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? $2 = 3.915(1.106)^d$ $7.830 = 3.915(1.106)^d$ $7.830 = 3.915(2)^d$ $2 = 1.106^d$ done

Explanation:

Step1: Understand Doubling Population

The original population is given by the initial value of the regression equation, which is \( 3.915 \) (when \( x = 0 \), \( y = 3.915(1.106)^0=3.915 \)). When the population doubles, the new population \( y \) will be \( 2\times3.915 = 7.830 \).

Step2: Substitute into Regression Equation

The regression equation is \( y = 3.915(1.106)^x \). For the number of days \( D \) (so \( x = D \)) when the population doubles, we substitute \( y = 7.830 \) into the equation. So we get \( 7.830=3.915(1.106)^D \).

Step3: Simplify the Equation

If we divide both sides of \( 7.830 = 3.915(1.106)^D \) by \( 3.915 \), we have \( \frac{7.830}{3.915}=(1.106)^D \). Since \( \frac{7.830}{3.915}=2 \), this simplifies to \( 2 = 1.106^D \).

Answer:

The two equations are \( 7.830 = 3.915(1.106)^D \) and \( 2 = 1.106^D \) (or in the options: the second option \( 7.830 = 3.915(1.106)^D \) and the fourth option \( 2 = 1.106^D \)).