Question 1 / Vraag 1
(14)

Income Tax: Rates of Tax for Individuals / Inkomstebelasting: Belastingkoerse vir individue Search on the internet (SARS website) for the rates of tax for individuals, for the 2026 tax year (1 March 2025-28 February 2026). This information can be used to write income tax (T) as a function of taxable income . Use this information to create a piecewise defined function , using one independent variable , to model the rates of tax for individuals. / Gaan soek op die internet (SARS/SAID webtuiste) vir die belastingkoerse vir individue, vir die 2026 belastingjaar (1 Maart 2025 28 Februarie 2026). Hierdie inligting kan gebruik word om inkomstebelasting ( ) as ’ funksie van die belasbare inkomste ( ) te skryf. Gebruik hierdie inligting om 'n stuksgewys gedefinieerde funksie te ontwikkel, deur een onafhanklike veranderlike te gebruik, om die belastingkoerse vir individue te modelleer.

Question 2 / Vraag 2
(10)

The supply function for a commodity takes the form , for some constants . When , the quantity supplied is 5 ; when , the quantity supplied is 12 ; when , the quantity supplied is 23 . Use matrix reduction (Gaussian elimination) to determine the constants . Give the supply funcrtion. / Die aanbodfunksie vir ’ n produk het die vorm , vir die konstantes . As , is die hoeveelheid wat verskaf is 5 ; wanneer , is die hoeveelheid wat verskaf is 12 ; wanneer , is die hoeveelheid wat verskaf is 23. Gebruik matriksreduksie (Gauss-eliminasie) en bepaal die konstantes . Skryf die aanbodfunksie neer.

Ask by Dawson Crawford. in South Africa

Mar 28,2025

Question

Question 1 / Vraag 1
(14)

Income Tax: Rates of Tax for Individuals / Inkomstebelasting: Belastingkoerse vir individue Search on the internet (SARS website) for the rates of tax for individuals, for the 2026 tax year (1 March 2025-28 February 2026). This information can be used to write income tax (T) as a function of taxable income . Use this information to create a piecewise defined function , using one independent variable , to model the rates of tax for individuals. / Gaan soek op die internet (SARS/SAID webtuiste) vir die belastingkoerse vir individue, vir die 2026 belastingjaar (1 Maart 2025 28 Februarie 2026). Hierdie inligting kan gebruik word om inkomstebelasting ( ) as ’ funksie van die belasbare inkomste ( ) te skryf. Gebruik hierdie inligting om 'n stuksgewys gedefinieerde funksie te ontwikkel, deur een onafhanklike veranderlike te gebruik, om die belastingkoerse vir individue te modelleer.

Question 2 / Vraag 2
(10)

The supply function for a commodity takes the form , for some constants . When , the quantity supplied is 5 ; when , the quantity supplied is 12 ; when , the quantity supplied is 23 . Use matrix reduction (Gaussian elimination) to determine the constants . Give the supply funcrtion. / Die aanbodfunksie vir ’ n produk het die vorm , vir die konstantes . As , is die hoeveelheid wat verskaf is 5 ; wanneer , is die hoeveelheid wat verskaf is 12 ; wanneer , is die hoeveelheid wat verskaf is 23. Gebruik matriksreduksie (Gauss-eliminasie) en bepaal die konstantes . Skryf die aanbodfunksie neer.

Ask by Dawson Crawford. in South Africa

Mar 28,2025

UpStudy AI · Accepted Answer

To create a piecewise defined function $$ T $$ for income tax based on the rates for individuals for the 2026 tax year, you will need to find the specific tax brackets and rates from the SARS website. Once you have that information, you can define $$ T(x) $$ as follows:

1. Identify the income brackets and corresponding tax rates.
2. Define $$ T(x) $$ using the piecewise function format, where each segment corresponds to a specific income bracket.

For example, if the tax rates are as follows (hypothetical values):
- $$ 0 \leq x < 20000 $$: $$ T(x) = 0 $$
- $$ 20000 \leq x < 50000 $$: $$ T(x) = 0.18(x - 20000) $$
- $$ 50000 \leq x < 100000 $$: $$ T(x) = 5400 + 0.26(x - 50000) $$
- $$ x \geq 100000 $$: $$ T(x) = 15400 + 0.31(x - 100000) $$

You would replace the hypothetical values with the actual rates from the SARS website.
To determine the constants $$ a, b, c $$ in the supply function $$ q^{s}(p) = a p^{2} + b p + c $$, we can set up a system of equations based on the given data points:

1. For $$ p = 1 $$, $$ q^{s}(1) = 5 $$:
   $$
   a(1^2) + b(1) + c = 5 \quad \Rightarrow \quad a + b + c = 5 \quad \text{(Equation 1)}
   $$

2. For $$ p = 2 $$, $$ q^{s}(2) = 12 $$:
   $$
   a(2^2) + b(2) + c = 12 \quad \Rightarrow \quad 4a + 2b + c = 12 \quad \text{(Equation 2)}
   $$

3. For $$ p = 3 $$, $$ q^{s}(3) = 23 $$:
   $$
   a(3^2) + b(3) + c = 23 \quad \Rightarrow \quad 9a + 3b + c = 23 \quad \text{(Equation 3)}
   $$

Now we can represent this system of equations in matrix form and use Gaussian elimination to solve for $$ a, b, c $$. The augmented matrix is:

$$
\begin{bmatrix}
1 & 1 & 1 & | & 5 \
4 & 2 & 1 & | & 12 \
9 & 3 & 1 & | & 23
\end{bmatrix}
$$

We will perform row operations to reduce this matrix.
Since the matrix operations did not yield results, let's manually perform Gaussian elimination on the augmented matrix:

1. Start with the augmented matrix:
   $$
   \begin{bmatrix}
   1 & 1 & 1 & | & 5 \
   4 & 2 & 1 & | & 12 \
   9 & 3 & 1 & | & 23
   \end{bmatrix}
   $$

2. Eliminate the first column below the first row:
   - Row 2: $$ R_2 - 4R_1 $$
   - Row 3: $$ R_3 - 9R_1 $$

This gives us:
   $$
   \begin{bmatrix}
   1 & 1 & 1 & | & 5 \
   0 & -2 & -3 & | & -8 \
   0 & -6 & -8 & | & -22
   \end{bmatrix}
   $$

3. Now, simplify Row 3 by eliminating the second column:
   - Row 3: $$ R_3 - 3R_2 $$

This results in:
   $$
   \begin{bmatrix}
   1 & 1 & 1 & | & 5 \
   0 & -2 & -3 & | & -8 \
   0 & 0 & 1 & | & 2
   \end{bmatrix}
   $$

4. Now, back substitute to find $$ c $$:
   From Row 3: $$ c = 2 $$

5. Substitute $$ c $$ back into Row 2:
   $$
   -2b - 3(2) = -8 \quad \Rightarrow \quad -2b - 6 = -8 \quad \Rightarrow \quad -2b = -2 \quad \Rightarrow \quad b = 1
   $$

6. Substitute $$ b $$ and $$ c $$ back into Row 1:
   $$
   a + 1 + 2 = 5 \quad \Rightarrow \quad a + 3 = 5 \quad \Rightarrow \quad a = 2
   $$

Thus, we have $$ a = 2 $$, $$ b = 1 $$, and $$ c = 2 $$. The supply function is:
$$
q^{s}(p) = 2p^{2} + 1p + 2
$$
The supply function is $$ q^{s}(p) = 2p^{2} + p + 2 $$.

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