Chapter 4.5, Problem 33E

(a)

To determine

To find:Validity of the model according to the data given.

Answer to Problem 33E

Fairly accurate

Explanation of Solution

Given information: Population: The population of Manhattan Island in New York City between 1890 to 1970 can be modeled by $P\left(x\right)=-0.78x^4+133x^3-7500x^2+147,500x+1,440,000,$ where $P\left(x\right)$ represents the population and x represents the number of years since 1890.

  

Calculation:

  

  

You can see from the graphs representing 1910 and 1930, that the graph is fairly accurate.

(b)

To determine

To use: The model to predict the population in 1980.

Answer to Problem 33E

  $-253,800$ , the population is actually negative

Explanation of Solution

Given information: Population: The population of Manhattan Island in New York City between 1890 to 1970 can be modeled by $P\left(x\right)=-0.78x^4+133x^3-7500x^2+147,500x+1,440,000,$ where $P\left(x\right)$ represents the population and x represents the number of years since 1890.

  

Calculation:

For 1980, $f\left(90\text{ years}\right)=-253,800$

Meaning the population is actually negative.

(c)

To determine

To find:happening between 1970 and 1980 according to the model.

Answer to Problem 33E

The population completely disappeared

Explanation of Solution

Given information: Population: The population of Manhattan Island in New York City between 1890 to 1970 can be modeled by $P\left(x\right)=-0.78x^4+133x^3-7500x^2+147,500x+1,440,000,$ where $P\left(x\right)$ represents the population and x represents the number of years since 1890.

  

Calculation:

According the graph,

Somewhere between 1970 and 1980,

The population completely disappeared.

(d)

To determine

To find:validity of this model for any time.

Answer to Problem 33E

No

Explanation of Solution

Given information: Population: The population of Manhattan Island in New York City between 1890 to 1970 can be modeled by $P\left(x\right)=-0.78x^4+133x^3-7500x^2+147,500x+1,440,000,$ where $P\left(x\right)$ represents the population and x represents the number of years since 1890.

  

Calculation:

No, obviously polynomial is reasonably accurate for the years requested,

However, outside the bounds of the years given,

It has no merit to accurately predict population.