Notice that x is still measured in whole hours. What exponent do you need to reflect a doubling every half hour?
C
c
Overtaking is another way of saying greater than.
D
d
Create functions of the left-hand and right-hand side, then solve for the point of intersection.
A
a
y=200 000(2)^x
B
b
y=50 000(4)^x
C
c
50 000(4)^x>200 000(2)^x
D
d
x > 2
Practice makes perfect
a
A general exponential function is written as y=a(b)^x, where a is the starting value and b is the rate of change. Examining the table, we see that the initial population is a= 200 000. Let's add this to the general form.
y= 200 000(b)^x
What about the rate of change? Well, we know that the population size doubles every hour. This is just another way of saying that it is multiplied by 2. Therefore, the rate of change is b= 2.
y=200 000( 2)^x
b
Just like in Part A, we can see from the table that the initial population is a=50 000. Let's add this to the general form of an exponential function.
y=50 000(b)^xAs for the rate of change, we know that it doubles every half hour. Therefore, if x is still measured in hours, we have to multiply x in the exponent by 2 to get a doubling every half hour.
y=50 000(2)^(2x)
By using the Power Rules we can rewrite this.
Overtaking is another way of expressing greater than, which mathematically can be written as >. With this information we can write an inequality.
Sample B > Sample A
If we substitute the expressions for these samples, we get an inequality that describes how many hours it will take for Sample B to overtake Sample A.
50 000(4)^x > 200 000(2)^x
d
To solve the inequality by graphing, we first have to create functions out of the equation's left-hand and right-hand side.
y=50 000(4)^x and y=200 000(2)^x
To enter them in your calculator, push Y= and write them in the first two rows.
With our functions, entered push GRAPH to draw them.
We can see there is one point of intersection. To find this point we can use the intersect option. Push 2nd and TRACE, then choose the list's fifth option. Now we have to select the two graphs and provide the calculator with a guess where the intersection might be.
The graphs intersect at x=2. Now we have to identify the x-values that makes the inequality true. To calculate this, we can substitute a test point that is on either side of x=2 and check if the inequality holds true.
