Question

regression equation : y = 3.915(1.106)^x
which two equations below could you solve to find d, the number of days it takes the water lily population to double?
x
2 = 3.915(1.106)^d

7.830 = 3.915(1.106)^d

x
7.830 = 3.915(2)^d

2 = 1.106^d

complete

solve either of the correct equations using any method. round to the nearest whole number.
d = (\boxed{quad}) days
done
image of water lilies

Explanation:

Step1: Choose the equation

To find the doubling time, we know that the initial population is \(3.915\) (from the regression equation \(y = 3.915(1.106)^x\)), and when the population doubles, \(y=2\times3.915 = 7.830\). So the equation \(7.830=3.915(1.106)^D\) (or simplifying by dividing both sides by \(3.915\) gives \(2=(1.106)^D\)) is appropriate. Let's use \(2=(1.106)^D\).

Step2: Take logarithm on both sides

Take the natural logarithm (ln) of both sides: \(\ln(2)=\ln((1.106)^D)\). Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get \(\ln(2)=D\ln(1.106)\).

Step3: Solve for D

Rearrange the formula to solve for \(D\): \(D = \frac{\ln(2)}{\ln(1.106)}\). Calculate \(\ln(2)\approx0.6931\) and \(\ln(1.106)\approx0.1008\). Then \(D=\frac{0.6931}{0.1008}\approx6.876\). Rounding to the nearest whole number, \(D\approx7\).

Answer:

7