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Solve for x in each of the following equations At least 2 of the 4 must be solved algebraically. When solving graphically, sketch or paste a digital image of the graph you used to solve for the question.

 

  1. e^(2x)-e^(x)-12=0
  2. 2(5^(x+2))=5(2^(x+5))
  3. The percentage of sunlight that penetrates the water of the ocean decays exponentially with depth and can be described by the equation L(x)=100e^((k)x) where L is the % of sunlight, x is the depth of water in metres, and k is a constant that varied depending upon water conditions (k=-0.2 for clear water).
    1. Assuming the water is clear, what percentage of the sun’s light penetrates water at 9.0m depth?
    2. How deep is clear water when only 5% of the sun’s light reaches it?
  4. A 200-g sample of a radioactive substance is placed in a chamber to be tested. After 3 h have passed only 140 g of the sample remains.
    1. Determine the half-life of this substance, to the nearest hundredth of an hour.
    2. How much of the substance will remain after 1.5 days?
 May 15, 2018
 #1
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4)

 

140 = 200 * (1/2)^(3/h), where h = half-life.

Divide both sides by 200

0.7 = 1/2^(3/h) 

Take the natural log of both sides

3/h =Ln(0.7) / Ln(1/2)

3/h =0.514573173

Cross multiply

0.514573173h = 3

Divide both sides by 0.514573173

h =5.83  hours - the half-life of this radioactive substance. 

 

1.5 days x 24 hours = 36 hours.

Use the same above formula:

R =200 * 1/2^(36/5.83)

R =200 * 1/2^(6.175)

R =200 *   0.0138401175...

R =2.768 grams will remain after 1.5 days or 36 hours.

 May 15, 2018
 #2
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3)  I modified your equation: e^-(xk)

 

L(x)=100e^-((k)x)

L(9) =100*e^-(0.2*9)

L(9) =100 * e^-1.8

L(9) =100 * 0.1653

L(9) =16.53% - percentage of sunlight that penetrates water at a depth of 9m.

 

5% = 100 * e^-(0.2x),        solve for x

0.05 =100 * e^-(0.2x)        divide both sides by 100

0.0005 = e^-(0.2x)             take the natural log of both sides

-0.2x = -7.6                        divide both sides by -0.2

x = 38 meters - at which only 5% of sunlight will penetrate the water.

 

 

 May 15, 2018

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