Factorize :
x^2+8x+15
x^2-2x-15
x^2+9x-36
x^2-12x-45

(a) Probability of exactly 16 enrolling in college is approximately 0.1487 or 14.87%.

(b) Probability of more than 14 enrolling in college is approximately 0.8622 or 86.22%.

(c) Probability of fewer than 10 enrolling in college is approximately 0.0003 or 0.03%.

(d) It would be considered unusual if more than 23 enrolled in college as the probability is approximately 0.0072 or 0.72%.

(a) To calculate the probability that exactly 16 of the 28 high school graduates enroll in college, we can use the binomial distribution formula.

P(X = 16) = (28 C 16) * (0.65)^16 * (1 - 0.65)^(28 - 16)

Using a binomial probability calculator or formula, we find that P(X = 16) ≈ 0.1487.

The probability that exactly 16 of the sampled high school graduates enroll in college is approximately 0.1487 or 14.87%.

(b) To calculate the probability that more than 14 of the 28 high school graduates enroll in college, we need to find the sum of probabilities for all values greater than 14.

P(X > 14) = 1 - P(X ≤ 14)

Using a binomial probability calculator or formula, we find that P(X > 14) ≈ 0.8622.

The probability that more than 14 of the sampled high school graduates enroll in college is approximately 0.8622 or 86.22%.

(c) To calculate the probability that fewer than 10 of the 28 high school graduates enroll in college, we need to find the sum of probabilities for all values less than 10.

P(X < 10) = P(X ≤ 9)

Using a binomial probability calculator or formula, we find that P(X < 10) ≈ 0.0003.

The probability that fewer than 10 of the sampled high school graduates enroll in college is approximately 0.0003 or 0.03%.

(d) To determine if it would be unusual for more than 23 of the 28 high school graduates to enroll in college, we can compare it to a certain threshold or criterion. A common threshold used is having a probability less than 5%.

P(X > 23) = 1 - P(X ≤ 23)

Using a binomial probability calculator or formula, we find that P(X > 23) ≈ 0.0072.

The probability that more than 23 of the sampled high school graduates enroll in college is approximately 0.0072 or 0.72%, which is less than the 5% threshold commonly used to define unusual events.

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The length of XY (height) is approximately 15.2 cm to 1 decimal place, according to Pythagoras' theorem.

To calculate the length of XY (height) using Pythagoras' theorem, we can use the formula:

[tex]\[XY = \sqrt{XZ^2 - ZY^2}\][/tex]

Given that XZ = 16 cm (hypotenuse) and ZY = 5 cm (base), we can substitute these values into the formula:

[tex]\[XY = \sqrt{16^2 - 5^2} = \sqrt{256 - 25} = \sqrt{231} \approx 15.2 \text{ cm}\][/tex]

Therefore, the length of XY (height) is approximately 15.2 cm to 1 decimal place.

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The shape that CANNOT possibly describe the cross section is a rectangle.

1. The given three-dimensional object consists of two congruent parallel trapezoids.

2. A plane intersects the object perpendicularly to the trapezoids, resulting in a cross section.

3. To determine the shape that cannot possibly describe the cross section, we need to examine the possible outcomes.

4. When a plane intersects the object perpendicularly to the trapezoids, the resulting cross section can be various shapes, such as a parallelogram, triangle, trapezoid, or rectangle.

5. Let's consider each of these possibilities:

  a. Parallelogram: If the cross section is a parallelogram, it means the intersecting plane is parallel to the bases of the trapezoids.

  b. Triangle: If the cross section is a triangle, it means the intersecting plane passes through one of the vertices of the trapezoids.

  c. Trapezoid: If the cross section is a trapezoid, it means the intersecting plane is parallel to one base of the trapezoids and intersects the other base.

  d. Rectangle: If the cross section is a rectangle, it means the intersecting plane is perpendicular to both bases of the trapezoids.

6. Out of these possibilities, the shape that cannot possibly describe the cross section is a rectangle because a perpendicular plane would not intersect both bases of the trapezoids at right angles simultaneously.

7. Therefore, the shape that cannot possibly be the cross section is a rectangle.

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The value of x in the secant line using the Intersecting theorem is 10.

Intersecting secants theorem states that " If two secant line segments are drawn to a circle from an exterior point, then the product of the measures of one  of secant line segment and its external secant line segment is the same or equal to the product of the measures of the other secant line segment and its external line secant segment.

From the diagram:

First sectant line segment = ( x + 6 )

External segement of the first secant line = 6

Second sectant line segment = ( 46 + 2 ) = 48

External segement of the second secant line = 2

Now, using the Intersecting secants theorem:

6 × ( x + 6 ) = 2 × 48

Solve for x:

6x + 36 = 96

6x = 96 - 36

6x = 60

x = 60/6

x = 10

Therefore, the value of x is 10.

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The fifth term of the geometric sequence is 324.

To find the fifth term of the geometric sequence with a first term of 4 and a common ratio of -3, we can use the formula for the nth term of a geometric sequence.

The formula for the nth term of a geometric sequence is given by:

[tex]a_n = a_1 \times r^{(n-1)[/tex]

Where:

[tex]a_n[/tex] represents the nth term of the sequence,

[tex]a_1[/tex] is the first term of the sequence,

r is the common ratio of the sequence, and

n is the position of the term we want to find.

In this case, we are looking for the fifth term (n = 5), the first term is 4 (a_1 = 4), and the common ratio is -3 (r = -3).

Plugging these values into the formula, we have:

[tex]a_5 = 4 \times (-3)^{(5-1)[/tex]

Simplifying the exponent:

[tex]a_5 = 4 \times (-3)^4[/tex]

Calculating the value of the exponent:

[tex]a_5 = 4 \times 81[/tex]

[tex]a_5 = 324[/tex]

Therefore, the fifth term of the geometric sequence is 324.

It's important to note that the common ratio being negative (-3) indicates that the terms in the sequence alternate in sign.

In this case, the sequence would be: 4, -12, 36, -108, 324.

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Answer:

The pattern in the given infinite sequence is that each term increases by 2 from the previous term.

Starting with -3, the subsequent terms are obtained by adding 2 to the previous term.

-3 + 2 = -1

-1 + 2 = 1

1 + 2 = 3

3 + 2 = 5

So, the sequence continues indefinitely with each term increasing by 2.

Answer:

[tex]\textsf{1.(a)} \quad x^2-14x+\boxed{49}=\left(x-\boxed{7}\right)^2[/tex]

[tex]\textsf{1.(b)} \quad 9x^2 + 30x +\boxed{25}= \left(3x +\boxed{5}\right)^2[/tex]

[tex]\begin{aligned}\textsf{2.(a)}\quad3x^2-24x+48&=3\left(x^2-\boxed{8}\:x+\boxed{16}\right)\\&=3\left(x-\boxed{4}\right)^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{2.(b)}\quad \dfrac{1}{2}x^2+8x+32&=\dfrac{1}{2}\left(x^2+\boxed{16}\:x+\boxed{64}\right)\\&=\dfrac{1}{2}\left(x+\boxed{8}\right)^2\end{aligned}[/tex]

Step-by-step explanation:

(a) When completing the square for a quadratic equation in the form ax² + bx + c where the leading coefficient is one, we need to add the square of half the coefficient of the x-term:

[tex]x^2-14x+\left(\dfrac{-14}{2}\right)^2[/tex]

[tex]x^2-14x+\left(-7\right)^2[/tex]

[tex]x^2-14x+49[/tex]

We have now created a perfect square trinomial in the form a² - 2ab + b². To factor a perfect square trinomial, use the following formula:

[tex]\boxed{a^2 -2ab + b^2 = (a -b)^2}[/tex]

Therefore:

[tex]a^2=x^2 \implies a=1[/tex]

[tex]b^2=49=7^2\implies b = 7[/tex]

Therefore, the perfect square trinomial rewritten as a binomial squared is:

[tex]x^2-14x+\boxed{49}=\left(x-\boxed{7}\right)^2[/tex]

(b) When completing the square for a quadratic equation where the leading coefficient is not one, we need to add the square of the coefficient of the x-term once it is halved and divided by the leading coefficient, and then multiply it by the leading coefficient:

[tex]9x^2 + 30x +9\left(\dfrac{30}{2 \cdot 9}\right)^2[/tex]

[tex]9x^2 + 30x +9\left(\dfrac{5}{3}\right)^2[/tex]

[tex]9x^2 + 30x +9 \cdot \dfrac{25}{9}[/tex]

[tex]9x^2 + 30x +25[/tex]

We have now created a perfect square trinomial in the form a² + 2ab + b². To factor a perfect square trinomial, use the following formula:

[tex]\boxed{a^2 +2ab + b^2 = (a +b)^2}[/tex]

Therefore:

[tex]a^2=9x^2 = (3x)^2 \implies a = 3x[/tex]

[tex]b^2=25 = 5^2 \implies b = 5[/tex]

Therefore, the perfect square trinomial rewritten as a binomial squared is:

[tex]9x^2 + 30x +\boxed{25}= \left(3x +\boxed{5}\right)^2[/tex]

[tex]\hrulefill[/tex]

(a) Factor out the leading coefficient 3 from the given expression:

[tex]3x^2-24x+48=3\left(x^2-\boxed{8}\:x+\boxed{16}\right)[/tex]

We have now created a perfect square trinomial in the form a² - 2ab + b² inside the parentheses. To factor a perfect square trinomial, use the following formula:

[tex]\boxed{a^2 -2ab + b^2 = (a -b)^2}[/tex]

Factor the perfect square trinomial inside the parentheses:

                        [tex]=3\left(x-\boxed{4}\right)^2[/tex]

(a) Factor out the leading coefficient 1/2 from the given expression:

[tex]\dfrac{1}{2}x^2+8x+32=\dfrac{1}{2}\left(x^2+\boxed{16}\:x+\boxed{64}\right)[/tex]

We have now created a perfect square trinomial in the form a² + 2ab + b² inside the parentheses. To factor a perfect square trinomial, use the following formula:

[tex]\boxed{a^2+2ab + b^2 = (a +b)^2}[/tex]

Factor the perfect square trinomial inside the parentheses:

                        [tex]=\dfrac{1}{2}\left(x+\boxed{8}\right)^2[/tex]

Step-by-step explanation:

Since ABCD is a rectangle

⇒ AB = CD and BC = AD

x + y = 30 …………….. (i)

x – y = 14 ……………. (ii)

(i) + (ii) ⇒ 2x = 44

⇒ x = 22

Plug in x = 22 in (i)

⇒ 22 + y = 30

⇒ y = 8

Answer:

Each mark on the top of the number line increases by 70, so the second mark is 140. Each mark on the bottom number increases by 20%. So 40% of 350 is 140.

Step-by-step explanation:

We can see that the number line is divided into 5 equal parts on both the top and bottom.

Let's do the bottom part of the number line first.We can divide 100 by 5 to get what each segment of the total number line is. 100/5=20. Each segment of the bottom number line is 20%. The rest of the bottom number line is adding by 20 until we have reached 100. 20%, 40%, 60%, 80%, 100%.

For the top: Having the bottom part done, we know that each small segment is 20%. We can multiply 350 (the total) by 0.2 which gives us 20% of 350. 350x0.2=70. Each part of the line segment for the top will increase by 70. You can keep adding 70 to each segment, which will fill out the number line just the same. But for more explanation, lets continue filling it out step by step. Now, we can multiply 350 by 0.4 (40%). 350x0.4=140 (part of your final answer). Continue doing so for the rest. An important thing to notice is that the numbers we are multiplying 350 by are all matching the percents that should be filled in at the bottom. 350x0.6=210. That's the third mark. Next, 350x0.8=280. That's the fourth mark. Now, we have 350, which is 100%.  

Also, when you're not asked to do a double number line, you can divide the part by the total. 140/350=0.4. 0.4=40%.

Hope this helped.

Answer: Expression: 7 × 8 - 2.

Step-by-step explanation:

The expression for the given statement "two less than a product of 7 and a number equal to 8" can be written as 7 × 8 − 2 = 54.

So, the expression is 7 × 8 - 2.

Answer:

1108.67 square centimeters

Step-by-step explanation:

Surface Area = 2πr² + 2πrh

Radius = diameter / 2 = 14 cm / 2 = 7 cm

Surface Area = 2π(7 cm)² + 2π(7 cm)(18.2 cm)

Surface Area ≈ 2π(49 cm²) + 2π(127.4 cm²)

Surface Area ≈ 307.86 cm² + 800.81 cm²

Surface Area ≈ 1108.67 cm²

Therefore, approximately 1108.67 square centimeters of metal are required to make the can.

The length of the minor arc CE in the given circle, with an angle CDE of 62 degrees and a radius CD of 8 units, is approximately 13.645 units, rounded to the nearest hundredth.

To find the length of arc CE (minor arc) in a circle with center D, where m∠CDE = 62 degrees and CD = 8 units (radius), we can use the formula for the length of an arc. The formula is L = rθ, where L represents the arc length, r represents the radius of the circle, and θ represents the central angle subtended by the arc.

In this case, the given angle m∠CDE = 62 degrees and the radius CD = 8 units.

First, we need to convert the angle from degrees to radians since the formula requires the angle in radians. To convert from degrees to radians, we use the conversion factor π/180. So, θ = 62 * (π/180) radians.

Next, we substitute the values into the formula for the length of an arc:

L = rθ

L = 8 * (62 * (π/180))

L ≈ 13.645 units (rounded to the nearest hundredth)

Therefore, the length of arc CE (minor arc) in the given circle, rounded to the nearest hundredth, is approximately 13.645 units.

It's important to note that the length of an arc is directly proportional to the central angle it subtends and the radius of the circle. In this case, the given angle of 62 degrees and the radius of 8 units determine the length of the arc CE.

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The interquartile range (IQR) of the given data set is 7.

To find the interquartile range (IQR) of the

dataset, we first need to sort the data in ascending order.

Then calculate the first quartile (Q1) and the third quartile (Q3) to determine the IQR.

Arrange the data in ascending order:

10, 12, 18, 19, 20, 21, 23, 24, 24, 25, 25, 26, 27, 28, 30

Then calculate Q1 and Q3 :

Q1: Median of the lower half of the dataset. In this case, we have 7 data points in the bottom half, so we take the average of the 4th and 5th values.

Q1 = (19 + 20) / 2 = 19.5

Q3: Top half median record.

Again, we have 7 data points in the top half, so we average the 4th and 5th values:

Q3 = (26 + 27) / 2 = 26.5

Finally, calculate the IQR :

IQR = Q3 - Q1 = 26.5 - 19.5 = 7

Therefore, the interquartile range (IQR) for the given dataset is 7.

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six-eighths times five is equal to nine and three-fifths which can also be expressed as 3 3/5. This is the final answer.I hope this helps. Let me know if you have any other questions!

To solve the expression six-eighths times five, we will first convert six-eighths into a fraction. To convert it into a fraction, we need to simplify it by dividing both the numerator and denominator by their greatest common factor (GCF) which is 2. This gives us the reduced fraction of three-fourths.So, six-eighths is equal to three-fourths. Now we can substitute this in the expression and multiply it by five.3/4 x 5 = 15/4We can convert this improper fraction into a mixed number by dividing the numerator (15) by the denominator (4). When we divide 15 by 4, we get a quotient of 3 and a remainder of 3. We then express the remainder as a fraction over the denominator. This gives us a mixed number of 3 and 3/4.

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The resulting polynomial of f(x) · g(x) is -42x³ + 48x² - 70x + 80.

The product of two polynomials, f(x) and g(x), we multiply each term of one polynomial by each term of the other polynomial and then combine like terms.

Given:

f(x) = 7x - 8

g(x) = -6x² - 10

To find f(x) · g(x), we substitute the expressions of f(x) and g(x) into the multiplication:

f(x) · g(x) = (7x - 8) × (-6x² - 10)

Now, we distribute each term of f(x) to each term of g(x):

f(x) · g(x) = (7x) × (-6x² - 10) + (-8) × (-6x² - 10)

Using the distributive property, we can simplify each part of the expression:

f(x) · g(x) = -42x³ - 70x + 48x² + 80

Now, we combine like terms to obtain the resulting polynomial:

f(x) · g(x) = -42x³ + 48x² - 70x + 80

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Answer:

The correct expanded form and value of (Four-fifths) cubed is:

(Four-fifths) cubed = (4/5)³

Expanded form: (4/5) × (4/5) × (4/5)

Value: (4/5) × (4/5) × (4/5) = 64/125

Therefore, the correct expanded form of (Four-fifths) cubed is (4/5) × (4/5) × (4/5), and its value is 64/125.

Answer: f(8) = 620.6

Step-by-step explanation:

To solve, you must first use substitution, and replace the x value in the given function, with 8.

[tex]f(8)=\frac{750}{1 + 74e^{-0.734(8)} }[/tex]

Next you follow PEMDAS to solve the rest of the equation to find f(8).

What is PEMDAS?

PEMDAS is the order of operations for mathematical expressions involving more than one operation.

You solve in the following order

In this case our next step is deal with the exponents.

And by doing so, you will be multiplying -0.734 by 8

[tex]f(8)=\frac{750}{1 + 74e^{-5.872} }[/tex]

And typical with exponents we would raise the number or variable given to the power of the exponent.

However we have a continuous number in this case, which is e.

What is e?

e is a mathematical constant approximately equal to 2.71828

We can now use a calculator to find what e (2.71....) is when raise to the exponent -5.872. Which results to 0.002817 (0.0028172332293320237716146943697).

When continuing to use numbers in calculators, don't use the rounded or shortened of the full number. Make sure to use the all of the numbers (given in the calculator) to its full extent to maintain accuracy.

[tex]f(8)=\frac{750}{1 + 74(0.002817) }[/tex]

Now multiply 74 and 0.002817.... (use the full number to multiply (0.0028172332293320237716146943697)). The product is (0.2084752589705697590994873833578).

[tex]f(8)=\frac{750}{1 + 0.2084 }[/tex]

Now lets add 1 + 0.2084.... and you get 1.2084

[tex]f(8)=\frac{750}{1.2084 }[/tex]

Last you need to divide:

and you get that

f(8) = 620.6167

Now lets go back to the question, and we see that is says for you to round your answer to the nearest tenth.

so your final answer is:

f(8) = 620.6

Answer:

a) 7.5 meters b) 588.75

Step-by-step explanation:

To calculate the volume you need to multiply the 3 values together so

3x5x0.5 = 7.5

That is the volume

Then you got to multiply that by 78.50 to get the answer for b

78.50x7.5 =588.75

Check the picture below.

so the parabolic path of the object is more or less like the one shown below in the picture, now this object has an initial of 1240 ft, as it gets thrown from the ski lift, so from 0 seconds is already higher than 1000 feet.

[tex]h=-16t^2+32t+1240\hspace{5em}\stackrel{\textit{a height of 1000 ft}}{1000=-16t^2+32t+1240} \\\\\\ 0=-16t^2+32t+240\implies 16t^2-32t-240=0\implies 16(t^2-2t-15)=0 \\\\\\ t^2-2t-15=0\implies (t-5)(t+3)=0\implies t= \begin{cases} ~~ 5 ~~ \textit{\LARGE \checkmark}\\ -3 ~~ \bigotimes \end{cases}[/tex]

now, since the seconds can't be negative, thus the negative valid answer in this case is not applicable, so we can't use it.

So the object on its way down at some point it hit 1000 ft of height and then kept on going down, and when it was above those 1000 ft mark  happened between 0 and 5 seconds.

The vertex is (2, 4) and (3, 3)

To graph the quadratic function

[tex]f(x) = -(x - 2)^2 + 4[/tex],

we'll first identify the vertex and then plot a second point on the parabola.

The vertex form of a quadratic function is given by

[tex]f(x) = a(x - h)^2 + k[/tex],

where (h, k) represents the vertex of the parabola.

Comparing the given function

[tex]f(x) = -(x - 2)^2 + 4[/tex]

to the vertex form, we can identify that the vertex is located at (2, 4).

Now, let's plot the vertex (2, 4) on a coordinate plane and find another point on the parabola.

We can choose a value for x and calculate the corresponding y-coordinate using the function.

Let's choose x = 3.

Substituting this value into the function, we get:

[tex]f(3) = -(3 - 2)^2 + 4[/tex]

[tex]= -(1)^2 + 4[/tex]

[tex]= -1 + 4[/tex]

[tex]= 3[/tex]

So, when x = 3, y = 3.

Now we have two points: the vertex (2, 4) and (3, 3).

Let's plot them on a coordinate plane and draw the parabola:

   |

   |

   |

   |     *

   |          *

   |

   |_______________

        2  3  4

In the graph, the vertex is denoted by '*' and the second point is marked as '.'.

The parabola opens downward because the coefficient of (x - 2)^2 is negative.

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The function for velocity is v(t) = 12t² + 84t + 120, the velocity equals zero at t = -2 and t = -5, and the function for acceleration is a(t) = 24t + 84.

To find the function for velocity, we need to take the derivative of the equation of motion, s(t), with respect to time, t. The derivative of s(t) gives us the rate of change of position, which is velocity.

Given: s(t) = 4t³ + 42t² + 120t

To find v(t), we differentiate s(t) with respect to t:

v(t) = 12t² + 84t + 120

So, the function for velocity is v(t) = 12t² + 84t + 120.

To find where the velocity equals zero, we set v(t) = 0 and solve for t:

12t² + 84t + 120 = 0

To solve this quadratic equation, we can factor out the greatest common factor (GCF), which is 12:

12(t² + 7t + 10) = 0

Now, we can factor the quadratic expression inside the parentheses:

12(t + 2)(t + 5) = 0

Setting each factor equal to zero, we get:

t + 2 = 0   -->   t = -2

t + 5 = 0   -->   t = -5

So, the velocity equals zero at t = -2 and t = -5.

To find the function for acceleration, we need to take the derivative of the velocity function, v(t), with respect to time, t. The derivative of v(t) gives us the rate of change of velocity, which is acceleration.

Given: v(t) = 12t² + 84t + 120

To find a(t), we differentiate v(t) with respect to t:

a(t) = 24t + 84

So, the function for acceleration is a(t) = 24t + 84.

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Answer: To find the absolute value of the expression 1 1/2 - 2/31, we first need to convert the mixed number 1 1/2 into an improper fraction.

1 1/2 can be written as (2 * 1 + 1) / 2, which is equal to 3/2.

Now we can subtract 2/31 from 3/2:

3/2 - 2/31 = (3 * 31 - 2 * 2) / (2 * 31) = (93 - 4) / 62 = 89/62.

The absolute value of a fraction is the positive value without considering its sign. So, the absolute value of 89/62 is 89/62.

Therefore, the absolute value of 1 1/2 - 2/31 is 89/62.

In the case of the given expression, we applied the same concept of multiplying two negatives to distribute -1 from each negative number, resulting in a positive simplified expression of 60/10.

To rewrite the expression "-60/-10" by distributing -1 from each negative number, we can perform the following steps:

Distribute -1 to the numerator (-60) and the denominator (-10) separately.

(-1) * (-60) = 60

(-1) * (-10) = 10

Rewrite the expression using the results of the distributions:

60/10

After distributing -1 to both the numerator and the denominator, we obtain the simplified expression of 60/10.

This simplification occurs because multiplying two negative numbers together yields a positive result. Therefore, the negative signs cancel out when distributing -1 from each negative number.

To understand this concept, consider the multiplication of two negative numbers, such as (-1) * (-2). When multiplying two negatives, the result is positive: (-1) * (-2) = 2.

This happens because multiplication is essentially repeated addition, and each negative value can be seen as an opposite or a reversal of a positive value. So, multiplying two negatives cancels out the reversal and returns a positive result.

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The car's speed is approximately 64.37 kilometers per hour, and it will travel 64.37 times the number of hours it is on the road in kilometers.

When converting miles per hour to kilometers per hour, we need to consider that 1 mile is approximately equal to 1.60934 kilometers. Therefore, to find the car's speed in kilometers per hour, we can multiply its speed in miles per hour by the conversion factor of 1.60934.

In this case, the car's speed of 40 miles per hour can be converted as follows:

40 miles/hour * 1.60934 kilometers/mile = 64.3736 kilometers/hour.

Thus, the car's speed is approximately 64.37 kilometers per hour.

To calculate the distance the car will travel in hours, we can use the formula distance = speed × time. Given that the car will be traveling for a certain number of hours, let's say 't' hours, the distance traveled in kilometers can be determined as:

Distance = 64.37 kilometers/hour × t hours = 64.37t kilometers.

Therefore, the car will travel 64.37 times the number of hours it travels in kilometers.

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The correct option is (C) A ∪ c = U.

Given,Consider an event B, non occurrence of event B is represented by c. intersection of A e. A is equal to zeroWe know that the intersection of any set with the null set results in a null set. In other words, if A is a null set, then the intersection of A with any other set B is also a null set, i.e., A ∩ B = ∅.

Here, A is the null set, i.e., A = ∅.Therefore, A ∩ E = ∅, where E is any other set.Since we know that the intersection of A with any other set is a null set, we can say that the intersection of c with A is also a null set, i.e., A ∩ c = ∅.Since A ∩ c = ∅, we can say that the union of A with c is equal to the universal set U, i.e., A ∪ c = U.

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Both |-38| and -(-38) are equal to 38, the correct symbol to insert between the pair of numbers is "=" (equals). Hence, the true statement is: |-38| = -(-38)

To determine the relationship between |-38| and -(-38),

let's evaluate each expression separately:

|-38| = 38 (since the absolute value of -38 is equal to 38)

-(-38) = 38 (since negating a negative number results in a positive number)

Therefore, |-38| = -(-38) = 38.

Since both |-38| and -(-38) are equal to 38, the correct symbol to insert between the pair of numbers is "=" (equals).

Hence, the true statement is:

|-38| = -(-38)

It's important to note that the absolute value function (denoted by | |) always yields a non-negative value, regardless of the sign of the number within the absolute value brackets.

In this case, |-38| evaluates to 38, and the negation of -38 also evaluates to 38, resulting in an equality between the two expressions.

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Option(A) is the correct answer : A. Two rays that share a common endpoint called the vertex.

An angle is a fundamental concept in geometry that helps us understand the relationship and measurement of lines and shapes. It is formed by two rays that share a common endpoint called the vertex. To fully grasp the concept of an angle, it's important to understand its components and properties.

The rays that form an angle are called the sides of the angle. They extend indefinitely from the vertex in opposite directions, defining the space between them. The endpoint where the rays meet is known as the vertex of the angle.

Angles can vary in size and measurement. The measure of an angle is typically expressed in degrees (°) or radians (rad). A full circle is divided into 360 degrees or 2π radians. Angles smaller than 90 degrees are considered acute angles, while angles exactly 90 degrees are right angles. Angles greater than 90 degrees but less than 180 degrees are called obtuse angles, and angles measuring exactly 180 degrees are straight angles.

Angles have various properties and relationships. When two angles share a common vertex and a common side, they are called adjacent angles. The sum of adjacent angles around a point is always 360 degrees. Angles that have the same measure are called congruent angles.

Understanding angles is crucial in geometry and various other fields, including trigonometry, physics, and engineering. They help us analyze and describe the orientation, position, and spatial relationships of objects in both two-dimensional and three-dimensional spaces.

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Answer:

A. Two rays that share a common endpoint called the vertex.

Step-by-step explanation:

By solving the system of linear equations using elimination x= 5 and y=2.

-5x + 3y = -19  is  equation 1

-x - 3y = -11  is equation 2

Add Equation 1 to Equation 2

-5x + 3y + -x - 3y = -19+ -11

(3y-3y=0 as a result y term can be eliminated)

gives -6x =  -30

Dividing,

  = x = 5

Put it in equation 1,  we get

-5*5 + 3y = -19

=-25 + 3y = -19

= 3y  = 25-19 = 6

= y = 2

Solving the equations we get x = 5  and y = 2.

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The probability of A and C, from the tree diagram above, is given as follows:

P(A and C) = 2/21.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The desired nodes on the tree diagram for A and C are given as follows:

2/7 and 1/3.

Hence the probability is given as follows:

P(A and C) = 2/7 x 1/3

P(A and C) = 2/21.

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The equations x² = 9, x³ = 27, x³ = -27, and (-x)² = 9 all have -3 as a solution. A, C, D, and F

To determine which equations have -3 as a solution, we can substitute -3 for x in each equation and see if the equation holds true.

A. x² = 9

Substituting -3 for x:

(-3)² = 9

9 = 9

B. x² = -9

Substituting -3 for x:

(-3)² = -9

9 ≠ -9

C. x³ = 27

Substituting -3 for x:

(-3)³ = 27

-27 = 27

D. x³ = -27

Substituting -3 for x:

(-3)³ = -27

-27 = -27

E. -x² = 9

Substituting -3 for x:

-(-3)² = 9

-9 = 9

F. (-x)² = 9

Substituting -3 for x:

(-(-3))² = 9

9 = 9

From the above calculations, we can see that equations A, C, D, and F hold true when x is replaced with -3. Therefore, the equations for which -3 is a solution are A, C, D, and F.

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The value of x can be any real number.

To find the value of x when the gradient of a line passes through points A(-3, -8) and H(x, -4), we can use the formula for calculating the gradient (slope) of a line:

Gradient = (change in y) / (change in x)

Given that the gradient is the same between points A and H, we can set up the following equation:

(-8 - (-4)) / (-3 - x) = (change in y) / (change in x)

Simplifying the equation, we have:

(-8 + 4) / (-3 - x) = (-4 - (-8)) / (x - (-3))

-4 / (-3 - x) = 4 / (x + 3)

To eliminate the fractions, we can cross-multiply:

-4(x + 3) = 4(-3 - x)

Simplifying further, we have:

-4x - 12 = -12 - 4x

Rearranging the terms, we get:

-4x + 4x = -12 + 12

0 = 0

The equation simplifies to 0 = 0, which means that the value of x can be any real number. In other words, there are infinitely many possible values of x for which the gradient of the line passing through points A and H remains the same.

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