A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

Home » A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

Application of Derivatives | Class 12 | Exercise 1 | Maths | NCERT Exemplar

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

We should consider that the organization expands the yearly membership by

$\mathbf{Rs}\text{ }\mathbf{x}.$

Along these lines, x is the quantity of supporters who end the administrations.

Absolute income,

$\mathbf{R}\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{500}\text{ }\text{ }\mathbf{x} \right)\text{ }\left( \mathbf{300}\text{ }+\text{ }\mathbf{x} \right)$

$=\text{ }\mathbf{150000}\text{ }+\text{ }\mathbf{500x}\text{ }\text{ }\mathbf{300x}\text{ }\text{ }\mathbf{x2}$

$=\text{ }-\text{ }\mathbf{x2}\text{ }+\text{ }\mathbf{200x}\text{ }+\text{ }\mathbf{150000}$

Separating the two sides w.r.t. x, we get R'(x)

$=\text{ }-\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{200}$

For LOCAL maxima and nearby minima, R'(x) = 0

$-\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{200}\text{ }=\text{ }\mathbf{0}\Rightarrow \mathbf{x}\text{ }=\text{ }\mathbf{100}$

$\mathbf{R}''\left( \mathbf{x} \right)\text{ }=\text{ }-\text{ }\mathbf{2}\text{ }<\text{ }\mathbf{0}\text{ }\mathbf{Maxima}$

Along these lines, R(x) is most extreme at

$\mathbf{x}\text{ }=\text{ }\mathbf{100}$

Along these lines, to get most extreme benefit, the organization should expand its yearly membership by

$\mathbf{Rs}\text{ }\mathbf{100}.$

i

Find the shortest distance between the given lines.
$\begin{array}{l} \square \vec{r}=(\hat{i}+\hat{j})+\lambda(2 \hat{i}-\hat{j}+\hat{k}) \\ \vec{r}=(2 \hat{i}+\hat{j}-\hat{k})+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k}) \end{array}$

If A(1, 2, 3), B(4, 5, 7), C(-4, 3, -6) and D(2, 9, 2) are four given points then find the angle between the lines AB and CD.

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$\begin{array}{l} \square \vec{r}=(\hat{i}+\hat{j})+\lambda(2 \hat{i}-\hat{j}+\hat{k}) \\ \vec{r}=(2 \hat{i}+\hat{j}-\hat{k})+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k}) \end{array}$

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$\begin{array}{l} \square \vec{r}=(\hat{i}+\hat{j})+\lambda(2 \hat{i}-\hat{j}+\hat{k}) \\ \vec{r}=(2 \hat{i}+\hat{j}-\hat{k})+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k}) \end{array}$

If A(1, 2, 3), B(4, 5, 7), C(-4, 3, -6) and D(2, 9, 2) are four given points then find the angle between the lines AB and CD.

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