Answer: [tex]r=-\frac{2}{3} \pm \frac{i\sqrt{23}}{3}[/tex]
Step-by-step explanation:
[tex]3r^2 +4r+9=0\\\\3\left(r^2 +\frac{4}{3} r \right)+9=0\\\\3\left(\left(r+\frac{2}{3} \right)^2 -\frac{4}{9} \right)+9=0\\\\3\left(r+\frac{2}{3} \right)^2 -\frac{4}{3}+9=0\\\\3\left(r+\frac{2}{3} \right)^2 =-\frac{23}{3}\\\\\left(r+\frac{2}{3} \right)^2 =-\frac{23}{9}\\\\r+\frac{2}{3}=\pm \frac{i\sqrt{23}}{3}\\\\r=-\frac{2}{3} \pm \frac{i\sqrt{23}}{3}[/tex]
15 milliliters of pure drug are in 500 mL of solution. What is the percent strength of the solution?
When 15 milliliters of pure drug are in 500 mL of solution, the percent strength of the solution is
How to find the percent strength of the solution'information from the problem
15 milliliters of pure drug are in 500 mL of solution
the percent strength of the solution = ?
The question is a percentage problem, percentage deals with a fraction hundred and used as follows
= volume of pure drug / volume of solution * 100
= 15 milliliters / 500 milliliters * 100
= 0.03
= 3%
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The population of a specific species of nocturnal mammal is decreasing at a rate of 3.5%/year. The graph models the number of mammals x years after they were originally counted.
Identify and interpret the key features of the exponential function modeled in terms of this situation.
Select each correct answer.
The y-intercept represents the number of mammals when they were originally counted.
The line y = 0 is an asymptote of the graph.
The y-intercept.is 75.
The y-intercept is 120.
The asymptote indicates that the number of mammals counted when the study began was 120.
The asymptote indicates that as years pass, the number of mammals will approach 0.
The line x = 0 is an asymptote of the graph.
Answer:
The y-intercept represents the number of mammals when they were originally counted.
The line y = 0 is an asymptote of the graph.
The y-intercept is 120.
The asymptote indicates that as years pass, the number of mammals will approach 0.
Two angles in triangle PQR are congruent, ∠P and ∠Q; ∠R measures 26.35°. What is the measure of ∠P?
The required measure of ∠P would be 76.825° in the given triangle PQR.
Given that two angles in triangle PQR are congruent, ∠P and ∠Q;
∠R measures 26.35°.
Let the measure of ∠P would be x
∠P = x = ∠Q
We know that the sum of interior angles is always 180 degrees in the triangle.
⇒ ∠P + ∠Q + ∠R = 180°
⇒ x + x + 26.35° = 180°
⇒ 2x + 26.35° = 180°
⇒ 2x = 180° - 26.35°
⇒ 2x = 153.65
⇒ x = 153.65/2
⇒ x = 76.825°.
Therefore, the required measure of ∠P would be 76.825° in the given triangle PQR.
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18. A random sample of n = 16 scores is obtained from a
population with a mean of μ = 45. After a treatment
is administered to the individuals in the sample, the
sample mean is found to be M = 49.2.
a. Assuming that the sample standard deviation is
s = 8, computer and the estimated Cohen's d to
measure the size of the treatment effect.
b. Assuming that the sample standard deviation is
s = 20, computer and the estimated Cohen's d to
measure the size of the treatment effect.
c. Comparing your answers from parts a and b, how
does the variability of the scores in the sample
influence the measures of effect size?
Through reporting statistical results, we found that [tex]r^{2}[/tex] is 0.22 and Cohen's [tex]d[/tex] is 0.52 when s = 8, and [tex]r^{2}[/tex] is 0.04 and Cohen's [tex]d[/tex] is 0.21 when s = 20. It is also observed that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.
It is given to us that -
A random sample of n = 16 scores
Population with a mean of μ = 45
Sample mean is found to be M = 49.2
In Reporting Statistical results, there are two different effect sizes, namely eta-squared [tex]r^{2}[/tex] and Cohen's [tex]d[/tex].
Eta-square, [tex]r^{2} = \frac{t^{2} }{t^{2}+df }[/tex]
where, [tex]df = n-1[/tex]
[tex]SEM = \frac{s}{\sqrt{n} }[/tex]
and, [tex]t=\frac{x_{2} -x_{1} }{SEM}[/tex]
Cohen's [tex]d=\frac{x_{2} -x_{1} }{s}[/tex]
a) Given that the sample standard deviation is s=8
We have n = 16
So, [tex]df = n-1 = 16-1 = 15[/tex]
[tex]SEM = \frac{s}{\sqrt{n} } = \frac{8}{\sqrt{16} } = \frac{8}{4} = 2[/tex]
We also have μ = 45 and M = 49.2. This implies
[tex]t=\frac{x_{2} -x_{1} }{SEM} = \frac{49.2-45}{2} = \frac{4.2}{2} =2.1[/tex]
Now,
[tex]r^{2} = \frac{t^{2} }{t^{2}+df } = \frac{(2.1)^{2} }{(2.1)^{2}+ 15 }= \frac{4.41}{4.41+15} \\= \frac{4.41}{19.41} = 0.22[/tex]
Cohen's [tex]d=\frac{x_{2} -x_{1} }{s} = \frac{49.2-45}{8} = \frac{4.2}{8} = 0.52[/tex]
b) Given that the sample standard deviation is s=20
We have n = 16
So, [tex]df = n-1 = 16-1 = 15[/tex]
[tex]SEM = \frac{s}{\sqrt{n} } = \frac{20}{\sqrt{16} } = \frac{20}{4} = 5[/tex]
We also have μ = 45 and M = 49.2. This implies
[tex]t=\frac{x_{2} -x_{1} }{SEM} = \frac{49.2-45}{5} = \frac{4.2}{5} =0.84[/tex]
Now,
[tex]r^{2} = \frac{t^{2} }{t^{2}+df } = \frac{(0.84)^{2} }{(0.84)^{2}+ 15 }= \frac{0.706}{0.706+15} \\= \frac{0.706}{15.706} = 0.04[/tex]
Cohen's [tex]d=\frac{x_{2} -x_{1} }{s} = \frac{49.2-45}{20} = \frac{4.2}{20} = 0.21[/tex]
c) Comparing a and b, we see that the variability of the scores in the sample, [tex]r^{2}[/tex] is 0.22 when s = 8 and [tex]r^{2}[/tex] is 0.04 when s = 20. Similarly, Cohen's [tex]d[/tex] is 0.52 when s = 8 and [tex]d[/tex] is 0.21 when s = 20.
Thus, we can see that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.
Therefore, through reporting statistical results, we found that [tex]r^{2}[/tex] is 0.22 and Cohen's [tex]d[/tex] is 0.52 when s = 8, and [tex]r^{2}[/tex] is 0.04 and Cohen's [tex]d[/tex] is 0.21 when s = 20. It is also observed that when the variability of the scores in the sample, [tex]r^{2}[/tex] decreases, there is simultaneous decrease in the measures of effect size, Cohen's [tex]d[/tex] as well.
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point r is located at (-5,-2), Point T is located at (2,5) the ratio of RS/ST is 2/5 Plot point S on RT to make the ratio true
The point S of line segment RT such that RS / ST = 2 / 5 is equal to (- 3, 0).
How to determine the coordinates of the point within a line point
Herein we find a line segment whose endpoints are known (R(x, y) = (- 5, - 2), T(x, y) = (2, 5)) and in which we must determine the coordinates of a point S within line segment RT such that the partition ratio is observed:
RS / ST = 2 / 5
[tex]\overrightarrow{RS} = \frac{2}{5} \cdot \overrightarrow {ST}[/tex]
S(x, y) - R(x, y) = (2 / 5) · [T(x, y) - S(x, y)]
(7 / 5) · S(x, y) = (2 / 5) · T(x, y) + R(x, y)
S(x, y) = (2 / 7) · T(x, y) + (5 / 7) · R(x, y)
Now we determine the location of point S:
S(x, y) = (2 / 7) · (2, 5) + (5 / 7) · (- 5, - 2)
S(x, y) = (4 / 7, 10 / 7) + (- 25 / 7, - 10 / 7)
S(x, y) = (- 21 / 7, 0)
S(x, y) = (- 3, 0)
The location of point S is (- 3, 0). A representation of the geometric system is shown in the image attached below.
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when he shoots a free throw he makes a basket 20% of the time Jai shoots 120 free throws in Use benchmarkercents of 1% and 10% to help you determine the answer
Benchmark percentage of 1% = 9.6 & 10% = 24.
What is Free Throw Percentage?
Free throw percentage (FT%) puts a player’s successful free throws in perspective to their total attempts.
In basketball, a free throw (or foul shot) is awarded to a player who has been fouled by the other team. The number of free throws depends on where on the court the player was while being fouled.
To find 1% of a number, we can move the decimal point of the number two places to the left.
Therefore, 1% of 120 is 1.2
To find 10% of a number, we can move the decimal point of the number one place to the left.
Therefore, 10% of 120 is 12
Since 10% of 120 is 12 and
20% = 2 x 10%
then 20% of 120 = 2 x 10% of 120
= 2 x 12 = 24
Since 1% of 120 is 1.2 and
8% = 8 x 1%
then 8% of 120 = 8 x 1% of 120
= 8 x 1.2 = 9.6
Hence the answer is, Benchmark percentage of 1% = 9.6 & 10% = 24.
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Drag drop the core in order from least to greatest on the number line -10 1 -31/3 0 8 1/4 9/8 -9/8
An arrangement of the order the numbers would appear on a number line from least to greatest is: -31/3, -10, -9/8, 0, 1, 1/4, 9/8, 8.
What is a number line?In Mathematics, a number line can be defined as a type of graph with a graduated straight line which is composed of both positive and negative numbers that are placed at equal intervals along its length.
Generally speaking, a number line typically increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).
In order to arrange the given numbers in order from least to greatest, we would convert them into a decimal number as follows:
-31/3 = -10.33 -10 = -10.0-9/8 = -1.12501/4 = 0.251 = 1.09/8 = 1.1258 = 8.0Read more on number line here: brainly.com/question/28032137
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Which equation defines the distance, d, between points (-3,3) and
(a,b)? Select all that apply.
d=√(a-3)² + (b+3)²
d √(-3-a)² + (3 — b)²
d=√(a+b)² + (−3 − 3)²
d=√(a+3)² + (b − 3)²
-
The distance is d = √(-3 - a)² + (3 - b)²
What is Distance between two points ?
Distance between two points is the length of the line segment that connects the two points in a plane.
The formula is d =√(x₁ - x₂)² + (y₁ - y₂)²
Points are (-3, 3) and (a, b)
We know, the distance formula is
d =√(x₁ - x₂)² + (y₁ - y₂)²
Let, (x₁, y₁) = (-3, 3)
(x₂, y₂) = (a, b)
Now, plug in the values in given formula
d = √(-3 - a)² + (3 - b)²
Hence, the distance is d = √(-3 - a)² + (3 - b)²
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Jack is 2 years older than Bob. What was the difference between their ages one year ago?
Need Help ASAP!
The difference between their ages one year ago was 2 years.
What is difference?Difference in maths, the result of one of the important mathematical operations, which is obtained by subtracting two numbers.
Given that, Jack is 2 years older than Bob,
Let Bob's age be x then, Jack's age will be (x+2)
Their ages before 1 year was =
Bob's = x-1
Jack's = (x+1)
Difference = x + 1 - x + 1 = 2
Hence, The difference between their ages one year ago was 2 years.
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Michelle is selling blueberries and raspberries at a farmers' market. She makes $1.75 for each pint (pt) of blueberries she sells and $2.40 for each pint of raspberries. At the end of the day, she has made exactly $116.65 by selling a total of 57 pt of fruit.
How many pints of blueberries did she sell?
The number of the blueberries is 31.
How many pints of blueberries did she sell?In this case, the only way that we can be able to obtain the number of the pints of the blueberries that she sold is by the use of a simultaneous equation.
Let the number of the blueberries be x and the number of the raspberries be y.
We have
x + y = 57 ------ (1)
1.75x + 2.40y = 116.65 ------- (2)
x = 57 - y ------ (3)
Then we have;
1.75(57 - y) + 2.40y = 116.65
99.75 - 1.75y + 2.40y = 116.65
Collecting like terms;
- 1.75y + 2.40y = 116.65 - 99.75
0.65y = 16.9
y = 16.9/0.65
y = 26
To obtain the number of blueberries
x + 26 = 57
x = 57 - 26
x = 31
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Translate the phrase into a math expression. Twelve more than the quotient of six divided by three. Responses (6÷3)+12 eft parenthesis 6 divided by 3 right parenthesis plus 12 6÷(3+12) 6 divided by left parenthesis 3 plus 12 right parenthesis (3÷6)+12 left parenthesis 3 divided by 6 right parenthesis plus 12 3÷(6+12) 3 divided by left parenthesis 6 plus 12 right parenthesis
The given phrase on translation to math expression is (6 ÷ 3) + 12 , the correct option is (a) .
In the question ,
an mathematical phrase is given , that is "Twelve more than the quotient of six divided by three" .
we have to translate it into mathematical expression ,
So , the term quotient of six divided by three is written as 6 ÷ 3 .
and phrase " more " is represented by " + " ,
Hence the given phrase, "Twelve more than the quotient of six divided by three" is (6 ÷ 3) [tex]+[/tex] 12 .
Therefore , The given phrase on translation to math expression is (6 ÷ 3) + 12 , the correct option is (a) .
The given question is incomplete , the complete question is
Translate the phrase into a math expression , "Twelve more than the quotient of six divided by three" .
(a) (6÷3)+12
(b) 6÷(3+12)
(c) (3÷6)+12
(d) 3÷(6+12)
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NO LINKS!! Please help me with this problem. Find a formula that expresses the fact that an arbitrary point P(x, y) is on the perpendicular bisector l of segment AB.
The point P is on the perpendicular line to AB that passes through its midpoint.
We know perpendicular lines have opposite-reciprocal slopes.
So the line we are looking for has a slope of 7/5.
Use the point-slope equation to find the line:
y - y₁ = m(x - x₁)y - (-1) = 7/5(x - 2)y + 1 = 7/5(x - 2) Point- slope formy = 7/5x - 19/5 Slope- intercept form5y = 7x - 197x - 5y = 19 Standard formChoose any form above of the same line.
Answer:
[tex]\textsf{Slope-intercept form}: \quad y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]
[tex]\textsf{Standard form}: \quad 7x-5y=19[/tex]
Step-by-step explanation:
A perpendicular bisector is a line that intersects another line segment at 90°, dividing it into two equal parts.
To find the perpendicular bisector of segment AB, find the slope of AB and the midpoint of AB.
Define the points:
Let (x₁, y₁) = A(-5, 4)Let (x₂, y₂) = B(9, -6)Slope of AB
[tex]\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-6-4}{9-(-5)}=\dfrac{-10}{14}=-\dfrac{5}{7}[/tex]
Midpoint of AB
[tex]\textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)=\left(\dfrac{9+(-5)}{2},\dfrac{-6+4}{2}\right)=(2,-1)[/tex]
If two lines are perpendicular to each other, their slopes are negative reciprocals.
Therefore, the slope of the line that is perpendicular to line segment AB is ⁷/₅.
Substitute the found perpendicular slope and the midpoint of AB into the point-slope formula to create an equation for the line that is the perpendicular bisector of line segment AB:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-(-1)=\dfrac{7}{5}(x-2)[/tex]
[tex]\implies y+1=\dfrac{7}{5}x-\dfrac{14}{5}[/tex]
[tex]\implies y=\dfrac{7}{5}x-\dfrac{14}{5}-1[/tex]
[tex]\implies y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]
Therefore, the formula that expresses the fact that an arbitrary point P(x, y) is on the perpendicular bisector of segment AB is:
[tex]\textsf{Slope-intercept form}: \quad y=\dfrac{7}{5}x-\dfrac{19}{5}[/tex]
[tex]\textsf{Standard form}: \quad 7x-5y=19[/tex]
2. Write the correct equation that you would use to solve for side x.
Z
X
520
y = 13cm
Z
Y
The correct equation that could be used to solve for side x is x = sin52° × (Hypotenuse)
Trigonometry: Determining the correct equation to solve for side xFrom the question, we are to determine the correct equation that could be used to solve for side x
From the given diagram, we observe that
Side x is the Opposite
Side z is the Adjacent
Side y is the Hypotenuse
Using SOH CAH TOA, we can write that
sin (angle) = Opposite / Hypotenuse
From the diagram,
Given angle = 52°
Hypotenuse = y = 13
Thus,
sin 52° = x/13
x = sin 52° × 13
OR
x = sin52° × (Hypotenuse)
Hence, the equation is x = sin52° × (Hypotenuse)
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A scientist invents a car that can travel for many hours without stopping for fuel. The
car travels around a track at 54 miles an hour for 24 hours. At the end of the 24
hours, how far has the car traveled?
o 1,286 miles
o 1, 296 miles
• 78 miles
• 1, 080 miles
Answer:
1,296 miles
Step-by-step explanation:
to answer this question, we have to apply the next formula:
Distance = Speed rate x time
speed rate = 54 miles per hour
time = 24 hours
Replacing with the values given and solving for D (distance)
D = 54 mph x 24 h = 1,296 m
The car traveled 1,296 miles
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Given f(x) = 2x^2 - 4x - 4, find the equation of the tangent line of f at the point
where x = -3.
The equation of the tangent line of f at the point x = -3 will be y = -16x-48.
According to the question,
We have the following information:
f(x) = [tex]2x^{2} -4x-4[/tex]
Now, we will first find the derivation of this function with respect to x:
Let's take its derivation to be f'(x).
f'(x) = 4x-4
Now, finding the slope of the equation when x = -3:
f'(-3) = 4*(-3)-4
f'(-3) = -12-4
f(-3) = -16
Now, we know that following formula is used to find the equation of a line:
(y-y') = m(x-x')
y-0 = -16{x-(-3)}
y = -16(x+3)
y = -16x-48
Hence, the equation of the tangent line of f at the point x = -3 will be y = -16x-48.
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A pole that is 2.5 m tall casts a shadow that is 1.65 m long. At the same time, a nearby building casts a shadow that is 50.25 m long. How tall is the building?
Round your answer to the nearest meter.
The building is 76.545 m tall.
Given:
A pole that is 2.5 m tall casts a shadow that is 1.65 m long. At the same time, a nearby building casts a shadow that is 50.25 m long.
Let x be the height of the building.
2.5/1.65 = x/50.25
2.5*50.25 = x * 1.65
25/10*5025/100 = x*165/100
25*5025/10 = 165x
25*5025 = 1650x
126300 = 1650x
x = 126300/1650
x = 76.545 m.
Therefore the building is 76.545 m tall.
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if the ( n+4)th terms of an A.P is 4n + 17 then find a. S10 b. An +1 c. Sn d. An
The measures of the arithmetic sequence are given as follows:
a. [tex]S_{10} = 350[/tex]
b. [tex]A_{n + 1} = 17 + 4n[/tex]
c. [tex]S_n = \frac{n(30 + 4n)}{2}[/tex]
d. [tex]A_n = 17 + 4(n - 1)[/tex]
What is an arithmetic sequence?An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.
The nth term of an arithmetic sequence is given by the rule shown below:
[tex]a_n = a_1 + (n - 1)d[/tex]
In which [tex]a_1[/tex] is the first term of the sequence.
The sum of the first n terms is given by the rule shown below:
[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]
The equation given for this problem is:
[tex]a_{n + 4} = 4n + 17[/tex]
Hence the sequence can be written as follows:
17, 21, 25, 29, 33.
Then the first term and the common ratio are given as follows:
[tex]a_1 = 17, d = 4[/tex]
Then the nth term is of:
[tex]A_n = 17 + 4(n - 1)[/tex]
The (n + 1)th term is of:
[tex]A_{n + 1} = 17 + 4(n + 1 - 1)[/tex]
[tex]A_{n + 1} = 17 + 4n[/tex]
The sum of the first n terms is of:
[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]
[tex]S_n = \frac{n(17 + 17 + 4(n - 1))}{2}[/tex]
[tex]S_n = \frac{n(30 + 4n)}{2}[/tex]
The sum of the first ten terms is of:
[tex]S_{10} = \frac{10(30 + 4(10))}{2} = 350[/tex]
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select all coordinate pairs that could represent a solution to 3x - 2y = 12
The coordinate pairs that could represent the equation 3x - 2y = 12 are
( x₁ , y₁ ) = ( 2 , -3 )
( x₂ , y₂ ) = ( 4 , 0 )
( x₃ , y₃ ) = ( 0 , -6 )
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be A
3x - 2y = 12
Now , Substituting the values of x and y in the above equation , we get
a)
The coordinate pair ( 2 , -3 )
So , the equation becomes
3 ( 2 ) - 2 ( -3 ) = 12
6 + 6 = 12
12 = 12
Therefore , the coordinate pair satisfies the given equation
b)
The coordinate pair ( 4 , 0 )
So , the equation becomes
3 ( 4 ) - 2 ( 0 ) = 12
12 = 12
Therefore , the coordinate pair satisfies the given equation
c)
The coordinate pair ( 0 , -6 )
So , the equation becomes
3 ( 0 ) - 2 ( -6 ) = 12
12 = 12
Therefore , the coordinate pair satisfies the given equation
Hence , the coordinate pairs that satisfies the the equation 3x - 2y = 12 are
( 2 , -3 ) , ( 4 , 0 ) and ( 0 , -6 )
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Describe the features of the function that can be easily seen when a quadratic function is given
in the form: y=ax^2+Bx+C and how they can be identified from the equation. How can this
form be used to find the other features of the graph?
Please help
When a quadratic equation is given in the form y = ax^2 + Bx + C the easily seen features are
the opening of the curve in the axis - open upwardthe x - coordinate of the vertex, v(h, k) h = -b/2a the axis of the parabola - y axisthe discriminant given by (b² - 4ac)How this form can be used to find other featuresEquation of a parabola is the equation used to trace the path of a parabola. this equation is a quadratic equation
The formula for the roots of the quadratic equation is derived from the equation of the form y = ax^2 + Bx + C to be
-b ± √{(b² - 4ac)2a)
the y intercepts is gotten by equating x = 0
The axis of symmetry refers to the axis where the parabolic curve can be bisected. this is in the y axis
"a" shows if the curve opens upwards or downwards.
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Determine the vertex of the graph of the following parabola. f(x)=−(x−2)2−3
The vertex of the Parabola is (2, -3)
What is a vertex?The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the coefficient of the x2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “ U ”-shape. If the coefficient of the x2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “ U ”-shape.
From the function f(x) =[tex]-(x-2)^{2}[/tex] - 3
The equation of parabola in vertex form is
y=[tex]a(x-h)^{2}[/tex]+k
where h and k are the coordinates of the vertex
comparing the two equations
h = 2, k = -3
The coordinates of the vertex is (2, -3)
In conclusion, the vertex is (2, -3).
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Customer: "If I purchase this product for $79.99 and two accessories for $9.99 and $7.00, how much would I owe after the 8.75% tax is applied?"
Employee: "Your total would be __________."
If a=-5xa=−5x and b=3x-4ib=3x−4i, then find the value of the a^{3}ba
3
b in fully simplified form.
The value of expression a³b will be;
⇒ - 75x⁴ + 100x³i
What is substitution method?
To find the value of any one of the variables from one equation in terms of the other variable is called the substitution method.
Given that;
The expression is,
⇒ a³b
The values are;
a = - 5x
b = 3x - 4i
Now,
Substitute the value of a and b, we get;
The expression is,
⇒ a³b
⇒ (-5x)³ × (3x - 4i)
⇒ - 25x³ (3x - 4i)
⇒ - 75x⁴ + 100x³i
Thus, The value of expression a³b will be;
⇒ - 75x⁴ + 100x³i
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Find the slope of the graph using points 1,2 and 5,10
Answer:
2
Step-by-step explanation:
The slope of a line is defined by the formula:
[tex]m=\dfrac{\Delta \, y}{\Delta \, x} = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
In this problem, we are given two points in the form [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex].
So, we can define the x's and y's as:
[tex]x_1 = 1[/tex], [tex]y_1 = 2[/tex], [tex]x_2 = 5[/tex], [tex]y_2 = 10[/tex].
Hence, the slope of the line can be solved for.
[tex]m = \dfrac{10-2}{5-1}[/tex]
[tex]m = \dfrac{8}{4}[/tex]
[tex]m = 2[/tex]
The slope of the line with points (1, 2) and (5, 10) is 2.
A chef uses ⅔ cup of flour for every ⅛ cup of water in a dough recipe. If the chef follows the recipe proportions, which statement is true?
If the chef follows the recipe proportions, then the true statement is;
A: The chef uses 3/16 cup of flour for every cup of water because 1/8 ÷ 2/3 = 3/16
How to solve fraction word problems?
Fractions can be presented in the form of word problems just like algebraic word problems.
Now, we are told that the chef uses 2/3 cup of flour for every 1/8 cup of water in a dough recipe.
This means that;
2/3 cup of flour = 1/8 cup of water
Thus;
1 cup of flour = (1 * 1/8)/(2/3) = 3/16 cups of flour for every cup of water
Looking at the given options, we can conclude that the only correct one based on our answer is Option A.
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Please help me I am a poor person and I’m not able for a reason but please
Answer:
d /c
Step-by-step explanation:
if its rotated at an 270 angle then its d or c cause they are the same
Sample space(3,4,5,6,7,8,9,10,11,12,13,14) event F(6,7,8,9,10) event G(10,11,12,13) outcomes are equally likely find P(ForG)
The probability of the sets P(F or G) is; P(F or G) = 0.67
How to Interpret Union of sets?We are given the following;
Sample Space; S = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
Set F = {6, 7, 8, 9, 10}
Set G = {10, 11, 12, 13}
Now, F or G, simple means; F ∪ G
Thus;
F or G = {6, 7, 8, 9, 10, 11, 12, 13}
Number of terms is (F or G) = 8
Number of terms is Sample space = 12
Thus;
P(F or G) = 8/12 = 2/3 = 0.67
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f(x)= 0.6x5-2x4+8x Describe the end behaviour of the polynomial function
Step-by-step explanation:
if I understand your typing correctly, and there is nothing missing, we have
f(x) = 0.6x⁵ - 2x⁴ + 8x
the "end behavior" means the general tendency of the result values for very large or very low values of x (going to +infinity and -infinity).
the higher (or lower in the negative direction) x gets, the more the highest exponent will dominate the result values.
it does not matter, that it has a diminishing factor (or coefficient) like 0.6.
the much stronger progression of x⁵ vs. smaller exponents like x⁴ or x will easily compensate for that with sufficiently large x.
so, ultimately, the term with the highest exponent (in our case 0.6x⁵) defines the end behavior.
with x going to +infinity, so does the function result (+infinity).
with x going to -infinity, so does the function result (-infinity, because an odd exponent number like 5 will maintain the sign of the argument).
x³ ÷ 3 = y; use x = 3, and y = 1
Step-by-step explanation:
If x = 3 :
x³ ÷ 3 = y
3³ ÷ 3 = y
27 ÷ 3 = y
y = 27 ÷ 3 = 9
What are the coordinates of point (1, 5) after dilating by 1/3 about (4,2)?
(2, 1)
(-1,-5)
(3, 2)
(3, 3)
The coordinates of point (1,5) after dilation is (3,3). Therefore, 4th option is correct.
It is given to us that -
The coordinates of the original point is (1,5)
=> [tex](x_{o},y_{o} )=(1,5)[/tex] ---- (1)
The dilating factor is 1/3
=> [tex]s=\frac{1}{3}[/tex] (say) ---- (2)
And, the center of dilation is at the point (4,2)
=> [tex](x_{cod},y_{cod})=(4,2)[/tex] ---- (3)
We have to find out the coordinates of point (1,5) after dilation.
Using the formula for dilation coordinates from original to image, we have
[tex][(x_{cod}+s(x_{o}-x_{cod}),y_{cod}+s(y_{o}-y_{cod})]\\=[4+\frac{1}{3}(1-4),2+\frac{1}{3}(5-2)]\\=[(4-1),(2+1)]\\=(3,3)[/tex]
Thus, the coordinates of point (1,5) after dilation is (3,3). Therefore, 4th option is correct.
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Find sin x/2, cos x/2, tan x/2. from the given information. tan x = [tex]\sqrt{2}[/tex]. 0° < x < 90°
The trigonometric measures for the given angle are as follows:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{6}}[/tex][tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 + \sqrt{3}}{6}}[/tex][tex]\tan{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{3 + \sqrt{3}}}[/tex]How to obtain the trigonometric measures?We are given the measure of the sine for the angle and we need to find the three measures, sine, cosine and tangent for half the angle.
These measures are dependent on the cosine of the function, hence we apply the definition of the tangent as follows:
[tex]\tan{x} = \frac{\sin{x}}{\cos{x}}[/tex]
Hence:
[tex]\sqrt{2} = \frac{\sin{x}}{\cos{x}}[/tex]
[tex]\sin{x} = \sqrt{2}\cos{x}[/tex]
The exact value of the cosine can be found applying the identity as follows:
sin²(x) + cos²(x) = 1.
Then, from the equation of the sine as a function of the cosine from the tangent relation, we have that:
[tex](\sqrt{2}\cos{x})^2 + \cos^2{x} = 1[/tex]
[tex]2^\cos^2{x} + \cos^2{x} = 1[/tex]
[tex]\cos^2{x} = \frac{1}{3}[/tex]
[tex]\cos{x} = \pm \sqrt{\frac{1}{3}}[/tex]
The angle is of the first quadrant, as 0° < x < 90°, hence the cosine is positive, thus:
[tex]\cos{x} = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}[/tex]
The identity that gives the sine for half the angle is:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \cos{x}}{2}}[/tex]
Hence:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \frac{\sqrt{3}}{3}}{2}}[/tex]
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{6}}[/tex]
The identity for the cosine is almost the same, just there is a plus instead of a minus, hence:
[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 + \cos{x}}{2}}[/tex]
[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 + \frac{\sqrt{3}}{3}}{2}}[/tex]
[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 + \sqrt{3}}{6}}[/tex]
The tangent is the sine divided by the cosine, hence the inserting the entire division and the same square root and simplify the common denominator, thus:
[tex]\tan{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{3 + \sqrt{3}}}[/tex]
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