To find the absolute maximum and minimum values of the function f(x) = x³ - 12x² - 27x + 9 over a given interval, we need to follow these steps:
1. Find the critical points of the function by setting its derivative f'(x) = 3x² - 24x - 27 equal to zero and solving for x. We get x = -3, 3, and 4 as critical points.
2. Evaluate the function at the critical points and the endpoints of the interval to find candidate points for the absolute max/min values.
f(-3) = -63, f(3) = -45, f(4) = 1, f(-infinity) = -infinity, and f(infinity) = infinity.
3. Compare the values of the function at the candidate points to determine the absolute maximum and minimum values.
The function has a local maximum at x = -3 and a local minimum at x = 4, but neither of these points is in the given interval. Therefore, we only need to consider the endpoints.
The absolute maximum value of the function over the interval (-infinity, infinity) is infinity, which occurs at x = infinity.
The absolute minimum value of the function over the interval (-infinity, infinity) is -infinity, which occurs at x = -infinity.
Explanation: We used the concept of critical points and candidate points to determine the absolute maximum and minimum values of the function over the given interval. The critical points are the points where the derivative of the function is zero or undefined, and the candidate points are the critical points and the endpoints of the interval. By evaluating the function at these points and comparing the values, we can identify the absolute max/min values. In this case, we found that the function has no absolute max/min values over the given interval, but has an absolute max of infinity at x = infinity and an absolute min of -infinity at x = -infinity over the entire domain of the function.
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PLSS HELP ASAP
A bag of marbles contains 7 red, 8 orange, 9 purple, 7 white, and 5 yellow. Mark reaches into the bag and randomly picks out 3 marbles all at once.
How many different 3-marble groups are possible?
What is the probability that he will select 2 red and 1 yellow marbles?
based on the number of marbles in the bag, the number of marble groups possible is 7,140 groups.
the probability of selecting 2 red and 1 yellow is 1/68.
what number of 3-marble groups can be made?first, find the number of marbles:
= 7 + 8 + 9 + 7 + 5
= 36
the number of marble groups are:
= 36 ! / (3 ! x 33 !)
= 7,140 groups
the probability of picking 2 red and 1 yellow is:
= (( (7! / (2! x 5!)) x ( ( 5! / (1! x 4!))) / 7,140
= 1 / 68
= 1.47%
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The difference between two number is 3.take the numbers as x and y form a equation?
The difference between two numbers are [tex]3[/tex] and numbers are [tex]x[/tex] and [tex]y[/tex] so the equation is [tex]x-y=3[/tex]
How to find the equation and what is the equation ?
equation, statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way.
if the difference of two numbers are [tex]3[/tex]
And one number is [tex]x[/tex] and another is [tex]y[/tex]
So we write the equation with operation of subtraction
[tex]x-y=3[/tex]
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Mr.david gave ross a number and asked him to divide it by 15. The quotient and remainder obtained by ross are 341 and 12 respectively. Find the number that teacher gave ross.
The number the teacher gave Ross is 5127
Word problems on linear equationsFrom the question, we are to determine the number that the teacher gave Ross
From the given information,
Mr. David gave Ross a number and asked him to divide it by 15.
Let the number be x
That is,
x/15
The quotient and remainder obtained by ross are 341 and 12 respectively
That is,
x/15 = 341 + 12/15
Now, we will solve the equation for x
x/15 = 341 + 12/15
Multiply through by 15
x = 15×341 + 12
x = 5115 + 12
x = 5127
Hence, the number the teacher gave Ross is 5127
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The equation t^3=a^2 shows the relationship between a planet’s orbital period, T, and the planet’s mean distance from the sun, A, in astronomical units, AU. If the orbital period of planet Y is twice the orbital period of planet X, by what factor is the mean distance increased?
Answer:
2√2
Step-by-step explanation:
We can find the relationship of interest by solving the given equation for A, the mean distance.
Solve for A[tex]T^3=A^2\\\\A=\sqrt{T^3}=T\sqrt{T}\qquad\text{take the square root}[/tex]
Substitute valuesThe mean distance of planet X is found in terms of its period to be ...
[tex]D_x=T_x\sqrt{T_x}[/tex]
The mean distance of planet Y can be found using the given relation ...
[tex]T_y=2T_x\\\\D_y=T_y\sqrt{T_y}=2T_x\sqrt{2T_x}=(2\sqrt{2})T_x\sqrt{T_x}\\\\D_y=2\sqrt{2}\cdot D_x[/tex]
The mean distance of planet Y is increased from that of planet X by the factor ...
2√2
Select the correct answer. malik is baking pumpkin bread and banana bread for friends and family. his pumpkin bread recipe calls for 4 eggs and cups of flour, and his banana bread recipe calls for 1 egg and cups of flour. malik has 14 eggs, 16 cups of flour, and plenty of other ingredients to make multiple loaves. what is one combination of breads malik can bake without getting more ingredients? a. 3 loaves of pumpkin bread and 3 loaves of banana bread b. 1 loaf of pumpkin bread and 9 loaves of banana bread c. 2 loaves of pumpkin bread and 6 loaves of banana bread d. 5 loaves of pumpkin bread and 1 loaf of banana bread
The correct answer is
2 loaves of pumpkin bread and 6 loaves of banana bread. Option C.
How to find a possible combination of bread?For the solution for the Pumpkin bread:
4 eggs
3 1/2 cups of flour
Banana bread:
1 egg
1 (0.5) cups of flour
Available ingredients:
14 eggs
16 cups of flour
Assume 2 loaves of pumpkin bread and 6 loaves of banana bread
For the solution for the Pumpkin bread:
PB=4 eggs* 2
PB= 8 eggs
PB=3 *0.5 * 2
PB = 7 cups of flour
For the solution of the Banana bread:
BB =1 *6
BB= 6 eggs
For cups
BB= 1 (0.5) * 6
BB= 9 cups of flour
In conclusion, The solution of the Total used ingredients
I=8 + 6
I= 14 eggs
For cups
IC=7 + 9
IC= 16 cups of flour
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Answer:
C 2 loaves of Pb and 6 loaves of BB
Step-by-step explanation:
20POINTS
help me please
Answer:
a. 50 students are randomly selected from the school; 11 drive to school
b. 407
Step-by-step explanation:
a. to get an accurate viewing, use random people that share 1 large common denominator (from the same school) that also has to do with the question
b. 11/50 is only between 50 students. make the denominator the same (1850) and multiply the same to the numerator
1850/50 = 37
11/50 x 37 = 407/1850
PLEASE HELP ME AS SOON AS POSSIBLE
Answer: [tex]f^{-1}(\text{x}) = (\text{x}-9)^2+4[/tex] when [tex]\text{x} \ge 9[/tex]
=================================================
Work Shown:
[tex]f(\text{x}) = 9 + \sqrt{\text{x} - 4}\\\\\text{y} = 9 + \sqrt{\text{x} - 4}\\\\\text{x} = 9 + \sqrt{\text{y} - 4}\\\\\text{x}-9 = \sqrt{\text{y} - 4}\\\\(\text{x}-9)^2 = (\sqrt{\text{y} - 4})^2\\\\(\text{x}-9)^2 = \text{y}-4\\\\(\text{x}-9)^2+4 = \text{y}\\\\f^{-1}(x) = (\text{x} - 9)^2+4[/tex]
Explanation:
I replaced f(x) with y. After that I swapped x and y, then solved for y to get the inverse.
The smallest that [tex]\sqrt{\text{x}-4}[/tex] can get is 0, which means the smallest f(x) can get is 9+0 = 9. The range for f(x) is [tex]\text{y} \ge 9[/tex]
Since x and y swap to determine the inverse, the domain and range swap roles. Therefore, the domain of the inverse [tex]f^{-1}(\text{x})[/tex] is [tex]\text{x} \ge 9[/tex]
So we will only consider the right half portion of the parabola.
The graph is below. The red curve mirrors over the black dashed line to get the blue curve, and vice versa.
Answer:
[tex]f^-^1(x)=(x-9)^2+4[/tex] OR [tex]x^2-18x+85[/tex] if you simplify it
Step-by-step explanation:
to find the inverse function you have to switch x and y with each other and solve for y.
[tex]y=9+\sqrt{x-4}\\x=9+\sqrt{y-4}[/tex] step 1: switch x and y with each other
[tex]x-9=\sqrt{y-4}[/tex] step 2: subtract 9 from both sides
[tex](x-9)^2=\sqrt{y-4}^2[/tex] step 3: square both sides to get rid of the square root
[tex](x-9)^2=y-4\\(x-9)^2+4=y\\[/tex] step 4: add 4 to both sides
you could leave the answer like this or you can simplify to get [tex]x^2-18x+85[/tex]
The two-way frequency table below shows data on playing a sport and playing a musical instrument for students in a class.
Complete the following two-way table of row relative frequencies.
(If necessary, round your answers to the nearest hundredth.)
Here is the completed table:
Plays a sport Doesn't play a sport
Plays a musical instrument 0.46 0.54
Doesn't play a musical instrument 0.73 0.27
What are the row frequencies?
Relative frequency measures how often a value appears relative to the sum of the total values.
Plays a sport Doesn't play a sport
Plays a musical instrument (6/13) = 0.46 (7/13) = 0.54
Doesn't play a musical instrument (8/11) 0.73 (3/11) =0.27
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A couple quick algebra 1 questions for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
The variation given in the first question is not a direct variation and there is no constant of variation.
Direct variationy = k × x
when
y = -40 and x = -0.5
y = k × x
-40 = k × -0.5
-40 = -0.5k
k = -40/-0.5
k = 80
When y = 8 and x = 2.5
y = k × x
8 = k × 2.5
8 = 2.5k
k = 8 / 2.5
k = 3.2
When y = 4 and x = -3
y = k × x
4 = k × -3
4 = -3k
k = -4/3
when y = 8 and x = -6
y = k × x
8 = k × -6
8 = -6k
k = 8/-6
k = -4/3
This is a direct variation and the constant of proportionality is -4/3
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I have no clue how to do this
The length and the width of the rectangle are 10 inches and 6 inches respectively.
In the question, we are given that the length of a rectangle exceeds its width by 4 inches, and the area is 60 square inches.
We are asked for the length and the width of the rectangle.
We assume the width (w) of the rectangle to be x inches.
Its length (l), exceeds its width (w) by 4 inches.
Thus, its length (l) = x + 4 inches.
Now, the area can be calculated using the formula, A = l*w, where A is its area, l is its length, and w is its width.
Thus, the area = (x + 4)x = x² + 4x.
But, we are given that the area is 60 square inches.
Putting the value, we get a quadratic equation:
x² + 4x = 60.
or, x² + 4x - 60 = 0,
or, x² + 10x - 6x - 60 = 0,
or, x(x + 10) - 6(x + 10) = 0,
or, (x - 6)(x + 10) = 0.
By the zero-product rule, we get:
Either, x - 6 = 0, or, x = 6,
or, x + 10 = 0, or, x = -10, but this is not possible as the width of a rectangle cannot be negative.
Thus, the width = x = 6 inches.
The length = x + 4 = 10 inches.
Thus, the length and the width of the rectangle are 10 inches and 6 inches respectively.
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Three consecutive positive prime numbers have a sum that is a multiple of . What is the least possible sum
Answer:
3, 7, 14
Step-by-step explanation:
small brain
help me i kinda stuck
For the given data,
Mean = 2.12
Median = 2
Mode = 2
Range = 3
Statistics
From the question we are to determine the mean
From the given table
Scoops of Ice cream Male Female Other Total frequencies
1 7 3 3 13
2 8 11 2 21
3 10 1 0 11
4 2 1 1 4
∴ [tex]Mean = \frac{(1 \times 13) + (2 \times 21) +(3 \times 11) + (4 \times 4)}{13 +21+11+4}[/tex]
[tex]Mean = \frac{(13) + (42) +(33) + (16)}{49}[/tex]
[tex]Mean = \frac{104}{49}[/tex]
Mean = 2.12
Median = 2
Mode = 2
Range = 4 - 1 = 3
Hence, for the given data
Mean = 2.12
Median = 2
Mode = 2
Range = 3
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A couple quick algebra 1 questions for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
Answer:
8. yes; 18
9. no
10. no
11. yes; -9
Step-by-step explanation:
The equation for inverse variation is y = k/x. The constant k will be the product of x and y: k = xy.
8.Products are (-3)(-6) = (2)(9) = (4.5)(4) = 18.
Inverse variation with k = 18.
9.Products are ...
(5)(4) ≠ (10)(8).
Not inverse variation.
10.Products are ...
(2)(25) ≠ (2.5)(16).
Not inverse variation.
11.Products are (3)(-3) = (9)(-1) = (12)(-.75) = -9.
Inverse variation with k = -9.
Tests show that the hydrogen ion concentration of a sample of apple juice is 0.0003 and that of ammonia is . find the approximate ph of each liquid using the formula , where is the hydrogen ion concentration.
The pH value of the apple juice is 3.5, is the correct answer.
The pH value of the ammonia is 8.9, is the correct answer.
The question seems to be incomplete and complete question is shown below
Tests show that the hydrogen ion concentration of a sample of apple juice is 0.0003 and that of ammonia is 1.3 x 10-9. Find the approximate pH of each liquid using the formula pH = -log (H+), where (H+) is the hydrogen ion concentration The pH value of the apple juice is___ The pH value of ammonia is____
1.pH of apple juice
A. 8.11
B. 1.75
C. 3.5
D. 2.1
2. pH of ammonia
A. 1.1
B. 7.0
C. 5.4
D. 8.9
The pH of a solution is defined as the logarithm of the reciprocal of the hydrogen ion concentration [H+] of the given solution.
From the formula;
pH = -log[ H⁺ ]
Given the data in the question.
For the Apple juice
hydrogen ion concentration H⁺ = 0.0003
pH of the apple juice pH = ?
pH = -log[ H⁺ ]
pH = -log[ 0.0003 ]
pH = 3.5
The pH value of the apple juice is 3.5
For the ammonia
hydrogen ion concentration H⁺ = 1.3 × 10⁻⁹
pH of the ammonia pH = ?
pH = -log[ H⁺ ]
pH = -log[ 1.3 × 10⁻⁹]
pH = 8.9
The pH value of the ammonia is 8.9
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division of 3 fraction
(2/5 ÷ 3/8 ) ÷ -3/5
Answer:
-16/9
Step-by-step explanation:
( [tex]\frac{2}{5}[/tex] ÷ [tex]\frac{3}{8}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= ( [tex]\frac{2}{5} * \frac{8}{3}[/tex] ) ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15}[/tex] ÷ [tex]-\frac{3}{5}[/tex]
= [tex]\frac{16}{15} * -\frac{5}{3}[/tex]
= -16/9
Use the base example of two people walking to describe a system of two linear equation has i.) 0 solution 2) 1 solution 3) an infinite solution
The system has:
No solutions for two parallel lines.1 solution for two nonparallel lines.Infinite solutions for two equations that describe the same line.How to identify the number of solutions for the systems?A system of two linear equations is given by:
y = a*x + b
y = c*x + d
1) The system has no solutions when both lines are parallel lines, this happens when both lines have the same slope and different y-intercept.
y = a*x + b
y = a*x + d
2) The system has one solution when the lines have different slopes:
y = a*x + b
y = c*x + d
3) The system has infinite solutions when both equations describe the same line:
y = a*x + b
y = a*x + b
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please help !!!!!!!!!
Answer:
D. completing the square
Step-by-step explanation:
The process shown rearranges the standard-form quadratic equation to vertex form. It involves factoring out the leading coefficient and distributing the constant so that the variable terms can be written as a perfect square.
What is a perfect square trinomial?A perfect square trinomial is the square of a binomial. It has the form ...
(x +b)² = x² +2bx +b²
Given two variable terms, x² and 2bx, the perfect square trinomial can be formed by adding b², which is the square of half the x-term coefficient.
When b² is added to the sum (x² +2bx), the square is complete. The sum (x² +2bx +b²) can be written as the square (x +b)².
Completing the squareThis process of adding b² to the sum (x² +2bx) will change the polynomial unless a similar quantity is subtracted. That is why the 3rd line of the problem statement show 4 being added and subtracted inside parentheses:
y = 5(x² +4x +4 -4) -17
When we're applying this transformation, we do not want to change the equation. We simply want to rearrange it.
In the next step, the subtracted term is brought outside the parentheses. This lets us write the quantity in parentheses as a perfect square.
y = 5(x² +4x +4) +5(-4) -17
y = 5(x² +4x +4) -20 -17
This process of adding and subtracting a suitable quantity and rearranging the equation so the variable terms make a perfect square is called ...
"completing the square."
__
Additional comment
The equation in this form is said to be in "vertex form."
y = a(x -h)² +k
In this form, the constant 'a' is a vertical scale factor; the ordered pair (h, k) identifies the vertex of the quadratic on a graph.
What is the quotient?
StartFraction t + 3 Over t + 4 EndFraction divided by (t squared + 7 t + 12)
(t + 3) squared
(t + 4) squared
StartFraction 1 Over (t + 4) squared EndFraction
StartFraction 1 Over (t + 3) squared EndFraction
The answer choice which represents the quotient of the expression given is; 1/(t+4)².
Which answer choice represents the quotient?It follows from the task content that the quotient which is to be evaluated is;
(t+3)/t+4) ÷ (t²+7t +12)
The quotient can therefore be evaluated as follows;
= (t+3)/t+4) ÷ (t+3)(t+4)
= (t+3)/t+4) × 1/(t+3)(t+4)
= 1/(t+4) × 1/(t+4)
= 1/(t+4)².
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Answer:
c
Step-by-step explanation:
c
Solve the inequality. 14 + 10y ≥ 3(y + 14)
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{14+10y\geq 3(y+14) } \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Simplify \ both \ sides \ of \ the \ inequality. }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{10y+14\geq 3y+14 } \end{gathered}$}[/tex][tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Subtract \ 3y \ from \ both \ sides. }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{10y+14-3y\geq 3y+42-3y } \end{gathered}$}\\\large\displaystyle\text{$\begin{gathered}\sf \bf{7y+14\geq 41 } \end{gathered}$}[/tex][tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{ Subtract \ 14 \ from \ both \ sides. }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{7y+14-14\geq 42-14 } \end{gathered}$}\\\large\displaystyle\text{$\begin{gathered}\sf \bf{7y\geq 28 } \end{gathered}$}[/tex][tex]\large\displaystyle\text{$\begin{gathered}\sf \boldsymbol{\sf{Divide \ both \ sides \ by \ 7. }} \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\frac{7y}{7}\geq \frac{28}{7} } \end{gathered}$}\\\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{y\geq 4} \end{gathered}$} }[/tex]A Ferris wheel is boarding platform is 2 meters above the ground, has a diameter of 68 meters, and rotates once every 9 minutes. How many minutes of the ride are spent higher than 37 meters above the ground
4.440 minutes of the ride is spent higher than 37 meters above the ground.
How to find the time spent on the ride?We have to find the amount of time spent by the ride 37 meters above the ground.
We know that 37 meters above the ground is actually one meter above the center.
We can find the angle above the center as shown below:
arcsin(1/34) = 0.0294 radians
Now we have to find the angle of rotation:
The angle of rotation of the Ferris wheel above 37 meters = π - (2*0.0294)
= 3.1 radians
The angular velocity of the Ferris wheel = Angle covered/Time
= 2π/9 rad/min.
Therefore time spent above 37 meters = 3.1*9/2π
= 3.1*9/6.28319
= 4.440 minutes
Therefore, we have found that 4.440 minutes of the ride is spent higher than 37 meters above the ground.
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Alan will be traveling out of town for business. during his travel, he is scheduled to have 4 dinners, 5 lunches, and 3 breakfasts. his employer will pay him for these meals at the following rate: $20.20 for dinner, $10.70 for lunch, and $6.93 for dinner. what is the total amount alan will be paid for these meals?
money paid by employer to Alan is $ 155.09
word problemA word problem is a type of mathematics exercise where the majority of the issue's context is supplied in spoken language as opposed to mathematical notation.
money paid for dinner by employer=$20.20
money paid for lunch by employer =$10.70
money paid for breakfast by employer =$6.93
according to question,
in this problem we will simply multiply quantity of dinners with amount
to get our answer.
so as given in problem Alan scheduled 4 dinners , 5 lunches and 3 breakfast
(4×$20.20)+(5×$10.70)+(3×$6.93)
80.8+53.5+20.79
=$ 155.09
hence money paid by employer to Alan is $ 155.09
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WILL GIVE BRAINLIEST!!!!!!!!!
Considering the given proportions, the correct statements are:
A person in the group who is more likely to read is a boy.A person in the group who is more likely to listen to music is a girl.What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The table gives these following proportions:
Boys and girls regarding books, boys are likelier to prefer books.Boys and girls regarding music, girls are likelier to prefer music.Hence the correct statements are:
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8 more than 3 times a number is the same as -5 times the number.
a. x = -1
b. x = 1
c. x = -2
Answer:
3x +8 = -5x
3x + 5x = -8
8x = -8x
x = -8 /8
x = -1
Hope it helps...
Please contact me for further explanation....
Using the unit circle, determine the value of sin(-240°).
Answer:
See my picture below for the answer.
Step-by-step explanation:
If you look at the unit circle and locate 240 degrees. You will see an ordered pair. The first number -1/2 is the cos of 240 and the second number is the sin.
Answer:
it’s the square root of 3, divided by 2 (sorry, can’t type any special characters at the moment)
Step-by-step explanation:
-240 = 120 because you measure negative angles clockwise starting at 0.
Using unit circle, value of sin for an angle is the 2nd number
The height of a certain species of plants is uniformly distributed on the interval of 5-12 cm. a plant was measured by jack. what is the probability that the height of this plant is between 6.3 and 7.8cm?
21 % is the answer.
Problem is that the height of this plant 6.3 and 7.8 cm
P (6.3 <x< 7.8)
12 -5 = 7cm
7.8 - 6.3 = 1.5cm
1.5 / 7 = 0.21
=21%
Probability is a field of mathematics that deals with the numerical description of the likelihood that an event will occur or that a statement is true. The probability of an event is a number between 0 and 1, with approximately 0 indicating the impossibility of the event and 1 indicating certainty.
Probability is a measure of the likelihood that an event will occur. Many events cannot be predicted with absolute certainty. You can only predict the probability that an event will occur.
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Determine the equation of a line that passes through (-2,5) and is parallel to the line whose equation is 5y +2x = 10.
Answer:
Step-by-step explanation:
Givens
line: 5y + 2x = 10
point (-2 , 5)
Discussion and Solution
You have to rearrange the given line to get the slope. It isn't nice, but it's not impossible.
5y + 2x = 10 Subtract 2x from both sides
5y + 2x - 2x = -2x + 10 Simplify
5y = - 2x + 10 Divide by 5
5y/5 = -2x/5 + 10/5 Simplify
y = -0.4x + 2
So the new equation is going to have a slope of - 0.4
So far what you have is
y = - 0.4x + b
Now use the point to find the y intercept. What you should be thinking about the point is that when x = -2 then y = 5. Substitute that into the new equation.
5 = -0.4*-2 + b
5 = 0.8 + b Subtract 0.8 from both sides
5 - 0.8 = 0.8 - 0.8 + b Simplify
4.2 = b
Answer
y = -0.4x + 4.2
Solve for X
7(8x + 6) = -1
9(5+ x) = 15 (10+ x)
Answer: 7(8x + 6) = -1: Exact form: x = -43/56, decimal form: x = -0.76785714
Thats all i have because you can't solve for x for the second one!
Someone please answer this for me (Algebra 2)
Using division of fractions, the equivalent expression is:
B. [tex]\frac{4}{d - 6}[/tex].
How to divide fractions?To divide two fractions, we multiply the first by the inverse of the second.
In this problem, we have that:
The first fraction is: [tex]\frac{2d - 6}{d^2 + 2d - 48} = \frac{2(d - 3)}{(d + 8)(d - 6)}[/tex].The second fraction is: [tex]\frac{d - 3}{2d + 16} = \frac{d - 3}{2(d + 8)}[/tex].Applying the multiplication, we have that:
[tex]\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3} = \frac{4}{d - 6}[/tex].
Hence option B is correct.
More can be learned about division of fractions at https://brainly.com/question/8433134
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Solve for w, where w is a real number.
Answer:
w= 9
Step-by-step explanation:
[tex] \sqrt{ - 4w + 61} = w - 4[/tex]
Square both sides:
-4w +61= (w -4)²
[tex]\boxed{(a - b)^{2} = a^2 -2ab + b^2 }[/tex]
Expand:
-4w +61= w² -2(w)(4) +4²
-4w +61= w² -8w +16
Simplify:
w² -8w +16 +4w -61= 0
w² -4w -45= 0
Factorize:
(w -9)(w +5)= 0
w -9= 0 or w +5= 0
w= 9 or w= -5 (reject)
Note:
-5 is rejected since we are only taking the positive value of the square root here. If the negative square root is taken into consideration, then w= -5 would give us -9 on both sides of the equation.
Why do we use negative square root?
When solving an equation such as x²= 4,
we have to consider than squaring any number removes the negative sign i.e., the result of a squared number is always positive.
In the case of x²= 4, x can be 2 or -2. Thus, whenever we introduce a square root, a '±' must be used.
However, back to our question, we did not introduce the square root so we should not consider the negative square root value.
Answer: w=9.
Step-by-step explanation:
[tex]\sqrt{-4w+61}=w-4[/tex]
Tolerance range:
[tex]\left \{ {{-4w+61\geq 0} \atop {w-4\geq 0}} \right. \ \ \ \ \left \{ {{4w\leq 61\ |:4} \atop {w\geq 4}} \right. \ \ \ \ \left \{ {{w\leq 15,25} \atop {w\geq 4}} \right. \ \ \ \ \Rightarrow\ \ \ \ \\w\in[4;15,25].[/tex]
Solution:
[tex](\sqrt{-4w+61})^2=(w-4)^2\\-4w+61=w^2-2*w*4+4^2\\-4w+61=w^2-8w+16\\w^2-4w-45=0\\D=(-4)^2-4*1*(-45)\\D=16+4*45\\D=16+180\\D=196.\\\sqrt{D}=\sqrt{196}\\\sqrt{D}=14 \\w_{1,2}=\frac{-(-4)б14}{2} \\w_1=\frac{4-14}{2} \\w_1=\frac{-10}{2} \\w_1=-5\ \notin tolerance\ range.\\w_2=\frac{4+14}{2}\\ w_2=\frac{18}{2} \\w_2=9\ \in tolerance \ range.[/tex]
describe how the graph is transformed from its parent function?
Answer:
It is stretched taller by a factor of 2.