Word Problem

Question

An oil spill in the ocean is increasing in its surface area by 21% each day. It begins at two square miles. How long before it reaches 10 square miles?

Answer 1

The oil spill grow will generate a series which is

A=Ao (1.21)^(n-1)

A=10

Ao=2

5=(1,21)^(n-1)

(n-1) = log5/log1,21

n-1 = 8,44

n= 7.44 days

 

 

 

Answer 2

Area of Oil Spill = 2 square miles when time (t) = 0

Area of Oil Spill after 1 day = 2 x (1.21)

STEP 1: Establish the formula
Area of Oil Spill after t days = 2 x (1.21)^t (raised to the power of t)

STEP 2: If the area of the oil spill is 10 square miles, set the formula equal to 10.
10 = 2 x (1.21)^t

STEP 3: Take the logarithm of both sides
log 10 = log [2 x(1.21)^t]
1 = log (2) + t x log (1.21)

REASON: 10^1 = 10
    the log of two amounts that are multiplied together equals the sum of the logs.
EXAMPLE: log (a x b) = log a + log b
   And the log of an amount raised to a power is equal to that power times the log
EXAMPLE: log b^2 = 2 x log b

STEP 4: Find the log values of the numbers
log(2) = .3010 log (1.21) = .0828

STEP 5: Plug the log values into the formula
1 = .3010 + t x (.0828)

STEP 6: Subtract .3010 from each side of the equation
.699 = t x (.0828)

Step 7: Divide each side by .0828
t = 8.442

 

  The answer is 8.442 days.

Answer 3

This is an exponential growth problem:

 

Area = (2 mi²)*(1.21)^t

 

Where t is the number of days after the spill.  Solve for t when the Area = 10 mi²:

 

10 = 2*(1.21)^t

 

5 = (1.21)^t

 

ln(5) = t*ln(1.21)

 

ln(5)/ln(1.21) = t

 

8.44 days = t