Answer:
x = 1
Step-by-step explanation:
Given exponential equation:
[tex]3^{2x}-9=0[/tex]
Add 9 to both sides of the equation:
[tex]\implies 3^{2x}-9+9=0+9[/tex]
[tex]\implies 3^{2x}=9[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{bc}=(a^b)^c:[/tex]
[tex]\implies \left(3^2\right)^{x}=9[/tex]
[tex]\implies9^{x}=9[/tex]
[tex]\textsf{Apply exponent rule} \quad a=a^1:[/tex]
[tex]\implies9^{x}=9^1[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{f(x)}=a^{g(x)} \implies f(x)=g(x):[/tex]
[tex]\implies x=1[/tex]
The values in the table represent a linear function. x 1 2 3 4 f(x) 1 4 7 10 What is the function rule? Write the rule in function notation, using the slope-intercept form of a line. Enter your answer in the box.
The rule in expressed in function notation and slope-intercept form of a line is f(x) = 3x -2
How to express the rule in function notation using the slope-intercept form of a line?The equation of line is an algebraic form of representing the set of points, which together form a line in a coordinate system
The equation of a line in slope-intercept form is y = mx + c.
Where m is the slope and c is the intercept
In order to obtain the equation of the line, we are going to pick any two convenient values of x and y from the table. Let's pick the first and last point. Thus:
point1: x₁ = 1, y₁ = 1
point2: x₂ = 4, y₂ = 10
Using the equation for 2 points:
y-y₁ / y₂-y₁ = x-x₁ / x₂-x₁
y-1 / 10-1 = x-1 / 4-1
y-1 / 9 = x-1 /3
3(y-1) = 9(x-1)
y-1 = 3(x-1)
y-1 = 3x -3
y = 3x -3+1
y = 3x - 2
Thus, f(x) = 3x -2
Therefore, the rule in function notation, using the slope-intercept form of a line is f(x) = 3x -2
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an ant walks along a straight path. After traveling for 1 meter it stops, turns through an angle of 90 degrees in an anticlockwise direction and sets off in a straight line covering a distance of half a meter. Again, the ant turns through an angle of 90 degrees in an anticlockwise direction and sets off in a straight line covering a quarter of a meter. The ant continues in this manner indefinitely.
a) How many turns will the ant have made after covering a distance of 63/32 meters?
b)How far will the ant "eventually" travel?
The description of the motion of the ant which is a geometric series is as follows;
a) After covering 63/32 meters, the ant will have made 6 turns
b) The distance the ant will eventually travel is 2 meters
What is a geometric series?Geometric series is an infinite sum of terms in which the ratio between consecutive terms is a constant.
The distances traveled by the ant is a geometric progression of the form;
aₙ = a·rⁿ⁻¹The sum of the distance is given by the sum of the geometric progression as follow;
[tex]S_n = \dfrac{a\cdot \left(1 - r^n \right)}{(1 - r)}[/tex]
Where;
aₙ = The distance traveled in the nth turn
Sₙ = The sum of the distances
a = The first term = 1 meter
r = The value of the common ratio = 0.5 meter ÷ 1 meter = 0.5
a) The number of turns the ant will have made after covering a distance of 63/32 meters is therefore;
[tex]S_n =\dfrac{63}{32} = \dfrac{1\times \left(1 - 0.5^n \right)}{(1 - 0.5)}[/tex]
31.5 = 32 × (1 - 0.5ⁿ)
32 × 0.5ⁿ = 32 - 31.5 = 0.5
0.5ⁿ = 0.5 ÷ 32
n = ln(0.5 ÷ 32) ÷ ln(0.5) = 6
After covering a distance of 63/62 meters, the ant will have made 6 turnsb) The distance the ant travels eventually is given by the formula for the sum to infinity of a geometric progression as follows;
[tex]S_{\infty} = \dfrac{a}{1-r}[/tex]
Which gives;
[tex]S_{\infty} = \dfrac{1}{1-0.5} = 2[/tex]
The distance the ant eventually travels is 2 meters
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Plot a point for some of the ordered pairs from the example problem shown at the right. Model the relationships by drawing a line from the y-axis through each point. Explain how the graphs show which relationship is proportional and which is not proportional.
Both the pairs of the points are linear.
Here i made the plot for the points and both the pairs of the points shows that the it follows a straight path which means it will be linear i nature.
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Solve the equation for w.
4w + 2 + 0.6w = −3.4w − 6
No solution
w = 0
w = 1
w = −1
Please help
Answer:
The answer is w= -1
Step-by-step explanation:
4w + 2 + 0.6w = -3.4w - 6
4.6w + 2 = -3.4w - 6
8w + 2 = -6
8w = -8
w = -1
For which equation will f(-2) = -6?
f(x) = x³ + x
f(x)=x² - 5x
f(x) = 4x³ + 6x² − x
f(x) = -3x³-4x² + 4x
The correct equation is f(x) = 4x³ + 6x² − x . A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
What is Function ?A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.
The value of the function is f(x). The slope of the line is m. b is either the function's value at x=0 or the position at which the line in the coordinate plane crosses the y-axis. x is the x-value. This form is called the slope-intercept form.
The slope of a linear function is calculated by rearranging the equation to its general form, f(x) = mx + c; where m is the slope.
in question f(-2) = - 6
so x= -2
In function f(x) = 4x³ + 6x² - x
F(-2) = 4(-2)³ + 6 (-2)² - (-2) = - 6
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Toby the Turkey needs you to buy place cards so that everyone will know where to sit during his Thanksgiving dinner. He would like to buy them in bulk. The card shop offers 3 different packs. Use the chart below to determine which pack offers Toby the best deal, then find the price per card.
Pack #1
Quantity 12 place cards
Cost $29 per pack
Pack #2
Quantity 24 place cards
Cost $89 per pack
Pack #3
Quantity 24 place cards
Cost $213 per pack
( This is confusing to me I don't even understand why my math teacher decided to give us it when she didn't explain it. currently gonna cry on this I need to show work what.)
The solution is Pack #1
The pack which offers Toby the best deal is Pack #1
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 price per card of each pack be = A
Now , the equation is
a)
Pack #1
The number of cards = 12 cards
The cost per pack = $ 29
The equation is
So , the cost of 1 card = cost per pack / number of cards
= $ 2.416
Therefore , the cost of one card in pack 1 is $ 2.416
b)
Pack #2
The number of cards = 24 cards
The cost per pack = $ 89
The equation is
So , the cost of 1 card = cost per pack / number of cards
= $ 3.708
Therefore , the cost of one card in pack 2 is $ 3.708
c)
Pack #3
The number of cards = 24 cards
The cost per pack = $ 213
The equation is
So , the cost of 1 card = cost per pack / number of cards
= $ 8.875
Therefore , the cost of one card in pack 3 is $ 8.875
And ,
cost of one card in pack 3 is $ 8.875 > cost of one card in pack 2 is $ 3.708 > cost of one card in pack 1 is $ 2.416
Therefore , the cost of one card in pack 1 is $ 2.416 and this is the best deal
Hence , The pack which offers Toby the best deal is Pack #1
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How to solve completing the square with the formula a (X+h)2 + k
The solutions of the given quadratic equation are
[tex]x = 3 \ or \ x = -\frac{1}{4}[/tex]
What is quadratic equation?
Equation shows the equality between two algebraic expressions by connecting the two algebraic expressions by an equal to sign.
A two degree equation is known as Quadratic equation.
The given quadratic equation is
[tex]4x^2 - 11x - 3[/tex]
Now,
[tex]4x^2 - 11x - 3 = 0\\\\(2x)^2 - 2 \times 2x \times \frac{11}{4} + (\frac{11}{4})^2 - (\frac{11}{4})^2 -3 = 0\\\\(2x - \frac{11}{4})^2 = \frac{121}{16} + 3\\\\(2x - \frac{11}{4})^2 = \frac{121+48}{16}\\\\(2x - \frac{11}{4})^2 = \frac{169}{16}\\\\2x - \frac{11}{4} = \sqrt{\frac{169}{16}}\\\\2x - \frac{11}{4} = \pm\frac{13}{4}\\\\2x - \frac{11}{4} = \frac{13}{4} \ or \ 2x - \frac{11}{4} = -\frac{13}{4}\\\\2x = \frac{13}{4} + \frac{11}{4} \ or \ 2x = \frac{11}{4} - \frac{13}{4}\\\\[/tex]
[tex]2x = 6 \ or \ 2x = -\frac{1}{2}\\x = 3 \ or \ x = -\frac{1}{4}[/tex]
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6x+5y=2
4x+2y=8
solve by elimination
Using the elimination method, the solution to the system of linear equations is: x = 4.5 and y = -5.
How to Solve a System of Linear Equations by Elimination?Solving a system of linear equations by elimination involves eliminating one of the variables, then solve to find the other variable.
Given the equations:
6x + 5y = 2 --> eqn. 1
4x + 2y = 8 --> eqn. 2
Multiply eqn. 1 by 2 and eqn. 2 by 5:
12x + 10y = 4 --> eqn. 3
20x + 10y = 40 --> eqn. 4
Subtract
-8x = -36
x = 4.5
Substitute x = 4.5 into eqn. 2
4(4.5) + 2y = 8
18 + 2y = 8
2y = 8 - 18
2y = -10
y = -5
The solution is: x = 4.5 and y = -5.
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Write2 1/4 as an improper fraction in its simplest form
Answer:
9/4
Step-by-step explanation:
Dados os polinomios:
X-2=
X-3=
X+1=
X+5=
Answer:
X-2=1
X-3=2
X+1=2
X+5=6
What is the y-intercept for x^3+8x^2+11x-20? How many zeros are their in the graph?
Cn anyone answer this question?
Against the Wind a small plane flew 858 miles in 2 hours and 10 minutes the return trip only took 1 hour and 50 minutes what was the speed of the wind?what was the speed of the plane in still air?
Let the speed of the small plane be 'R'.
Let the speed of the wind be 'S'.
Plane's time against wind = 2 hours and 10 minutes = 13/6 hours.
Plane's time with wind = 1 hour and 50 minutes = 11/6 hours
Use formula;
Distance = speed x time
Against the wind:
858 = (13/6) (R-S) = (13/6) R - (13/6)S
With wind:
858 = (11/6)(R + S) = (11/6)R + (11/6)S
Multiplies each by 6.
5148 = 13R - 13S
5148 = 11R + 11S
Use elimination method
56628 = 143R - 143S (multiplies by 11)
66924 = 143R + 143S (multiplies by 13)
Further solving;
123552 = 276R
R = 432 miles/ hr. (speed of small plane )
And, S = 36 miles/ hr. (speed of wind)
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What id an equation of a line in point-slope form that passes theough (1,-7) and has a slope of -2/3?
(y + 7) = -2/3(x - 1) is the point-slope form of the equation of a line passing through (1, -7) and having a slope of -2/3.
How to calculate a line's equation in point-slope formThe query revealed information include
a slope of -2/3
point (1, -7)
Describe a linear function.Those functions with exponents of 1 are known as linear functions.
The form represents the equation of a line in point-slope form.
(y - y₁) = m (x - x₁)
Variables in the y direction are described by (y and y₁).
variables in the x direction are described by (x and x₁).
m = slope
substituting into the slope and point (1, -7) equation gives
()y + 7) = -2/3 (x - 1)
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Pls. Help. Lots of points plus Brainliest!
What is 12*12/2+13?
Step-by-step explanation:
[tex]\sf 12 \times \cfrac{12}{2} + 13[/tex]
First, let's divide 12 by 2:-
[tex]\sf 12 \times 6 + 13[/tex]Multiply 12 × 6 :-
[tex]\sf 72 + 13[/tex]Add 72 + 13:-
[tex]\sf 85[/tex]Therefore, your answer is 85!!
____________________
Hope this helps!
Have a great day!!
for a lower tail test, the test statistic z is determined to be zero. what is the p-value for this test?
0.5 is the p-value for this Z -test .
What is the Z test's p-value?
For each test, the p-value is the likelihood that the null hypothesis will be correct, which is expressed as the probability of receiving a z-value . an absolute value at least as great as the one we actually observed in the sample data.P-value, or probability value, is the likelihood that a statistic will be at least this distant from the mean if the null hypothesis is assumed to be true. So it can be thought of as a conditional probability, to put it another way.For a lower tail test, the test statistic z is determined to be zero that means
P(z<0)
Then p-value is
p-value = P(z<0) = 0.5 (From z score table )
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given that AC is 38. Find bc
Answer:
x = 12
Step-by-step explanation: You start off with this
3x + 2
Then since AC is 38 you make it
3x+2=38
Then you do -2 getting rid of the 2 and doing 38 - 2 which is 36
you now have:
3x = 36
then you do 3x divided by 3 on both sides
you do 36 divided by 3 then you get
x = 12
What is the rate of change of the function y = 5x + 4?
Answer:
The slope of y = 5x - 4 is 5.
Notice that the equation, y = 5x - 4, of the line is written in the form y = mx + b. Thus, it is in slope-intercept form. When the equation of a line is in slope-intercept form, we know that the slope of the line is equal to m, the coefficient of x.
In y = 5x - 4, the coefficient of x is 5. Thus, the slope of the line, y = 5x - 4, is 5.
Please help!! I don't get this
5. Based on arc and central angle relationship, m∠BCD is: B. 108°.
6. The length of arc PQ is: H. 3.14 m
7. Length of CF is: A. 6 cm.
What is the Formula for the Length of an Arc?To find the length of an arc, the formula to use is:
Arc length = ∅/360 × 2πr
5. Measure of arc AB = 72°
Measure of arc BD = 180 - measure of arc AB
Measure of arc BD = 180 - 72
Measure of arc BD = 108°
Measure of angle BCD = measure of arc BD [central angle theorem]
Measure of angle BCD = 108°
The answer is: B. 108°
6. ∅ = 60°
r = 3 m
Arc length = 60/360 × 2 × π × 3
Arc length = 3.14 m
The answer is: H. 3.14 m
7. Since the distance between the two chords, AB and CD are equal, therefore, AB = CD.
AB = CD = 12 cm
CF = 1/2(CD)
CF = 1/2(12)
CF = 6 cm.
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a sample of 1,400 american households was asked if they planned to buy a new car next year. of the respondents, 34% indicated they planned to buy a new car next year. construct a 90% confidence interval of the proportion of american households who expect to buy a new car next year.
The proportion of American households who expect to buy a new car next year is[0.3191, 0.3609].
The Four answer choices:
[0.3152, 0.3648]
[0.3191, 0.3609]
[0.3394, 0.3406]
[0.3354, 0.3446]
According to the survey, 1,400 American households are expected to buy a new car next year. According to respondents, 34% plan on buying a new car next year.
The confidence interval for a population proportion is given by the following equation for large random samples
Sample proportion ± z × [tex]\frac{\sqrt{(sample proportion(1-sample proportion)}}{n}[/tex]
whereas z is a multiplier number that comes from the normal curve and determines the level of confidence.
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a polling organization took a random sample of 1600 high school students to determine how many high school students have a part-time job. suppose 43% of all high school students have a part itme job. if the total population of high shool students is 15 million, is the 10% condition met? justify your answer
Based on this if we find the 10% of the population we have:
15000000 × 0.1 = 1500000 And we see that our sample 1600 & lt;
1500000 and we satisfy the 10% condition.
For this case we have the following data given:
n = 1600 represent the sample size select.
dp = 0.43, which, represent the estimated proportion of high school students that have a part time job
N= 15000000 represent the total population size
And we want to check if the 10% condition is satisfied. So we can use the following definitions for the 10% condition: “We need to have Bernoulli trials must be independent.
If that assumption is not satisfied, it is still okay to proceed if we have that the sample is smaller than 10% of the population.”"One of the conditions to be checked before using the normal model for sample proportions is, “The sample size, n, must be no larger than 10% of the population.
”So based on this if we find the 10% of the population we have:
15000000 × 0.1 = 1500000
And we see that our sample 1600 & lt;
1500000 and we satisfy the 10% condition.
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Suppose you have 50 blueberry scones and 75 cranberry scones you wanted to make as many identical birds of scones as possible each bread should have an equal number of blueberry scones and an equal number of cranberries called what is the greatest number of bags you can fill explain
He can fill a total of 25 bags.
The values are,
50 blueberry scones total.
75 cranberry scones total.
For each bag to have an equal quantity of blueberry scones & cranberry scones, he must determine the maximum amount of bags he can fill.
H.C.F. of 50 & 75 is thus 25.
As a result, he can fill 25 bags.
There would be scones with blueberries in each bag of
50 ÷ 25 = 2
There would be cranberry scones for each bag, for a total of
75 ÷ 25 = 3
As a result, he may fill a total of 25 bags.
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Simplify the expression:
(–5.5 + 2p)(–2) =
The expression, (–5.5 + 2p)(–2), after simplification is 11-4p.
According to the question,
We have the following expression:
(–5.5 + 2p)(–2)
Now, in order to simplify this expression, we will multiply -2 into the bracket:
(Note that when a number or a variable is multiplied into the bracket then it is multiplied into every number or term within the bracket.)
-5.5*-2+2p*(-2)
We know that when two negative integers are multiplied then the result is positive and when one negative and one positive integer is multiplied then the result is negative.
11-4p
Hence, the expression, (–5.5 + 2p)(–2), after simplification is 11-4p.
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The image of the point (9, -3)(9,−3) under a translation is (4, 1)(4,1). Find the coordinates of the image of the point (0, 2)(0,2) under the same translation.
Solve for
x
xx.
x
=
x=x, equals
Two triangles A B C and A D E. Triangle A B C is on top of Triangle A D E at point A. Side A C is smaller than A E and side A B is smaller than side A D. Side C B has a length of three. Side E D has a length of x. Side A B has a length of nine. Segment B D has a length of six. Angle A B C and A D E are both ninety degrees.
A
A
E
E
D
D
B
B
C
C
9
9
3
3
6
6
x
x
The ΔABC placed on top of ΔADE, and having angles ∠ABC and ∠BAC congruent to ∠ADE and ∠DAE in ΔADE is similar to ΔADE. The side CB is 3 indicating that the side ED, which has a length of x = 5
What are similar triangles?Similar triangles are triangles that have similar (the same shape), such that one triangles has side lengths that are a constant multiple of the other.
Please find attached a possible drawing that is based on the parameters of the question
From the question and the drawing created with MS Word, we have;
Length of AB = 9
Side CB = 3
Segment BD = 6
∠ABC = ∠ADE = 90°
∠ABC ≅ ∠ADE by definition of congruency
From the drawing, ∠BAC ≅ ∠BAC by reflexive property of equality
Triangle ΔBAC is similar to triangle ΔDAE by AA similarity postulate
Segment AD = Segment AB + Segment BD by segment addition postulate
AD = 9 + 6 = 15
The pair of corresponding sides of similar triangles are proportional, therefore;
[tex]\dfrac{\overline{AB}}{\overline{AD}} = \dfrac{\overline{BC}}{\overline{DE}}[/tex]
Which gives;
[tex]\dfrac{9}{15} = \dfrac{3}{x}[/tex]
9 × x = 15 × 3 = 45
x = 45 ÷ 9 = 5
The length of side ED, x = 5
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Answer: 12
Step-by-step explanation: Because 9 x 2 is 18, and 7 x 2 is 14, we want to multiply 6 by 2. This gives us the answer 12.
9 x 2 = 18
7 x 2 = 14
6 x 2 = 12
Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
The equations that have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p are (b) 2.3p – 10.1 = 6.49p – 4 and (c) 230p – 1010 = 650p – 400 – p
How to determine the equations with the same solution?The equation is given as
2.3p – 10.1 = 6.5p – 4 – 0.01p
Evaluate the like terms on the right-hand side
So, we have the following representation
2.3p – 10.1 = 6.49p – 4
The above equation is indicated in option (b)
Multiply through the equation by 100
So, we have:
100(2.3p – 10.1 = 6.5p – 4 – 0.01p)
Evaluate
230p – 1010 = 650p – 400 – p
The above equation is indicated in option (c)
Hence, the equations with the same solution are (b) and (c)
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8.
y=
Which word describes the connection between the shape and its reflection?
[] congruent
[] similar
[ ] translation
[] corresponding
The corresponding sides and angles of both the triangles [pre-image and the image] are congruent
what is congruency?Congruent in geometry refers to being the same size and shape. Congruence can be used with figures, angles, and line segments. If any two line segments are equal in length, they are said to be congruent. If two angles have the same measure, they are said to be congruent. If the matching sides and angles of two triangles are the same, they are said to be congruent. In this post, we will learn more about congruence and congruent figures.The word 'congruent' means 'exactly equal' in terms of shape and size. Even when we turn, flip, or rotate the shapes, they remain equal.Hence,The corresponding sides and angles of both the triangles [pre-image and the image] are congruent
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what 5x5(45x5)x3(56x34)
Answer: 32130000
Step-by-step explanation:
Multiply all of your numbers
Standing on top of a 200 m building with the estimation angle of depression from the top of the building to a park bench on the ground to be 54°
How far is the park bench is from the foot of the building
Distance of Park bench from building is 275 m when angle of depression top of the building to a park bench on the ground.
What is the elevation and depression angle?
The angle between the horizontal and an item is referred to as its elevation. The line of sight of an observer would be upward. Angle of depression refers to the downward angle from the horizontal to an item.Angle of depression, θ = 54°
The height, h = 200
The distance of park bench from the building, d is given by
Tan θ = opposite / Adjacent
Tan 54 = d / 200
d = 200 * Tan 54
d = 200 * 1.3764
d = 275.28
D = 275 M
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On a coordinate plane, a line goes through (negative 2, negative 4) and (2, 2). A point is at (negative 3, 1).
What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point (−3, 1)?
y – 1=Negative three-halves(x + 3)
y – 1=Negative two-thirds(x + 3)
y – 1= Two-thirds(x + 3)
y – 1= Three-halves(x + 3)
The equation of the line that is parallel to the given line and passes through the point (-3,1) is y - 1 = 3/2(x + 3)
The line is passing through (-2, -4) and (2, 2)
The slope of the line is change in y coordinate with respect to the change in x coordinate of the line
Slope of the line m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Substitute the values in the equation
Slope of the line = (2--4) / (2--2)
= 6/2
= 3/2
The point slope form is
[tex]y-y_1=m(x-x_1)[/tex]
The point is given that (-3, 1)
The slope of the line = 3/2
because both are parallel lines
Substitute the both values in the equation
y - 1 = 3/2( x - (-3))
y - 1 = 3/2(x + 3)
Hence, the equation of the line that is parallel to the given line and passes through the point (-3,1) is y - 1 = 3/2(x + 3)
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Answer:
y – 1 = 3/2(x + 3)
Step-by-step explanation: