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Alona [7]
2 years ago
7

Let h(x)=20e^kx where k ɛ R (Picture attached. Thank you so much!)

Mathematics
1 answer:
zloy xaker [14]2 years ago
3 0

Answer:

A)

k=0

B)

\displaystyle \begin{aligned} 2k + 1& = 2\ln 20 + 1 \\ &\approx 2.3863\end{aligned}

C)

\displaystyle \begin{aligned} k - 3&= \ln \frac{1}{2} - 3 \\ &\approx-3.6931 \end{aligned}

Step-by-step explanation:

We are given the function:

\displaystyle h(x) = 20e^{kx} \text{ where } k \in \mathbb{R}

A)

Given that h(1) = 20, we want to find <em>k</em>.

h(1) = 20 means that <em>h</em>(x) = 20 when <em>x</em> = 1. Substitute:

\displaystyle (20) = 20e^{k(1)}

Simplify:

1= e^k

Anything raised to zero (except for zero) is one. Therefore:

k=0

B)

Given that h(1) = 40, we want to find 2<em>k</em> + 1.

Likewise, this means that <em>h</em>(x) = 40 when <em>x</em> = 1. Substitute:

\displaystyle (40) = 20e^{k(1)}

Simplify:

\displaystyle 2 = e^{k}

We can take the natural log of both sides:

\displaystyle \ln 2 = \underbrace{k\ln e}_{\ln a^b = b\ln a}

By definition, ln(e) = 1. Hence:

\displaystyle k = \ln 2

Therefore:

2k+1 = 2\ln 2+ 1 \approx 2.3863

C)

Given that h(1) = 10, we want to find <em>k</em> - 3.

Again, this meas that <em>h</em>(x) = 10 when <em>x</em> = 1. Substitute:

\displaystyle (10) = 20e^{k(1)}

Simplfy:

\displaystyle \frac{1}{2} = e^k

Take the natural log of both sides:

\displaystyle \ln \frac{1}{2} = k\ln e

Therefore:

\displaystyle k = \ln \frac{1}{2}

Therefore:

\displaystyle k - 3 = \ln\frac{1}{2} - 3\approx-3.6931

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This question is difficult to give a definite answer to because it's an approximation, but I estimate the solutions to be around x = 2.6 and x = -2.6.

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