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1 Pure Mathematics

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Question 1107

The intensity of light, III lumens, penetrating a deep lake decreases with depth, ddd metres, according to the exponential model:

I=I0kd I = I_0 k^d I=I0​kd

where:

  • I0I_0I0​ is the intensity of light at the surface.
  • kkk is a constant, with 0<k<10 < k < 10<k<1.
  • ddd is the depth below the surface in metres.
a.

Sketch the graph of III against ddd for a day where the surface intensity I0=1200I_0 = 1200I0​=1200.

[2]
b.

Given that the light intensity is halved at a depth of 8 metres, show that the constant kkk is approximately 0.9170040.9170040.917004 to 6 decimal places.

[2]
c.

On a brighter day, the surface intensity is 5000 lumens. Calculate, to 3 significant figures, the light intensity at a depth of 20 metres.

[2]
d.

A specific sensor requires at least 312.5 lumens to operate. If the surface intensity is 2500 lumens, use logarithms to calculate the maximum depth, to the nearest metre, at which the sensor will function.

[3]
Markscheme

1 Pure Mathematics Questions

  1. A Level
  2. /Maths
  3. /1 Pure Mathematics

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