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Challenge: Suppose you know the initial dose of a drug is 500 mg, and 3 hours later, there are 325 mg left, but you don't know the base multiplier [tex]\(b\)[/tex]. Use this information in the equation [tex]y = a \cdot b^x[/tex] to solve for [tex]\(b\)[/tex].

First, substitute [tex]\(a = 500\)[/tex], and use [tex]\(x = 3\)[/tex] with [tex]\(y = 325\)[/tex].

The equation will look like:
[tex]\[ 325 = 500 \cdot b^3 \][/tex]

Now solve for [tex]\(b\)[/tex]:

First, divide both sides by 500:
[tex]\[ \frac{325}{500} = b^3 \][/tex]
[tex]\[ 0.65 = b^3 \][/tex]

To undo a third power, take the cube root of both sides:
[tex]\[ \sqrt[3]{0.65} = \sqrt[3]{b^3} \][/tex]
[tex]\[ 0.866 = b \][/tex]

If your calculator doesn't have a cube root function, you can raise both sides of the equation to the [tex]\(\frac{1}{3}\)[/tex] power:
[tex]\[ (0.65)^{\frac{1}{3}} = \left(b^3\right)^{\frac{1}{3}} \][/tex]
[tex]\[ 0.866 = b \][/tex]

So the equation for this problem is:
[tex]\[ y = 500(0.866)^x \][/tex]

Sagot :

GinnyAnswer
Let's go through the solution step-by-step:

1. Given Information:
- Initial dose of the drug, [tex]\(a\)[/tex]: 500 mg
- Remaining dose after 3 hours, [tex]\(y\)[/tex]: 325 mg
- Time, [tex]\(x\)[/tex]: 3 hours

We are to use the exponential decay model [tex]\(y = a \cdot b^x\)[/tex] to find the base multiplier [tex]\(b\)[/tex].

2. Substitute the given values into the equation [tex]\(y = a \cdot b^x\)[/tex]:
[tex]\[ 325 = 500 \cdot b^3 \][/tex]

3. Solve for [tex]\(b\)[/tex]:
- First, divide both sides of the equation by 500 to isolate [tex]\(b^3\)[/tex]:
[tex]\[ \frac{325}{500} = b^3 \][/tex]
[tex]\[ 0.65 = b^3 \][/tex]

- To find [tex]\(b\)[/tex], we need to take the cube root of 0.65. In mathematical terms, this involves raising 0.65 to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ b = (0.65)^{\frac{1}{3}} \][/tex]

4. Calculate the cube root of 0.65:
- Using a calculator, you can evaluate [tex]\((0.65)^{\frac{1}{3}}\)[/tex]:
[tex]\[ b \approx 0.866239 \][/tex]

5. Result:
- The base multiplier [tex]\(b\)[/tex] is approximately [tex]\(0.866239\)[/tex].

Hence, the exponential decay model for this problem is:
[tex]\[ y = 500 \cdot (0.866)^x \][/tex]

This model describes how the drug's concentration decreases over time. The base multiplier [tex]\(b \approx 0.866239\)[/tex] indicates that approximately 86.62% of the drug's concentration remains each hour.
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