David is playing a trivia game where he gains points for correct answers and loses points for incorrect answers. At the start of round 3 his score is −1500 points. During round 3 he answered five 1000 point questions correctly and three 500 points questions incorrectly. What is his score at the end of round 3?
A) −2000
B) −500
C) 2000
D) 3500

The volume of a pyramid with a rectangular base and height of 2h is given by the formula V = (1/3) * Base area * Height. The base area for this pyramid is 32b² (derived from 4b * 8b), and the height is 2h. Hence, the volume equates to 64/3 b²h.

The volume of a pyramid is calculated with the formula V = (1/3) * Base area * Height. For the pyramid in question, the base is a rectangle with a length of 4b and a width of 8b, so the area of the base (A) is 4b * 8b = 32b². The height of the pyramid is given as 2h. So, by plugging these values into the formula, we obtain the volume of the pyramid as V = (1/3) * 32b² * 2h = 64/3 b²h.

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An investment that grows by 5% per year for 20 years will increase by approximately 165.33% over the 20-year period. This calculation is based on the principle of compound interest.

In the given scenario, the growth of the investment is calculated using the principle of

compound interest

- a concept in Mathematics. Compound interest takes into account the phenomenon that, over time, your investment will grow not only from the initial amount invested but also from any interest or earnings that have been accumulated. Therefore, the 5% growth rate per year is being applied to an increasingly larger amount because the interest is being compounded, i.e., you're earning 'interest on interest'.

To calculate the overall percent increase after 20 years, we use the formula for compound interest: A = P(1 + r/n)^(nt), where:

In this case, we can assume that P = 1 (representing 100% of the initial investment), r = 0.05 (5% growth per year), n = 1 (interest is compounded once per year), and t = 20 years. Substituting these values in, we get A = 1(1 + 0.05/1)^(1*20) = 2.6533 approximately. Therefore, the investment increased by about 165.33% over the 20-year period.

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The simplified expression of 225/400 is 9/16

From the question, we have the following parameters that can be used in our computation:

225/400

Divide 225 and 400 by 25

so, we have the following representation

225/400 = 9/16

This cannot be further simplified

Hence, the simplified expression is 9/16

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The price per ticket (before any additional charges) is approximately $90.79. Let's assume the price of each ticket before any additional charges (handling charge and GST) is "x" dollars.

Given that a handling charge of $5 is charged per ticket, the total handling charge for 2 tickets will be $5 * 2 = $10.

Now, let's calculate the total cost of the tickets after adding the handling charge:

Total cost of tickets (before GST) = 2x + 10

Now, the GST of 5% is charged on the ticket price and handling charges. The GST amount can be calculated as:

GST amount = 5% of (2x + 10) = 0.05 * (2x + 10) = 0.1x + 0.5

The total charge for 2 tickets, including the handling charge and GST, is given as $201.16. So, we can set up the equation:

Total cost of tickets (including GST) = Total cost of tickets (before GST) + GST amount

$201.16 = (2x + 10) + (0.1x + 0.5)

Now, solve for "x":

2x + 10 + 0.1x + 0.5 = 201.16

2.1x + 10.5 = 201.16

2.1x = 201.16 - 10.5

2.1x = 190.66

x = 190.66 / 2.1

x ≈ 90.79

So, the price per ticket (before any additional charges) is approximately $90.79.

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the correct number label for point A on the number line is [tex]\( \frac{1}{20} \).[/tex]

Let's break this down step by step:

1. Determine the time available for each chore: Amy has 1/4 hour to do 5 chores. To find out how much time she can spend on each chore, divide the total time available (1/4 hour) by the number of chores (5).

[tex]\( \frac{1}{4} \div 5 = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20} \)[/tex]  hour per chore.

2. Representing the fraction of an hour on a number line:The number line represents the whole hour. Since Amy has 1/20 hour per chore, each point on the number line represents 1/20 hour.

  - Let's label point A as [tex]\( \frac{1}{20} \).[/tex]

So, the correct number label for point A on the number line is [tex]\( \frac{1}{20} \).[/tex]

Amy has 1/4 hour to complete 5 chores. To find out how much time she can spend on each chore, we divide 1/4 by 5, which gives us 1/20 hour per chore. On the number line representing an hour, each point represents 1/20 hour, so point A is labeled as 1/20. This means that Amy can spend 1/20 hour on each chore to complete all five chores within the 1/4 hour time frame she has.

complete question

Amy has 1/4 hour to do 5 chores. She plans to spend the same fraction of an hour on each chore. She wants to use the number line to help her determine what fraction of an hour she can spend on each chore. What is the correct number label for point A?

To order the numbers √7, 2, √8/2 from least to greatest, simplify √8/2 to √2. Then compare √2, 2, and √7 to find the order.

To order the numbers √7, 2, √8/2 from least to greatest, first simplify √8/2. We know that √8 = √(4*2) = 2√2, so √8/2 = (2√2)/2 = √2. Now we have √7, 2, and √2. Comparing these numbers, √2 < 2 < √7.

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The actual length of the living room is 6 meters.

Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors.

Here the unit of lengths, millimeter is converted to meter based on the given scale

For the given situation,

In blue print the house is represented as follows,

44 millimeters =  8 meters

⇒ [tex]1 millimeter = \frac{8}{44} meters[/tex]

⇒ [tex]1 millimeter = 0.18 meters[/tex]

The length of the living room = 33 millimeters

The actual length of the living room in meters = [tex]33(0.18)[/tex]

⇒ [tex]5.94[/tex] ≈ [tex]6 meters[/tex]

Hence we can conclude that the actual length of the living room is           6 meters.

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There are 60 people enrolled in the program, then 24 people will be sent to the new plant.

The percentage is defined as a ratio expressed as a fraction of 100.

To find the number of people sent to the new plant, we can multiply the percentage of people sent to the new plant by the total number of people enrolled in the training program. The percentage is given as a decimal by dividing the percentage by 100, so 40% is equal to 0.40.

The result is then multiplied by the number of people enrolled in the training program, which is 60.

40% of those enrolled in the training program will be sent to the new plant, which is 40/100 x 60 people = 24 people.

The final result is 24 people.

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The mouse ran a distance of 11 feets before being caught by the cat

Using the information on the graph ;

Y - axis = Distance covered by the mouse and cat

X - axis = Time taken to cover Distance

The intersection point of the two lines is the point at which the mouse was caught .

Tracing the point to the y-axis , we get the required Distance value

Hence, from the graph, the distance is 11 feets

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Before being caught by the cat, the mouse sprinted 11 feet.

How can we apply the graph to our solution?

Using the data from the graph;

The distance traveled by the mouse and cat is represented by the Y axis.

X axis = Distance traveled in time

The spot where the two lines connect is where the mouse was trapped.

We get the requisite Distance value by tracing the point to the y-axis.

As a result of the graph, the distance is 11 feet.

To solve the surface integral of a scalar field as given, parametrize the surface S, compute the gradient of the scalar field f, and setup the integral related to the given plane equation and limits of the unit disk. Details of the setup and a specific numerical solution are not provided.

The problem you've asked about, i.e., evaluating the integral i = z s f ds when f(x, y, z) = z 2 + 3xy and s is the portion of the plane x + 2y + 2z = 0 above the unit disk x 2 + y 2 ≤ 1 in the xy-plan, falls under the subject of vectors and calculus, particularly triple integrals. It includes a surface integral of a scalar field.

In general, to evaluate a surface integral of a scalar field, first, you should parametrize the surface S with vector function r(u, v). Then you calculate the cross product of partial derivatives of r with respect to u and v to find the surface element dS. In other words, you compute the gradient of the scalar field f.

In this case, you would need to set up the integral with f(x, y, z) and ds related to the given plane equation x + 2y + 2z = 0 and the limits to the unit disk. Solve this integral by any standard method (like substitution or by parts) as needed depending on the complexity of f.

Note: Since the specific setup and solution to this integral could be complex and calculation heavy, a detailed step by step solution is not provided in this formatted answer.

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The total length of the curve with parametric equations give is 23.5 units.

A parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

Given that, a parametric equation, x = sin(t), y = sin(2t), z = sin(3t),

Finding the first derivative,

x' = cost

y' = 2cos2t

z' = 3cos3t

The parametric integral for length =

L = [tex]\int\limits^{t_2}_{t_1} \sqrt{ {x'^2+y'^2+z'^2} \, dx[/tex]

t₁ = 0, t₂ = 2π

T = [tex]\int\limits^{2\pi}_{0} \sqrt{ {(cost)^2+(2cos2t)^2+(3cos3t)^2} \, dt[/tex]

We will solve the integral by trapezoidal rule,

[tex]\int\limits^a_bf ({x}) \, dx = \frac{f(a)+f(b)}{2} (b-a)[/tex]

Therefore,

[tex]\int\limits^{2\pi}_{0} \sqrt{ {(cost)^2+(2cos2t)^2+(3cos3t)^2} \, dt[/tex]

= [tex]\frac{\sqrt{(cos2\pi)^2+(2cos2\pi)^2+(3+cos3\pi)^2} +\sqrt{(cos0)^2+(2cos0)^2+(3+cos0)^2}}2} (2\pi)[/tex]

= 23.5

Hence, the total length of the curve with parametric equations give is 23.5 units.

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The dimensions of the rectangle are length = 22 meters and width = 13 meters

[tex]A=l\times w[/tex], where 'l' is the length and 'w' is the width of the rectangle.

For given question,

Suppose 'l' is the length and 'w' is the width of the rectangle.

The length of a rectangle is 4 meters less than twice the width.

So, we get an equation,

⇒ l = 2w - 4

The area of the rectangle is 286 square​ meters.

⇒ A = 286 sq. m.

Using the formula for the area of the rectangle,

[tex]\Rightarrow A=l\times w\\\\\Rightarrow 286=(2w-4)\times w\\\\\Rightarrow 286=2w^2-4w\\\\\Rightarrow 2w^2-4w-286=0[/tex]

Now, we solve the quadratic equation [tex]2w^2-4w-286=0[/tex]

[tex]\Rightarrow 2w^2-4w-286=0\\\\\Rightarrow w^2-2w-143=0\\\\\Rightarrow (w-13)(w+11)=0\\\\\Rightarrow w-13=0~~~or~~~w+11=0\\\\\Rightarrow w=13~~~or~~~w=-11[/tex]

w = -11 is not possible.

So, the width of the rectangle is 13 meters.

And the length of the rectangle would be,

[tex]\Rightarrow l \\= 2w - 4\\=(2\times 13)-4\\=22[/tex]

Therefore, the dimensions of the rectangle are length = 22 meters and width = 13 meters

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Answer:

X+2y=12

Step-by-step explanation:

The series ∑∞ n=1 an with terms [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]diverges. Hence the correct option is 2.

Ratio Test:

For the series ∑∞ n=1 an with terms an, let:

lim n->∞ |an+1 / an| = L

If L < 1, the series converges.

If L > 1, the series diverges.

If L = 1 or the limit diverges, the test is inconclusive.

Applying the Ratio Test:

Here, [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]

Let's analyze the limit as n approaches infinity for the ratio of successive terms:

Simplifying the numerator and denominator, we get:

[tex]lim n- > ∞ | 7^(^n^+^1^) * (n+1)! / (7^n * n!) | = lim n- > ∞ | 7 * (n+1) |[/tex]

As n approaches infinity, the constant term 7 becomes insignificant. Therefore:

lim n->∞ |an+1 / an| = lim n->∞ |n+1|

Since the limit approaches positive infinity as n approaches infinity, L > 1.

Conclusion:

Based on the ratio test, the series ∑∞ n=1 an with terms [tex]a_n = (1 * 8 * 15 * ... * (7n-6)) / (7^n * n!)[/tex]diverges. Hence the correct option is 2.

Complete question:

Use either the ratio test or the root test as appropriate to determine whether the series ∑∞, n=1 an with given terms an converges, or state if the test is inconclusive. an= 1⋅8⋅15⋯(7n−6)/7^n n!

The series converges.

The series diverges.

The test is inconclusive.

Answer: For those in the future!

The answer to this is D: (4g+2)(16g^2minus8g+4)

Step-by-step explanation:

Answer:

The answer is option C on plato :)

Step-by-step explanation:

Answer:

D) [tex]8.7 * 10^{10}[/tex]

Step-by-step explanation:

The question is correctly written as below:

Which number is one million times larger than [tex]8.7 * 10^{4}[/tex]?

A) [tex]8.7 * 10^{7}[/tex]

B) [tex]8.7 * 10^{8}[/tex]

C) [tex]8.7 * 10^{9}[/tex]

D) [tex]8.7 * 10^{10}[/tex]

Consider [tex]8.7 x 10^{4}[/tex] to be number A and number B is one million times ([tex]10^{6}[/tex]) greater than number A

[tex]B = A * 10^{6}[/tex]

[tex]B = 8.7 * 10^{4} * 10^{6}[/tex]

[tex]B = 8.7 * 10^{4+6}[/tex]

[tex]B = 8.7 * 10^{10}[/tex]

D) [tex]8.7 * 10^{10}[/tex]

Final answer:

To express 2.6x10^-6 in standard decimal form, the decimal point is moved 6 places to the left, giving us 0.0000026.

Explanation:

To write 2.6x10^-6 in standard decimal form, start by understanding the exponent. The notation 10^-6 means that we are moving the decimal point 6 places to the left because it is a negative exponent. Therefore, we start with 2.6 and move the decimal point to the left 6 places, which means we have to add zeros to fill in the gaps.

The final expression in standard form will be 0.0000026.