the data show that the population of lily pads has an increasing growth rate.
an exponential function would be most suitable to model these data.
the general form of an exponential function is $y = ab^x$. use the regression calculator to find the values of $a$ and $b$ for the water lily population growth. round to the nearest thousandth.
$a =$ and $b =$

table:
$x$ (days) | $y$ (water lilies)
0 | 4
5 | 7
10 | 10
15 | 15
20 | 33
25 | 51
30 | 79

Question

the data show that the population of lily pads has an increasing growth rate.
an exponential function would be most suitable to model these data.
the general form of an exponential function is $y = ab^x$. use the regression calculator to find the values of $a$ and $b$ for the water lily population growth. round to the nearest thousandth.
$a =$   and $b =$

table:
$x$ (days) | $y$ (water lilies)
0 | 4
5 | 7
10 | 10
15 | 15
20 | 33
25 | 51
30 | 79

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Exponential Growth Calculation", "Exponential Function Graphs" ], "new_concepts": [ "Exponential Regression", "Least Squares Regression" ], "current_concepts": [ "Exponential Growth Calculation", "Exponential Function Graphs", "Exponential Regression" ] } ## Set up the exponential regression model Using the *Exponential Growth Calculation* and *Exponential Function Graphs* knowledge points $$ \begin{aligned} &y = a \cdot b^x \implies \ln(y) = \ln(a) + x \ln(b) \ &\text{Let } Y = \ln(y), \quad A = \ln(a), \quad B = \ln(b) \implies Y = A + Bx \end{aligned} $$ ## Transform the data table Using the *Exponential Growth Calculation* knowledge point $$ \begin{aligned} (x_i, y_i) &\to (x_i, \ln(y_i)) \ (0, 4) &\to (0, 1.3863) \ (5, 7) &\to (5, 1.9459) \ (10, 10) &\to (10, 2.3026) \ (15, 15) &\to (15, 2.7081) \ (20, 33) &\to (20, 3.4965) \ (25, 51) &\to (25, 3.9318) \ (30, 79) &\to (30, 4.3694) \end{aligned} $$ ## Calculate linear regression parameters Using the *Exponential Growth Calculation* knowledge point $$ \begin{aligned} n &= 7 \ \sum x &= 105, \quad \bar{x} = 15 \ \sum x^2 &= 2275 \ \sum Y &= 20.1406, \quad \bar{Y} \approx 2.8772 \ \sum xY &= 365.1765 \ B &= \frac{\sum xY - n\bar{x}\bar{Y}}{\sum x^2 - n\bar{x}^2} = \frac{365.1765 - 7(15)(2.8772)}{2275 - 7(15)^2} \approx 0.1011 \ A &= \bar{Y} - B\bar{x} = 2.8772 - 0.1011(15) \approx 1.3607 \end{aligned} $$ ## Convert back to exponential parameters Using the *Exponential Growth Calculation* knowledge point $$ \begin{aligned} a &= e^A = e^{1.3607} \approx 3.899 \ b &= e^B = e^{0.1011} \approx 1.106 \end{aligned} $$ The general form of an exponential function is $y = ab^x$. Use the regression calculator to find the values of a and b for the water lily population growth. Round to the nearest thousandth. $a =$ 3.899 and $b =$ 1.106 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Regression" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the exponential regression model

Transform the data table

Calculate linear regression parameters

Convert back to exponential parameters

Answer: