Unit 9
9 marks · NormalA worker in Kalculus has an annual salary given by dollars. The after-tax salary can be written as a function of , based on the income tax rates:
| Band | Tax Rate |
|---|---|
| $0 – $50,000 | 20% |
| >$50,000 | 40% (on excess) |
Write the piecewise function for after-tax salary , for all .
A company wishes to set annual after-tax salaries between $40,000 and $80,000 for its staff.
If the company can only offer salaries in increments of $5,000, determine all possible gross salaries that would give after-tax salaries in the required range.
Explain, using algebraic reasoning, why there are more possible gross salaries in the upper range ($>50,000) compared to the lower range ($\leq 50,000), for the same after-tax interval.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) for ; for · must show piecewise definition
(b) For $40,000 ≤ after-tax ≤ $80,000: For x ≤ 50,000: x in [50,000]; for x > 50,000: solve 0.6x + 10,000 in [40,000, 80,000]. So x = 50,000 ($40,000), x = 50,000, 55,000, ..., 116,666.67 ($80,000). Thus, possible values: x = 50,000, 55,000, …, 116,667 (rounded to nearest $5,000 and checking endpoints). · must show all valid salary increments listed
(c) Gradient in higher band is less (0.6), so each $5,000 increment gives a smaller after-tax increase, allowing for more increments within the fixed interval. In lower band, only one value possible ($40,000 = 0.8*50,000), but in upper band, many more increments fit within $40,000 to $80,000. · must show explanation linked to gradients and interval
Common mistakes: Not adjusting for tax band; Incorrect piecewise structure
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Piecewise formula or clear explanation for both income regions
A1 (a) · 2 marks · All correct expressions and boundaries stated clearly (follow-through)
M2 (b) · 2 marks · Substitutes after-tax values into each region, solves for x
A2 (b) · 2 marks · Lists all increment values for each band, with correct boundaries and step size (follow-through)
R1 (c) · 2 marks · Explains in algebraic terms why gradient change leads to more salary points in upper band
Worked solution
(a) After-tax salary function: A(x) = 0.8x for 0 ≤ x ≤ 50,000; A(x) = 0.6x + 10,000 for x > 50,000.
(b) For after-tax in [$40,000, $80,000]: For x ≤ 50,000: 0.8x = 40,000 → x = 50,000. Only one possible. For x > 50,000: 0.6x + 10,000 = 40,000 → x = 50,000; 0.6x + 10,000 = 80,000 → x = 116,666.67. So x = 55,000, 60,000, ..., 115,000 (values every $5,000). Possible: x = 50,000; x = 55,000, 60,000, ..., 115,000.
(c) In higher band, the gradient (after-tax per pre-tax) is smaller (0.6), so more x-values fit within the after-tax interval: the same interval contains more possible gross salaries due to the shallower slope.