Question

The growth rate of a bacterial culture is given by \( x \left( 1 - \frac{x}{100} \right) \), where \( x \) is the density of the culture. The growth rate is maximum when the density is equal to ............ (round off to 1 decimal place)

Hint:

To find the maximum of a function, take the derivative and set it equal to zero. Solve for \( x \) to find the critical point.

Answer 1

Correct Answer: 49.9

Step 1: Take the derivative of the growth rate function.
The growth rate is given by: \[ G(x) = x \left( 1 - \frac{x}{100} \right) \] To find the value of \( x \) that maximizes the growth rate, we take the derivative of \( G(x) \) with respect to \( x \): \[ \frac{dG(x)}{dx} = 1 - \frac{2x}{100} \] Step 2: Set the derivative equal to zero.
To find the maximum, set the derivative equal to zero: \[ 1 - \frac{2x}{100} = 0 \] Solving for \( x \): \[ \frac{2x}{100} = 1 \quad \Rightarrow \quad x = 50 \] Step 3: Conclusion.
Thus, the density at which the growth rate is maximum is \( \boxed{50.0} \).