Your yacht sails in the direction S 71° W for 141 miles. It then turns


and sails in the direction N 63º E 213 miles. Find its distance from


the starting point. Round answer to nearest hundredth.

The measure of x in the intersected chord is 16.

If two chords intersect in a circle, then the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Using the chord intersection angle theorem,

5x - 7  = 1 / 2 (119 + 27)

5x - 7 = 1 / 2 (146)

5x - 7 = 73

add 7 to both sides of the equation

5x - 7 = 73

5x - 7 + 7 = 73 + 7

5x = 80

divide both sides of the equation by  5

x = 80 / 5

x = 16

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lets assume.

Algebraic expression: c - 2.5. The variable is c, which represents Julie's initial weight in kilograms.

Algebraic equation: Jhon's height = 2m + 5. The variables are m, which represents Peter's height in centimeters, and the height of Jhon, which is represented by the equation.

Algebraic equation: Ador's age = 3(Emy's age - d). The variables are Ador's age and Emy's age, which is d years less than 9.

Algebraic expression: n + 7. The variable is n, which represents Jupiter's current age in years.

Algebraic equation: Anna's age = 3p + 4. The variables are p, which represents Anna's sister's age in years, and Anna's age, which is represented by the equation.

SO ANNA  current age is P=3+4

and p=7

Anna's age = 3p + 4, where p is the age of Anna's sister in years, and Anna is 4 years older than thrice the age of her sister.

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To solve this problem, we first need to understand what "at most 4 of them were successes" means. This includes the cases where there are 0, 1, 2, 3, or 4 successful serves out of the 6 selected.

We can use the table to find the probabilities for each of these cases.

For 0 successful serves, the probability is 0.0002.

For 1 successful serve, the probability is 0.004.

For 2 successful serves, the probability is 0.033.

For 3 successful serves, the probability is 0.132.

For 4 successful serves, the probability is 0.297.

To find the probability of at most 4 successful serves, we add up these probabilities:
[tex]0.0002 + 0.004 + 0.033 + 0.132 + 0.297 = 0.466[/tex]

So the probability of at most 4 successful serves is 0.466.

Therefore, the answer is 0.466 and it is found by adding up the probabilities for the cases where there are 0, 1, 2, 3, or 4 successful serves out of the 6 selected from the table.

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The  expected value and standard error for the number of winning plays is $3.1 and $ 2.21.

The population mean's likelihood to differ from a sample mean is indicated by the standard error of a mean, and simply standard error.

It reveals how much what the sample mean will change if a study were to be repeated with fresh samples drawn from a single population.

Chance of winning $3 = 2/10

chance of losing $2 = 3/10

chance of losing $1 = 5/10

Average of tickets. = - $2.50

SD of tickets = $1.80.

The box model for net gain has 2 tickets labeled $3, 3 tickets labeled $2, 5 tickets labeled

a) Expected value for the net gain.

The Expected value for net = ∑ x.p(x)

Here

can take value $1, $2 and $3

Here p(x) us the probability of winning respectively.

So, Now, Expected gain is,

(2 x 3 x 2/10) + (3 x -2 x 7/10) + (5 x -1 x 5/10)

= 12/10 - 18/10 - 25/10

= -31/10 = -$3.1.

b) Standard error of the net gain,

S.D = [tex]\sqrt{E(x^2) - [E(x)]^2}[/tex]

Now E(x²) = (2 x 3² x 2/10) + (3 x -2² x 7/10) + (5 x -1² x 5/10)

= 36/10 + 84/10 + 25/10 = 145/10

= $ 14.5

SD = [tex]\sqrt{14.5 - 3.1}\\[/tex]

SD = $ 2.21

c) Chance that the net gain is $15

P(X=15) = (z = 15-(-2.50)/1.80

= P(z=9.72) = 0.99.

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In general, it is best to determine the equation of the line of best fit exactly rather than relying on estimation by inspection. This is because an exact equation allows for more precise predictions and calculations.

Estimation by inspection can be useful in situations where the data is relatively simple and a rough estimate is sufficient. However, in more complex datasets, it is important to use statistical methods to determine the line of best fit accurately.

For example, in some cases, it may be more important to prioritize the accuracy of the slope of the line over the accuracy of the intercept. In such cases, certain methods, such as minimizing the sum of the squares of the vertical deviations, may be more appropriate than others.

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The best estimate of the number of ants in Mrs. Brown's yard is 15,116,596.

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.

Mrs. Brown's yard has three ant hills, each with a different number of ants. To estimate the total number of ants in the yard, we simply add up the number of ants in each hill.

The first hill has 4,867,190 ants, the second has 6,256,304, and the third has 3,993,102. When we add these numbers together, we get a total of 15,116,596 ants in Mrs. Brown's yard. Of course, this is just an estimate, as there may be other ant hills or individual ants scattered around the yard.

However, this calculation gives us a good approximation of the number of ants in the yard based on the information given.

To estimate the total number of ants in Mrs. Brown's yard, we can add up the number of ants in each of the three ant hills:4,867,190 + 6,256,304 + 3,993,102 = 15,116,596.

Therefore, the best estimate of the number of ants in Mrs. Brown's yard is 15,116,596.

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The required area of the shaded part of the circle is 50.24 sq. cm

A circle of radius r has an area of r2 in geometry. Here, the Greek letter  denotes the constant ratio of a circle's diameter to circumference, which is roughly equivalent to 3.14159.

Given data:

Radius of small circle = 6/2 = 3 cm

Radius of big circle = 10/2 = 5 cm

then.

Area of shaded part = area of big circle - area of small circle

Area of shaded part = π(5)² -  π(3)²

Area of shaded part = 25π - 9π

Area of shaded part = 16π

Area of shaded part = 50.24 sq. cm

Thus, required area of the shaded part of the circle is 50.24 sq. cm

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The bases could be circled 40 times in 50 minutes at the same rate.

For this issue's solution, let's use unit rates.

In order to calculate the unit rate,

Considering that the player went 12 times around the bases in 15 minutes, the unit rate is 12/15, =  0.8 times per minute.

In a minute, the player would have circled the bases 0.8 times. By dividing the unit rate by the number of minutes, we can calculate how many times the bases could be circled in 50 minutes:

50 minutes x  0.8 times each minute = 40 times.

Therefore, the bases could be circled 40 times in 50 minutes at the same rate.

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The volume of the cone is 57 cubic inches. To find the volume of the cone, we use the formula: V = (1/3)π[tex]r^{2}[/tex]h, where r is the radius of the cone, h is the height of the cone, and π is approximately 3.14.

Given that the radius of the cone is 3 inches and the height is 6 inches, we can substitute these values into the formula and solve for V: V = (1/3)π([tex]3^{2}[/tex])(6), V = (1/3)π(9)(6), V = (1/3)(3.14)(54), V = 56.52 cubic inches

Rounding to the nearest whole number, the volume of the cone is 57 cubic inches.

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The number of type 2 cabinets produced last week is 80. The number of type 2 cabinets produced last week exceeded the number of type 1 cabinets produced during the week by 50.

Using the system of equations given, we can solve for the number of type 1 and type 2 cabinets produced.

x + y = 110 represents the total number of cabinets produced, where x is the number of type 1 cabinets and y is the number of type 2 cabinets produced.

y = 2x + 20 represents the relationship between the number of type 1 and type 2 cabinets produced. This equation tells us that the number of type 2 cabinets produced exceeds twice the number of type 1 cabinets produced by 20.

To solve for y, we substitute the value of y from the second equation into the first equation:

x + (2x + 20) = 110

Simplifying this equation:

3x + 20 = 110

3x = 90

x = 30

Therefore, the number of type 1 cabinets produced last week is 30.

To find the number of type 2 cabinets produced, we substitute x = 30 into the second equation:

y = 2x + 20 = 2(30) + 20 = 80

The number of type 2 cabinets produced last week exceeds the number of type 1 cabinets produced during the week by:

80 - 30 = 50.

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The integer that represents Andrews depth in ¼ of an hour is 60 feet.

Word problems are sentences describing a 'real-life' situation where a problem needs to be solved by way of a mathematical calculation e.g. calculation of length and depth.

If Andrew descends at a rate of 4 feet per minute and we want to find his depth in ¼ of an hour.

1/4 of an hour = (1/4 * 60) minutes = 15 minutes

Thus, the integer that represents Andrews depth in ¼ of an hour will be:

(4 feet per minute) * (15 minutes) = 60 feet

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The remaining distance after 33 minutes of driving = 58.8 miles.

Here, the slope of a linear function the remaining distance to drive (in miles) is -0.85

For this situation, we can write a linear equation as,

remaining distance = (slope)(drive time) + (intercept)

remaining distance = -0.85(drive time) + (intercept)      

y =  -0.85x + c        ..........(1)

where y represents the remaining distance

x is the drive time

and c is the y-intercept

Here, Laura has 52 miles remaining after 41 minutes of driving.

i.e., x = 41 and y = 52

Substitute these values in equation (1)

52 =  -0.85(41) + c    

c = 52 + 34.85

c = 86.85

So, equation (1) becomes,

y =  -0.85x + 86.85

Now, we need to find the remaining distance after 33 minutes of driving.

i.e., the value of y for x = 33

y =  -0.85(33) + 86.85

y =  -28.05 + 86.85        

y = 58.8

This is the remaining distance 58.8 miles.

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Answer: 4.5

Step-by-step explanation:

The minimum sample size which is need to to create the given confidence interval is equal to 467.

Sample size n

z = z-score for the desired confidence level

From attached table,

For 85% confidence level, which corresponds to a z-score of 1.44.

Maximum error or margin of error E = 0.06

Population standard deviation σ = 0.9

Minimum sample size required to construct a 85% confidence interval with an error of no more than 0.06,

Use the formula,

n = (z / E)^2 × σ^2

Plugging in the values, we get,

⇒ n = (1.44 / 0.06)^2 × 0.9^2

⇒ n = 466.56

Rounding up to the next integer, we get a minimum sample size of 467.

Therefore, the minimum sample size required to construct a 85% confidence interval with an error of no more than 0.06 is 467.

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To prove that the expression (36^5−6^9)(38^9−38^8) is divisible by 30, we need to show that it is divisible by both 2 and 3.

First, we can factor out a 6^9 from the first term:

(36^5−6^9)(38^9−38^8) = 6^9(6^10-36^5)(38^9-38^8)

Notice that 6^10 can be written as (2*3)^10, which is clearly divisible by both 2 and 3. Also, 36 is divisible by 3, so 36^5 is divisible by 3^5. Thus, we can write:

6^9(6^10-36^5) = 6^9(2^10*3^10 - 3^5*2^10) = 6^9*2^10*(3^10 - 3^5)

Since 2^10 is divisible by 2, and 3^10 - 3^5 is clearly divisible by 3, the whole expression is divisible by both 2 and 3, and therefore divisible by 30.

To prove that the expression is divisible by 37, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if p is a prime number and a is any positive integer not divisible by p, then a^(p-1) is congruent to 1 modulo p, which can be written as a^(p-1) ≡ 1 (mod p).

In this case, p = 37, and 36 is not divisible by 37. Therefore, by Fermat's Little Theorem:

36^(37-1) ≡ 1 (mod 37)

Simplifying the exponent gives:

36^36 ≡ 1 (mod 37)

Similarly, 38 is not divisible by 37, so:

38^(37-1) ≡ 1 (mod 37)

Simplifying the exponent gives:

38^36 ≡ 1 (mod 37)

Now we can use these congruences to simplify our expression:

(36^5−6^9)(38^9−38^8) ≡ (-6^9)(-1) ≡ 6^9 (mod 37)

We know that 6^9 is divisible by 3, so we can write:

6^9 = 2^9*3^9

Since 2 and 37 are relatively prime, we can use Euler's Totient Theorem to simplify 2^9 (mod 37):

2^φ(37) ≡ 2^36 ≡ 1 (mod 37)

Therefore:

2^9 ≡ 2^9*1 ≡ 2^9*2^36 ≡ 2^(9+36) ≡ 2^45 (mod 37)

Now we can simplify our expression further:

6^9 ≡ 2^45*3^9 ≡ (2^5)^9*3^9 ≡ 32^9*3^9 (mod 37)

Notice that 32 is congruent to -5 modulo 37, since 32+5 = 37. Therefore:

32^9 ≡ (-5)^9 ≡ -5^9 ≡ -1953125 ≡ 2 (mod 37)

So:

6^9 ≡ 2*3^9 ≡ 2*19683 ≡ 39366 ≡ 0 (mod 37)

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Answerx = 131.25

Step-by-step explanation:f x varies directly as T, then we can use the formula for direct variation:

x = kT

where k is the constant of proportionality.

To find k, we can use the given values:

x = 105 when T = 400

105 = k(400)

k = 105/400

k = 0.2625

Now that we have the value of k, we can use the formula to find x when T = 500:

x = kT

x = 0.2625(500)

x = 131.25

Therefore, when T = 500, x is equal to 131.25.

Minimizing the sum of squared residuals.

The OLS (Ordinary Least Squares) estimator is derived by minimizing the sum of squared residuals. This method aims to find the line of best fit that minimizes the vertical distance between the observed data points (Yi) and the predicted values on the regression line. The residuals represent the differences between the observed values and the predicted values.

By minimizing the sum of squared residuals, the OLS estimator ensures that the line fits the data as closely as possible. This approach is based on the principle of least squares, which seeks to find the parameters that minimize the overall discrepancy between the observed data and the predicted values.

Connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi or ensuring that the standard error of the regression equals the standard error of the slope estimator are not the steps involved in deriving the OLS estimator. The OLS method specifically focuses on minimizing the sum of squared residuals.

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The best estimate for the volume of the tank in cubic feet is 5.5 cubic feet.

The volume of the tank is:

V = l x w x h

where l is the length, w is the width, and h is the height.

Substituting the given values, we get:

V = 25 x 34.5 x 11 = 9547.5 cubic inches

To convert cubic inches to cubic feet, we divide by (12 x 12 x 12), since there are 12 inches in a foot and 12 x 12 x 12 cubic inches in a cubic foot:

V = 9547.5 / (12 x 12 x 12) cubic feet

V ≈ 5.5 cubic feet

Therefore, 5.5 cubic feet is the best estimate for the tank's cubic foot capacity.

Hence , the volume of the rectangular glass tank is 5.490 feet³

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To use Lagrange multipliers, we need to define the Lagrangian function:

L(x, y, z, λ) = xyz + λ(x + y + z - 9)

Now, we need to find the partial derivatives of L with respect to x, y, z, and λ and set them equal to 0:

∂L/∂x = yz + λ = 0
∂L/∂y = xz + λ = 0
∂L/∂z = xy + λ = 0
∂L/∂λ = x + y + z - 9 = 0

From the first three equations, we can see that:

yz = -λ
xz = -λ
xy = -λ

Multiplying these equations together, we get:

(xyz)^2 = (-λ)^3

Substituting λ = -yz into the fourth equation, we get:

x + y + z - 9 = 0

Substituting λ = -yz into the first equation and solving for x, we get:

x = -λ/yz = (yz)^2/(-yz) = -y^2z^2

Similarly, we can solve for y and z:

y = -x^2z^2
z = -x^2y^2

Substituting these expressions into the constraint equation, we get:

(-y^2z^2) + (-x^2z^2) + (-x^2y^2) - 9 = 0

Simplifying and solving for xyz, we get:

xyz = sqrt(9/(x^2 + y^2 + z^2))

To maximize xyz, we need to minimize x^2 + y^2 + z^2. Therefore, we can set:

x^2 + y^2 + z^2 = 3

Substituting this into the expressions for x, y, and z, we get:

x = -y^2z^2
y = -x^2z^2
z = -x^2y^2

Substituting these expressions into xyz, we get:

xyz = sqrt(9/3) = 3

Therefore, the maximum value of f(x, y, z) = xyz subject to the constraint x + y + z - 9 = 0 is 3.
To solve this problem using Lagrange multipliers, we first set up the Lagrangian function L(x, y, z, λ) with the constraint function g(x, y, z) = x + y + z - 9.

L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z))

L(x, y, z, λ) = xyz - λ(x + y + z - 9)

Now we take the partial derivatives with respect to x, y, z, and λ, and set them equal to 0:

∂L/∂x = yz - λ = 0
∂L/∂y = xz - λ = 0
∂L/∂z = xy - λ = 0
∂L/∂λ = x + y + z - 9 = 0 (the constraint)

From the first three equations, we get:

yz = xz = xy

Since x, y, and z are positive, we can divide the first two equations:

y/z = x/z => y = x
x/z = y/z => x = y

So x = y = z. Now we can use the constraint equation:

x + x + x - 9 = 0 => 3x = 9 => x = 3

Thus, x = y = z = 3. Now we can find the maximum value of f(x, y, z):

f(3, 3, 3) = 3 * 3 * 3 = 27

So the maximum value of f(x, y, z) = xyz subject to the constraint x + y + z - 9 = 0 is 27, and this occurs at the point (3, 3, 3).

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The correct answer for an angle coterminal to -497° within the interval 0°≤θ≤360° is 223°.

What are Coterminal angles?

Coterminal angles are angles that share the same initial and terminal sides when drawn in the standard position (starting from the positive x-axis) on the coordinate plane. In other words, coterminal angles are angles that differ by an integer multiple of 360° or 2π radians.

To find an angle coterminal to within the interval use the fact that to add or subtract a multiple of to an angle does not change its position on the unit circle.

To make the angle positive, add  360°  repeatedly until an angle within the desired interval is obtained:

= -497° +360°

= -137°

adjust this angle to be within the interval 0°≤θ≤360°, and add another 360°:

= -137° + 360°

= 223°

The required angle is 223°.

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An angle θ coterminal to -497 degrees, where 0 ≤ θ < 360 degrees, is 223 degrees.

Given that; the angle is, -497 degrees.

Now, for an angle coterminal to -497 degrees within the range 0≤θ<360 degrees, add or subtract multiples of 360 degrees until we get an angle within the desired range.

Now, add multiples of 360 degrees until we get a positive angle:

-497 + 360 = -137

Now we have an angle of - 137 degrees, but it is still not within the desired range of 0 ≤ θ < 360 degrees.

To adjust the angle, add 360 degrees to it:

-137 + 360 = 223

Now an angle of 223 degrees, which is within the desired range.

Therefore, an angle θ coterminal to -497 degrees, where 0 ≤ θ < 360 degrees, is 223 degrees.

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The value of W to the nearest tenth of degree is 15.8°

The sine rule states that if a, b and c are the lengths of the sides of a triangle, and A, B and C are the angles in the triangle; with A opposite a, etc., then a/sinA=b/sinB=c/sinC.

Sine rule can be used to find unknown side or angle In a triangle.

w/sinW = v/sinV

4.7/sinW = 2.4 / sin8

2.4sinW = 4.7 sin8

2.4sinW = 0.654

sinW = 0.654/2.4

sinW = 0.273

W = sin^-1( 0.273)

W = 15.8° ( nearest tenth)

therefore the value of W is 15.8°

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The total cost after discount and tax is $46.28.

To find the total cost after the discount and tax, we need to first find the subtotal before tax.

Fred bought two pairs of jeans for $17.99 each, so the cost of the jeans is $17.99 x 2 = $35.98.

He also bought three shirts at $5.99 each, so the cost of the shirts is $5.99 x 3 = $17.97.

The pack of socks costs $3.

The subtotal before discount is $35.98 + $17.97 + $3 = $57.95.

With the 25% off coupon, Fred gets a discount of $57.95 x 0.25 = $14.49.

The new subtotal after discount is $57.95 - $14.49 = $43.46.

Finally, we need to add the 6.5% sales tax.

The sales tax is $43.46 x 0.065 = $2.82.

The total cost after discount and tax is $43.46 + $2.82 = $46.28.

Therefore, the closest answer is C: $45.49.

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As the x-values go to either positive or negative infinity, the function decreases towards negative infinity.

Step 1: Identify the degree and leading coefficient.
In f(x) = -x² - 1, the degree is 2 (the highest power of x), and the leading coefficient is -1.
Step 2: Determine the end behavior based on the degree and leading coefficient.
Since the degree is even (2) and the leading coefficient is negative (-1), we know that both ends of the graph will point in the same direction.

Step 3: Identify the specific end behavior.
Because the leading coefficient is negative, the graph of the function will open downward. As x approaches positive infinity, f(x) will decrease towards negative infinity. Similarly, as x approaches negative infinity, f(x) will also decrease towards negative infinity.

Step 4: Write the end behavior in a concise format.
The end behavior of f(x) = -x² - 1 can be written as:
As x → ±∞, f(x) → -∞.
In summary, the function f(x) = -x² - 1 has a downward-opening parabola due to its even degree and negative leading coefficient.

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Answer: 8 cups

Step-by-step explanation: 16 cups in a gallon, Half of 16 is 8. (16 divided by 2 = 8)

If Franco's Pizza Parlor knows that the marginal cost of the 500th pizza is $3.50 and that the average total cost of making 499 pizzas is $3.30, then Average total costs are rising at Q = 500. The correct answer is (c)

The marginal cost is the additional cost of producing one more unit. In this case, the marginal cost of the 500th pizza is $3.50.

The average total cost is the total cost of producing all units up to a certain level, divided by the number of units produced. In this case, the average total cost of making 499 pizzas is $3.30.

If the marginal cost of producing the 500th pizza is greater than the average total cost of making the first 499 pizzas, then the average total cost will increase when the 500th pizza is produced.

Therefore, the correct answer is (c) average total costs are rising at Q = 500.

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The expression for the calculation of adding 8 to the sum of 23 and 10 is 8 + (23 + 10)

To calculate expression parentheses the sum of 23 and 10, we add them together, which gives us 33. Then, we add 8 to that result, giving us a final answer of 41. So, the expression 8 + (23 + 10) equals 41.

This expression follows the order of operations, which states that we should first perform the addition inside the parentheses and then add the result to 8.

expressions are made up of numbers and symbols, and they represent a mathematical relationship or operation. In this case, the expression includes addition and parentheses, which tell us to perform the addition inside them first. The parentheses clarify which numbers should be added together first before adding 8.

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The position of the particle at time t = 1 is x = 4 units.

To find the position of the particle at time t = 1, we need to integrate the given velocity function v(t) with respect to time from 0 to 1:

x(t) = ∫v(t)dt (from t = 0 to t = 1)

= ∫([tex]3t^2[/tex]+ 6t)dt (from t = 0 to t = 1)

= ([tex]t^3 + 3t^2[/tex]) (from t = 0 to t = 1)

[tex]= (1^3 + 3(1^2)) - (0^3 + 3(0^2))[/tex]

= 4

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The trapezoidal approximation for | Vx dx using a regular partition with n=6 is approximately 0.1901.

To compute the trapezoidal approximation for | Vx dx using a regular partition with n=6, we can use the formula:

Tn = (b-a)/n * [f(a)/2 + f(x1) + f(x2) + ... + f(xn-1) + f(b)/2]

where Tn is the trapezoidal approximation, n=6 is the number of partitions, a and b are the limits of integration, and x1, x2, ..., xn-1 are the partition points.

In this case, we have | Vx dx as the function to integrate. Since there are no given limits of integration, we can assume them to be 0 and 1 for simplicity.

So, a=0 and b=1, and we need to find the values of f(x) at x=0, 1/6, 2/6, 3/6, 4/6, and 5/6 to use in the formula.

We can calculate these values as follows:

f(0) = | V0 dx = 0

f(1/6) = | V1/6 dx = V(1/6) - V(0) = sqrt(1/6) - 0 = 0.4082

f(2/6) = | V2/6 dx = V(2/6) - V(1/6) = sqrt(2/6) - sqrt(1/6) = 0.2317

f(3/6) = | V3/6 dx = V(3/6) - V(2/6) = sqrt(3/6) - sqrt(2/6) = 0.1547

f(4/6) = | V4/6 dx = V(4/6) - V(3/6) = sqrt(4/6) - sqrt(3/6) = 0.1104

f(5/6) = | V5/6 dx = V(5/6) - V(4/6) = sqrt(5/6) - sqrt(4/6) = 0.0849

Now we can substitute these values in the formula and simplify:

T6 = (1-0)/6 * [0/2 + 0.4082 + 0.2317 + 0.1547 + 0.1104 + 0.0849/2]
   = 0.1901

Therefore, the trapezoidal approximation for | Vx dx using a regular partition with n=6 is approximately 0.1901.

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There exists at least one c in the open interval (a, b) such that f'(c) = 0.

There are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.

To prove that the given equation has exactly 3 distinct real solutions, let's follow the steps mentioned in the question.

First, consider the polynomial p(x) = (x-a)(x-b)(x-c)(x-d). Since a, b, c, and d are distinct real numbers, p(x) has 4 distinct real roots, namely a, b, c, and d.

Now, let's find the derivative p'(x) using logarithmic differentiation. Taking the natural logarithm of both sides, we have:

[tex]ln(p(x)) = ln((x-a)(x-b)(x-c)(x-d))[/tex]

Differentiating both sides with respect to x, we get:

[tex]p'(x)/p(x) = 1/(x-a) + 1/(x-b) + 1/(x-c) + 1/(x-d)[/tex]

Multiplying both sides by p(x) and simplifying, we have:

[tex]p'(x) = (x-b)(x-c)(x-d) + (x-a)(x-c)(x-d) + (x-a)(x-b)(x-d) + (x-a)(x-b)(x-c)[/tex]

Now, we apply Rolle's Theorem, which states that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.

Since p(x) has 4 distinct real roots, there must be 3 intervals between these roots where the function p(x) satisfies the conditions of Rolle's Theorem. Therefore, there are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.

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