Question

(Look at picture) and ty btw <3

Answer 1

Answer: Choice A

No because the bike order does not meet the restrictions of
`4c+6a \le 120 \ \text{ and } \ 4c+4a \le 100\\\\`

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Step-by-step explanation:

c = number of child bikes

a = number of adult bikes

Each child bike takes 4 hours to build. This means 4c represents the total time spent building all c number of child bikes. Eg: if c = 2, then 4c = 4*2 = 8 hours is spent total building these two child bikes.

Meanwhile, each adult bike takes 6 hours to build. That means 6a represents the total time spent building these subset of bikes.

The expression 4c+6a is the total time spent building all of the bikes, child and adult.

The company has 120 hours of building time. The total time (4c+6a) cannot exceed this 120 hour ceiling.

This is how we arrive at
`4c+6a \le 120` which is one of the constraints or restrictions.

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Now onto the testing side of things.

Each child bike takes 4 hours to test. That gives us 4c.

Each adult bike takes 4 hours to test. We get 4a.

4c+4a = total testing time for all of the bikes

`4c+4a \le 100` is the second restriction we place, since we have at most 100 hours of testing time total. We cannot exceed this.

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To summarize so far:

• 
  `4c+6a \le 120` is the restriction for build time
• 
  `4c+4a \le 100` is the restriction for testing time

Let's see if we can build c = 20 child bikes and a = 6 adult bikes

Plug this pair of values into the first inequality

`4c+6a \le 120\\\\4*20+6*6 \le 120\\\\80+36 \le 120\\\\116 \le 120`

The last inequality is true, which means the inequality
`4c+6a \le 120` is also true for the values c = 20 and a = 6.

So far it looks like we can build these number of bikes.

But we need to check the other inequality.

`4c+4a \le 100\\\\4*20+4*6 \le 100\\\\80+24 \le 100\\\\104 \le 100\\\\`

We run into a problem. The last inequality is false, which means
`4c+4a \le 100` is false when c = 20 and a = 6 are plugged into it.

Meaning it is impossible to test 20 child bikes and 6 adult bikes in a week.

This is why choice A is the final answer.