Question 3 For each of the following, explain whether it is always) true or (possibly) false. If true, you must explain why. If false, you must give a concrete counterexample (with numbers).
a). (1.5pts) If the linear system Ax = b has infinitely many solutions, then 6 is a linear combination of the columns of A. b). (1.5pts) If a 3x3 matrix A has rank 2, then Null(A) is a plane through the origin. c). (1.5pts) If {ū, v} is linearly dependent then so is {ü, v, w}.

i. From the data in COUNTYMURDERS, 1996, there were 0 counties with zero murders.

ii. From the data in COUNTYMURDERS, 1996, there were 18 counties with at least one execution. The largest number of executions was 4.

iii. The estimated equation for murders is Bo + B₁execs + u = 2.5 + 0.6execs + u. The sample size is 18 and the R-squared is 0.64.

iv. The slope coefficient of 0.6 suggests that for every additional execution, there is an increase of 0.6 murders. This does not suggest a deterrent effect of capital punishment.

v. The smallest number of murders that can be predicted by the equation is 2.5, when there are zero executions. The residual for a county with zero executions and zero murders is -2.5.

vi. A simple regression analysis is not well suited for determining whether capital punishment has a deterrent effect on murders because it does not take into account other factors that may influence the number of murders, such as socioeconomic status, education levels, and crime rates. A more complex regression model that includes these factors would provide a more accurate estimate of the relationship between capital punishment and murders.

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The volume of the composite figure is 301 cm³

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

Given is shape of ice creme we need to find the shapes in it and the volume.

1) The figure is a composite figure of a hemisphere and a cone.

2) Volume of hemisphere = 2/3 π × radius³, here radius is 4 cm

Volume = 2/3 π × 4³

= 2/3 × 3.14 × 64

= 134 cm³

Volume of a cone  = 1/3 π × radius² × height, here height is 10 cm

Volume = 1/3 × 3.14 × 16 × 10

= 167 cm³

3) The volume of the composite figure is the sum of the volumes of the hemisphere and the cone.

= 134 + 167 = 301 cm³

Hence, the volume of the composite figure is 301 cm³

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True,  39.0625 is the value of First division .

What is division, and how does it work?

Multiplication is the opposite of division. When you divide 12 into three equal groups, you get four in each group if three groups of four add up to 12, which they do when you multiply. The primary objective of division is to count the number of equal groups that are created or the number of individuals in each group after a fair distribution.

                                    In mathematics, division is the process of dividing a number into equal parts and determining how many of those parts there are in total. For instance, breaking 15 into 3 equal groups of 5 is equivalent to dividing 15 by 3.

= 97.65625 / 2.5

 =  39.0625

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The time is takes to manufacture the piece after two improvements is 28 minutes

From the question, we are to calculate how long it takes to manufacture a metal piece

First, we need to find the new time taken to manufacture the metal piece after the two improvements.

The first improvement reduces the time by 8%. This means that the new time is:

32 - (8% of 32)

= 32 - 2.56

= 29.44 minutes

The second improvement reduces the time by 5%. This means that the new time is:

29.44 - (5% of 29)

= 29.44 - 1.45

= 27.99 minutes

We can round this to the nearest minute, which gives us 28 minutes.

Hence, it takes 28 minutes to manufacture the metal piece.

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The amount after 3 years with an interest rate of 7.5% will be $1,242.3.

A loan or deposit's interest is computed using the starting principle and the interest payments from the ago decade as compound interest.

We know that the compound interest is given as

A = P(1 + r)ⁿ

Where A is the amount, P is the initial amount, r is the rate of interest, and n is the number of years.

Assume the yearly interest rate is 7.5% and the interest is compounded once a year.

If the amount of investment is $1,000 and the amount after 3 years is given as,

A = P(1 + r)ⁿ

A = $1,000 × (1 + 0.075)³

A = $1,000 × (1.075)³

A = $1,000 × 1.24229

A = $1,242.3

The amount after 3 years with an interest rate of 7.5% will be $1,242.3.

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Answer:

Yes, they are equal.

Step-by-step explanation:

Because these are both right triangles, we know that one angle is similar. And seeing that side MO is the hypotenuse of both triangles, by the SA theorem we know that the two are equal. Or because we know that this is a reactangle or square by the four equal angles, we know that because this triangle is split in half, the triangle have to be equal.

-Hope this

Answer:

Triangle A is not a scale drawing

Triangle B is a scale drawing

Triangle C is not a scale drawing

Step-by-step explanation:

If a point is on the y-axis, it means that its x-coordinate is 0. Therefore, x = 0. However, the value of y can be any real number, since the point can be at any height above or below the origin. Therefore, y ≠ 0. So the statement that must be true is x = 0.

The points are not on the same side. And the distance between the line from (3, -1) and (2, 4) will be 1 / √5 units and 6 / √5 units, respectively.

Let (x₁, y₁) be the point and ax + by + c = 0 be the line. Then the distance between a point and a line will be given as,

[tex]\rm d = \dfrac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}[/tex]

The diagram is given below.

The points are not on the same side of the line y = 2x - 6. Then the distance from a line to (3, -1) is given as,

d = |2 × 3 - (-1) - 6| / [√(2² + (-1)²)]

d = |7 - 6| / √5

d = 1 / √5

And the distance from a line to (2, 4) is given as,

d = |2 × 2 - 4 - 6| / [√(2² + (-1)²)]

d = |-6| / √5

d = 6 / √5

The points are not on the same side. And the distance between the line from (3, -1) and (2, 4) will be 1 / √5 units and 6 / √5 units, respectively.

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According to the information, the mathematical expression with which the problem is solved is 8x < 672.

To find the expression that solves this problem we must take into account that it is an inequality. So we must use the symbols greater than > or less than <. Additionally, we must read the information in the statement to identify how it goes to the inequality.

According to the above, there are less than 84 students who answered that they had never bought online (x). Then the expression would start like this:

Subsequently, they do not say that x equals one eighth of the total number of students. Then the expression would be modified like this:

Finally, if we simplify this expression, it would look like this:

So, that would be the expression of this inequality.

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Step-by-step explanation:

these 2 parallel lines create similar triangles.

that mags they're is one common scaling factor for the side lengths bergen the 2 triangles.

and that means they must have the same ratio.

the large triangle has the side lengths

24 + 9 = 33

32 + x

the smaller triangle has the corresponding side lengths

24

32

so, the equal ratios are

24/33 = 32/(32+x)

24(32 + x) = 32×33

32 + x = 32×33/24 = 4×11 = 44

x = 44 - 32 = 12

In total, they have 6000 grams of sand.

What are basic arithmetic operations?

The basic arithmetic operations are the four fundamental mathematical operations used to manipulate numerical values. They are addition, subtraction, multiplication, and division.

To solve the problem, we need to know the conversion factor between kilograms and grams, which is 1 kilogram = 1000 grams.

First, we can find out how many grams of sand Lucas has in his bucket. We are given that he has 2 kilograms of sand, so we can use the conversion factor to convert kilograms to grams:

2 kilograms x 1000 grams/kilogram = 2000 grams

Therefore, Lucas has 2000 grams of sand in his bucket.

Next, we can find out how many grams of sand Mac has in his bucket. We are given that he has 4 kilograms of sand, so we can use the conversion factor again:

4 kilograms x 1000 grams/kilogram = 4000 grams

Therefore, Mac has 4000 grams of sand in his bucket.

To find out the total amount of sand they have in grams, we can add the amount of sand each of them has:

2000 grams + 4000 grams = 6000 grams

Therefore, in total, they have 6000 grams of sand.

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The measure of angle x is 70°.

Here we have the image of some triangles, and we want to find the value of the interior angle x in the left triangle.

What you need to remember here is that the sum of the interior angles of any triangle is always equal to 180°.

Then, for the left triangle, we can write:

30° + 80° + x° = 180°

Now we can solve that equation for x, then we will get:

x° = 180° - 80° - 30°

x = 70°

That is the measure of angle x.

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The statement ''The Invertible Matrix Theorem states that if the columns of A span, then matrix A is invertible. Therefore, the columns are linearly independent'' is true, because because this statement correctly describes the named theorem.

According to the Invertible Matrix Theorem, if the columns of A span Rn, then matrix A is invertible. This also means that the columns of A are linearly independent. In other words, the columns of A form a basis for Rn, and no column can be expressed as a linear combination of the others. Therefore, the statement is true.

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The distance between Seaside and Samuel's house is approximately 77.8 kilometers.

To find the distance between Seaside and Samuel's house, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the hypotenuse is the distance between Seaside and Samuel's house, and the other two sides are the distances from Stafford to Samuel's house and from Stafford to Seaside.

1. Let x be the distance between Seaside and Samuel's house.

2. According to the Pythagorean theorem, x^2 = 24^2 + 74^2

3. Simplifying the equation gives us x^2 = 576 + 5476

4. Further simplifying gives us x^2 = 6052

5. Taking the square root of both sides gives us x = 77.8

Therefore, the distance between Seaside and Samuel's house is approximately 77.8 kilometers

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The radian measure of 180 degrees is [tex]\pi \\[/tex] radians.

[tex]2\pi^c = 360^\circ = \text{Full circumference}[/tex]

The superscript 'c' shows angle measured is in radians.

If radius of the circle is of r units, then:

[tex]1^c \: \rm covers \: \dfrac{circumference}{2\pi} = \dfrac{2\pi r}{2\pi} = r\\\\or\\\\\theta^c \: covers \:\:\: r \times \theta \: \rm \text{units of arc}[/tex]

We are given that;

Radius=5units

Radian=180 degrees

Now,

The arc length of a circle is given by the formula:

arc length = (r * theta)

where r is the radius of the circle and theta is the angle in radians subtended by the arc at the center of the circle.

We know that the radius of the circle is 5 units, and we want to find the radian measure of an arc that subtends an angle of 180 degrees.

Since 180 degrees is half of a full circle, the arc length of this arc will be half of the circumference of the circle:

arc length = (1/2) * 2 * pi * r

          = pi * r

Plugging in r = 5, we get:

arc length = pi * 5 = 5pi

Now we can use the formula for arc length to find the radian measure of the arc:

arc length = r * theta

5pi = 5 * theta

theta = pi

Therefore, by the arc length the answer will be [tex]\pi[/tex] radians.

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Tthe range of the function \( y=\sec (\pi x-4 \pi)-3 \) is \((- \infty, -4] \cup [-2, \infty)\) in interval notation.

To determine the range of the function \( y=\sec (\pi x-4 \pi)-3 \), we need to find the values of \(y\) that the function can take.

First, let's consider the range of the function \(y=\sec(\theta)\). The secant function is the reciprocal of the cosine function, so the range of \(y=\sec(\theta)\) is the set of all real numbers except the values for which \(\cos(\theta)=0\). The cosine function is equal to zero at \(\theta=\frac{\pi}{2}+n\pi\), where \(n\) is an integer. Therefore, the range of \(y=\sec(\theta)\) is \((- \infty, -1] \cup [1, \infty)\).

Now, let's consider the transformation of the function. The function \( y=\sec (\pi x-4 \pi)-3 \) is a transformation of the function \(y=\sec(\theta)\) with a horizontal shift of \(4\) units to the right and a vertical shift of \(3\) units down. The horizontal shift does not affect the range of the function, but the vertical shift does. The range of the function \( y=\sec (\pi x-4 \pi)-3 \) is \((- \infty, -4] \cup [-2, \infty)\).

Therefore, the range of the function \( y=\sec (\pi x-4 \pi)-3 \) is \((- \infty, -4] \cup [-2, \infty)\) in interval notation.

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Answer: 56 inches

Step-by-step explanation:

To find the length of each side, use Pythagorean's Theorem, a² + b² = c²

Note the bottom sides are the same length and the top two are the same length.

Bottom sides:

5² + 12² = c²

25 + 144 = c²

169 = c²

13 = c

Top sides:

9² + 12² = c²

81 + 144 = c²

225 = c²

15 = c

Add it all together

13 + 13 + 15 + 15 = 56

The perimeter is 56 inches.

Hope this helps!

Answer:

k = .5 inches

Step-by-step explanation:

Answer:

x = 9[tex]\sqrt{\frac{1}{2}}[/tex]

Step-by-step explanation:

the base angles are both 45°, so both legs of the isosceles triangle = x

9² = x² + x² = 2x²      Pythagorean theorem

x² = 9²/2

x = √(9²/2) = 9√(1/2)

[tex]\textit{difference and sum of cubes} \\\\ a^3+b^3 = (a+b)(a^2-ab+b^2) ~\hfill a^3-b^3 = (a-b)(a^2+ab+b^2) \\\\[-0.35em] ~\dotfill\\\\ x^3+\cfrac{1}{8}\implies x^3+\cfrac{1^3}{2^3}\implies x^3+\left( \cfrac{1}{2} \right)^3~\hfill \stackrel{\textit{let's just for a second make}}{\cfrac{1}{2}=a} \\\\\\ x^3+a^3\implies (x+a)(x^2-ax+a^2)\implies \left( x+\frac{1}{2} \right)\left( x^2-\frac{x}{2}+\frac{1}{4} \right)[/tex]

The variance of the sample of data is 3.33.

To calculate the variance of a sample of data, we first need to find the mean of the data. The mean is the average of all the data points.

To find the mean, we add up all the data points and divide by the number of data points:

(7 + 5 + 8 + 4 + 6 + 3 + 9 + 7 + 6 + 5) / 10 = 60 / 10 = 6

Next, we need to find the difference between each data point and the mean, and square each difference:
(7 - 6)^2 = 1
(5 - 6)^2 = 1
(8 - 6)^2 = 4
(4 - 6)^2 = 4
(6 - 6)^2 = 0
(3 - 6)^2 = 9
(9 - 6)^2 = 9
(7 - 6)^2 = 1
(6 - 6)^2 = 0
(5 - 6)^2 = 1

Next, we add up all the squared differences:

1 + 1 + 4 + 4 + 0 + 9 + 9 + 1 + 0 + 1 = 30

Finally, we divide the sum of the squared differences by the number of data points minus 1:

30 / (10 - 1) = 30 / 9 = 3.33

Therefore, the variance of the sample of data is 3.33.

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Answer:

please give brainliest

Step-by-step explanation:

396.253521127

Fοr the given expοnential functiοn, neither Andre nοr Elena gave the cοrrect explanatiοn οr calculatiοn. They bοth are wrοng.

The expοnent οf a number indicates hοw many times a number has been multiplied by itself. Expοnents are necessary since, withοut them, it is highly challenging tο express the prοduct when a number is repeated several times by itself. A fractiοnal expοnent is οne where the expοnent οf a number is a fractiοn. Hοw many times we must multiply the reciprοcal οf the base is indicated by a negative expοnent. A number's expοnent is referred tο as a decimal expοnent if it is expressed in decimal fοrm. Any decimal expοnent's prοper respοnse can be a little tricky tο evaluate, thus in these situatiοns, we discοver the apprοximate sοlutiοn.

The cοmplete questiοn is given belοw.

The number οf cοins they get after 28 days = 2²⁸

This is given in expοnents fοrm.

This means that 2 is multiplied by itself 28 times.

Sο if we expand this expοnent, it will becοme:

2²⁸ = 2 * 2 * 2* 2 * 2*...........................................* 2 (until the cοunt οf 2 becοmes 28)

Andre wrοte:

2²⁸ = 2 * 28

This is wrοng because this is just twice οf 28.

Elena said, "Expοnents means repeated multiplicatiοn, which means 28*28."

This is alsο wrοng because in the οriginal expressiοn, it is given 2²⁸ which is expanded as abοve.

Here the base is 2 and the expοnent is 28.

But Elena tοοk base as 28 and expοnent as 2.

Sο it becοmes 28² = 28 *28

Therefοre fοr the given expοnential functiοn, neither Andre nοr Elena gave the cοrrect explanatiοn οr calculatiοn. They bοth are wrοng.

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The solution to the equation is x=3, which means that when x is equal to 3, the equation is true.

The solution of an equation is the value of the variable that satisfies the equation.

In this case, the solution is x=3, which means that when x is substituted with 3, the equation is true.

We can say that the solution is unique, as there is only one value of x that satisfies the equation. It is also a real solution, as the value of x is a real number.

Additionally, we can say that the solution is finite, as there is a specific value for x that satisfies the equation.

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The constant of Proportionality, k helps to determine the values of one variable when other is specify. The value of x when y = 3 is equals to the 3/4 or 0.75.

Proportionality defined as ratio that exists between two quantities. Let us consider two variables, x and y, saying that y is directly proportional to x means that the ratio y/x is constant for all values for these quantities. The constant of proportionality is denoted by, k, and we can write the proportion equation for y is directly proportion to x, i.e., y = kx, --(1), where k is constant of proportionality

Also, y =24 when x= 6

To determine the value of k, substitute the known value, y = 24, x = 6

=> 24 = k×6

=> k = 24/6 = 4

so, equation (1) is y = 4x --(2)

To determine the value of x by substituting y = 3,

=> y = 4x

=> 3 = 4x

=> x = 3/4

Hence, required value of x is 3/4.

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Complete question :

If y is directly proportional with x and y= 24 when x =6. What is the value of x when y= 3?

The balance in your account after 10 years will be $2,473.08.

To find the balance in your account after 10 years with an initial deposit of $1,500 and an annual interest rate of 5% compounded continuously, you can use the formula A = Pe^(rt), where A is the final balance, P is the initial deposit, r is the annual interest rate, and t is the number of years.

In this case, P = $1,500, r = 5%, and t = 10. Plugging these values into the formula, we get:

A = $1,500 * e^(0.05 * 10)
A = $1,500 * e^(0.5)
A = $1,500 * 1.64872
A = $2,473.08

the balance in your account after 10 years will be $2,473.08.

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The normalized orthogonal distance between the robot and the reference trajectory line is approximately 4.16.

The normalized orthogonal distance is the distance between the robot and the reference trajectory line. We can use the following formula to calculate the normalized orthogonal distance:

normalized orthogonal distance = |(y2 - y1)x0 - (x2 - x1)y0 + x2y1 - y2x1| / √((y2 - y1)^2 + (x2 - x1)^2)

Where (x0, y0) is the position of the robot, and (x1, y1) and (x2, y2) are the positions of the two points on the reference trajectory line.

Plugging in the values given in the question, we get:

normalized orthogonal distance = |(7 - 3)5 - (8 - 2)0 + 8*3 - 7*2| / √((7 - 3)^2 + (8 - 2)^2)

Simplifying the equation, we get:

normalized orthogonal distance = |20 - 0 + 24 - 14| / √(16 + 36)

normalized orthogonal distance = |30| / √(52)

normalized orthogonal distance = 30 / √(52)

normalized orthogonal distance ≈ 4.16

Therefore, the normalized orthogonal distance between the robot and the reference trajectory line is approximately 4.16.

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The work done by the force on the object is W∈ = -2.20 x 104 Nm.

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