Question

the half life of a certain substance is about 4 hours. the graph shows the decay of a 50 gram sample of the substance that is measured every hour for 9 hours.

which function can be used to determine the approximate number of grams of the sample remaining after \\(t\\) hours?

\\(y = 25(0.15)^t\\)
\\(y = 25(0.85)^t\\)
\\(y = 50(0.15)^t\\)
\\(y = 50(0.85)^t\\)

Explanation:

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<reasoning>

Identify the initial value

Using the Exponential Functions knowledge point
\[
a = 50
\]

Determine the decay factor

Using the Exponential Decay and Half-Life Formula knowledge points
\[

$$\begin{aligned} y &= a \cdot b^t \\ y &= 50 \cdot (0.5)^{t/4} \\ b &= (0.5)^{1/4} \approx 0.8409 \approx 0.85 \end{aligned}$$

\]

Formulate the final function

Using the Exponential Functions knowledge point
\[
y = 50(0.85)^t
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 25(0.15)^t\)</mcq-option>
<mcq-option>(B) \(y = 25(0.85)^t\)</mcq-option>
<mcq-option>(C) \(y = 50(0.15)^t\)</mcq-option>
<mcq-correct>(D) \(y = 50(0.85)^t\)</mcq-correct>
</answer>

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]
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Answer:

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<reasoning>

Identify the initial value

Using the Exponential Functions knowledge point
\[
a = 50
\]

Determine the decay factor

Using the Exponential Decay and Half-Life Formula knowledge points
\[

$$\begin{aligned} y &= a \cdot b^t \\ y &= 50 \cdot (0.5)^{t/4} \\ b &= (0.5)^{1/4} \approx 0.8409 \approx 0.85 \end{aligned}$$

\]

Formulate the final function

Using the Exponential Functions knowledge point
\[
y = 50(0.85)^t
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 25(0.15)^t\)</mcq-option>
<mcq-option>(B) \(y = 25(0.85)^t\)</mcq-option>
<mcq-option>(C) \(y = 50(0.15)^t\)</mcq-option>
<mcq-correct>(D) \(y = 50(0.85)^t\)</mcq-correct>
</answer>

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"Algebra",
"Exponential Functions"
]
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