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 46

which solution method did you use?

• logarithms
• graphing \\(y = 3.915(1.106)^x\\) and tracing
• graphing \\(y = 3.915(1.106)^x\\) and \\(y = 400\\), then finding the x-value of the intersection

by the \\(46^{\text{th}}\\) day, there are 400 water lilies in the pond. the estimate you made

• close to 46 (or exactly right) because it was estimated well.
• too low because the quick exponential growth was not accounted for.
• too high because the exponential growth was overestimated.

Explanation:

Set up the exponential equation

Using the Solving Exponential Equations knowledge point

$$ 3.915(1.106)^x = 400 $$

Solve for x algebraically

Using the Solving Exponential Equations Algebraically knowledge point

$$
LATEXBLOCK0
$$

Evaluate the estimation accuracy

Using the Exponential Growth and Decay Behavior knowledge point
The algebraic solution yields \(x \approx 46.01\), which is extremely close to the actual value of 46. Therefore, the estimate of 46 days is highly accurate because the exponential regression model was well-fitted to the data.

Answer:

Question 1

• (A) Logarithms (Correct answer)
• (B) Graphing \(y = 3.915(1.106)^x\) and tracing
• (C) Graphing \(y = 3.915(1.106)^x\) and \(y = 400\), then finding the x-value of the intersection

Question 2

• (A) close to 46 (or exactly right) because it was estimated well. (Correct answer)
• (B) too low because the quick exponential growth was not accounted for.
• (C) too high because the exponential growth was overestimated.