1) The intensity of light, I, when the light passes through water is given by the equation, I=10^1-0.13x where 'x' is the depth of water in metres. Most Merepeople require a light intensity of 4.2 units to grow. Determine the maximum depth of water where Merepeople receive the required amount of light.

To determine the maximum depth of water where Merepeople receive the required amount of light, we need to solve for \( x \) in the equation given that the intensity \( I \) is 4.2 units.<br /><br />The equation for the intensity of light is:<br />\[ I = 10^{1 - 0.13x} \]<br /><br />We are given that \( I = 4.2 \). Therefore, we can set up the equation:<br />\[ 4.2 = 10^{1 - 0.13x} \]<br /><br />To solve for \( x \), we first take the logarithm (base 10) of both sides of the equation:<br />\[ \log_{10}(4.2) = \log_{10}(10^{1 - 0.13x}) \]<br /><br />Using the property of logarithms that \(\log_{10}(a^b) = b \cdot \log_{10}(a)\), we get:<br />\[ \log_{10}(4.2) = 1 - 0.13x \]<br /><br />Next, we calculate \(\log_{10}(4.2)\):<br />\[ \log_{10}(4.2) \approx 0.6232 \]<br /><br />So the equation becomes:<br />\[ 0.6232 = 1 - 0.13x \]<br /><br />Now, we solve for \( x \):<br />\[ 0.13x = 1 - 0.6232 \]<br />\[ 0.13x = 0.3768 \]<br />\[ x = \frac{0.3768}{0.13} \]<br />\[ x \approx 2.8985 \]<br /><br />Therefore, the maximum depth of water where Merepeople receive the required amount of light intensity of 4.2 units is approximately \( 2.90 \) meters.