Example
A population of 75 foxes in a wildlife preserve quadruples in every 15 years.
a) Assign variables to the quantities and write an exponential equation to represent the future population in terms of time.
b) How many foxes will be in the population after 45 years?
c) How many foxes will be in the population after 3 years?
d) After how many years will the population double?
e) After how many years will the foxes reach a population of 100,000
Solution
a) The fox population can be written as a function of time. Let P represent the turtle population. Let t represent the number of years.
b) To the find the population after 45 years, substitute t=45 into the equation
In 45 years, the population of foxes would be 4,800.
c) The question asks how many foxes will there be in 3 years
The population will be 98 in 3 years.
d) The initial population is 75, and double that would be 150, hence we substitute the left hand side, P, for 150 and thus need to solve one of the following equations:
Now divide both sides by 75, 150 becomes 2, and the coefficient cancels out. Our equation becomes:
First of all, notice that this is always the case in this type of situation. That is, if we know the future population is related to the initial population by a factor, then the initial population will always cancel out. Do not hesitate to show the work, however, which helps in preventing errors on this type of equation. Now we can convert to the logarithmic form using the following formula:
Which is already solved for the decade equation, and only requires one more step on the year equation, multiplying both sides by 15.
Now, simply apply the Change of Base property so we can type this in our calculator to get a decimal approximation:
The population of foxes would double in 7 years and 6 months.
e) To find when the population will reach 100,000, we simply substitute 100,000 into the P variable of the equation,
1. Just like in the doubling problem, we have to divide both sides by the coefficient
2. Just like in part (d), we convert to the logarithmic form 3. Just like in Part (d), the variable is already solved for in the decade equation, and the years equation only requires one more step, multiplying both sides by 15. 4. Again, like in Part (d), we apply the Change of Base property so we can get decimal approximations. |
The fox population will reach 100,000 in 78 years.
EQUATION, TABLE, AND GRAPH
Exponential trend-line used
Reflection
In this example, what we're trying to find out this the population, of 75 foxes that quadruple in 15 year periods, after a certain amount of years. To find the population after a certain amount of years or amount of years for this amount of foxes by using the exponential formula. 'P' representing the population of the foxes and 'y' for the amount of years. This problem is what exponential growth is. Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative. The equation that is used to find the population of the foxes is y=a*b^x. In this case, the formula is P=a*b^x/15. The equation this is used to find the amount of years for a certain population is P=B^E which is converted to log subscript B (P)=E. This was one of the more advance problems in Algebra that
are more intimidating. Overall the problem is a well made difficult problem as
an exponential property.