the table shows the estimated number of bees, $y$, in a hive $x$ days after a pesticide is released near the hive.

bee population over time
\begin{array}{|c|c|}
\hline
\text{number of days} & \text{estimated number of bees} \\
\hline
0 & 10,000 \\
10 & 7,500 \\
20 & 5,600 \\
30 & 4,200 \\
40 & 3,200 \\
50 & 2,400 \\
\hline
\end{array}

which function best models the data?

$y = 9,958(0.972)^x$
$y = 0.972(9,958)^x$
$y = 9,219x - 150$
$y = -150x + 9,219$

Question

the table shows the estimated number of bees, $y$, in a hive $x$ days after a pesticide is released near the hive.

bee population over time
\begin{array}{|c|c|}
\hline
\text{number of days} & \text{estimated number of bees} \\
\hline
0 & 10,000 \\
10 & 7,500 \\
20 & 5,600 \\
30 & 4,200 \\
40 & 3,200 \\
50 & 2,400 \\
\hline
\end{array}

which function best models the data?

$y = 9,958(0.972)^x$
$y = 0.972(9,958)^x$
$y = 9,219x - 150$
$y = -150x + 9,219$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Decay", "Exponential Modeling" ], "new_concepts": [], "current_concepts": [ "Exponential Decay", "Exponential Modeling" ] } ## Evaluate initial value $$ \begin{aligned} &\text{At } x = 0, \text{ the table gives } y = 10,000.\ &\text{For } y = 9,958(0.972)^x: y(0) = 9,958(0.972)^0 = 9,958 \approx 10,000.\ &\text{For } y = 0.972(9,958)^x: y(0) = 0.972(9,958)^0 = 0.972 e 10,000. \end{aligned} $$ ## Test exponential decay model at subsequent points $$ \begin{aligned} &y(10) = 9,958(0.972)^{10} \approx 9,958 \times 0.7518 \approx 7,486 \approx 7,500\ &y(20) = 9,958(0.972)^{20} \approx 9,958 \times 0.5652 \approx 5,628 \approx 5,600\ &y(30) = 9,958(0.972)^{30} \approx 9,958 \times 0.4249 \approx 4,231 \approx 4,200\ &y(40) = 9,958(0.972)^{40} \approx 9,958 \times 0.3194 \approx 3,181 \approx 3,200\ &y(50) = 9,958(0.972)^{50} \approx 9,958 \times 0.2401 \approx 2,391 \approx 2,400 \end{aligned} $$ ## Compare with linear models $$ \begin{aligned} &\text{For } y = -150x + 9,219:\ &y(0) = 9,219 e 10,000\ &y(50) = -150(50) + 9,219 = 1,719 e 2,400 \end{aligned} $$

(A) $y = 9,958(0.972)^x$

(B) $y = 0.972(9,958)^x$

(C) $y = 9,219x - 150$

(D) $y = -150x + 9,219$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Modeling" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate initial value

Test exponential decay model at subsequent points

Compare with linear models

Answer: