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secondary 4 | A Maths

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Hi please assist TIA! :)

N is the number of germs at first.

After 1 minute, 21% of germs are killed. So 79% of germs are left alive.

79% = 79/100 = 0.79

So number of germs after 1 minute

= 0.79N

(this is the same as saying 79/100 of N or 79% of N)

Now let's say another minute passes. Again, 21% of the remaining 0.79N germs are killed. 79% are left alive.

Number of germs after 2 minutes

= 0.79 (0.79N)

= (0.79)²N

Likewise,after 3 minutes,

0.79(0.79)²N = (0.79)³N

After 4 minutes,

0.79(0.79)³N = (0.79)⁴N

And so on.

The power/index of 0.79 in the expression is always the same as the number of minutes that have passed.

So , after n minutes

Number of germs left = (0.79)ⁿN

So after 20 minutes,

Number of germs left = (0.79)²⁰N

≈ 0.00896N

Percentage of germs left

= Number of germs left ÷ total number of germs at first × 100%

= 0.00896N / N × 100%

= 0.896%

Percentage of germs killed

= 100% - 0.896%

= 99.104%

= 99% (2s.f)

So x = 99

We already know (0.79)ⁿN is the number of germs left after n minutes. So equate this to Neᵏⁿ

(0.79)ⁿN = Neᵏⁿ

(0.79)ⁿ = eᵏⁿ ( or (eᵏ)ⁿ )

Since the index/power is the same for both single terms on the left and right hand side, comparing both bases,

0.79 = eᵏ

k = ln 0.79

k ≈ -0.23572

k = -0.236 (3s.f)

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