Detailed Answer

a) w = 0.39

b) We know that a has to be between 1 and 10, so the only possibility is:

\[ 0.3862901 = 3.862901 * 10^{-1}\]

That means that k = -1.

c) Using the formula for percentage error, we get that:

\[ ε= |\frac{v_a - v_e}{v_e}| * 100\% = |\frac{0.3862901- 0.427}{0.427}| * 100\% = 9.53%\]

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Detailed Answer

a)

\[ p = \frac{8cos(45°)}{3x + 2y} + z = \frac{8cos⁡(45°)}{3*6+2*9} + 4 = \frac{8 * \frac{\sqrt{2}}{2}}{18 + 18} = \frac{\sqrt{2}}{9} + 4\]

b) Using GDC we can see that:

\[ \frac{\sqrt{2}}{9} + 4 \approx 4.157\]

c) Since k = 2, we know that:

\[ 4.157 = 0.04157 * 10^2\]

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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:

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

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Detailed Answer

a)

\[ q = \frac{12x^2sin(α)}{5x^2 + 2y} = \frac{12 * 9 * \frac{\sqrt{3}}{2}}{45 + 18} = \frac{54\sqrt{3}}{63} = \frac{6\sqrt{3}}{7}\]

b)

\[\frac{6\sqrt{3}}{7} \approx 1.48\]

c) We know that k = 4, so:

\[1.48 = 0.000148*10^4\]

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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:

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

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Detailed Answer

a) By plugging the values into the formula, we get:

\[ V = \frac{1}{2}(2.4 + 3.6) * 3.1 * 8 = 74.4cm^3\]

b)

\[ V = 74.4cm^3 \approx 74cm^3\]

c) Knowing that a has to be between 1000 and 10000, we know that the only possible value it can take is 7440, so:

\[ 74.4 = 7440 * 10^{-2}\]

That means that \( k = -2 \)

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Detailed Answer

a)

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.

b)

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.

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Detailed Answer

a)

\[\text{Lower bound: } 0.305 - 0.0005 = 0.3045 \, \text{meters}\]

\[\text{Upper bound: } 0.305 + 0.0005 = 0.3055 \, \text{meters}\]

b) The dimensions of the pyramid are:

- Base side length: \(620 \, \text{feet}\)

- Height: \(\frac{2717}{2} = 1358.5 \, \text{feet}\)

To find the minimum possible volume, convert the bounds of \(1 \, \text{foot}\) into meters and calculate the volume:

\[ \text{Minimum side length: } 620 \times 0.3045 = 188.79 \, \text{meters}\]

\[ \text{Minimum height: } 1358.5 \times 0.3045 = 413.53725 \, \text{meters}\]

Volume of the pyramid:

\[ V = \frac{1}{3} \times (\text{side length})^2 \times \text{height}\]

\[ V = \frac{1}{3} \times (188.79)^2 \times 413.53725 \]

\[ V \approx 4913052 \, \text{m}^3\]

c) \( 4913052 = 4.913052 \times 10^6 \)

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