Catherine and Ana wanted to know if student athletes (students on at least one varsity team) have faster reaction times than non-athletes. They took separate random samples of 33 athletes and 30 non-athletes from their school and tested their reaction time using an online reaction test, which measured the time in seconds) between when a green light went on and the subject pressed a key on the computer keyboard. A 95% confidence interval for the difference (Non-athlete - Athlete) in the mean reaction time was 0.018 + 0.034 seconds. Does the interval provide convincing evidence of a difference in the true mean reaction time of athletes and non-athletes? Explain your answer. Yes, because the conditions are met. No, the interval does not include a difference of 0 (no difference) as a plausible value. Yes, because the interval includes a difference of 0 (no difference) as a plausible value. Yes, because the sample sizes are both at least 30. No, because the interval includes a difference of 0 (no difference) as a plausible value.

The matrix A for L with respect to {e1, e2} is [-1 0; 0 1], and the matrix B for L with respect to {u1, u2} is [-1 1; 1 1].

a) For the linear operator \(L(x) = (-x1, x2)^T\), we need to determine the matrix A representing L with respect to {e1, e2} and the matrix B representing L with respect to\({u1 = (1,1)^T, u2 = (-1,1)^T}.\)

To find the matrix A, we apply L to each basis vector e1 and e2 and express the results in terms of the basis {e1, e2}:

L(e1) =\((-e1, 0)^T = -e1e1^T + 0e2e2^T = -e1e1^T\)

L(e2) =\((0, e2)^T = 0e1e1^T + e2e2^T = e2e2^T\)

Therefore, the matrix A representing L with respect to {e1, e2} is:

A = [ -1  0 ]

      [ 0  1 ]

To find the matrix B, we apply L to each basis vector u1 and u2 and express the results in terms of the basis {u1, u2}:

L(u1) =\((-1, 1)^T = -u1u1^T + u2u2^T\)

L(u2) =\((1, 1)^T = u1u1^T + u2u2^T\)

Therefore, the matrix B representing L with respect to {u1, u2} is:

B = [ -1  1 ]

      [ 1  1 ]

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Full question : For each of the following linear operators L on R^2, determine the matrix A representing L with respect to {e1,e2} and the matrix B representing L with respect to\({u1 = (1,1)^T, u2=(-1,1)^T\)

a) \(L(x) = (-x1,x2)^T\)

b) \(L(x) = (x2,x1)^T\)

c) \(L(x) = x2e2\)

Answer:

B= 3xy

Step-by-step explanation:

6xy^3= 2ₓ3ₓ x ₓyₓyₓy

9x^2y= 3ₓ3ₓ x ₓ x ₓ y

common factors = 3, x , y

HCF:  3 ₓ x ₓ y

     =3xy

The angles A and C are complementary angles.

In this problem we find six possible angles, from which we must determine two angles that are complementary angles. Two angles are complementary when the total measure is equal to 90°. The algebraic formula representing the situation is shown below:

α + β = 90°

Where α, β are angles.

If we know that α = 61° and β = 29°, then the result is:

m = α + β

m = 61° + 29°

m = 90°

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Answer:

A and C

Step-by-step explanation:

a complementary angle is when 2 angles add up to 90 degrees

As for A and C, A’s degree is 29 when C’s 61

29+61 is 90

From the analysis of the above math, the amount she invested in the CD account is 5,000. See the analysis below.

Let A be the amount invested in the CD.

Let B be the amount invested in the savings bond.

A + B = 15, 000

.04A + .07B = 900

Multiply the first equation by -.04 and add it to the second equation to get rid of the A term.

-.04A-.04B = -600  add this equation to the second equation.

.03B = 300  multiplying by 100.  3B = 30,000

dividing by 3 on both sides: B = 10,000.  So A = 5,000

So she invested 5,000 in the CD account

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The solution of   \(\rm log(2t+4)=log(14-3t)\)  is t=2

It is given that:

\(\rm log(2t+4)=log(14-3t)\)

It is required to find the value of 't'.

It is another way to represent the power of numbers ie.

\(a^b=c\\log_ac=b\)

We have:

\(\rm log(2t+4)=log(14-3t)\)   \(\rm (Taking \ log_1_0 \ base \ and \ removing \ the \ log_1_0 \ from \ both \ the\ side)\)

We will get:

\(\rm 2t+4=14-3t\\\rm 5t=10\\\rm t=2\)

Therefore the solution of   \(\rm log(2t+4)=log(14-3t)\)  is t=2

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The graph of y = x(x² - 2)(x² + x + 1) intersects the x-axis in three different points.

The answer is (C) Three.To determine the number of points where the graph of the equation y = x(x² - 2)(x² + x + 1) intersects the x-axis, we need to find the x-values that make the equation equal to zero.

Setting y = 0, we have:

0 = x(x² - 2)(x² + x + 1)

Since the product of three factors is zero, at least one of the factors must be zero.

1. Setting x = 0:

0 = 0(x² - 2)(x² + x + 1)

This gives us one solution: x = 0.

2. Setting x² - 2 = 0:

x² = 2

Taking the square root of both sides:

x = ±√2

This gives us two additional solutions: x = √2 and x = -√2.

3. Setting x² + x + 1 = 0:

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = 1, and c = 1. Substituting these values into the quadratic formula:

x = (-1 ± √(1² - 4(1)(1))) / (2(1))

Simplifying:

x = (-1 ± √(-3)) / 2

Since the discriminant is negative, there are no real solutions for this quadratic equation.

In summary, we have found three different x-values where the equation intersects the x-axis: x = 0, x = √2, and x = -√2.

Therefore, the graph of y = x(x² - 2)(x² + x + 1) intersects the x-axis in three different points. The answer is (C) Three.

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Answer:

\(g(x)=2\,|x-5|\)

Step-by-step explanation:

Recall that a horizontal translation (shift) in 3 units to the right involves directly subtracting 3 from the variable x (horizontal axis variable) , and that moving the function up 5 units involves adding to the whole function 5 units. That is:

\(g(x)=2\,|x-3-2|- 5+ 5\\g(x)=2\,|x-5|+0\\g(x)=2\,|x-5|\)

Answer:

V = four-third (27 pi)

Step-by-step explanation:

Complete question;

A sphere and a cylinder have the same radius and height. The volume of the cylinder is 27 pi feet cubed.

A sphere with height h and radius r. A cylinder with height h and radius r.

Which equation gives the volume of the sphere?

V = two-thirds (27 pi)

V = four-thirds (27)

V = two-thirds (27)

V = four-third (27 pi)

Volume of the cylinder V = πr²h

If Volume is 27pi feet cubed, hence;

27pi = πr²h

27 = r²h

Since r = h

27 = r³

r = ∛27

r = 3

Since the volume of the sphe4/3πr³re = 4/3πr³

Volume of sphere = 4/3π(27)

Volume of sphere = 4/3(27π)

Hence the correct option is V = four-third (27 pi)

The circumference of the circle in this problem is given as follows:

C = 32.42 cm.

The circumference of a circle of radius r is given by the equation presented as follows:

\(C = 2\pi r\)

First we must obtain the diameter of the circle, given by segment MON, appying the Pythagorean Theorem, as follows:

d² = 3.5² + 9.7²

d = square root of (3.5² + 9.7²)

d = 10.31 cm.

The radius of a circle has half the length of the diameter, hence it is given as follows:

r = 0.5 x 10.31

r = 5.16 cm.

Then the circumference of the circle is given as follows:

C = 2 x 3.1415 x 5.16

C = 32.42 cm.

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The values of x and y the triangles to the fight congruent are x = 3 and

y = 1.

What is congruent triangles?

Triangles with the same size and shape are said to be congruent. This implies that the corresponding sides and angles are both equal. Without comparing every angle and side of the two triangles, we can determine whether two triangles are congruent.

If given these triangles are congruent then all corresponding side must be equal

As we can see that both triangle are right triangle

Therefore hypotenuse of one triangle equal to another

x+1  = 4y   ..... (A)

And lenght ( height)  of one  must equal to another triangle

x = y +2          .... (B)

therefore

Plug the value of x = y + 2 in equation A

y + 2 + 1 = 4y

        3y = 3

          y = 1

and x = y+2

x = 1 + 2

x = 3

Hence, the values of x and y the triangles to the fight congruent are

x = 3 and y = 1.

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Complete question:

Complete question is attached below.

Answer:

$19.68 for each foot

Step-by-step explanation:

Answer:

$19.68

Step-by-step explanation:

no <3

There are 7200 ways for the senate committee to sit around a circular table.

To solve this problem, we can use the formula for circular permutations: (n-1)!/d, where n is the total number of people and d is the number of ways in which they can be distinguished. In this case, n is 11 (5 democrats + 5 republicans + 1 independent) and d is 1 (since all the members of each party sit together, they are not distinguishable from one another). Plugging these values into the formula, we get

This means that there are 7200 ways for the senate committee to sit around a circular table.

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An estimate for the mean age of the 50 people on the bus is 34 years.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Let us calculate the sum of the products of the midpoints and their respective frequencies, and divide by the total frequency to obtain the estimated mean age.

The midpoint and frequency data for each age group can be represented as follows:

Age group      Midpoint        Frequency

0 < h ≤ 20           10                    20

20 < h ≤ 40         30                    10

40 < h ≤ 60         50                     10

60 < h ≤ 80         70                     10

To calculate the estimated mean age, we can use the formula:

Estimated mean age = (Σ(midpoint × frequency)) / total frequency

Estimated mean age = ((10 × 20) + (30 × 10) + (50 × 10) + (70 × 10)) / (20 + 10 + 10 + 10)

Estimated mean age = (200 + 300 + 500 + 700) / 50

= 1700 / 50

= 34

Therefore, an estimate for the mean age of the 50 people on the bus is 34 years.

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Answer:

45  1/2

Step-by-step explanation:

(g) and Theorem 5 are not equivalent.

Are (g) and Theorem 5 equivalent?

You are correct. Statement (g) in Theorem 8, which states that the equation Ax = b has at least one solution for each b in R", includes both the case of a unique solution and the case of infinitely many solutions.

Therefore, (g) is not equivalent to Theorem 5, which specifically states that the equation Ax = b has a unique solution x = A^(-1)b when A is an invertible matrix. The equivalence mentioned in the text seems to be an error or a misinterpretation.

The correct interpretation is that Theorem 5 implies statement (g) in Theorem 8, but the converse is not necessarily true.

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Answer:

one solution

Step-by-step explanation:

To arrive at the objective function, we must make the main assumption that the returns are normally distributed

The objective function is a mathematical expression used to represent the goal of a problem in terms of one or more variables. In finance, the objective function is used to represent the goal of maximizing returns.

The distribution of returns is an important factor to consider when creating an objective function for an investment portfolio. Returns are the gains or losses an investor makes on his investments. The distribution of returns refers to the way in which these returns are distributed in the portfolio.

To arrive at the objective function, we must make the main assumption that the returns are normally distributed. A normal distribution is a bell-shaped curve that represents the distribution of data in a population. In finance, normal distributions are used to model portfolio returns because they ensure a good approximation of how returns are distributed across a wide range of investments.

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The volume of the solid generated by revolving the region enclosed by the triangle about the y-axis is 9π cubic units.

To find the volume of the solid generated by revolving the region enclosed by the given triangle about the y-axis, we can use the washer method.

The first step is to determine the limits of integration.

The triangle is bounded by the vertical lines x = 4, x = 5, and the line connecting the points (4, 1) and (5, 2).

We need to find the y-values that correspond to these x-values on the triangle.

At x = 4, the corresponding y-value on the triangle is 1.

At x = 5, the corresponding y-value on the triangle is 2.

So, the limits of integration for y will be from y = 1 to y = 2.

Now, let's consider an arbitrary y-value between 1 and 2. We need to find the corresponding x-values on the triangle.

The left side of the triangle is a vertical line segment, so for any y-value between 1 and 2, the corresponding x-value is x = 4.

The right side of the triangle is a line connecting the points (4, 2) and (5, 2).

This line has a constant y-value of 2, so for any y-value between 1 and 2, the corresponding x-value is given by the equation of the line: x = 5.

Now, we can set up the integral using the washer method. The volume can be calculated as follows:

V = ∫[1,2] π(\(R^2 - r^2\)) dy,

where R is the outer radius and r is the inner radius.

Since we are revolving the region about the y-axis, the outer radius R is the distance from the y-axis to the right side of the triangle, which is x = 5.

Thus, R = 5.

The inner radius r is the distance from the y-axis to the left side of the triangle, which is x = 4.

Thus, r = 4.

Substituting these values into the integral, we have:

V = ∫[1,2] π(5^2 - 4^2) dy.

Simplifying the integral:

V = ∫[1,2] π(25 - 16) dy

= ∫[1,2] π(9) dy

= 9π ∫[1,2] dy

= 9π [y] [1,2]

= 9π (2 - 1)

= 9π.

Therefore, the volume of the solid generated by revolving the region enclosed by the triangle about the y-axis is 9π cubic units.

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Answer: Mean≈73.2   Median=78.5   Mode=85    Range=47

Step-by-step explanation:

85+58+72+85+46+93=439.

Let's arrange the series:

46  58  72  85  85  93

\(\displaystyle\\Mean=\frac{46+58+72+85+85+93}{6} \\\\Mean=\frac{439}{6}\\ \\Mean\approx73.2\\\\Median=\frac{72+85}{2} \\\\\Median=\frac{157}{2}\\ \\Median=78.5\\\\Mode=85\\\\Range=93-46\\\\Range=47\)

the value of  gs(4, 2) = 5 * 2 = 10.

gr(4, 2) = 30 and gs(4, 2) = 10.

To calculate gr(4, 2) and gs(4, 2) using the table of values provided, we need to use the chain rule of differentiation.

Let's start with gr(4, 2):

gr(4, 2) represents the partial derivative of g with respect to r at the point (4, 2).

Using the chain rule, we have:

gr(4, 2) = (d/dx) [f(5x - y,\(y^2\)- 7x)] * (d/dx) [5r - s]

The first part, (d/dx) [f(5x - y, \(y^2\) - 7x)], represents the partial derivative of f with respect to x, evaluated at (5r - s, \(s^2\) - 7r).

Looking at the given table, we can see that fx = 6 at the point (4, 2). Therefore, (d/dx) [f(5x - y, \(y^2\) - 7x)] = 6.

The second part, (d/dx) [5r - s], represents the partial derivative of 5r - s with respect to r.

Taking the derivative with respect to r, we get:

(d/dx) [5r - s] = 5.

Therefore, gr(4, 2) = 6 * 5 = 30.

Now, let's calculate gs(4, 2):

gs(4, 2) represents the partial derivative of g with respect to s at the point (4, 2).

Using the chain rule, we have:

gs(4, 2) = (d/dx) [f(5x - y,\(y^2\) - 7x)] * (d/dx) [\(s^2\) - 7r]

Using the given table, fy = 5 at the point (4, 2). Therefore, (d/dx) [f(5x - y, \(y^2\) - 7x)] = 5.

Taking the derivative of \(s^2\) - 7r with respect to s, we get:

(d/dx) \([s^2\) - 7r] = 2s.

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Answer:

2(3x+2)(x+4)

Step-by-step explanation:

6x^2+28x+16

first : 2 as a common factor:

2(3x²+14x+8)   then factorize

2(3x+2)(x+4)

Answer:

10% of 210 is 21

Step-by-step explanation:

The 1 in 10% is much help. Move the decimal point one place to the right.

I know this is not much, but I hope this helps :)

The rule y = -3 - x is an illustration of a linear equation

The points on the coordinate plane with the rule y = -3 - x are (0,-3), (1,-4) and (2,-5)

The rule of the points is given as:

y = -3 - x

When x = 0, we have:

y = -3 - 0

y = -3

When x = 1, we have:

y = -3 - 1

y = -4

When x = 2, we have:

y = -3 - 2

y = -5

From the above computation, we have the following points

(0,-3), (1,-4) and (2,-5)

Hence, the points on the coordinate plane with the rule y = -3 - x are (0,-3), (1,-4) and (2,-5)

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Answer:

4.167%

Step-by-step explanation:

Given: It was predicted that it will take 23 minutes to drive to grandma's house but it actually takes 24 minutes.

To find: percent error

Solution:

Predicted time = 23 minutes

Actual time = 24 minutes

Error is the difference between Approximate and Exact Value. Percentage Error is used to compare an estimate value to an exact value.

Error = 24 - 23 = 1 minute

Percent error = (error/actual time) × 100

= 4.167%

Answer:

-3/14

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

m=(2-5)/(11-(-3))

m=-3/(11+3)

m=-3/14

Answer:

56.5 in^2

Step-by-step explanation:

The volume of a cone is π\(r^{2}\)h/3. The radius is d/2=6/2=3

V=(3.14 x \(3^{2}\) x 6)/3

 =(3.14 x 54)/3

 =169.56/3

 =56.52