Question

The population of a certain inner-city area is estimated to be declining according to the model P(t) = 342,000e^-0.023t, where t is the number of years from the present. What does this model predict the population will be in 14 years? Round to the nearest person.

Answer 1

The model predicts that the population of the inner-city area will be approximately 267,315 people in 14 years.

Step-by-step explanation:

The population model
`\(P(t) = 342,000e^(-0.023t)\)` describes the population
`(\(P\))` of the inner-city area as a function of time
`(\(t\)).` To find the population in 14 years
`(\(t = 14\))`, we substitute
`\(t = 14\)` into the model:

`\[P(14) = 342,000e^(-0.023 * 14)\]`

Calculating this expression yields
`\(P(14) \approx 267,315\)`. Therefore, the model predicts that the population will be approximately 267,315 people in 14 years.

The exponential decay model is commonly used in demographic studies to describe population decline over time. In this case, the model incorporates a negative exponent (-0.023t), signifying a decrease in population. The rate of decline is determined by the coefficient of the exponent, where a higher absolute value leads to a faster decline. Understanding and using such population models aids in predicting demographic trends and assists urban planners and policymakers in making informed decisions regarding resource allocation and community development.

The rounded prediction of 267,315 people in 14 years provides valuable insights for urban planning and resource management in the inner-city area. This estimate helps anticipate future needs related to housing, healthcare, and infrastructure, allowing for proactive measures to address the challenges associated with a declining population.