Regression equation: [tex]y=3.915(1.106)^x[/tex]
Which two equations below could you solve to find [tex]D[/tex], the number of days it takes the water lily population to double?
* [tex]2=3.915(1.106)^D[/tex]
* [tex]7.830=3.915(1.106)^D[/tex]
* [tex]7.830=3.915(2)^D[/tex]
* [tex]2=1.106^D[/tex]
Solve either of the correct equations using any method. Round to the nearest whole number.
[tex]D=[/tex] days
Answer (1)
Set up the equation 2 = 1.10 6 D to find the number of days it takes for the water lily population to double.
Take the natural logarithm of both sides: ln ( 2 ) = D ln ( 1.106 ) .
Solve for D : D = l n ( 1.106 ) l n ( 2 ) .
Calculate D and round to the nearest whole number: D ≈ 7 days.
Explanation
Understanding the Problem We are given the regression equation y = 3.915 ( 1.106 ) x , where y represents the population of water lilies and x represents the number of days. We want to find the number of days, D , it takes for the population to double. This means we want to find D such that the population is twice the initial population.
Setting up the Equation The initial population is 3.915 (when x = 0 ). So, we want to find D such that y = 2 × 3.915 = 7.830 . Thus, we need to solve the equation 7.830 = 3.915 ( 1.106 ) D . Dividing both sides by 3.915 , we get 2 = ( 1.106 ) D .
Applying Logarithms To solve for D , we can take the natural logarithm of both sides of the equation 2 = ( 1.106 ) D . This gives us ln ( 2 ) = ln (( 1.106 ) D ) . Using the power rule of logarithms, we have ln ( 2 ) = D ln ( 1.106 ) .
Isolating D Now, we can solve for D by dividing both sides by ln ( 1.106 ) : D = ln ( 1.106 ) ln ( 2 )
Calculating D Using a calculator, we find that D = ln ( 1.106 ) ln ( 2 ) ≈ 0.1006 0.6931 ≈ 6.8798
Final Answer Rounding to the nearest whole number, we get D = 7 days.
Examples
Imagine you are tracking the growth of a bacterial colony in a petri dish. The regression equation helps you model how the colony's size changes over time. By finding the doubling time, you can predict when the colony will reach a certain size, which is crucial for planning experiments or controlling contamination. This type of exponential growth model is also used in finance to calculate compound interest or in environmental science to model population growth.
