Chapter SH, Problem 9.19EP

To determine

To calculate the time when there are 200 cells in the Petri dish.

Answer to Problem 9.19EP

The time when the number of cells is 200 is after $4.97$ hours.

Explanation of Solution

Given information :

The function provided is $n=6t^2-4.5t+74$

n is the number of cells and t is the time that the cells have been in the Petri dish.

Formula used :

To calculate the time when $n=200$ , use the formula to find roots of a quadratic equation. This can be found by using $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ . Here ‘a’, ‘b’ and ‘c’ are the co-efficient of the given equation.

Here, $a=6,b=-4.5\text{ and}c=-126$

Calculation :

The given quadratic equation.

  $n=6t^2-4.5t+74$

Putting the value of $n=200$ in the equation,

  $200=6t^2-4.5t+74$

Simplifying the expression.

  $6t^2-4.5t-126=0$

Putting the values in the formula.

  $t=\frac{-(-4.5)\pm \sqrt{{(-4.5)}^2-4\times 6\times (-126)}}{2\times 6}$

Simplifying the expression.

  $t=\frac{4.5\pm \sqrt{20.25+3024}}{12}$

Further simplification

  $t=\frac{4.5\pm \sqrt{3044.25}}{12}$

Removing the square root

  $t=\frac{4.5\pm 55.17}{12}$

First square root is

  $t=\frac{59.67}{12}$

Second square root is

  $t=-\frac{50.67}{12}$

Therefore in decimal form $t=4.97$

Neglecting the negative root as time cannot be negative.

Thus, the number of cells is 200 at $t=4.97$ hours.