The population of Japan, p, in millions of people at time t, where negative values of t represent a number of years before January 1^("st "), 2000 and positive values of t represent a number of years after January 1^("st "),2000 is projected as: p=128-0.012(t-9.17)^(2) According to this projection, during which year does Japan reach its maximum population?

Question

The population of Japan,  p, in millions of people at time  t, where negative values of  t represent a number of years before January  1^("st "), 2000 and positive values of  t represent a number of years after January  1^("st "),2000 is projected as:  p=128-0.012(t-9.17)^(2) According to this projection, during which year does Japan reach its maximum population?

Bytelearn · Accepted Answer

Population Projection Function: The given population projection function is: p = 128 - 0.012(t - 9.17)² To find the year when Japan reaches its maximum population, we need to identify the vertex of the parabola represented by this function, since the vertex will give us the maximum value of p.Identifying Parabola Vertex: The function is in the form of a parabola with a negative coefficient for the squared term, which means it opens downwards. The maximum value of p occurs at the vertex of the parabola. The vertex form of a parabola is given by: p = a(t - h)² + k where (h, k) is the vertex of the parabola.Comparing with Vertex Form: Comparing the given function with the vertex form, we can see that h = 9.17 and k = 128. Since the coefficient a is negative (-0.012), the vertex represents the maximum point of the parabola.Finding Maximum Population Time: The vertex (h, k) corresponds to the time t = h and the maximum population p = k. Therefore, the maximum population occurs at t = 9.17 years.Calculating Actual Year: To find the actual year when the maximum population is reached, we need to add the value of t to the base year, which is 2000. Year of maximum population = 2000 + t Year of maximum population = 2000 + 9.17Calculating the year: Year of maximum population ≈ 2000 + 9.17 Year of maximum population ≈ 2009.17 Since we count full years, we round down to the nearest whole year. Year of maximum population = 2009

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