tex]. Since each week leaves [tex]$112$[/tex] kg unused, the total leftover after [tex]$n$[/tex] weeks is
[tex]$$112 \times n.$$[/tex]
We set up the inequality to determine [tex]$n$[/tex]:
[tex]$$112 \times n \ge 400.$$[/tex]
Dividing both sides by [tex]$112$[/tex] we get:
[tex]$$n \ge \frac{400}{112} \approx 3.57.$$[/tex]
Since [tex]$n$[/tex] must be a whole number (you can’t have a fraction of a week in this context), we round up to the next whole number. Thus, [tex]$n = 4$[/tex].
To confirm, let’s look at the cumulative totals:
- After Week 1: [tex]$112$[/tex] kg
- After Week 2: [tex]$112 \times 2 = 224$[/tex] kg
- After Week 3: [tex]$112 \times 3 = 336$[/tex] kg
- After Week 4: [tex]$112 \times 4 = 448$[/tex] kg
At Week 4, the cumulative leftover is [tex]$448$[/tex] kg, which is enough to meet the [tex]$400$[/tex] kg threshold.
Therefore, the factory does not need to take delivery of any more tomatoes after [tex]$\boxed{4}$[/tex] weeks.
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