Chapter 1.1, Problem 61E

a.

To determine

Sketch the graph of the tax rate $R$ as a function of the income $I$ .

Answer to Problem 61E

  

Explanation of Solution

Given information:

In a certain country, income tax is assessed as follows. There is no tax on income up to $\$10,000$ . Any income over $\$10,000$ is taxed at rate of $10\%$ , up to an income of $\$20,000$ . Any income over $\$20,000$ is taxed at $15\%$ .

Sketch the graph of the tax rate $R$ as a function of the income $I$ .

Calculation:

Suppose $R$ be the rate of tax and $I$ be the income, so $R$ is the function of $I$ .

Now we have been given that, there is no tax income up to $\$10,000$

So $R(I)=0$ for $I\le \$10,000$

If the income $I$ is in the interval $\$10000<I\le \$20000$ , then the tax rate is $10\%$

So $R(I)=10$ (%) for $\$10000<I\le \$20000$

If the income is more than $\$20000$ then the tax rate is $15\%$

Then $R(I)=15$ (%) for $I>\$20000$

Now we can write the tax rate $R$ as a function of $I$ as follows

  $R(I)=\{\begin{array}{l}0(\%),for,I\le \$10000\\ 10(\%),for,\$10000<I\le \$20000\\ 15(\%),for,I>\$20000\end{array}$

Now we can draw a graph,

  

b.

To determine

How much tax is assessed on an income of $\$14,000$ on $\$26,000$ ?

Answer to Problem 61E

  $\boxed{\$14000=\$400}$ and $\boxed{\$\text{26}000=\$\text{19}00}$

Explanation of Solution

Given information:

In a certain country, income tax is assessed as follows. There is no tax on income up to $\$10,000$ . Any income over $\$10,000$ is taxed at rate of $10\%$ , up to an income of $\$20,000$ . Any income over $\$20,000$ is taxed at $15\%$ .

How much tax is assessed on an income of $\$14,000$ on $\$26,000$ ?

Calculation:

We have to calculate tax on income of $\$14,000$

There is no tax if income $I\le \$10000$

So non-taxable income is = $\$10000$

Then taxable income is = $140000-10000$ dollars= $\$4000$

Since the total income is in the interval $(10000,20000)$

So the income tax = $10\%$ of $4000$

  $=4000\times \frac{10}{100}=\$400$

Hence, $\boxed{\text{tax on income of }\$\text{14}000=\$\text{4}00}$

Now, we have to calculate the tax on income of $\$26,000$

First we have to break this

  $\$26000=\$10000+\$10000+\$6000$

Tax on first $\$10000=0$

Tax on second $\$10000=10\%of\$10000$

Tax on last $\$6000=15\%of\$6000$

Then total tax on income of $\$26000$

  $=10000\times 0+10000\times 0.1+6000\times 0.15$ dollars

  $=\$1900$

Hence, the $\boxed{\text{tax on income of }\$\text{26}000=\$\text{19}00}$

c.

To determine

Sketch the graph of the total assessed tax $T$ as a function of the income $I$ .

Answer to Problem 61E

  

Explanation of Solution

Given information:

In a certain country, income tax is assessed as follows. There is no tax on income up to $\$10,000$ . Any income over $\$10,000$ is taxed at rate of $10\%$ , up to an income of $\$20,000$ . Any income over $\$20,000$ is taxed at $15\%$ .

Sketch the graph of the total assessed tax $T$ as a function of the income $I$ .

Calculation:

Since there is no tax if income $I\le \$10000$ , so graph of tax T(I) will start from the point $(10000,0)$

If the income $I>\$10000$ and $I\le \$20000$ , then tax rate is $10\%$

Maximum taxable income in the interval is $\$20000-\$10000=\$10000$

Then maximum tax in this interval = $10000\times 0.1=\$1000$

Hence curve passes through the point $(20000,1000)$

Now if income tax $I>\$20000$ , then the tax rate is $15\%$

So tax T (I) = $(1-20000)\times 0.15+1000$ dollars

Or $T(I)=0.15I-2000$ dollars

Hence the graph of tax $T(I)$ is as follows