Question

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

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

(b) How much tax is assessed on an income of \(14,000? On \)26,000?

(c) Sketch the graph of the total assessed tax T as a function of the income I.

Short answer

1. The graph of the tax rate R is obtained.
2. The income tax assessed on an income of $14,000 is $400 and on an income of $26,000 is $1900.
3. The graph of the total assessed tax T for \(x > 20,000\) is the ray with the initial point \(\left( {20,000,1000} \right)\) that passes through \(\left( {30,000,2500} \right)\).

Step-by-step solution

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

a)

Sketch the graph of the tax rate R as a function of the income I as shown below:

Determine the tax assessed on incomes of $14,000 and $26,000

b)

The income tax assessed on an income of $14,000 is

\(10\% \left( {\$ 4000} \right) = \$ 400\).

The income tax assessed on an income of $26,000 is

\(\begin{aligned}10\% \left( {\$ 10,000} \right) + 15\% \left( {\$ 6000} \right) &= \$ 1000 + \$ 900\\ &= \$ 1900.\end{aligned}\)

Thus, the income tax assessed on an income of $14,000 is $400 and on an income of $26,000 is $1900.

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

c)

According to part (b), the tax assessed on an income of $20,000 is $1000. Therefore, the graph of the line segment is from the point \(\left( {10,000,0} \right)\) to \(\left( {20,000,1000} \right)\).

The income tax assessed on an income of $30,000 is $2500. Therefore, the graph of the total assessed tax T for \(x > 20,000\) is the ray with the initial point \(\left( {20,000,1000} \right)\) that passes through \(\left( {30,000,2500} \right)\).