The problem asks us to solve the equation $60 = \frac{75}{1 + 5e^{-0.3x}}$ for $x$, and to give the answer correct to 3 decimal places.

1. Problem Description

The problem asks us to solve the equation $60 = \frac{75}{1 + 5e^{-0.3x}}$ for $x$, and to give the answer correct to 3 decimal places.

2. Solution Steps

We are given the equation $60 = \frac{75}{1 + 5e^{-0.3x}}$. We want to solve for $x$.

First, multiply both sides by $1 + 5e^{-0.3x}$:

$60(1 + 5e^{-0.3x}) = 75$

$60 + 300e^{-0.3x} = 75$

Subtract 60 from both sides:

$300e^{-0.3x} = 15$

Divide both sides by 300:

$e^{-0.3x} = \frac{15}{300} = \frac{1}{20} = 0.05$

Now, take the natural logarithm of both sides:

$\ln(e^{-0.3x}) = \ln(0.05)$

$-0.3x = \ln(0.05)$

Now, divide both sides by -0.3:

$x = \frac{\ln(0.05)}{-0.3}$

Using a calculator, we find that $\ln(0.05) \approx -2.99573227$.

$x = \frac{-2.99573227}{-0.3} \approx 9.98577423$

Rounded to three decimal places, we have $x \approx 9.986$.

3. Final Answer

$x = 9.986$