Question

The number P, in hundreds of bacteria in a sample, can be modeled by the Equation P= T^4+ 5T^3 + 5T^2 + 6t where t is measured in weeks. explain how to determine the number of weeks which the population would be greater than 10,000

Answer 1

The values of T which satisfies the inequality are:
`(8.8, \infty)`

Explanation:

In the equation
`P= T^4+ 5T^3 + 5T^2 + 6T`

• Represent P by 10,000
• Write an inequality where the expression is greater than 10000:
  `T^4+ 5T^3 + 5T^2 + 6T>10000`
• Get 0 on one side of the inequality.

`T^4+ 5T^3 + 5T^2 + 6T-10000>0`

• Graph the polynomial function.
• We have real x-intercepts of 8.84 and -11.38.
• Determine intervals where the graph is above the x-axis.
• Since negative values of x in this situation are irrelevant, the values of T which satisfies the inequality are:
  `(8.8, \infty)`
• We now test a value from the set of solution to see if it is valid.
• Let T=9

`T^4+ 5T^3 + 5T^2 + 6T>10000\\9^4+ 5(9)^3 + 5(9)^2 + 6(9)>10000\\10665>10000`

Since 10665 is grater than 10000, the result is reasonable.