if r(t) = 6t, 5t2, 5t3 , find r'(t), t(1), r''(t), and r'(t) × r ''(t).
r'(t) = ...
T(1) = ...
r''(t) = ...

Answer:

Q + O equals œ

Step-by-step explanation:

Answer:

\(m = \frac{4}{5} \)

Step-by-step explanation:

The slope of a graph is also known as its gradient, which is the steepness of the graph.

If we are given two points on the line, we can find the slope by taking rise/ run, which is the ratio of the change in y-coordinate against the change in the x-coordinate. This can also be written as a formula:

\(\boxed{slope = \frac{y1 - y2}{x1 - x2} }\)

☆ (x₁, y₁) is the first coordinate and (x₂, y₂) is the second coordinate

In this question, we are given the equation of the line. This equation is already in the slope-intercept form (y= mx +c) since the coefficient of y is 1 and all the other terms are on the other side of the equal sign. In the slope-intercept form, m is the slope while c is the y-intercept.

m= ⅘ since the coefficient of x is ⅘ in the given equation (when the equation is in the slope-intercept form).

We have proven that if the inequality holds true for k, it also holds true for k + 1. By the principle of mathematical induction, the inequality 2^n > n^2 holds true for all n > 4.

To prove the inequality 2^n > n^2 using induction with a basis greater than 4, we will use n=5 as the basis.

Basis: n = 5
2^5 = 32 and 5^2 = 25. Since 32 > 25, the inequality holds true for n = 5.

Now, we need to assume that the inequality holds true for an arbitrary positive integer k > 4, and then prove that it holds true for k + 1.

Assume: 2^k > k^2 for k > 4.

Prove: 2^(k + 1) > (k + 1)^2

Step 1: Multiply both sides of the assumption by 2.
2 * (2^k) > 2 * (k^2)

Step 2: Simplify the left side of the inequality.
2^(k + 1) > 2k^2

Step 3: Show that 2k^2 is greater than (k + 1)^2 for k > 4.
2k^2 > (k + 1)^2
2k^2 > k^2 + 2k + 1

Step 4: Rearrange the inequality to show that it holds true for k > 4.
k^2 - 2k - 1 > 0

Since k > 4, it is clear that k^2 - 2k - 1 > 0, as the left side increases as k increases.

Therefore, we have proven that if the inequality holds true for k, it also holds true for k + 1. By the principle of mathematical induction, the inequality 2^n > n^2 holds true for all n > 4.

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The problem statement requires to construct a 99% confidence interval for the population mean of gas mileages of 25 randomly selected sports cars, with the given assumption that mileages are not normally distributed.

Also, the problem requires us to justify the choice of distribution and interpret the results.Part 1: Which distribution should be used to construct the confidence interval?Solution: The given sample size is n = 25, and the standard deviation of the population is not known.

The population's non-normality is also given. Therefore, we will have to use the t-distribution for constructing the confidence interval. The t-distribution is more suitable when the population standard deviation is unknown, and n is less than 30.

Since n = 25, we can use the t-distribution for constructing the confidence interval. Therefore, we should use the following distribution to construct the confidence interval: A t-distribution should be used to construct the confidence interval, since sigma is unknown, and the population is not normally distributed.Part 2: Constructing the 99% confidence interval.Solution:

We need to find the 99% confidence interval, which is given by the formula below:\[\bar{x}-t_{\frac{\alpha}{2},n-1}\frac{s}{\sqrt{n}}<\mu<\bar{x}+t_{\frac{\alpha}{2},n-1}\frac{s}{\sqrt{n}}\]Where, \[\bar{x}\] is the sample mean, s is the sample standard deviation, n is the sample size, \[\frac{\alpha}{2}\] is the significance level, and \[t_{\frac{\alpha}{2},n-1}\] is the t-value from the t-distribution with (n - 1) degrees of freedom and a probability of \[\frac{\alpha}{2}\] in each tail.Given the data, we can find the sample mean and sample standard deviation as follows:Sample mean, \[\bar{x}\] = 18.548Sample standard deviation, s = 4.379Now, we need to find the t-value for the given significance level and degrees of freedom.

Since the significance level is 99%, the \[\frac{\alpha}{2}\] level of significance is 0.5%.

The degrees of freedom is n - 1 = 25 - 1 = 24. Therefore, the t-value from the t-distribution table with 0.5% significance level and 24 degrees of freedom is 2.797 (approximately).Substituting the values in the formula, we get the 99% confidence interval for the population mean as:\[18.548 - 2.797\frac{4.379}{\sqrt{25}} < \mu < 18.548 + 2.797\frac{4.379}{\sqrt{25}}\]\[16.474 < \mu < 20.622\]Hence, the 99% confidence interval for the population mean of gas mileages of 25 randomly selected sports cars is (16.474, 20.622).Interpretation: We are 99% confident that the true population mean of gas mileages of 25 randomly selected sports cars lies between 16.474 and 20.622 miles per gallon.

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Answer:

(a)  See attachment.

(b)  r = 0.8765 (4 d.p.)

(c)  Strong positive correlation.

(d)  See explanation.

Step-by-step explanation:

Correlation measures how closely two variables are linked.

When creating a scatter plot with bivariate data:

Plot a scatter diagram using the data from the given table and the following variables.

(See attachment).

The correlation coefficient (r) measures the strength of the linear correlation between two variables.

Using a calculator and the given data, calculate r:

As r is close to +1, this suggests that the data has a strong positive correlation.  

This suggests that the metabolic rate increases as the body mass increases.

We cannot use a regression line to predict a value of the explanatory variable. Therefore, as the body mass is the explanatory variable, we cannot say that a person's body mass will increase if their metabolic rate increases.  However, we can say that according to the scatter plot and correlation coefficient, it is highly likely that a person's metabolic rate increases as their body mass increases.  

Remember to apply caution when using values of of x outside the range of the original data to predict corresponding values of y, as these predications can be unreliable since there is no evidence that the relationship is true for all values of x.

0.33333 as a fraction is = 1/3

0.33333 = 33333/100000 ≈ 1/3

We can follow the steps mentioned below to write 0.33333 as a fraction.

Determine the number of digits after the decimal point in the decimal number.

Here, there are 5 digits after the decimal point.

Remove the decimal point and divide the number by the power of 10 which is equal to the number of digits after the decimal point.

So, here we will divide 33333 by 105.

Express the fraction in simplified form.

Since the HCF of both numerator and denominator is 1, Therefore 0.33333 as a fraction is,

0.33333 = 33333/100000 ≈ 1/3

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Answer:

The center of circle is: (-7,4) and Radius is 7 units

or (-4,7) and Radius is 7 units

We have to compare the given equation of circle with standard equation of circle

Given equation is:

2nd pic

down below

Standard equation of circle is:

3rd pic

down below

Here

h and k are coordinates of center of circle

So,

comparing

1st pic

down below

Hence,

The center of circle is: (-7,4) and Radius is 7 units

Keywords: Circle, radius

Answer:

Step-by-step explanation:

First expand the equation

That's

( x - 4)² + ( y - 7)² = 49

x² - 8x + 16 + y² - 14y + 49 - 49 = 0

x² + y² - 8x - 14y + 16 = 0

Comparing with the general equation of a circle

x² + y² + 2gx + 2fy + c = 0

2g = - 8 2f = - 14

g = - 4 f = - 7 c = 16

Center of a circle is ( - g , - f)

( --4 , --7)

Which is ( 4 , 7)

The radius of the circle is given by

r = √g² + f² - c

Where r is the radius

r = √ (-4)² + (-7)² - 16

= √16 + 49 - 16

= √49

Hope this helps you

Symbolic Form:

P: National elections deteriorate into TV popularity contests

Q: Smooth-talking morons will get elected

The argument can be expressed as:

If P then Q

Therefore, if not P then not Q

This is a valid argument as it follows the logical form of Modus Ponens. We can verify this using a truth table:

P | Q | P → Q | ¬P | ¬P → ¬Q

T | T | T       | F  |  T

T | F | F       | F  |  T

F | T | T       | T  |  T

F | F | T       | T  |  T

The truth table shows that the argument is valid because the final column (¬P → ¬Q) is true for all four cases. This indicates that if the symbolic form of premise (P) is false, then the conclusion (Q) must also be false.

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Answer:

x = -2, y = -4

coordinate: (-2, -4)

Step-by-step explanation:

Given system of equations:

a) 4x - 7y = 20

b) x - 3y = 10

1. Make x the subject in the second equation:

⇒ x - 3y = 10 [add 3y to both sides]

⇒ x - 3y + 3y = 10 + 3y

⇒ x = 3y + 10

2. Substitute the given value of x into first equation:

⇒ 4x - 7y = 20

⇒ 4(3y + 10) - 7y = 20 [distribute 4 through the parentheses]

⇒12y + 40 - 7y = 20 [combine like terms]

⇒ 5y + 40 = 20 [subtract 40 from both sides]

⇒ 5y - 40 - 40 = 20 - 40

⇒ 5y = -20 [divide both sides by 5]

⇒ 5y ÷ 5 = -20 ÷ 5

⇒ y = -4

3. Find the value of x by substituting the given value of y into x = 3y + 10:

⇒ x = 3y + 10

⇒ x = 3(-4) + 10 [multiply]

⇒ x = -12 + 10 [add]

⇒ x = -2

4. Check your work:

a) 4x - 7y = 20

⇒ 4(-2) - 7(-4) = 20

⇒ -8 + 28 = 20

⇒ 20 = 20 ✔

b) x - 3y = 10

⇒ -2 - 3(-4) = 10

⇒ -2 + 12 = 10

⇒ 10 = 10 ✔

This system of equations has a solution at (-2, -4).

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The correct solution after solving linear using the substitution method is x = -2, y = -4

Therefore, coordinate: (-2, -4)

How do we apply the substitution method?

To solve utilizing the substitution method, we first want to isolate any of the variables from one of two equations, and then you take the value of it and put it into the other equation.

Step-by-step explanation:

Given system of equations are as follows:

4x - 7y = 20    ..........(i)

x - 3y = 10       ..........(ii)

STEP 1: Make x the subject in the second equation:

⇒ x - 3y = 10         [add 3y to both sides]

⇒ x - 3y + 3y = 10 + 3y

⇒ x = 3y + 10

STEP 2:  Substitute the given value of x into first equation:

⇒ 4x - 7y = 20

⇒ 4(3y + 10) - 7y = 20         [distribute 4 through the parentheses]

⇒12y + 40 - 7y = 20            [combine like terms]

⇒ 5y + 40 = 20                   [subtract 40 from both sides]

⇒ 5y - 40 - 40 = 20 - 40

⇒ 5y = -20                          [divide both sides by 5]

⇒ 5y ÷ 5 = -20 ÷ 5

⇒ y = -4

STEP 3: Find the value of x by substituting the given value of y into x = 3y + 10:

⇒ x = 3y + 10

⇒ x = 3(-4) + 10     [multiply]

⇒ x = -12 + 10        [add]

⇒ x = -2

STEP 4: Check your work:

4x - 7y = 20  .............(i)

⇒ 4(-2) - 7(-4) = 20

⇒ -8 + 28 = 20

⇒ 20 = 20     [verifird]

x - 3y = 10   .............(ii)

⇒ -2 - 3(-4) = 10

⇒ -2 + 12 = 10

⇒ 10 = 10      [verifird]

This system of equations has a solution at (-2, -4).

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Answer:

The equation is

an=25n+55

Since the p - value is less than significance level . So, we will accept the null hypothesis.

​(a) State the appropriate null and alternative hypotheses to assess whether women are taller today.

Definition of null hypothesis : The null hypothesis attempts to show that no variation exists between variables or that a single variable is no different than its mean. it is denoted

Alternative Hypothesis : In statistical hypothesis testing, the alternative hypothesis is a position that states something is happening, a new theory is true instead of an old one (null hypothesis).

We are given that The mean height of women 20 years of age or older was 63.7 inches.

So, null hypothesis :  H₀ : μ = 63.7

Alternative Hypothesis :  H₁ : μ > 63.7

b)The​ P-value for this test is 0.06.

The p-value represents the probability of getting a sample mean height of 20 years old women is  64.1 inches.

c) Write a conclusion for this hypothesis test assuming an alpha equals 0.10 level of significance.

p- value = 0.06

Since the p - value is less .

So, we will accept the null hypothesis

So,  

Hence The mean height is 63.7 inches

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The measure of convex arc upper Z is 243⁰ Option D

We should be aware that a convex angle is the angles that are irregular in nature

Assume that  the radius of the given circle is r.

The central angle of the lower arc ZWX = 41+180 = 221⁰

Note that then central angle of the upper arc ZWX = 360- 221⁰

This implies that the central angle of the upper arc ZWX = 360-221 = 139⁰

Recall that Lenght of arc = A/360 *2*π*r

Where A is the angle made by the radii, r= radius of the circle

Arc length of upper ZWX:

Arc length of upper ZWX = 139π/180*r

Arc length of upper ZWX = 139πr/180

The lenght is (139*22)/ 180*7

= 3058/2360

= 245⁰

In conclusion the measure of the convex upper is 245⁰

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 f(t) = L^(-1){1 / (s^2 + s)} Inverse Laplace transform tables or techniques, determine the time-domain function f(t) that satisfies the given integral equation.

The Laplace transform is a powerful mathematical tool that can be used to solve complex integral equations, like the one you've provided: f(t) + t * ∫(t - τ)f(τ)dτ = t.

To solve this equation using the Laplace transform, follow these steps:

1. Apply the Laplace transform to both sides of the equation. The Laplace transform of f(t) is F(s), and the Laplace transform of t is 1/s^2. The integral equation becomes:

  L{f(t)} + L{t * ∫(t - τ)f(τ)dτ} = L{t}

  F(s) + L{t * ∫(t - τ)f(τ)dτ} = 1/s^2

2. Next, apply the convolution theorem to the integral term. The convolution theorem states that L{f(t) * g(t)} = F(s) * G(s). In this case, f(t) = t and g(t) = (t - τ)f(τ):

  F(s) + L{t} * L{(t - τ)f(τ)} = 1/s^2

3. Now, substitute the known Laplace transforms for t and f(t):

  F(s) + (1/s^2) * F(s) = 1/s^2

4. Combine the terms containing F(s):

  F(s) * (1 + 1/s^2) = 1/s^2

5. Isolate F(s) by dividing both sides of the equation by (1 + 1/s^2):

  F(s) = (1/s^2) / (1 + 1/s^2)

6. Simplify the expression for F(s):

  F(s) = 1 / (s^2 + s)

7. Finally, apply the inverse Laplace transform to F(s) to obtain the solution for f(t):

  f(t) = L^(-1){1 / (s^2 + s)}

Using inverse Laplace transform tables or techniques, you can determine the time-domain function f(t) that satisfies the given integral equation.

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Answer:

58\(\sqrt{65}\)

Step-by-step explanation:

you can find the height of this trapezoid by using the pythagorean theorem where the hypotenuse is 18 and one leg is 8 (33-25)

let x = height

\(8^{2}\) + \(x^{2}\) = \(18^{2}\)

\(x^{2}\) = 324 - 64

\(x^{2}\) = 260     therefore, x = \(\sqrt{260}\)

\(\sqrt{260}\) = \(\sqrt{4}\)· \(\sqrt{65}\)  or  2\(\sqrt{65}\)

A = \(\frac{1}{2}\)h x (sum of the bases)        

bases are 25 and 33

A = (2\(\sqrt{65}\)) ÷ 2)58

Answer:

neither

Step-by-step explanation:

A perfect square is a number that can be written in the form s² where s is an integer (integers are the numbers ....-2, -1, 0, 1, 2...). Listing some perfect squares, we have 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. We see that 200 is not part of this list, hence, it is not a perfect square.

A perfect cube is a number that can be written in the form s³ where s is, again, an integer. Listing some perfect cubes, we have 1, 8, 27, 64, 125, 216. Again, 200 is not part of this list, hence, it isn't a perfect cube either.

Therefore, the answer is neither.

I would classify it as a perfect square

The mean score in the fifth game that will allow you to advance is 73.

The mean score in the fifth game that will allow you to advance can be gotten this way:

Assign the mean score, the variable, and x.

Compute the scores with remarks as follows:

95 + 91 + 77 + 89 + x  ≥ 85

              5

352 + x ≥ 85

      5

85 × 5 = 425

352 + x ≥ 425

x = 425 - 352

x = 73.

So, the score in the fifth game that will allow you to advance will be 73.

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The magnitude of the correlation coefficient, regardless of the sign, provides information about the strength of the linear relationship between variables.

The sample correlation coefficient, r, ranges between -1 and 1. A value of 1 or -1 indicates a perfect linear relationship, where all data points lie precisely on a straight line. On the other hand, a value close to 0 indicates a weak or no linear relationship.

In the given scenario, r = .40 indicates a moderate positive linear relationship. Although the correlation is not perfect (not equal to 1), it still suggests a moderate degree of association between the variables. The positive sign indicates that as one variable increases, the other tends to increase as well, but not necessarily in a strictly linear fashion.

On the other hand, r = -.60 indicates a stronger linear relationship, albeit in the negative direction. The negative sign signifies an inverse relationship, meaning that as one variable increases, the other tends to decrease, but again, not necessarily in a perfectly linear manner. The magnitude, which is the absolute value of the correlation coefficient, indicates a stronger relationship compared to r = .40.

Therefore, it is important to consider both the magnitude and the sign of the correlation coefficient to assess the strength and direction of the linear relationship between variables.

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Answer:

$1.56

Step-by-step explanation:

1.50 times 0.04 is 0.06

1.50+0.06

$1.56

Answer:

ANswer is $2.10.

Step-by-step explanation:

The tax is $.60 and you add that to $1.50.

Answer:

0.267\(mm^3\)

Step-by-step explanation:

Radius of the Bubble = 0.7 mm

Volume of the bubble is given by the formula :

\(r=\sqrt[3]{\frac{3V}{4\pi } }\)

Taking cube on both sides

\(V=\frac{4\pi\times r^3 }{3}\\\\ V=\frac{4\times3.14\times(0.7)^3}{3}\\\\V=\frac{12.56\times 0.064}{3}\\\\ V=\frac{0.804}{3}\\\\V=0.267mm^3\)

Answer:

I believe it is 40% since there are 4 marbles within the bag!

Step-by-step explanation:

Answer:

6y + 11x

Step-by-step explanation:

Let's simplify step-by-step.

6y+4x+7x

Combine Like Terms:

=6y+4x+7x

=(4x+7x)+(6y)

=11x+6y

Answer:The answer would be the first one 6y + 11x

Step-by-step explanation:All you would have to do is do 4x + 7x which would equal 11x. So the answer would be 6y +11x

A sequence of transformation that would move ΔABC onto ΔDEF is: D. a dilation by a scale factor of 1/2, centered at the origin, followed by a 90° clockwise rotation about the origin.

In Geometry, a dilation is a type of transformation which typically changes the size of a geometric object, but not its shape.

In this scenario an exercise, we would dilate the coordinates of the pre-image by applying a scale factor of 1/2 that is centered at the origin as follows:

Ordered pair B (-4, 2) → Ordered pair B' (-4 × 1/2, 2 × 1/2) = Ordered pair B' (-2, 1).

In Mathematics and Geometry, a rotation can be defined as 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;

(x, y)                               →            (y, -x)

Ordered pair B' (-2, 1)   →          E (1, 2)

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The equation y = mx + b, when isolated for the variable m, gives us m in terms of x, y, and b, as m = (y - b)/x.

What is the isolation of a variable in an equation?

Equations are relations between two quantities involving variables raised to multiple powers, making up terms along with numerals.

Isolation is the process of moving one variable to one-side of the equation and everything else on the other side of the equation.

How to solve the question?

In the question, we are given an equation, y = mx + b, and are asked to write m in terms of x, y, and b.

This means that we need to isolate the variable m, to get an expression equal to m.

The isolation of the variable m can be done as follows:

y = mx + b,

or, y - mx = mx + b - mx {Subtracting mx from both sides},

or, y - mx = b {Simplifying},

or, y - mx - y = b - y {Subtracting y from both sides},

or, -mx = b - y {Simplifying},

or, (-mx)/(-x) = (b - y)/(-x) {Dividing both sides by (-x)}

or, m = (y - b)/x {Simplifying}.

Thus, the equation y = mx + b, when isolated for the variable m, gives us m in terms of x, y, and b, as m = (y - b)/x.

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The equation y = mx + b, when isolated for the variable m, gives us m in terms of x, y, and b, as m = (y - b)/x.

What is the isolation of a variable in an equation?

Equations are relations between two quantities involving variables raised to multiple powers, making up terms along with numerals.

Isolation is the process of moving one variable to one-side of the equation and everything else on the other side of the equation.

How to solve the question?

In the question, we are given an equation, y = mx + b, and are asked to write m in terms of x, y, and b.

This means that we need to isolate the variable m, to get an expression equal to m.

The isolation of the variable m can be done as follows:

y = mx + b,

or, y - mx = mx + b - mx {Subtracting mx from both sides},

or, y - mx = b {Simplifying},

or, y - mx - y = b - y {Subtracting y from both sides},

or, -mx = b - y {Simplifying},

or, (-mx)/(-x) = (b - y)/(-x) {Dividing both sides by (-x)}

or, m = (y - b)/x {Simplifying}.

Thus, the equation y = mx + b, when isolated for the variable m, gives us m in terms of x, y, and b, as m = (y - b)/x.

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Answer:

x³+5x²+2

Step-by-step explanation:

DM me if you have other questions!

Answer:

Area= 108ft²

Step-by-step explanation:

To find the area of a rectangle, you must do the following formula:

Area= Length × Width

A represents Area

L represents Length

W represents Width

Because the length (length is always longer than width) is 12 ft and the width (width is always shorter than length) is 9 ft. Your equation should be:

A= L × W

= 12ft × 9ft

= 108 ft²

Remember: The answer to a question asking for the area of a shape that is 2D, is always squared (let x represents the answer: x²). And the question asking the area of a shape that is 3D always cubed (let x represents the answer: x³). Always write the unit of measurement (let x represent the answer and cm as the example of unit of measurement: x cm²)

I hope this helps! I'm sorry if it's too complicated.

Answer:

Quadrant 1

Step-by-step explanation:

Quadrant 1 has positive x and y.