Question

regression equation: $y = 3.915(1.106)^x$
the pond can hold 400 water lilies. by what day will the pond be full? write and solve an equation.
the pond will be full by the end of day
retry

Explanation:

Step1: Set up the equation

We know the regression equation is \( y = 3.915(1.106)^x \), and we want to find \( x \) when \( y = 400 \). So we set up the equation:
\( 400 = 3.915(1.106)^x \)

Step2: Divide both sides by 3.915

To isolate the exponential term, divide both sides of the equation by 3.915:
\( \frac{400}{3.915}=(1.106)^x \)
Calculating \( \frac{400}{3.915}\approx102.17 \), so we have \( 102.17=(1.106)^x \)

Step3: Take the natural logarithm of both sides

Using the property of logarithms \( \ln(a^b)=b\ln(a) \), take the natural log of both sides:
\( \ln(102.17)=\ln((1.106)^x) \)
Which simplifies to \( \ln(102.17)=x\ln(1.106) \)

Step4: Solve for x

Divide both sides by \( \ln(1.106) \) to solve for \( x \):
\( x = \frac{\ln(102.17)}{\ln(1.106)} \)
Calculating the values: \( \ln(102.17)\approx4.627 \), \( \ln(1.106)\approx0.1008 \)
Then \( x=\frac{4.627}{0.1008}\approx45.9 \)
Since we are dealing with days (a whole number context, and it will be full by the end of the day), we round up to the next whole number, so \( x = 46 \)

Answer:

46