integral of e to -x cos2x from 0 to infinity

The velociraptor's speed is 2 m/s, and its velocity is -0.3 m/s [w].

To calculate the speed and velocity of the velociraptor, we need to first find the total distance it covered and the displacement.

Total distance = 5m + 10m + 3m + 2m = 20m

Displacement = Final position - Initial position

Displacement = (2m [w]) - (5m [e])

Displacement = -3m [w]

Velocity is defined as displacement per unit time, while speed is defined as distance per unit time.

Velocity = Displacement / Time

Velocity = -3m [w] / 10s

Velocity = -0.3 m/s [w]

Speed = Total distance / Time

Speed = 20m / 10s

Speed = 2 m/s

Therefore, the velociraptor's speed is 2 m/s, and its velocity is -0.3 m/s [w].

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If Ann takes a boat then the distance between Ann's home and the market across the lake is approximately 29 km.

To find the distance from Ann's home to the market, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

In this case, Ann's home, the market, and the point where she crosses the lake form a right triangle, with the distance she travels across the lake being the hypotenuse.

To calculate the distance, we can use the following formula:

c^2=a^2+b^2

where c is the distance from Ann's home to the market, a is the distance from her home to the point where she crosses the lake, and b is the distance from the market to the point where she crosses the lake.

We know that a = 20 km and b = 21 km, so we can plug these values into the equation:

c^2=20^2+21^2

c^2=400+441

c^2=841

To solve for c, we take the square root of both sides of the equation:

c=sqrt(841)

c=29

Therefore, the distance from Ann's home to the market is approximately 29 km, when she takes the shortest path across the lake directly toward the market.

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Let's assume that the pieces am and N have heights of h_am and h_N respectively, and let k be the number of times the height of piece N fits into the height of piece am (i.e., k is the ratio of the height of piece am to the height of piece N).

We know that the volume of each stack is equal. Let's use the following variables:
- h for the height of each piece of A
- k for the height of each piece of N
- 10 for the number of pieces in stack A
- 8 for the number of pieces in stack N

The equation for the volume of each stack is:
Volume of stack A = h x 10
Volume of stack N = k x 8

Since we know the volumes are equal, we can set the two equations equal to each other:
h x 10 = k x 8

To create an equation relating h and k, we can solve for one variable in terms of the other:
h = (8/10)k
or
k = (10/8)h

Either equation shows how h and k are related to each other. For example, if we know the value of h, we can use the first equation to find k.

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The measures of the arc angle AP is 63° using the Angles of Intersecting Chords Theorem

The Angles of Intersecting Chords Theorem states that the angle formed by the intersection of the chords is equal to half the sum of the intercepted arcs, and conversely, that the measure of an intercepted arc is half the sum of the two angles that intercept it.

109° = (AP + RQ)/2

109 = (AP + 155)/2

AP + 155 = 2 × 109 {cross multiplication}

AP + 155 = 218

AP = 218 - 155 {collect like terms}

AP = 63°

Therefore, the measures of the arc angle AP is 63° using the Angles of Intersecting Chords Theorem

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The expression represented by the given number line is f(x) = -k(x+2.5)(x-3) where k > 0.

The expression represented by the given number line can be determined by identifying the values that correspond to the endpoints of each arrow and the direction of the arrow.

Starting from the left endpoint, the arrow goes from -2.5 to -1. This means that the expression is positive between -2.5 and -1. To determine the exact expression, we need to know the interval of the arrow.

The arrow starts at 0 and ends at 3, which means the expression is positive between 0 and 3. Finally, the arrow goes from 3 to -2.5, which means the expression is negative between 3 and -2.5.

Putting all of this information together, we can write the expression as:

f(x) = k(x+2.5)(x-3)

where k is a constant that determines the overall scale of the expression. Since the expression is positive between -2.5 and -1, we know that k must be negative. Since the expression is negative between 3 and -2.5, we know that k must be positive.

Therefore, the expression represented by the given number line is:

f(x) = -k(x+2.5)(x-3) where k > 0.

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The approximate number of items that should be manufactured so that the profit, P(x) = R(x) - C(x), is maximum is 47 items (option C).

To find the approximate number of items that should be manufactured to maximize profit, we need to first find the profit function P(x) by subtracting the total cost, C(x), from the total revenue, R(x). Then, we need to find the critical points of P(x) and determine which one corresponds to the maximum profit.

Step 1: Find the profit function P(x) = R(x) - C(x)
Given R(x) = 25 ln(6x + 1) and C(x) = ∫x, let's find P(x):
P(x) = R(x) - C(x)
P(x) = 25 ln(6x + 1) - ∫x
Step 2: Find the critical points of P(x)
To find the critical points, we need to take the derivative of P(x) and set it equal to 0:
P'(x) = d/dx [25 ln(6x + 1) - ∫x]
Since the derivative of ln(6x + 1) is (6/(6x + 1)), and the derivative of ∫x is x:
P'(x) = 25 [tex]\times[/tex] (6/(6x + 1)) - x
Now, set P'(x) = 0 and solve for x:
25 [tex]\times[/tex] (6/(6x + 1)) - x = 0
Step 3: Determine which critical point corresponds to the maximum profit
The approximate number of items that should be manufactured so that the profit, P(x) = R(x) - C(x), is maximum is 47 items (option C).

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The value of n which is the number of frogs Heather has is 6.

The number of frogs Heather has is calculated as follows;

let the number of frogs Heather has = n

So Ava has 2 fogs, which is equal to 1/3 n.

The value of n which is the number of frogs Heather has is calculated as follows;

(1/3) n = 2

multiply both sides by 3;

n = 3 x 2

n = 6

The division using a diagram, is determined as;

    0         0

    I           I

    I           I

    I            I

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The daily newspaper has the most users among those surveyed.

To determine which news source has the most users, we need to compare the percentages of those who use each source.

According to the survey:

55% get news from local television station

60% get news from daily newspaper

40% get news from the internet

To compare these percentages, we can either convert them to fractions or decimals. Let's convert them to decimals:

55% = 0.55

60% = 0.60

40% = 0.40

Now we can compare them directly. We see that the source with the highest percentage is the daily newspaper, with 60% of those surveyed saying they get news from it. Therefore, the daily newspaper has the most users among the surveyed population.

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It takes approximately 10 hours for the drug concentration to decrease.

We can use exponential decay to model the concentration of the drug in the bloodstream. Let C(t) be the concentration of the drug in milligrams at time t in hours since the person took the drug. Then we have:

[tex]C(t) = 125(0.7)^t[/tex]

where 0.7 is the factor by which the concentration decreases each hour.

We want to find the time t such that C(t) = 1. Substituting this into the equation above, we get:

[tex]1 = 125(0.7)^t[/tex]

Dividing both sides by 125, we get:

[tex]0.008 = (0.7)^t[/tex]

Taking the logarithm of both sides with base 0.7, we get:

[tex]t = log(0.008) / log(0.7)[/tex]

Using a calculator, we can evaluate this expression to get:

t ≈ 10.07

Therefore, it will take approximately 10.07 hours for the concentration of the drug in the bloodstream to be 1 milligram.

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The expression that denotes the number of slices of pizza eaten by Harris is: (N/4) - 1

The parameters are given as:

Slices of pizza = n

Number of friends = 4

Slices of pizza evenly divided among friends = Total number of slices/number of friends = N/4

Now, this value will represent the number of slices each friend got.

Since, Harris had 1 slice lesser than what he received, then we can say that:

Number of slices of pizza eaten by Harris = (N/4) -1

This is because it's evenly divided into 4 people and as such we divide the total (N) by 4.

Since he at one less, you subtract one.

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Complete question is:

4 friends evenly divided up an n-slice pizza. One of the friends, Harris, ate 1 fewer slice than he received. How many slices of pizza did Harris eat? Write your answer as an expression.

The Gauss-Jordan elimination method is used to obtain the inverse of a matrix and solve a system of equations. The solution for the given system is x1 = 2/3, x2 = 5/3, x3 = 2/3.

We can represent the given system of equations in the matrix form as

| 4   3   1 |   | x1 |   | 2 |

| 1   1   1 | * | x2 | = | 3 |

| 2   5   2 |   | x3 |   | 1 |

Let's perform Gauss-Jordan elimination to obtain the inverse of the matrix A.

Augment the matrix with an identity matrix of the same order:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |

| 1   1   1 | * | 0  1  0 | = | ? ? ? |

| 2   5   2 |   | 0  0  1 |   | ? ? ? |

Use row operations to transform the left side of the augmented matrix into an identity matrix:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |   | 1  0  0 |

| 1   1   1 | * | 0  1  0 | = | ? ? ? | =>| 0  1  0 |

| 2   5   2 |   | 0  0  1 |   | ? ? ? |   | 0  0  1 |

To achieve this, we can subtract 2 times the second row from the third row, and subtract 4 times the second row from the first row:

| 4   3   1 |   | 1  0  0 |   | ? ? ? |   | 1  0  0 |

| 1   1   1 | * | 0  1  0 | = | ? ? ? | =>| 0  1  0 |

| 0   3   0 |   | 0 -2  1 |   | ? ? ? |   | 0  0  1 |

Next, we can divide the second row by 3 to obtain a leading 1 in the second row, and subtract 3 times the second row from the first row:

| 1   0  -1 |   | 1 -1  1/3 |   | ? ? ? |   | 1  0  0 |

| 0   1  0  | * | 0  1  0   | = | ? ? ? | =>| 0  1  0 |

| 0   0  1  |   | 0 -2  1   |   | ? ? ? |   | 0  0  1 |

Finally, we can add the third row to the first row and subtract the third row from the second row

| 1   0  0 |   | 1 -1  4/3 |   | ? ? ? |   | 1  0  0 |

| 0   1  0 | * | 0  1  2   | = | ? ? ? | =>| 0  1  0 |

| 0   0  1 |   | 0 -2  1   |   | ? ? ? | =>| 0  0  1 |

Hence, we have obtained the inverse of the matrix A as

| 1 -1  4/3 |

| 0  1  2   |

| 0 -2  1   |

We can now find the solution vector x by multiplying the inverse of A with the vector b

| x1 |   | 1 -1  4/3 |   | 2 |

| x2 | = | 0  1  2   | * | 3 |

| x3 |   | 0 -2  1   |   | 1 |

Performing the matrix multiplication, we get

| x1 |   | 2/3 |

| x2 | = | 5/3 |

| x3 |   | 2/3 |

Therefore, the solution of the given system of equations is

x1 = 2/3

x2 = 5/3

x3 = 2/3

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Answer:

a) £54

b) £1908

Step-by-step explanation:

a) use £1800 × 3% = £54

b) use £1800 + ( 54 × 2) = £1908

Answer: Debnil has 2 Tablespoons of salt.

Step-by-step explanation:

3/1 is the ratio for teaspoons to tablespoons.

Substitute the 1 with the 6. What is six divided by three? 2.

The factored form of the polynomial x⁴ - 2x³- 16x² + 2x + 15 is [x³(x - 2) - (4x + 3)(4x - 5)].

Factoring the polynomial?

x⁴- 2x³ - 16x² + 2x + 15.

First, look for any common factors among the terms. In this case, there are none.

Next, try factoring by grouping. To do this, group the first two terms and the last three terms: (x⁴ - 2x³) - (16x² - 2x - 15).

Factor out the greatest common factor from each group: x³(x - 2) - 1(16x² - 2x - 15).

Now, we have a difference of two expressions, but there isn't a common factor to factor further. Therefore, we must use other methods to factor the quadratic expression 16x²- 2x - 15.

Factor the quadratic expression using the "ac method." Multiply the leading coefficient (16) by the constant term (-15) to get -240. Find two numbers that multiply to -240 and add up to the linear coefficient (-2). These numbers are 12 and -20.

Rewrite the middle term using the two numbers found: 16x² + 12x - 20x - 15.

Group the terms in pairs: (16x² + 12x) + (-20x - 15).

Factor out the greatest common factor from each group: 4x(4x + 3) - 5(4x + 3).

Factor out the common binomial factor: (4x + 3)(4x - 5).

Now, put everything together: x³(x - 2) - (4x + 3)(4x - 5).

So, the factored form of the polynomial x⁴ - 2x³- 16x² + 2x + 15 is [x³(x - 2) - (4x + 3)(4x - 5)].

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The decay of the radioactive substance can be modeled by the exponential decay function:

y = a(1 - r)^t

where:

- y is the amount of substance present after t hours

- a is the initial amount of substance (in grams), which is 120 grams in this case

- r is the decay rate per hour, which is 18% or 0.18 in decimal form

- t is the time elapsed in hours

Plugging in the values we get:

y = 120(1 - 0.18)^t

Simplifying:

y = 120(0.82)^t

So this is the equation that represents the number of grams, y, present after t hours, given the initial amount of 120 grams and a decay rate of 18% per hour.

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She would have to work for 11 hours to pay these bills without her FSA/HSA.

Emma went to the doctor 9 times last month, which means she had to pay a total of 9 x $20 = $180 in copays.

Without her FSA/HSA, she would have to work to earn $180 after taxes. Since she pays 17 percent in taxes, the amount she would have to earn before taxes is $180 / (1 - 0.17) = $216.87.

To earn $216.87, she would have to work for $216.87 / $20 per hour = 10.84 hours.

Rounding up, she would have to work for 11 hours to pay these bills without her FSA/HSA.

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The value of measure of RU is,

⇒ 149°

We have to given that;

In circle,

m RS = 88 degree

m ST = 35 degree

Hence, We can formulate;

The value of measure of RU is,

⇒ 360° - ( 88° + 35° + 88°)

⇒ 360° - 211°

⇒ 149°

Thus, The value of measure of RU is,

⇒ 149°

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Latoya may experience some fluctuations in her winnings and losses in the short run, in the long run, her average winnings will approach $0.50 per selection.

What is the expected value of playing the game and what Latoya can expect after playing the game many times?

(a) To find the expected value of playing the game, we need to multiply the amount that Latoya can win or lose by the probability of each outcome, and then add up the results. Let p(i) be the probability of selecting the ball with the number i. Since there are 8 balls in total and each ball is equally likely to be selected, we have:

p(i) = 1/8 for i = 1, 2, ..., 8

Now we can calculate the expected value:

E(X) = ∑[i=1 to 5] (i * p(i)) + ∑[i=6 to 8] (-1 * p(i))

= (1/8)(1) + (1/8)(2) + (1/8)(3) + (1/8)(4) + (1/8)(5)

- (1/8)(1) - (1/8)(1) - (1/8)(1)

= 0.5

Therefore, the expected value of playing the game is $0.50.

(b) In the long run, after playing the game many times, Latoya can expect to break even (neither gain nor lose money) on average per selection. This means that over a large number of selections, she can expect to win some money on some selections and lose some money on others, but on average, her total winnings and losses will balance out to zero.

To see why this is the case, consider that the expected value of a single selection is $0.50. If Latoya plays the game many times, the law of large numbers tells us that the average winnings per selection will converge to the expected value of $0.50. So even though Latoya may experience some fluctuations in her winnings and losses in the short run, in the long run, her average winnings will approach $0.50 per selection.

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To test the claim that the outcomes of rolling the modified die are not equally likely, we can use a chi-square goodness-of-fit test. We will use a significance level of 0.05.

The null hypothesis is that the outcomes are equally likely. The alternative hypothesis is that the outcomes are not equally likely.

First, we need to calculate the expected frequencies assuming that the outcomes are equally likely.

Since the die has six sides, each outcome has a probability of 1/6. Therefore, the expected frequency for each outcome is 200/6 = 33.33.

To calculate the test statistic [tex]x^2[/tex], we can use the formula:

[tex]x^2 = Σ (observed frequency - expected frequency)^2 / expected frequency[/tex]

where Σ is the sum over all outcomes.

Using the observed and expected frequencies given in the problem, we get:

[tex]x^2 = (27 - 33.33)^2 / 33.33 + (31 - 33.33)^2 / 33.33 + (42 - 33.33)^2 / 33.33 + (40 - 33.33)^2 / 33.33 + (28 - 33.33)^2 / 33.33 + (32 - 33.33)^2 / 33.33[/tex]

[tex]x^2 = 3.02[/tex]

The degrees of freedom for this test is 6 - 1 = 5 (since there are 6 sides on the die).

Using a chi-square distribution table (or calculator), we can find the critical value for a significance level of 0.05 and 5 degrees of freedom to be 11.070.

Since the test statistic x^2 = 3.02 is less than the critical value of 11.070, we fail to reject the null hypothesis.

Therefore, we do not have enough evidence to conclude that the outcomes of rolling the modified die are not equally likely.

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The equation of the circle with a center at (-3, 6) and passing through the point (9, 1) is (x + 3)^2 + (y - 6)^2 = 169.

To write the equation of a circle with a center at the point (-3, 6) and passing through the point (9, 1), we can use the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle, and r is the radius.

1. Identify the center (h, k) as (-3, 6).
2. Calculate the radius using the distance formula between the center and the given point (9, 1):
  r = √((x2 - x1)^2 + (y2 - y1)^2)
  r = √((9 - (-3))^2 + (1 - 6)^2)
  r = √((12)^2 + (-5)^2)
  r = √(144 + 25)
  r = √169
  r = 13

3. Substitute the values of h, k, and r into the equation of a circle:
  (x - (-3))^2 + (y - 6)^2 = 13^2
  (x + 3)^2 + (y - 6)^2 = 169

So, the equation of the circle with a center at (-3, 6) and passing through the point (9, 1) is (x + 3)^2 + (y - 6)^2 = 169.

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To solve this problem, we need to use related rates. Let's start by drawing a diagram:

```
        /\
       /  \
      /    \
     /      \
    /        \
   /__________\
```

We know that the radius of the conical pile is always two times its height, so we can label the diagram as follows:

```
        /\
       /  \
      /    \
     /      \
    /        \
   /__________\
  /|    r=2h   \
 / |___________\
```

Now we need to find an equation that relates the height of the pile to its radius. We can use the formula for the volume of a cone:

```
V = (1/3)πr^2h
```

We want to solve for h in terms of r:

```
V = (1/3)πr^2h
3V/πr^2 = h
```

Now we can differentiate both sides of this equation with respect to time:

```
d/dt (3V/πr^2) = d/dt h
0 = (3/πr^2) dV/dt - (2/πr^3) dr/dt
```

We're given that the height is increasing at a rate of 2 cm/s when the pile is 11 cm high, so we know that:

```
dh/dt = 2 cm/s
h = 11 cm
```

We want to find the rate at which sand is leaving the bin, which is given by `dV/dt`. We can solve for this using the equation we derived:

```
0 = (3/πr^2) dV/dt - (2/πr^3) dr/dt
dV/dt = (2/3)πr^2 (dh/dt) / r
```

Now we just need to plug in the values we know:

```
dh/dt = 2 cm/s
h = 11 cm
r = 2h = 22 cm

dV/dt = (2/3)π(22)^2 (2) / 22
dV/dt = 264π/3
```

So the rate at which sand is leaving the bin when the pile is 11 cm high is `264π/3 cm^3/s`.

To solve this problem, we can use the relationship between the radius and height of the conical pile, as well as the given rate of height increase.

Since the radius (r) is always two times the height (h), we have r = 2h. The volume (V) of a cone is given by the formula V = (1/3)πr^2h. We can substitute r with 2h, so V = (1/3)π(2h)^2h.

Now, let's differentiate both sides with respect to time (t):

dV/dt = (1/3)π(8h^2)dh/dt

When the height is 11 cm, the rate of height increase (dh/dt) is 2 cm/s. We can substitute these values into the equation:

dV/dt = (1/3)π(8(11)^2)(2)

Solving for dV/dt:

dV/dt ≈ 2046.92 cm³/s

At that instant, the sand is leaving the bin at a rate of approximately 2046.92 cm³/s.

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The volume of the table tent 3 cmH8 cm12 cm is 288 cubic cm.

The volume of the table tent can be found by multiplying the length, width, and height of the tent.

Volume = length x width x height

In this case, the length is 12 cm, the width is 8 cm, and the height is 3 cm.

Volume = 12 cm x 8 cm x 3 cm

Volume = 288 cubic cm

Therefore, the volume of the table tent is 288 cubic cm.

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A sequence of transformations that maps figure 1 onto figure 2 and then figure 2 onto figure 3 include the following: D. a translation followed by a rotation.

In Mathematics and Geometry, a translation can be defined as a type of rigid transformation which moves every point of the object in the same direction, as well as for the same distance.

This ultimately implies that, a translation is a type of rigid transformation  that does not change the orientation of the original geometric figure (pre-image).

In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

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Complete Question:

Which sequence of transformations maps figure 1 onto figure 2 and then figure 2 onto figure 3?

a reflection followed by a translation

a rotation followed by a translation

a translation followed by a reflection

a translation followed by a rotation

Answer:

0.91

30.4

850

Step-by-step explanation:

The 1,000 feet and 200 feet distances of the top and the base of the skyscraper from the point on the ground, indicates, using Pythagorean Theorem that the height of the skyscraper is 400·√6 feet

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the other two sides.

The distance of the top of the skyscraper from a point on the ground = 1000 feet

The distance of the base of the skyscraper from the same point = 200 feet

Therefore, according to the Pythagorean Theorem, in the right triangle formed by the ray from the top of the skyscraper to the point on the ground, the height, h, of the skyscraper, and the distance of the point on the ground from the skyscraper, we get;

1000² = h² + 200²

h² + 200² = 1000²

h² = 1000² - 200² = 960,000

h = √(960,000) = 400·√6

The height of the skyscraper is 400·√6 feet

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The equation that can be used to decode the secret code is m = c - 2

For you to decode the secret message, you need to turn the the encoding process around. Find the inverse.

Since the encoding process uses the equation c = m + 2, to decode the message, all that need to be found is the value of m. This can be done by rearranging the encoding equation to solve for m

move 2 to c side. it becomes m = c-2

The above answer is in response to the full question below;

Suppose M and C each represent the position number of a letter in the alphabet, but M represents the letters in the original message and C represents the letters in a secret code. The equation c=m+2 is used to encode a message.

Write an equation that can be used to decode the secret code into the original message.

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The rate of change of a function as x approaches infinity is determined by the leading term in the function.

For f(x) = 2x - 10, the leading term is 2x.

For g(x) = 16x - 4, the leading term is 16x.

For h(x) = 3x^2 - 7x + 8, the leading term is 3x^2.

Since the coefficient of the leading term in h(x) is positive, and it has a higher degree than the leading terms of f(x) and g(x), h(x) has the greatest rate of change as x approaches infinity.

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The coordinates of the triangle after the dilation are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

The perimeter of the triangle increases, as the side lengths are multiplied by 3, hence the perimeter is also multiplied by 3.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The scale factor for this problem is given as follows:

k = 3.

The scale factor is greater than 1, meaning that the figure is an enlargement, and thus the perimeter increases.

The original vertices of the triangle are given as follows:

a(0, 4), b(−2, 5), and c(3, 7)

Hence the vertices of the dilated triangle are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

(each coordinate of each vertex is multiplied by the scale factor of 3).

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Answer:

3. 254.34 mm^2

4. 615.44 cm^2

5. 314 in^2

6. 7.065 in^2

7. 3.14 cm^2

8. 1.76625 ft^2

Step-by-step explanation:

AREA FORMULA: π * r^2

This question is asking to use 3.14 or 22/7 for x.

The following steps will use 3.14.

3. r = 9 mm (r^2 = 81 mm)

A = 81 * 3.14 = 254.34 mm^2

4. r = 14 cm (r^2 = 196 cm)

A = 196 * 3.14 = 615.44 cm^2

5. r = 10 in (r^2 = 100 in)

A = 100 * 3.14 = 314 in^2

Questions 6-8 show the diameter of the circle.

Divide by 2 to find the radius, then plug that into the area formula

6. r = 1.5 in (r^2 = 2.25 in)

A = 2.25 * 3.14 = 7.065 in^2

7. r = 1 cm (r^2 = 1 cm)

A = 1 * 3.14 = 3.14 cm^2

8. r = 0.75 ft (r^2 = 0.5625 ft)

A = 0.5625 * 3.14 = 1.76625 ft^2

The coordinates of point M are:

Coordinates of M = (4.5 + sqrt(34)/5, 5)
Coordinates of M = (5.86, 5)

To find the coordinates of point M, we first need to find the coordinates of the midpoint of segment PQ.

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).

Using this formula, we can find the midpoint of PQ as follows:

Midpoint = ((6 + 3)/2, (2 + 8)/2)
Midpoint = (4.5, 5)

Now we can use the fact that PM/MQ = 2/3 to find the coordinates of point M.

Let's start by finding the distance between P and the midpoint:

Distance between P and midpoint = sqrt((4.5 - 6)^2 + (5 - 2)^2)
Distance between P and midpoint = sqrt(4.25)

Since PM/MQ = 2/3, we know that PM is 2/5 of the total distance between P and Q, and MQ is 3/5 of the total distance.

Let's call the total distance between P and Q "d". Then we have:

PM = (2/5)d
MQ = (3/5)d

We can use these expressions to find the distance between M and the midpoint:

Distance between M and midpoint = PM - MQ
Distance between M and midpoint = (2/5)d - (3/5)d
Distance between M and midpoint = -(1/5)d

Finally, we can use the distance formula to find the coordinates of point M:

Coordinates of M = (4.5, 5) + (Distance between M and midpoint in the x direction, Distance between M and midpoint in the y direction)

In other words, the x-coordinate of M is 4.5 plus the distance between M and the midpoint in the x direction, and the y-coordinate of M is 5 plus the distance between M and the midpoint in the y direction.

We already know that the distance between M and the midpoint in the y direction is 0 (since M lies on the same horizontal line as the midpoint), so we only need to find the distance between M and the midpoint in the x direction:

Distance between M and midpoint in the x direction = sqrt((1/5)d^2)

Since d is just the distance between P and Q, we can find it using the distance formula:

d = sqrt((6 - 3)^2 + (2 - 8)^2)
d = sqrt(34)

So we have:

Distance between M and midpoint in the x direction = sqrt((1/5)(sqrt(34))^2)
Distance between M and midpoint in the x direction = sqrt(34)/5

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