A certain country taxes the first $20,000 of an individual's income at a rate of 15%, and all income over $20,000 is taxed at 20%.
a. Al makes $16,000. Betty makes $36,000. How much is each taxed?
b. Write a piecewise function T that specifies the total tax on an income of x dollars.
c. Catina is taxed $5000. What is her income.
if i </= 20,000
tax = .15 * i
if i> 20,000
tax = 3,000 + 0.20 ( i - 20,000)
To find out how much Al and Betty are taxed, we can apply the tax rates for the specific income range for each individual.
a. Al makes $16,000:
Since Al's income is less than $20,000, he falls into the first income range. Therefore, his entire income is taxed at a rate of 15%.
Tax paid by Al = (Income * Tax Rate) = ($16,000 * 0.15) = $2,400
Betty makes $36,000:
For Betty, the first $20,000 is taxed at a rate of 15%, and the remaining amount ($36,000 - $20,000 = $16,000) is taxed at a rate of 20%.
Tax paid by Betty = (Tax on first $20,000 + Tax on remaining $16,000)
= ([$20,000 * 0.15] + [$16,000 * 0.20])
= ($3,000 + $3,200)
= $6,200
Therefore, Al is taxed $2,400, and Betty is taxed $6,200.
b. To write a piecewise function T that specifies the total tax on an income of x dollars, we break it down into two segments based on the income thresholds:
For income x ≤ $20,000, the tax rate is 15%.
For income x > $20,000, the tax rate is 20%.
Hence, the piecewise function T(x) can be written as:
T(x) = 0.15x, if x ≤ $20,000
T(x) = 0.20x, if x > $20,000
c. Catina is taxed $5,000:
To find Catina's income, we need to determine the income range based on the given tax amount.
If Catina's tax is $5,000, we can set up an equation using the piecewise function for the tax rates:
T(x) = 0.15x, if x ≤ $20,000
T(x) = 0.20x, if x > $20,000
Now we set the equation equal to $5,000 and solve for x:
0.15x = $5,000
Divide both sides by 0.15:
x = $5,000 / 0.15
Thus, Catina's income is $33,333.33 (rounded to two decimal places).