Calculate the value of \(k\) .

Find the intensity of light at a depth \(25{\text{ m}}\) below the surface.

To be able to see clearly, a diver needs the intensity of light to be at least \(65\% \).

Using your graphic display calculator, find the greatest depth below the surface at which she can see clearly.

The table below gives the intensity of light (correct to the nearest integer) at different depths.

Using this information draw the graph of \(I\) against \(d\) for \(0 \leqslant d \leqslant 100\) . Use a scale of \(1{\text{ cm}}\) to represent 10 metres on the horizontal axis and 1 cm to represent \(10\% \) on the vertical axis.

Some sea creatures have adapted so they can see in low intensity light and cannot tolerate too much light.

Indicate clearly on your graph the range of depths sea creatures could inhabit if they can tolerate between \(5\% \) and \(35\% \) of the light intensity at the surface.

\(d = 0\), \(k = 100\)     (M1)(A1)(G2)

Note: Award (M1) for \(d = 0\) seen.

\(I = 100 \times {(1.05)^{ - 25}} = 29.5(\% )\) (\(29.5302 \ldots \))     (M1)(A1)(ft)(G2)

\(65 = 100 \times {(1.05)^{ - d}}\)     (M1)

Note: Award (M1) for sketch with line drawn at \(y = 65\) .

\(d = 8.83{\text{ (m)}}\) (\(8.82929 \ldots \))     (A1)(ft)(G2)

(A1) for labels and scales
(A2) for all points correct, (A1) for 3 or 4 points correct
(A1) for smooth curve asymptotic to the \(x\)-axis     (A4)

Lines in approx correct positions on graph     (M1)
The range of values indicated (arrows or shading) \(22\)–\(60{\text{ m}}\)     (A1)