Final answer:
The population of the city reaches 312,000 after 8 years from the year 2000, according to the model P(t)=0.5t²+10t+200. This corresponds to the year 2008.
Step-by-step explanation:
A student has inquired about when the population of a city will reach 312,000 according to a given model: P(t)=0.5t²+10t+200, where P(t) is the population in thousands and t represents time in years since the year 2000. To solve for t when the population is 312,000, we convert this value into 'thousands' as the function P(t) is defined in thousands.
The equation would be:
0.5t² + 10t + 200 = 312
Simplifying and solving for t, we get:
- 0.5t² + 10t + 200 - 312 = 0
- 0.5t² + 10t - 112 = 0
- t² + 20t - 224 = 0 (Multiplying through by 2 to eliminate the fraction)
- Factoring yields (t + 28)(t - 8) = 0
- t = -28 or t = 8
We disregard the negative value as time cannot be negative and determine that t = 8 years. This means that according to the model, the population will reach 312,000 in the year 2000 + 8 = 2008. Therefore, the correct answer is B. t=8 years.