When t million dollars is invested in technology for a manufacturing plant, the plant needs w(t) workers to maximize production. Labor costs are L(w) million dollars when w workers are employed. When $5 million is invested in technology, 2400 workers are needed to maximize production, and labor needs are increasing by 200 workers per million dollars. It costs $32 million to employ 2400 workers. At 2400 workers, labor costs are increasing by approximately $0.024 per worker. Evaluate w(t) when $5 million is invested in technology, and write a sentence interpreting the value.
A) workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2400 workers to minimize production.
B) workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production.
C) workers When $5 billion is invested in technology for a manufacturing plant, labor needs are increasing by 200 workers per million dollars.
D) workers When $5 billion is invested in technology for a manufacturing plant, labor needs are decreasing by 400 workers per million dollars.
E) workers When $5 billion is invested in technology for a manufacturing plant, the plant needs 2600 workers to maximize production.

Question

When t million dollars is invested in technology for a manufacturing plant, the plant needs w(t) workers to maximize production. Labor costs are L(w) million dollars when w workers are employed. When $5 million is invested in technology, 2400 workers are needed to maximize production, and labor needs are increasing by 200 workers per million dollars. It costs $32 million to employ 2400 workers. At 2400 workers, labor costs are increasing by approximately $0.024 per worker. Evaluate w(t) when $5 million is invested in technology, and write a sentence interpreting the value.
A) workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2400 workers to minimize production. 
B) workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production. 
C) workers When $5 billion is invested in technology for a manufacturing plant, labor needs are increasing by 200 workers per million dollars. 
D) workers When $5 billion is invested in technology for a manufacturing plant, labor needs are decreasing by 400 workers per million dollars. 
E) workers When $5 billion is invested in technology for a manufacturing plant, the plant needs 2600 workers to maximize production.

Quizplus · Accepted Answer

From the given information, we have:
- At $5 million investment, the plant needs 2400 workers to maximize production.
- Labor needs are increasing by 200 workers per million dollars. 
- Labor costs are increasing by approximately $0.024 per worker at 2400 workers.
- It costs $32 million to employ 2400 workers.

Let's find the labor cost function, L(w), at 2400 workers:
32 = L(2400)
So, L(w) = 32 + 0.024(w-2400)
Now, we can find the number of workers needed at $5 million investment as follows:
5 = (2400 + 200t)/1000
t = 2.5
So, at $5 million investment, the number of workers needed to maximize production is:
w(5) = 2400 + 200(2.5) = 2200

Therefore, the best choice is B, which states that when $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production.

When T Million Dollars Is Invested in Technology for a Manufacturing