No such vector exists because the directional cosines must be unit vector, with a magnitude of 1.
The directional cosines of a vector are ratios of the components of the vector in the x, y, and z directions. Each directional cosine is a unit vector, meaning it has a magnitude of 1. Therefore, if the directional cosines of a vector are 5 and 6, as in the question, the magnitude of the vector would be the square root of (5^2 + 6^2), which is 7.81. Similarly, if the directional cosines are 2 and 3, the magnitude of the vector will be the square root of (2^2 + 3^2), which is 3.61. Therefore, the magnitude of the vector would be neither 5 nor 6, so no such vector exists.
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given a j = 3 2 and b j = −3 4 , where j = −1 , find (use just pencil and paper, not software or calculators)
Value of the following expressions for a = 3 + 4j and b = 3 - 4j , where j = √−1 are as follow :
1. a + b = 6
2. a × b = 25
Calculation of the value for the required expression by substituting the given values are:
1. a + b
= ( 3 + 4 j ) + ( 3 - 4j )
Take like terms together we get,
= ( 3 + 3 ) + ( 4j - 4j )
= 6 + 0j
= 6
To get the value of second expression substitute the value of 'a' and 'b' we get :
2. a × b
= ( 3 + 4 j ) × ( 3 - 4j )
Apply the formula : ( x + y ) (x - y ) = x ² - y²
= ( 3 )² - ( 4j )²
= 9 - 16j²
Here j = √-1 ⇒ j² = -1
= 9 - 16 (-1 )
= 9 + 16
= 25
Therefore, the value of 1. a + b = 6 and 2. a × b = 25.
The above question is incomplete, the complete question is:
Given a = 3 + 4j and b = 3 - 4j , where j = √−1 , find (use just pencil and paper, not software or calculators) Find the value of the following :
1. a + b 2. a × b
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A tree that is 12 feet tall is growing at a rate of 2 1/2
feet each year. A tree that is 15 feet tall is growing at a rate of 1 foot each year.
Enter the number of years it will take the two trees to reach the same height.
HELP please
The number of years taken for the tree to reach the same height found using mathematical expression is 2 years.
What is an expression?
Mathematical statements are called expressions if they have at least two terms that are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.
Let x be the number of years taken to reach the same height.
Height of the first tree after x years,
12 + 2.5(x)
Height of the second tree after x years,
15 + 1(x)
Given that both heights are the same after x years.
So,
12 + 2.5x = 15 + x
1.5x = 3
x = 2
Therefore the number of years taken for the tree to reach the same height is 2 years.
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5g > 45 as a true and false statement
The true statement of the inequality is that g is greater than 9 and as a false statement, g is less than 9.
What are inequalities and their types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
Given, An inequality of 5g > 45.
Dividing both sides by 5 we have (5/5)g > 45/5.
g > 9.
Now, The true statement is 'g' is greater than 9 and as a false statement 'g' is less than 9.
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Given the series − 20/8 + 80/64 − 320/512 + 1280/4096 ⋯ Does this series converge or diverge?If the series converges, find the sum: ____________
The series is the converge series and the sum of the converges is 55/16
The term converge series in math is defined as a series is the sum of the terms of an infinite sequence of numbers
Here we have know that the series 20/8 + 80/64 − 320/512 + 1280/4096.
When we simplify this series the we get,
=> 20/8 + 80/64 − 320/512 + 1280/4096
=> 5/2 + 5/4 - 5/8 + 5/16
As per the following rule If ∑an is conditionally convergent and r is any real number then there is a rearrangement of ∑an whose value will be r.
Based on these we have to simplify then we get, here with unlike denominators find the Least Common Denominator (LCD)
=> LCD = 16
Multiplying numerators and denominators to get the LCD in all fraction denominators
Then we get,
=> 40/16 + 20/16 - 10/16 + 5/16
=> 55/16
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how to prove 431 divides 2^43 -1
By using Fermat's Little Theorem, we were able to prove through a series of mathematical steps that 431 divides 2⁴³⁻¹
In mathematics, proving that a number divides another number means finding out if it can evenly divide the other number with no remainder.
Here we need to prove that 431 divides 2⁴³⁻¹, we can use the mathematical property of modular arithmetic.
In modular arithmetic, we are interested in finding the remainder when a number is divided by another number.
In this case, we want to find the remainder when 2⁴³⁻¹ is divided by 431.
If the result is 0, then we can conclude that 431 divides 2^43 -1.
In order to do this, we can use Fermat's Little Theorem.
Since 431 is a prime number, we can use Fermat's Little Theorem to calculate
=> 2⁴³⁰ mod 431.
If the result is 1, then we know that
=> 2⁴³ mod 431
=> 2^(43 mod 430) * 2⁴³⁰ mod 431 = 2³ mod 431.
Finally, we can calculate 2³ mod 431 and if the result is 1, we can conclude that
=> 2⁴³⁻¹ mod 431 = 1.
This means that 2 is divisible by 431 with no remainder, so we can conclude that 431 divides 2⁴³⁻¹.
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Two classes of fifth graders are raising money for a field trip. The two classes bake cherry cherry pies that are all the same size to sell at the winter festival. Each cherry pie is cut in eight equal pieces. The fifth graders sell each piece of cherry pie for sixty cents. At the end of the bake each sale class reports how much cherry pie was sold. 4 2/8 5 1/2
A circular pie cut in 8 pieces and sold at a fractional form 4(2/8) and 5(1/2), yields a total sale of 78 pieces of pie.
What is a circle?
A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle has rotational symmetry around the centre.
A pie is made in a circular shape.
Number of pieces in one pie = 8 pieces
Number of pieces of pie sold by first class of fifth grades = 4(2/8)
Number of pieces of pie sold by second class of fifth grades = 5(1/2)
The total number of pieces sold is = Pieces sold by first class + Pieces sold by second class
To find the pieces sold by first class multiply the fractional form by number of pieces -
4(2/8) × 8
34/8 × 8
34
The first class sold 34 pieces of pie.
To find the pieces sold by second class multiply the fractional form by number of pieces -
5(1/2) × 8
11/2 × 8
11 × 4
44
The second class sold 44 pieces of pie.
The total number of pies sold is -
The total number of pieces sold is = 34 + 44
The total number of pieces sold is = 78
Therefore, 78 pies were sold.
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Please solve 1 SIMPLE question!
The value of tanФ = -1/4 in the III quadrant is -14⁰:
How to find the tan?A trigonometric ratio is a ratio between two sides of a right triangle. The tangent ratio is just one of these ratios. In this tutorial, you'll see how to find the tangent of a particular angle in a right triangle.
Tangent = opposite /Adjacent
Tan ∅ = -1/4
Tan ∅= -0.25
Taking the tangent inverse to get ∅
∅=Tan⁻0.25
∅=-14
This is to say that in the III quadrant, Tan ∅=-1/4 is -14⁰
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Anne is painting her house light blue. To make the color she wants, she must add 3 cans of white paint to every 2 cans of blue paint.
1. How many cans of white paint will she need to mix with 6 cans of blue?
2. Rate needed (white/blue)
3. What is the unit rate?
4. Interpretation of unit rate?
1. 9 cans of white paint will she need to mix with 6 cans of blue
2. The rate of white to blue is 9 : 6.
3. The unit rate is 3/2.
What is Ratio?Given:
Anne must add 3 cans of white paint to every 2 cans of blue paint.
So, the ratio of white to blue = 3:2
and, for 6 cans of blue paint the white paint needed
= 6 x 3/2
= 3x 3
= 9 cans
2. The rate of white to blue is 9 : 6.
3. The unit rate is 3/2.
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Based on these data, estimate the number of days that the dow decreased by more than 1% in these 61 days.
Number of the 61 days in 2009 the Dow decreased by more than 1% is 0.514.
What is normal distribution?A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.
Given:
61 days in 2009 the Dow decreased by more than 1%.
In 2009
z = (1- (-0.198))/2.331
z = 0.514
In 2010
z = (1-0.078)/0.821
z = 1.123
Thus, number of the 61 days in 2009 the Dow decreased by more than 1% is 0.514.
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Write sin t in terms of cos t using the Pythagorean Identity for:
The solution to sin(t) is sin(t) = -√[1 - cos²(t)]
How to determine the solution to sin(t)From the question, we have the following parameters that can be used in our computation:
The angle t is the quadrant IV
This means that
sin²(t) + cos²(t) = 1
Subtract cos²(t) from both sides of the equation
So, we have the following representation
sin²(t) = 1 - cos²(t)
Take the square root of both sides
sin(t) = ±√[1 - cos²(t)]
sin(t) is negative in quadrant IV
So, we have
sin(t) = -√[1 - cos²(t)]
Hence, the solution is sin(t) = -√[1 - cos²(t)]
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How many miles is equal to 13 kilometers?
Step-by-step explanation: 1 kilometer is equal to 0.621371 miles. To convert 13 kilometers to miles, you can multiply 13 by 0.621371.
13 kilometers * 0.621371 miles/kilometer = 8.042443 miles
So 13 kilometers is equal to 8.042443 miles.
if a1 = 4 and an =2an-1-1 then find the value of a4
The value of [tex]a_{4} = 37[/tex]
We need to find value of [tex]a_{4}[/tex].
What is Sequence?A sequence is a collection of elements that are arranged according to a specific pattern.
Arithmetic sequence is a sequence where each consequtive term has a common constant difference.
Now,
Given that value of [tex]a_{1} = 3[/tex]
[tex]a_{n} = 4 a_{n-1} -1[/tex]
Then to find value of [tex]a_{4}[/tex], we can find [tex]a_{2} , a_{3}[/tex]as;
[tex]a_{2} = -2(a_{1} ) - 1\\a_{2} = -2(4) - 1\\a_{2} = -9[/tex]
[tex]a_{3} = -2(a_{2} ) - 1\\a_{3} = -2(9) - 1\\a_{3} = -19[/tex]
Hence, the value of [tex]a_{4}[/tex] as;
[tex]a_{4} = -2(a_{3} ) - 1\\a_{4} = -2(-19) - 1\\a_{4} = 37[/tex]
Therefore, The value of [tex]a_{4}[/tex]= 37.
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Special right triangle
Using properties of right angle, The value of a will be 32 units.
What is the special right triangle rule?Particular Right Triangle Formula
The right triangle formula, which states that the square of a right-angled triangle's right triangle is equal to the product of its other two squares, is the fundamental Pythagoras theorem formula.
How long are the other two legs if a hypotenuse of a triangle with a 45°, 45°, and 90° angle is 32 units.
We are aware that Leg: Ankle: Length x = x: x: x2 is the formula for a triangle with a 45°, 45°, and 90° angle.
We can state that x2 = 32 because it is a known that the hypotenuse equals 32. As a result, x = 3 is determined.
After replacing this number of x = 3, the lengths of the empty sides may now be determined.
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Complete question -
Find values of a and b -
What is the solution to this system of equations?
Negative 5.9 x minus 3.7 y = negative 2.1. 5.9 x + 3.7 y = 2.1.
What is the solution to this system of equations?
Negative 5.9 x minus 3.7 y = negative 2.1. 5.9 x + 3.7 y = 2.1.
(0, 2.1)
(0, -2.1)
no solution
infinitely many solutions
The system of equations has an infinite number of solutions. The correct answer would be an option (D).
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.
The system of equations is given in the question, as follows:
-5.9x - 3.7y = -2.1 .....(i)
5.9x + 3.7y = 2.1 .....(ii)
As we know that when a solution set of infinite points exists, a system of linear equations has an infinite number of solutions.
Here the system of equations has an infinite number of solutions because both equations are identical.
Hence, the correct answer would be an option (D).
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Please someone help!
The area of parallelogram
ABCD is 24 square units.
Find x.
The value of x of the parallelogram is 7 units
How to find the side x of the parallelogram?The product of the base and height gives the total area of the parallelogram.
The area of a parallelogram can be calculated using the formula:
Area = base * height
A = b * h
where the base is the length of one side of the parallelogram, and height is the perpendicular distance between the two parallel sides.
Given that A = 24 square units, b = 8
A = b * h
28 = 8 * h
28 = 8h
h = 28/8
h = 3.5 units
sin 30 °= h/x (opposite / hypotenuse)
0.5 = 3.5/x
0.5x = 3.5
x = 3.5/0.5
x = 7 units
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please help in due in like 30 mins and I dont know how to work it out
The fraction by which the income decreased in August is given as follows:
1/5.
How to obtain the fraction?The total income is given by a proportional relationship of the cost per item and the number of items sold, hence:
y = kx.
In which:
y is the total income.k is the price per item.x is the number of items sold.For August, the parameters are given as follows:
k: 4k/3, as 1/3 + 1 = 4/3.x: 3x/5, as 1 - 2/5 = 3/5.Hence the August expression is given as follows:
y = 4k/3 x 3x/5
y = 12kx/15
y = 4kx/5.
Hence the decrease is given as follows:
1 - 4/5 = 5/5 - 4/5 = 1/5.
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3/1 = d/9
d=
Help please!
[tex] \frac{3}{1} = \frac{d}{9} \\ [/tex]
cross multiply...
[tex]d = 27[/tex]
a hypothesis is a a) prediction of results. b) tentative statement that something may be true. c) fact. d) all of these
A hypothesis is a) prediction of results and b) tentative statement that something maybe true.
A hypothesis is a statement or assumption made about a phenomenon or relationship, used as a starting point for further investigation. It is a tentative explanation for an observation, phenomenon or problem that can be tested through experimentation or additional data collection.
So a hypothesis can be true of false. A hypothesis is not certainly a fact too since it need to be tested.
If a hypothesis is proven to be true through experimentation and data analysis, it becomes a confirmed explanation or prediction. This means that the evidence supports the hypothesis and it can be used as a basis for further study or explanation. However, a true hypothesis is always subject to revision or rejection if new information or evidence becomes available.
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decrease 90 by 65%
pls do simple working out :)
Answer:31.5
Step-by-step explanation: Since we are decreasing 90 by 65%, we want to find 35% of 90. To do this, simply multiply 90 by 0.35 to get 31.5.
Trey drove 315 miles in 5 hours.
At the same rate, how many miles would he drive in 11 hours?
Within 5 hours, Trey covered 315 miles. He would need to travel 504 miles in 11 hours at the same speed.
How to calculate speed?Speed exist calculated as distance times speed. You need to be aware of the units for distance and time in order to calculate the units for speed. The units in this example will be metres per second (m/s), since the distance is measured in metres (m) and the time is measured in seconds (s). Speed (or rate, r) is a scalar number that represents the amount of time (Δt) spent traveling over the distance (d), as shown by the equation r = d/Δt.In order to compute speed, the distance traveled must be divided by the amount of time spent traveling.You must ascertain the unit rate. Divide the miles by the hours to arrive at that.
The unit rate should be found to be 63 miles per hour.
You need to double that by 8 (because it takes 8 hours),
Which is 504 miles.
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use the number 8, 15, and 23, to make two adition facts and two subtratio9n facts
Answer:
8-15
8-23
and
15+23
15+8
Step-by-step explanation:
Caron want to put up a glow in the dark contellation diplay in hi bedroom he need a pack of two ided tape which cot $12 the tar ticker cot $14 per pack and he want to buy 3 pack what i a good etimation of how much change Caron will get back if he give $75?
If Caron gives $75, he will get $21 as a change, when he buys a constellation display.
To estimate the amount of change Caron will get back, we need to calculate the total cost of the two-sided tape and the star stickers. The two-sided tape costs $12 and Caron wants to buy 3 packs of star stickers which cost $14 each, so the total cost of the star stickers is 3 * $14 = $42. The total cost of both the two-sided tape and the star stickers is $12 + $42 = $54.
If Caron gives $75, he will get $75 - $54 = $21 as change. This is a rough estimate of the amount of change Caron will get back, and it's always a good idea to have a little extra money in case the actual cost is slightly higher.
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find the value of m for which the simultaneous equations 3x my = 5 (m 2)x 5y = m a have infinitely many solutions b have no solution.
In the simultaneous equations the value of m is 3 and the equation have many solution.
The term called equation in math is known as a condition on a variable (or variables) such that two expressions in the variable have equal value.
Here we have know that the general solution is calculated as,
=> x = (m-5)/(m-3)
and
=> y = 2/(m-3)
When we equate the equations then we get,
=> (m - 5) / (m -3) = 2/(m - 3)
=> m - 5 = 2
=> m = 3
So this dependent system has infinitely many solutions.
When m=3 the equations become look like 3x+3y=5 and 5x+5y=3.
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What is the slope of the line passing through the points 12/7 and 16/7
Answer:
(12,7) and (16,7)
There is no slope as both points are on the same line. So the equation would be:
y=7
Step-by-step explanation:
in a chi-square test of goodness of fit, the p-value of the test is 3.2578x10-13 and the level of significance of the test α=0.05. what statistical decision could you make from this information?
In this particular case, the p-value is 3.2578x10⁻¹³, which is much smaller than the level of significance of 0.05.
A chi-square test of goodness of fit is a statistical test used to compare the observed data to a theoretical distribution.
In mathematical terms, the null hypothesis states that the observed data follows the theoretical distribution, and is represented by
H0= The observed data follows the theoretical distribution. The alternative hypothesis, which is the opposite of the null hypothesis, states that the observed data does not follow the theoretical distribution, and is represented by
HA= The observed data does not follow the theoretical distribution.
Based on the p-value of 3.2578x10⁻¹³ and the level of significance of 0.05, we can make the following statistical decision:
p-value < α (3.2578x10-13 < 0.05),
Therefore reject H0 and accept HA.
n other words, we can conclude that the observed data does not follow the theoretical distribution, and there is strong evidence to support this conclusion. The chi-square test of goodness of fit is a useful tool for making statistical decisions about the distribution of data.
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how many solutions does -x+y=-5,3x+5y=15 have
Answer: It could be 7 or 10 or 113
Step-by-step explanation: Just to make sure that you don't think it's only for 5. If I just get something, that something is equal to itself, which is just going to be true no matter what x you pick.
can someone please help
The length of the other side of the triangle is 30 cm. Thus, option b is correct.
What is Pythagoras theorem?The hypotenuse's square is equal to the sum of the squares of the other two sides if a triangle has a straight angle (90 degrees), according to the Pythagoras theorem. Keep in mind that BC² = AB² + AC² in the triangle ABC signifies this. Base AB, height AC, and hypotenuse BC are all used in this equation. The longest side of a right-angled triangle is its hypotenuse, it should be emphasized.
Here given that hypotenuse = 32 and one side = 16
Using Pythagoras theorem
32² = 16² + x²
x = [tex]\sqrt{32^2 - 16^2}[/tex]
x = 30
Hence, the length of the other side of the triangle is 30 cm.
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The heptagon at the right ha been diected into five triangle with angle labeled a hown ue the five triangle to prove that the um of the interior angle of any heptagon i alway 900°
If the Heptagon has been dissected into five triangle , the the proof that sum of the interior angle of any heptagon is always 900° is shown below .
To prove that the sum of the interior angles of a heptagon is 900°, we can use the five triangles that have been dissected from the Heptagon, as follows: let the 5 triangles be A , B , C , D and E .
we know that the sum of interior angle of triangle is always 180° ;
Triangle A: The sum of the angles of triangle A is 180°.
Triangle B: The sum of the angles of triangle B is 180°.
Triangle C: The sum of the angles of triangle C is 180°.
Triangle D: The sum of the angles of triangle D is 180°.
Triangle E: The sum of the angles of triangle E is 180°.
So , the sum of all the angles in the five triangles is 180°×5 = 900°, which is also equal to the sum of the interior angles of the heptagon.
Therefore , we can conclude that the sum of the interior angles of any heptagon is always 900°.
The given question is incomplete , the complete question is
The Heptagon has been dissected into five triangle with angle labeled , use the five triangle to prove that the sum of the interior angle of any heptagon is always 900° .
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Start at 2⅓, add 2¼. I need the first 4 multiples
The multiple of 2⅓, add 2¼. is 2 7/12.
What are multiple?A multiple in mathematics is created by multiplying any number by an integer. In other words, if b = na for some integer n, known as the multiplier, it can be said that b is a multiple of a given two numbers, a and b. This is equivalent to stating that b/a is an integer if an is not zero.
In other words, the product of 4 and any natural integer is the multiple of 4. For instance, the result of multiplying 4 by 4 is 16, making 16 a multiple of 4. Examples of multiples of four are 4, 12, 20, 24, and so forth.
Start at 2⅓, add 2¼. I need the first 4 multiples?
→2 1/3+2 1/4
⇒2 3+4/12
⇒2 7/12
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Which substitution should best be used in solving the recurrence T(n) = 5T(4n/3) + 2?a) n = (3/4)^kb) n = (4/3)^kc) n = (15/4)^kd ) n = (20/3)^ke) n = (3/20)^kf ) n = (3/15)^kg ) n = 2^kh ) n = 3^ki ) n = 4^kj ) n = 5^k
The substitution that should best be used in solving the recurrence T(n) = 5T(4n/3) + 2 is "n = (4/3)^k".
The idea behind choosing a substitution for a recurrence relation is to simplify the relation and make it easier to solve. In this case, we want to simplify T(4n/3) to something involving T(n). By setting n = (4/3)^k, we can rewrite T(4n/3) as T((4/3)^(k+1)).
This substitution allows us to express T(4n/3) in terms of T(n) and simplify the relation.
T(n) = 5T(4n/3) + 2
becomes
T(n) = 5T((4/3)^(k+1)) + 2
We can then use this substitution to solve the relation using, for example, the Master Theorem.
In conclusion, the substitution "n = (4/3)^k" is the best option because it simplifies the relation and makes it easier to solve.
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