Detailed Answer

a) The volume of a cuboid can be calculated by multiplying the area of its base by its length, so:

\[ 4.8 * 5.8 * 9.1 = 253.344 \ cm^3\]

b) As we know that $ 1dm = 10cm $, the conversion factor is $ 10^3 = 1000 $, so we get:

\[253.344 \ cm^3 = 0.253344 \ dm^3\]

c) 250 cm³

d) Let's start by calculating the area of the base:

\[ A_{base} = 4.8 * 5.8 = 27.84 \ cm^2\]

Now, the area of the larger side:

\[ A_{large \ side} = 5.8 * 9.1 = 52.78 \ cm^2\]

Finally, the area of the smaller side

\[ A_{small \ side} = 4.8 * 9.1 = 43.68 \ cm^2\]

We know that we have two of each side, so all of them have to be multiplied by two.

Let's calculate how much will be spent on the top and bottom of the cuboid. We know that it costs 0.15$ per cm², so:

\[ C_{top + bottom} = 27.84 * 2 * 0.15 = $8.352\]

Now, the cost for the sides:

\[ C_{sides} = (52.78 * 2 + 43.68*2)*0.12 = $23.1504\]

So, the total cost is:

\[ C_{total} = 23.1504 + 8.352 = 31.5024 \approx $31.50\]

Detailed Answer

a) To find the volume we need to apply the volume formula:

\[ V = \pi r^2 * h\]

As it can be seen from the figure, the radius will be a half of 8.4cm, so 4.2cm.

\[ V = \pi 4.2^2 * 12.5 = 692.721... \approx 693cm^3\]

b) The area of a cylinder is given by the formula:

\[ A = 2 \pi rh + 2 \pi r^2\]

We know that the top paert should not be paint, meaning that the formula we will actually use is:

\[ A = 2 \pi rh + \pi r^2\]

\[ A = 2 \pi 4.2 * 12.5 + \pi 4.2^2 = 385.285... \approx 385cm^2\]

c) To calculate the percentage error we first need to find the exact value, so the area will need to be multiplied by the dollar amount per each cm². It is important to here take the exact value, not the rounding to 3 significant figures:

\[ 0.05 * 385.285... \approx 19.26\]

Now, using the formula for percentage error, we get that:

\[ ε= |\frac{v_a - v_e}{v_e}| * 100\% = |\frac{20 - 19.26}{19.26}| * 100\% = 3.84\%\]

Detailed Answer

Given: Point A: (3, 5), Point B: (6, 1)

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substitute the coordinates of points A and B into the formula:

\[ \text{Distance}_{AB} = \sqrt{(6 - 3)^2 + (1 - 5)^2} \]

\[ = \sqrt{3^2 + (-4)^2} \]

\[ = \sqrt{9 + 16} \]

\[ = \sqrt{25} \]

\[ = 5 \]

So, the distance between points A and B is 5.

We first need to find the distance between points C and D. So, given: Point C: (2, 4), Point D: (8, 9)

Using the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substitute the coordinates of points C and D into the formula:

\[ \text{Distance}_{CD} = \sqrt{(8 - 2)^2 + (9 - 4)^2} \]

\[ = \sqrt{6^2 + 5^2} \]

\[ = \sqrt{36 + 25} \]

\[ = \sqrt{61}\]

\[ \approx 7.81\]

Now, let's check Alex's claim that the distance between points C and D is twice the distance between points A and B:

Alex's claim: $\text{Distance}_{CD} = 2 \times \text{Distance}_{AB}$

\[ 7.81 = 2 \times 5 \]

Since $7.81 \neq 10$, Alex's claim is incorrect.

Question 1

[Maximum mark: 6]

The answer found to a given problem is $w=0.3862901$.

a) State value $w$ correct to 2 significant figures.

b) Write down the exact value of $w$ in the form $a * 10^k$, where $1 < a < 10$ and state the value of k.

c) Provided the correct answer to the value of $w$ is 0.427, what is the percentage error (to 2 decimal places) in the answer found?

Answers and Explanations

Show Answer

Explanations

Question 2

[Maximum mark: 8]

Given that $p = \frac{8cos(θ)}{3x + 2y} + z$, where $θ=45°, \ x=6, \ y=9, \ and \ z=4$. Calculate the value of $p$ in formats given below.

a) Calculate the exact value of $p$.

b) Write down your answer from part (a) rounded to 3 decimal points.

c) Write down your answer from part (a) in the form $a * 10^k$, where $k=2$

Answers and Explanations

Show Answer

Explanations

Question 3

[Maximum mark: 12]

A cuboid has the following dimensions: length = 4.8 cm, height = 9.1 cm, width = 5.8 cm.

a) Calculate the exact volume of the cuboid in cm³.

b) Give the value obtained in part (a) to dm³.

c) Write the answer for part (a) correct to 2 significant figures.

Mark is instructed to paint the cuboid using black and white paint. Black paint is used for the top and bottom (4.8cm x 5.8cm), and white paint is used for the sides. The cost of covering 1cm² with black paint is 0.15$, and with white paint it is 0.12$.

d) How much will mark have to spend to paint the entire cuboid? Round your answer to 2 decimal points.

Answers and Explanations

Show Answer

Explanations

Question 4

[Maximum mark: 9]

Given that $q = \frac{12x^2sin(α)}{5x^2 + 2y}$ where $α=60°, \ x=3, \ y=9$.

a) Calculate the value of $q$ in the simplest form of $\frac{v\sqrt{3}}{w}$, where $v$ and $w$ are positive integers.

b) Find the value of $q$ rounded to 3 significant figures.

c) Write down your answer to part (b) in the form $a * 10^k$, where $k=4$

Answers and Explanations

Show Answer

Explanations

Question 5Register

[Maximum mark: 7]

Consider a cylinder, as shown on the figure below:

a) Find the volume of the cylinder (to 3 significant figures).

Blue paint is being used to paint the entire cylinder, apart from its top.

b) Calculate the area of the cylinder painted in blue to 2 decimal places.

The cost of 1cm² of the paint is $0.05. Sara would estimate the cost of having painted the cylinder blue to be $20.

c) Using your answer to part (b), what is the percentage error in Sara’s estimate? Round your answer to two decimal places.

Answers and Explanations

Question 6Register

[Maximum mark: 9]

The volume of a trapezoidal is given by:

\[ V = \frac{1}{2}(a + b) * h * l\]

Where a = short base, b = long base, h = height, l = length.

Consider a trapezoidal prism in which a = 2.4cm, b = 3.6cm, h = 3.1cm, and l = 8cm.

a) Calculate its volume.

b) Round your answer to the neareast integer.

c) Write down the exact value for the volume in the form $a * 10^k$, where $1000 < a < 10000$ and state the value of k.

Answers and Explanations

Question 7Premium

[Maximum mark: 6]

The formula for the distance between two points is given by:

\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Consider point A with coordinates (3,5), and points B with coordinates (6,1).

a) Find the distance between points A and B.

Now, consider point C with coordinates (2,4), and points D with coordinates (8,9). Alex believes that the distance between points C and D is twice the distance between points A and B.

Anonymous

Number Skills

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Detailed Answer

Detailed Answer

Detailed Answer

Detailed Answer

Detailed Answer

Detailed Answer

Detailed Answer

Question 1

[Maximum mark: 6]

The answer found to a given problem is \(w=0.3862901\).

a) State value \(w\) correct to 2 significant figures.

b) Write down the exact value of \(w\) in the form \(a * 10^k\), where \(1 < a < 10\) and state the value of k.

c) Provided the correct answer to the value of \(w\) is 0.427, what is the percentage error (to 2 decimal places) in the answer found?

Answers and Explanations

Question 2

[Maximum mark: 8]

Given that \(p = \frac{8cos(θ)}{3x + 2y} + z\), where \(θ=45°, \ x=6, \ y=9, \ and \ z=4\). Calculate the value of \(p\) in formats given below.

a) Calculate the exact value of \(p\).

b) Write down your answer from part (a) rounded to 3 decimal points.

c) Write down your answer from part (a) in the form \(a * 10^k\), where \(k=2\)

Answers and Explanations

Question 3

[Maximum mark: 12]

A cuboid has the following dimensions: length = 4.8 cm, height = 9.1 cm, width = 5.8 cm.

a) Calculate the exact volume of the cuboid in cm3.

b) Give the value obtained in part (a) to dm3.

c) Write the answer for part (a) correct to 2 significant figures.

Mark is instructed to paint the cuboid using black and white paint. Black paint is used for the top and bottom (4.8cm x 5.8cm), and white paint is used for the sides. The cost of covering 1cm2 with black paint is 0.15$, and with white paint it is 0.12$.

d) How much will mark have to spend to paint the entire cuboid? Round your answer to 2 decimal points.

Answers and Explanations

Question 4

[Maximum mark: 9]

Given that \(q = \frac{12x^2sin(α)}{5x^2 + 2y}\) where \(α=60°, \ x=3, \ y=9\).

a) Calculate the value of \(q\) in the simplest form of \(\frac{v\sqrt{3}}{w}\), where \(v\) and \(w\) are positive integers.

b) Find the value of \(q\) rounded to 3 significant figures.

c) Write down your answer to part (b) in the form \(a * 10^k\), where \(k=4\)

Answers and Explanations

Question 5Register

[Maximum mark: 7]

Consider a cylinder, as shown on the figure below:

a) Find the volume of the cylinder (to 3 significant figures).

Blue paint is being used to paint the entire cylinder, apart from its top.

b) Calculate the area of the cylinder painted in blue to 2 decimal places.

The cost of 1cm2 of the paint is $0.05. Sara would estimate the cost of having painted the cylinder blue to be $20.

c) Using your answer to part (b), what is the percentage error in Sara’s estimate? Round your answer to two decimal places.

Answers and Explanations

Question 6Register

[Maximum mark: 9]

The volume of a trapezoidal is given by:

\[ V = \frac{1}{2}(a + b) * h * l\]

Where a = short base, b = long base, h = height, l = length.

Consider a trapezoidal prism in which a = 2.4cm, b = 3.6cm, h = 3.1cm, and l = 8cm.

a) Calculate its volume.

b) Round your answer to the neareast integer.

c) Write down the exact value for the volume in the form \(a * 10^k\), where \(1000 < a < 10000\) and state the value of k.

Answers and Explanations

Question 7Premium

[Maximum mark: 6]

The formula for the distance between two points is given by:

\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Consider point A with coordinates (3,5), and points B with coordinates (6,1).

a) Find the distance between points A and B.

Now, consider point C with coordinates (2,4), and points D with coordinates (8,9). Alex believes that the distance between points C and D is twice the distance between points A and B.

b) Find out whether Alex was correct.

Answers and Explanations

a) Calculate the exact volume of the cuboid in cm³.

b) Give the value obtained in part (a) to dm³.

Mark is instructed to paint the cuboid using black and white paint. Black paint is used for the top and bottom (4.8cm x 5.8cm), and white paint is used for the sides. The cost of covering 1cm² with black paint is 0.15$, and with white paint it is 0.12$.

The cost of 1cm² of the paint is $0.05. Sara would estimate the cost of having painted the cylinder blue to be $20.