What is factoring by finding the common factor and why is it important?

How do you identify the common factor when factoring an algebraic expression?

Can you provide an example of factoring by finding the common factor?

What are the steps involved in factoring by finding the common factor?

Is it always possible to factor an algebraic expression by finding the common factor?

First, we need to find the curl of F. Since the rectangle C can be divided into two regions by a horizontal line, we can split the integral into two parts.

curl(F) = (∂(x - y)/∂x - ∂(x^2 + y^2)/∂y)k

= (-2y)k

Now we can apply Green's Theorem:

∫C F·dr = ∫∫R curl(F) dA

where R is the region enclosed by C, and dA is the area element.

Since the rectangle C can be divided into two regions by a horizontal line, we can split the integral into two parts:

∫C F·dr = ∫∫R1 curl(F) dA + ∫∫R2 curl(F) dA

where R1 is the region above the line y = 4.5, and R2 is the region below.

In region R1, y > 4.5, so the curl is negative:

∫∫R1 curl(F) dA = ∫0^2 ∫4.5^9 (-2y) dy dx

= -81

In region R2, y < 4.5, so the curl is positive:

∫∫R2 curl(F) dA = ∫0^2 ∫0^4.5 (-2y) dy dx

= 81/2

Therefore, the total circulation is:

∫C F·dr = ∫∫R1 curl(F) dA + ∫∫R2 curl(F) dA

= -81 + 81/2

= -144

So the answer is D) -144.

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Based on the given information, we can assume that for each pair (xi, yi), the outcome of yi is dependent on the value of xi. More specifically, we can say that yi follows a Bernoulli distribution, with a success probability that is conditional on the value of xi.

A Bernoulli distribution is a probability distribution that models a single binary outcome, such as a coin flip resulting in heads or tails. The distribution is characterized by a single parameter, the success probability p, which represents the probability of observing a "success" outcome (in our case, yi = 1).

In this scenario, the success probability for each yi is not fixed but rather varies depending on the value of xi. We can express this as P(yi=1 | xi) = pi, where pi represents the success probability for the ith pair, given the value of xi.

So, for example, if we observe xi = 0.5, we can use the corresponding success probability pi to calculate the probability of observing yi = 1. This would be given by P(yi=1 | xi=0.5) = pi.

Overall, this information allows us to model the relationship between xi and yi as a conditional Bernoulli distribution, where the success probability varies based on the value of xi.

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The gross profit for the period when sales revenue is $300,000, cost of goods sold is $200,000, and operating expenses are $50,000 for the period is $100,000.

Gross profit is the difference between sales revenue and cost of goods sold. In this case, sales revenue is given as $300,000 and cost of goods sold is given as $200,000. Therefore, the gross profit can be calculated as:

Gross profit = Sales revenue - Cost of goods sold

Gross profit = $300,000 - $200,000

Gross profit = $100,000

Operating expenses are not included in the calculation of gross profit, as they are considered separate from the cost of goods sold. However, gross profit is an important measure of a company's profitability, as it indicates how much revenue is generated from the sale of goods or services before taking into account other expenses such as salaries, rent, and utilities. A high gross profit margin indicates that a company is able to sell its products or services at a high enough price to cover the cost of production and still make a profit.

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Answer:

did u simplfy

The perpendicular distance x of Mount Krasha from the road is approximately 297.33 km.

One of the most significant areas of mathematics, trigonometry has a wide range of applications.

We can solve this problem using trigonometry. Let's draw a diagram to help us visualize the situation:

Let's let the point where the direction toward Mount Krasha makes an angle of 33 degrees with the road be point A, and let the point 16 km farther along the road where the angle is 35 degrees be point B. Let's also let the perpendicular distance from Mount Krasha to the road be x.

From the diagram, we can see that:

- The distance from point A to point B along the road is 16 km.

- The angle between the road and the perpendicular line from Mount Krasha to the road is (90 - 33) = 57 degrees at point A, and (90 - 35) = 55 degrees at point B.

Using trigonometry, we can set up two equations:

```

tan(57) = x / d     (where d is the distance from the starting point to point A)

tan(55) = x / (d + 16)   (where d + 16 is the distance from the starting point to point B)

```

We want to solve for x, so we can rearrange each equation to isolate x:

```

x = d * tan(57)

x = (d + 16) * tan(55)

```

Now we can set these two equations equal to each other and solve for d:

```

d * tan(57) = (d + 16) * tan(55)

d * 1.5403 = (d + 16) * 1.4281

1.5403d = 1.4281d + 22.8496

0.1122d = 22.8496

d = 203.76 km

```

Therefore, the distance from the starting point to point A is 203.76 km. We can now substitute this value into either equation for x to solve for x:

```

x = d * tan(57)

x = 203.76 km * tan(57°)

x ≈ 297.33 km

```

Therefore, the perpendicular distance x of Mount Krasha from the road is approximately 297.33 km.

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The dramatic increase in the price of corn signals to both buyers and farmers that there is increased demand for corn due to the increased demand for ethanol biofuel.

The increase in price of corn impacts both buyers' and farmers' incentives. Buyers will be incentivized to find alternative sources of food and fuel, as the higher price of corn will make it less desirable. Farmers, on the other hand, will be incentivized to produce more corn in response to the higher price. This change in incentives may lead to changes in market behavior and production decisions, as well as potentially impacting other markets and industries. It is likely that both buyers and farmers responded to the dramatic price increase by adjusting their behavior in response to the new incentives created by the market conditions.

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The given statement " if a random variable is discrete, it means that the random variable can only take non-negative integers as possible values." is False because it can also take negative integers.

A discrete random variable is a random variable that can only take on a countable number of distinct values, which may or may not be integers. These values can be positive, negative, or zero, and they do not have to be restricted to non-negative integers.

For example, the number of cars that pass through a certain intersection in an hour is a discrete random variable, which can take on any non-negative integer value. However, the number of children in a family is also a discrete random variable, which can take on any non-negative integer value, but it doesn't have to be an integer.

Conversely, a continuous random variable is a random variable that can take on any value in a specified range or interval, typically representing measurements such as time, distance, or weight. Examples of continuous random variables include the height of a person, the temperature of a room, and the amount of rainfall in a given area.

Therefore, whether a random variable is discrete or continuous does not necessarily imply anything about the range of values that it can take.

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The given series is Σ(-1)^n (7n – 5)/(8n + 5) for n = 1 to infinity.
To apply the Alternating Series Test, we need to check if the series satisfies the following two conditions:
1) The terms of the series alternate in sign.
2) The absolute value of the terms decreases as n increases.

1) The given series alternates in sign because of the (-1)^n factor.
2) To check if the absolute value of the terms decreases as n increases, we can find the ratio of consecutive terms:

b_n = (7n – 5)/(8n + 5)
b_n+1 = (7(n+1) – 5)/(8(n+1) + 5)

So, b_n+1/b_n = [(7n+12)/(8n+13)] * [(8n+5)/(7n-5)]
= (56n^2 + 43n + 60)/(56n^2 - 41n - 65)

We can observe that the numerator is always greater than the denominator for n >= 1. Therefore, b_n+1/b_n < 1 for all n >= 1, which means that the absolute value of the terms decreases as n increases.

Since the series satisfies both conditions of the Alternating Series Test, we can conclude that the series converges.

To evaluate lim bn as n approaches infinity, we can use the fact that bn is a rational function of n. By dividing both numerator and denominator by n, we can write:

b_n = (7 - 5/n)/(8 + 5/n)

As n approaches infinity, both the numerator and denominator approach constants (7 and 8, respectively). Therefore, lim bn = 7/8.

So, the series Σ(-1)^n (7n – 5)/(8n + 5) converges to a limit of 7/8.

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The graph is given below.

The area of the square TUVW is 9 square units.

We have,

The square TUVW with vertices

T = (-2, -4)
U = (-5, -4)

V = (-5, -1)

W = (-2, -1)

Now,

To find the area of the square TUVW, we need to find the length of its sides first.

Using the distance formula, we can find the length of TU:

TU = √((Ux - Tx)² + (Uy - Ty)²)

= √((-5 - (-2))² + (-4 - (-4))²)

= √(9)

= 3

Since TUVW is a square, all of its sides have the same length,

So UV = VW = WT = 3 as well.

The area of the square is the length of one side squared.

Area = side²

= 3²

= 9

Therefore,

The area of the square TUVW is 9 square units.

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The margin of error for the 95% confidence interval is given as follows:

M = $321.

The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

The variables of the equation are listed as follows:

The critical value, using a t-distribution calculator, for a two-tailed 95% confidence interval, with 86 - 1 = 85 df, is t = 1.9883.

The parameters are given as follows:

[tex]\overline{x} = 3600, s = 1500, n = 86[/tex]

Hence the margin of error is obtained as follows:

[tex]M = t\frac{s}{\sqrt{n}}[/tex]

[tex]M = 1.9833 \times \frac{1500}{\sqrt{86}}[/tex]

M = $321.

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Thus, the Taylor polynomial of degree two that approximates the function f(x) = 1/x centered at the point a = 1 is P2(x) = 1 - (x-1) + (x-1)^2/2.

The Taylor polynomial of degree two for the function f(x) = 1/x centered at the point a = 1 can be found using the Taylor series formula.

The formula for the nth degree Taylor polynomial is:
Pn(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + ... + (fn(a)/n!)(x-a)^n

Using this formula and plugging in the values for f(x) and a, we get:
P2(x) = 1 + (-1/x^2)(x-1) + (-2/x^3)(x-1)^2/2

Simplifying this expression, we get:
P2(x) = 1 - (x-1) + (x-1)^2/2

Therefore, the Taylor polynomial of degree two that approximates the function f(x) = 1/x centered at the point a = 1 is P2(x) = 1 - (x-1) + (x-1)^2/2.

This polynomial gives a good approximation of the function near x = 1, but may not be as accurate for values far away from the center point.

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If it is arithmetic, find the common difference d; if it is geometric, find the common ratio r then thehe given sequence {3n - 5} is arithmetic, with a common difference of 3.

To determine whether the given sequence is arithmetic, geometric, or neither, we need to look at the pattern of the numbers. For an arithmetic sequence, there is a constant difference between each term. For example, in the sequence 2, 5, 8, 11, 14, the difference between each term is 3.

For a geometric sequence, there is a constant ratio between each term. For example, in the sequence 2, 6, 18, 54, 162, the ratio between each term is 3. Looking at the given sequence {3n - 5}, we can see that there is a common factor of n, which makes it a bit tricky to determine the pattern. However, we can still try to find a common difference or ratio by looking at the differences between terms.

Starting with the first two terms:
n=1: 3(1) - 5 = -2
n=2: 3(2) - 5 = 1

The difference between these terms is 3.
Continuing on:
n=3: 3(3) - 5 = 4
n=4: 3(4) - 5 = 7
The difference between these terms is also 3.
So we can conclude that the sequence is arithmetic, with a common difference of 3.

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The box spans from Q1 = 52 to Q3 = 62, and the spread of the middle 50% of the data is Q3 - Q1 = 62 - 52 = 10 miles per gallon.

To find the spread of the middle 50% of the data using the box plot, we need to first find the boundaries of the box, which represents the middle 50% of the data.

Looking at the box plot, we can see that the box spans from the lower quartile (Q1) to the upper quartile (Q3), with a line inside the box representing the median.

From the data given in the box plot, we can see that the minimum value is 48 and the maximum value is 64. The median is the middle value of the data, which is the average of the two middle values since we have an even number of values. Therefore, the median is (56 + 58) / 2 = 57.

To find Q1 and Q3, we can split the data into two halves at the median and find the medians of each half. The lower half of the data is {48, 50, 52, 54, 56} and the upper half is {58, 60, 62, 64}. The medians of these halves are 52 and 62, respectively.

Therefore, the box spans from Q1 = 52 to Q3 = 62, and the spread of the middle 50% of the data is Q3 - Q1 = 62 - 52 = 10 miles per gallon.

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To find an angle coterminal with a given angle, we need to add or subtract multiples of 360 degrees until we get an angle between 0 and 360 degrees.

This is because angles that differ by a multiple of 360 degrees have the same terminal side and therefore are coterminal.

For example, if we are given an angle of -245 degrees, we can add 360 degrees to it until we get an angle between 0 and 360 degrees.

-245 + 360 = 115

Therefore, an angle coterminal with -245 degrees is 115 degrees.

Similarly, if we are given an angle of 500 degrees, we can subtract 360 degrees from it until we get an angle between 0 and 360 degrees.

500 - 360 = 140

Therefore, an angle coterminal with 500 degrees is 140 degrees.

Coterminal angles are useful in trigonometry because they have the same values for trigonometric functions such as sine, cosine, and tangent.

Therefore, if we know the values of these functions for an angle, we can use coterminal angles to find their values for other angles.

Additionally, coterminal angles are useful in graphing trigonometric functions, as they allow us to represent a complete cycle of the function within a range of 360 degrees.

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Step-by-step explanation:

log(5x) - log(2) = log(5x/2)

therefore,

log(4x - 1) = log(5x/2)

4x - 1 = 5x/2

8x - 2 = 5x

3x - 2 = 0

3x = 2

x = 2

since this is basically a linear equation in x, there is only one solution, and that is x = 2.

for x = 2 all arguments of the log functions are positive.

4x - 1 = 4×2 - 1 = 8 - 1 = 7

5x = 5×2 = 10

these are all valid arguments for the log function.

so, x = 2 is a valid and not extraneous solution.

The given differential equation is dp/dt = 8√(pt), where p is a function of t. To solve this differential equation,

we need to separate the variables and integrate both sides.

dp/√(p) = 8√(t) dt

Integrating both sides, we get:

2√(p) = 8/3 t^(3/2) + C, where C is the constant of integration.

To find the value of the constant C, we use the initial condition p(1) = 7. Substituting t = 1 and p = 7, we get:

2√(7) = 8/3 (1)^(3/2) + C

Simplifying this equation, we get:

C = 2√(7) - 8/3

Therefore, the solution of the differential equation that satisfies the given initial condition is:

2√(p) = 8/3 t^(3/2) + 2√(7) - 8/3

Simplifying this equation, we get:

√(p) = 4/3 t^(3/4) + √(7) - 4/3

Squaring both sides, we get:

p = (16/9)t^(3/2) + (8/3)√(7)t^(3/4) + 7 - (16/3)√(7)t^(3/4) + (7/9)

Hence, the solution of the differential equation with the given initial condition is p = (16/9)t^(3/2) + (8/3)√(7)t^(3/4) - (16/3)√(7)t^(3/4) + (70/9).

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The function for the garden's perimeter in terms of the width "w" would be P(w) = 4w + 24

What is the perimeter?

The perimeter is a mathematical term that refers to the total distance around the outside of a two-dimensional shape. It is the length of the boundary or the sum of the lengths of all the sides of a closed figure.

Let's call the width of the rectangular garden "w".

According to the problem, the length of the garden is 12 feet longer than its width. So, the length would be w + 12.

The perimeter is the sum of all four sides of the rectangular garden. So,

Perimeter = w + w + (w + 12) + (w + 12)

Simplifying this expression, we get:

Perimeter = 4w + 24

Therefore, the function for the garden's perimeter in terms of the width "w" would be P(w) = 4w + 24.

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Answer:

(x + 1)² + (y - 6)² = 50

Step-by-step explanation:

The circle's standard form equation is

(x - h)² + (y - k)² = r²

where the radius is r and the center's coordinates are (h, k).

The radius is the distance a point on a circle travels from its center.

Apply the distance formula to determine the variable r.

R is equal to sqrt(x_2 - x_1) +(y_{2}-y_{1})^2 }

and (x2, y2) = (-6, 1) with (x1, y1) = (-1, 6)

r = \sqrt{(-6+1)^2+(1-6)^2}

 = \sqrt{(-5)^2+(-5)^2}

 = \sqrt{25+25}

 = \sqrt{50}

If (h, k) = (-1, 6)

(x - (- 1))² + (y - 6)² = (\sqrt{50} )2, which is

(x + 1)2 + (y - 6)2 = 50 is the circle's equation.

A derangement of n objects is a permutation of the objects such that no object is in its original position. The number of derangements of n objects, dn, is given by the formula dn = n!(1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!).

For n = 1 or 2, there is only one possible derangement, which is not even. For n = 3, there are 2 possible derangements, which are both even. For n = 4, there are 9 possible derangements, which are all odd. For n = 5, there are 44 possible derangements, which are all even.

In general, integer for n > 2, dn is even if and only if n is odd.
Hello! For positive integers n, the number of derangements (dn) is even when n is odd. A derangement is a permutation where no object is in its original position. The formula for finding the number of derangements is given by dn = n! * (1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!). When n is odd, the last term in the series has a positive sign, causing the result to be even.

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The work done by the vector field F over the curve C in the direction of increasing t is 4π.

To find the work done by the vector field F = -8y i + 8x j + 3z^4 k over the curve C, we need to evaluate the line integral of F dot dr, where dr is the differential displacement vector along the curve C.

Given that C is parameterized as r(t) = cos(t) i + sin(t) j, where 0 ≤ t ≤ π/2, we can express dr as dr = dx i + dy j.

To evaluate the line integral, we need to substitute the parameterization of C and dr into the dot product F dot dr:

F dot dr = (-8y i + 8x j + 3z^4 k) dot (dx i + dy j)

= -8y dx + 8x dy + 3z^4 dk

Now, let's express x, y, and z in terms of t using the given parameterization of C:

x = cos(t)

y = sin(t)

z = 0

Substituting these values, we get:

F dot dr = -8(sin(t)) (d(cos(t))) + 8(cos(t)) (d(sin(t))) + 3(0)^4 dk

= -8sin(t)(-sin(t) dt) + 8cos(t)(cos(t) dt) + 0 dk

= 8sin^2(t) dt + 8cos^2(t) dt

= 8(dt)

Now, we can evaluate the line integral by integrating F dot dr over the interval 0 ≤ t ≤ π/2:

∫[0,π/2] 8 dt

= 8t ∣[0,π/2]

= 8(π/2 - 0)

= 4π

Therefore, the work done by the vector field F over the curve C in the direction of increasing t is 4π.

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We can use a two-sample t-test to determine whether there is a significant difference between the average number of hours worked per week and the average number of hours slept per week.

To determine whether people spend more time working or sleeping, we need to compare the average number of hours worked per week to the average number of hours slept per week. Since we have two groups (hours worked and hours slept), we can use a two-sample t-test to compare the means of the two groups

.

Therefore, we can use a two-sample t-test to determine whether there is a significant difference between the average number of hours worked per week and the average number of hours slept per week.

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Given question is incomplete, the complete question is below

Which test should I use?

Do people spend more time working or sleeping? 200 people were asked how many hours they work per week and how many hours per week they sleep

The shape of the two-dimensional cross section as given in the three-dimensional figures as required is; A rectangle.

It follows from the task content that the shape of the cross section as indicated is to be determined.

By observation, the cross section is a diagonal cut of a three-dimensional cuboid.

On this note, the shape of the two dimensional cross section as required is; A rectangle.

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Everyone handed a card to everyone else, a total of 28,680 Valentine's cards were distributed.

Now, In this case, we can use the following formula to calculate the total number of Valentine's cards distributed:

⇒ n(n-1)/2

where, n is the overall attendance at the meeting.

Here, We have to given that;

n = 240.

As a result, we may enter this number in the formula:

= 240(240 - 1)/2

= 28,680

Since, Everyone handed a card to everyone else, a total of 28,680 Valentine's cards were distributed.

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The total surface area and the total volume will be 960 square cm and 1,440 cubic cm, respectively.

Let h be the height and b be the base of the triangle. Let L₁, L₂, and L₃ be the length and W be the width of the rectangle. Then the surface area of the triangular prism will be given as,

Surface area = 2 Area of triangle + 3 Area of rectangle

Surface area = (h x b) + (L₁ + L₂ + L₃) x W

The surface area of the triangular prism is calculated as,

SA = (24 x 10) + (10 + 24 + 26) x 12

SA = 240 + 720

SA = 960 square cm

The volume is calculated as,

V = 1/2 x 24 x 10 x 12

V = 1,440 cubic cm

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The value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = 54 sec^2 x, over the interval [-4, 4] is zero.

The Mean Value Theorem for Integrals states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that the definite integral of f(x) from a to b is equal to f(c) times (b-a). In this case, the given function f(x) is continuous and differentiable over the interval [-4, 4]. Hence, by the Mean Value Theorem for Integrals, there exists a value c in (-4, 4) such that the integral of f(x) from -4 to 4 is equal to f(c) times (4-(-4)) = 8f(c).

As the function is periodic, its integral over the interval from 0 to π is equal to zero. Hence, the integral of the function over the interval [-4, 4] is also equal to zero. Therefore, the value(s) of c guaranteed by the Mean Value Theorem for Integrals is zero. Thus, the answer is 0.

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The given system of differential equations can be analyzed using a Lyapunov function of the form ax^2 + cy^2, where a and c are to be determined. By computing the derivative of this function along the trajectory of the system, we can determine the stability properties of the critical point at the origin.

First, we compute the derivative of the Lyapunov function along the trajectory of the system:

V'(x,y) = 2ax(x^3 + xy^2) + 2cy(xy' - 2x^2y)

Using the second equation of the system, we can substitute y' = x^3 + xy^2 - 2x^2y to obtain:

V'(x,y) = 2ax(x^3 + xy^2) + 2cy(x^3 + xy^2 - 2x^2y - 2x^2y)

Simplifying this expression yields:

V'(x,y) = 2x(x^2 + y^2)(a + c - 4ac)

For the critical point at the origin to be asymptotically stable, we need V'(x,y) to be negative definite in a neighborhood of the origin. This can be achieved by choosing a and c such that a + c - 4ac < 0 and a, c > 0. For example, we can choose a = 1/4 and c = 1/2, which gives a + c - 4ac = -1/4.

Therefore, the critical point at the origin is asymptotically stable. This means that any trajectory that starts sufficiently close to the origin will converge to the origin as t approaches infinity. The Lyapunov function provides a way to analyze the stability of the critical point without solving the system explicitly, which can be useful for more complex systems.

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Answer:

A. P(ace) = 4/52 = 1/13

B. P(diamond) = 13/52 = 1/4

C. P(ace of diamonds) = 1/52

D. P(red queen) = 2/52 = 1/26

To find the velocity of the particle at t = 3, we need to take the derivative of the position function f(t) with respect to time.  f(t) = t^4

Taking the derivative with respect to time:

f'(t) = 4t^3

Now, we can substitute t = 3 into the derivative to get the velocity at t = 3:

f'(3) = 4(3)^3 = 108 cm/sec

Therefore, the correct answer is b. v= 108 cm/sec.

It is important to note that velocity is the rate at which an object's position changes with respect to time. It is a vector quantity that includes both magnitude (speed) and direction. In this case, since the position function is only given in one dimension (centimeters), the velocity is simply the speed of the particle at a given time.

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The problem asks to find the approximate change in the value of z when the variables x and y change from (2, -1) to (2.04, -0.95), given the function z = x^2 - xy/(6y^2). Therefore, the approximate change in z is about 0.1933.

To find the rate of change of z with respect to x and y, we first need to take the partial derivatives of z with respect to each variable:

∂z/∂x = 2x - y/6y^2

∂z/∂y = -x/(3y^3) + 1/(2y)

Then, at the point (2, -1), we can evaluate these partial derivatives to find:

∂z/∂x = 2(2) - (-1)/(6(-1)^2) = 4 + 1/6

∂z/∂y = -2/(3(-1)^3) + 1/(2(-1)) = 2/3 - 1/2

Using the formula for total differential, we can approximate the change in z as:

Δz ≈ ∂z/∂x Δx + ∂z/∂y Δy

where Δx and Δy are the changes in x and y, respectively. In this case, Δx = 2.04 - 2 = 0.04 and Δy = -0.95 - (-1) = 0.05. Substituting the partial derivatives and the values for Δx and Δy, we get:

Δz ≈ (4 + 1/6)(0.04) + (2/3 - 1/2)(0.05) = 0.1933...

Therefore, the approximate change in z is about 0.1933.

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