Question

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the function \\(w\\) is given by \\(w(t) = \frac{65,000}{1 + 0.5e^{kt}}\\), where \\(k\\) is a constant. if \\(w(5) = 20,000\\), what is the value of \\(w(10)\\)?

\\(\bigcirc\\) 5,900
\\(\bigcirc\\) 7,900
\\(\bigcirc\\) 15,900
\\(\bigcirc\\) 21,900

Explanation:

Set up the equation for \(W(5)\)

We are given the logistic function:

$$W(t) = \frac{65,000}{1 + 0.5e^{kt}}$$

Using the given condition \(W(5) = 20,000\), we substitute \(t = 5\):

$$20,000 = \frac{65,000}{1 + 0.5e^{5k}}$$

Solve for the exponential term \(e^{5k}\)

We solve for \(e^{5k}\) by rearranging the equation:

$$1 + 0.5e^{5k} = \frac{65,000}{20,000}$$

$$1 + 0.5e^{5k} = 3.25$$

$$0.5e^{5k} = 2.25$$

$$e^{5k} = 4.5$$

Express \(e^{10k}\) in terms of \(e^{5k}\)

We need to find \(W(10)\), which involves \(e^{10k}\).
Using exponent rules, we relate \(e^{10k}\) to \(e^{5k}\):

$$e^{10k} = (e^{5k})^2$$

Substituting the value of \(e^{5k}\):

$$e^{10k} = (4.5)^2 = 20.25$$

Calculate the value of \(W(10)\)

Substitute \(e^{10k} = 20.25\) back into the function for \(t = 10\):

$$W(10) = \frac{65,000}{1 + 0.5e^{10k}}$$

$$W(10) = \frac{65,000}{1 + 0.5(20.25)}$$

$$W(10) = \frac{65,000}{1 + 10.125}$$

$$W(10) = \frac{65,000}{11.125} \approx 5842.7$$

Match with the closest option

The calculated value is approximately \(5,843\).
Comparing this to the given choices:

• \(5,900\)
• \(7,900\)
• \(15,900\)
• \(21,900\)

The closest value is \(5,900\).

Answer:

• (A) 5,900 (Correct answer)
• (B) 7,900
• (C) 15,900
• (D) 21,900