Question

The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model R = 150 for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model C = 50 + 0.4t2, where t is the time (in years). Approximate the profit over the 10-year period. (Round your answer to two decimal places.)

Answer 1

To approximate the profit over 10 years, compute the total revenue by multiplying the annual revenue by 10 years and summing the yearly costs based on the cost formula. Then, subtract the total cost from the total revenue to find the profit. The profit is approximately $846 million.

Step-by-step explanation:

To approximate the profit over the 10-year period for the manufacturing process, we use the given revenue and cost models. The revenue model is R = 150 (in millions of dollars per year), and it is constant over the 10 years. On the other hand, the cost model is C = 50 + 0.4t2, where t is the time in years.

To determine the total profit, we calculate the difference between the total revenue and the total costs over the 10 years:

1. 
   
3. Calculate the total revenue over 10 years: Rtotal = R × 10 = 150 × 10 = 1500 million dollars.
4. 
   
6. Calculate the total cost over 10 years:
   
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   • For each year, the cost is C = 50 + 0.4t2.
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   • Sum the costs for each year from t = 1 to t = 10.
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7. 
   
9. Subtract the total cost from the total revenue to get the total profit.

To find the total cost, we need to sum the cost for each year:

Ctotal = Σ(50 + 0.4t2) over t from 1 to 10. Calculating this, we find:

Ctotal = 50(10) + 0.4(12 + 22 + … + 102)
= 500 + 0.4(385)
= 500 + 154
= 654 million dollars.

Now we can calculate the profit:

Profit = Rtotal - Ctotal
= 1500 million dollars - 654 million dollars
= 846 million dollars.

The approximate profit over the 10-year period is $846 million when rounded to the nearest million dollars.