Answer:
The value of x is 2[tex]\sqrt{29}[/tex] .
Step-by-step explanation:
Given as :
The area of Trapezoid = x² unit²
The measure of base 1 = [tex]b_1[/tex] = - 19 unit
The measure of base 2 = [tex]b_2[/tex] = 10 unit
The height of the mid segment = 8 unit
Now, From the Area of Trapezoid
i.e Area = [tex]\dfrac{1}{2}[/tex] × (sum of opposite base) × height
Or, x² = [tex]\dfrac{1}{2}[/tex] × (19 + 10) × 8
Or, x² = [tex]\dfrac{8}{2}[/tex] × (29)
Or, x² = 4 × (29)
Or, x² = 116
∴ x = [tex]\sqrt{116}[/tex]
I.e x = 2[tex]\sqrt{29}[/tex]
Hence, The value of x is 2[tex]\sqrt{29}[/tex] . Answer
Final answer:
Using the properties of a trapezoid's midsegment being the average length of the bases, the equation (10 + (x² - 19))/2 = 8 was solved to find that x equals 5.
Explanation:
The question revolves around the properties of a trapezoid and its midsegment. In a trapezoid, the midsegment (also called the median) is the segment that joins the midpoints of the non-parallel sides and is parallel to both bases.
The length of the midsegment is the average of the length of the two bases, which can be expressed as (base1 + base2) / 2.
Given that the midsegment is 8 units long, and one base is given as 10 units, let the other base be expressed as x² − 19 units. The formula for the midsegment becomes:
(10 + (x² − 19))/2 = 8
Now we solve for x:
10 + x² − 19 = 16
x² − 9 = 16
x² = 25
x = ± 5
Since the measure of a base cannot be negative, the positive root is the realistic solution:
x = 5
Leanne, owner of The Candy Shoppe, wants to make 16-ounce gift bags with a mix of hard peppermint candies and chocolate mints. If peppermints cost 6¢ per ounce and the chocolate mints 22¢ per ounce, how many of each are in a gift bag that costs $2.72?
The gift bag contains 5 ounces of peppermint candies and 11 ounces of chocolate mint.
Step-by-step explanation:
Cost of peppermint candies per ounce = 6 cents
Cost of chocolate mint = 22 cents
Cost of bag = $2.72 = 2.72*100 = 272 cents
Weight of bag = 16 ounce
Let,
x be the ounces of peppermint candies
y be the ounces of chocolate mint
x+y=16 Eqn 1
6x+22y=272 Eqn 2
Multiplying Eqn 1 by 6
[tex]6(x+y=16)\\6x+6y=96\ \ \ Eqn\ 3[/tex]
Subtracting Eqn 3 from Eqn 2
[tex](6x+22y)-(6x+6y)=272-96\\6x+22y-6x-6y=176\\16y=176[/tex]
Dividing both sides by 16
[tex]\frac{16y}{16}=\frac{176}{16}\\y=11[/tex]
Putting y=11 in Eqn 1
[tex]x+11=16\\x=16-11\\x=5[/tex]
The gift bag contains 5 ounces of peppermint candies and 11 ounces of chocolate mint.
Keywords: linear equation, subtraction
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Type the correct answer in the box. Round your answer to the thousandth place.
The board of directors of a company knows that the probability that carbon emissions from the company’s factory exceed the permissible level is 35%. They hire a consultant who uses a carbon footprint calculator to test the emissions level. The accuracy of the test is 85%.
The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is __________
Answer:
The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is 19.338% (Rounding to the next thousandth place)
Step-by-step explanation:
1. Let's review the information provided to us to answer the question correctly:
Probability that carbon emissions from the company’s factory exceed the permissible level = 35% = 0.35
Accuracy of the test of emissions level = 85% = 0.85
2. The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is?
These two events, carbon emissions from the company’s factory and the accuracy of the test are independent events, therefore:
Probability that carbon emissions from the factory are within the permissible level = 1 - 0.35 = 0.65
Probability that the test predicts the opposite to be true = 0..35 * 0.85 = 0.2975 (The opposite is that the carbon emissions from the company exceed the permissible level)
Probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is:
0.65 * 0.2975 = 0.193375
The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is 19.338% (Rounding to the next thousandth place)
Final answer:
The probability that the emissions are within the permissible level and the test incorrectly predicts the opposite is 0.098 or 9.75% after rounding to the thousandth place.
Explanation:
The probability that carbon emissions from the factory are within the permissible level is the complement of the probability that they exceed the permissible level, which is 100% - 35% = 65% or 0.65. Given that the test has an accuracy of 85%, the probability that the test incorrectly predicts the emissions to be over the permissible level when they are not (a false positive) is 15% or 0.15. Therefore, the probability that emissions are within the permissible level and the test incorrectly predicts the opposite is the product of the two probabilities: 0.65 * 0.15 = 0.0975 or 9.75% when expressed as a percentage. To provide the answer to the thousandth place as required, we express this as 0.098.
Solve this linear equation simultaneously using the sub method.
2x+y=4
x+y=10
Answer:
(-6,16), also can be written as x=-6, y=16.
Step-by-step explanation:
By 'sub method' I assume you mean substitution of one solved equation into another. Let's begin by solving x+y=10 for y.
[tex]x+y=10[/tex]
-x on both sides
[tex]y=10-x[/tex]
Now we have an equation that tells us y=10-x. We can now replace 'y' in the other equation with this new definition of y, that being 10-x.
[tex]2x+y=4[/tex]
Substitute 10-x for y
[tex]2x+(10-x)=4[/tex]
When can remove the parentheses as they are unnecessary.
[tex]2x+10-x=4[/tex]
Now we solve for x. We can first simplify, by combining all like terms on each side, those being 2x and -x on the left side. 2x+-x=x.
[tex]x+10=4[/tex]
Now we remove 10 from both sides, to isolate x.
[tex]x=4-10[/tex]
We can again simplify, 4-10=-6
[tex]x=-6[/tex]
Now we have the x coordinate of the point that solves our set of equations. We can plug this number back as an x value of either equation to get the y value for the solution. In this example I will use x+y=10, as it is a simpler equation.
[tex]x+y=10[/tex]
Substitute x for -6.
[tex]-6+y=10[/tex]
Isolate y by adding 6 to both sides, cancelling out the -6 on the left side.
[tex]y=10+6[/tex]
Simplify 10+6=16.
[tex]y=16[/tex]
Now we have an x value, -6, and a y value, 16. These are the x and y values of our solution, therefore the solution is (-6,16).
This is the solution using the sub method. There are other ways to solve this question (although the answer, if done correctly, will always be the same), including plotting both lines on a graph, as shown in the attached image.
How to find the unknown measure of a rectangle whose area is 91 and width is 7.
Answer:
Length of rectangle= 13
Step-by-step explanation:
Area= Length × width
91= length × 7
length= 91 ÷7 = 13
What is the value of x?
6x - 3
3x + 6
Enter your answer in the box.
BA
- 5x
Answer:
The correct answer is x = 3
Complete question and statement:
Given the equilateral triangle Δ ABC in the graph attached,
AB = 6x - 3
AC = 3x + 6
What is the value of x?
Source: Previous question that can be found at brainly
Step-by-step explanation:
Let's recall that an equilateral triangle has its three sides equal. therefore:
AB = AC
6x - 3 = 3x + 6
3x = 9 (Common terms)
x = 9/3 (Dividing by 3 at both sides)
x = 3 ⇒ AB = 15 units, AC = 15 units, BC = 15 units
Is 4/19 a rational number?
Yes, 4/19 is a rational number because it can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero.
Explanation:Yes, 4/19 is a rational number. In mathematics, a rational number is defined as a number that can be expressed as the quotient or fraction p/q of two integers, where the denominator q is not equal to zero. Here, 4 and 19 are both integers, and 19 (the denominator) is not zero, so 4/19 meets the criteria for being a rational number. An example of an irrational number would be the square root of 2, because it cannot be exactly expressed as a fraction.
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Marla has budgeted $65 per day on food during a business trip. Write a function that is a model for the situation.
A. Not Enough Information
B. f(x) = 65x
C. f(x) = 65
D. f(x) = 65y
Answer:
A
Step-by-step explanation:
That seems like it would be the correct equation.
Answer:
a
Step-by-step explanation:
it is the tell how many days
following frequency distribution shows the daily expenditure on milk of 30 households in a locality:Daily expenditure on milk(in Rs):0-30,30-60,60-90,90-120,120-150 No.of households:5,6,9,6,4
Note: As you missed to identify what we have to find in this question. But, after a little research, I am able to find that we had to find the Mode for the data given in your question. So, I am assuming we have to calculate the the Mode. Hopefully, it would clear your concept regarding this topic.
Answer:
The mode of the data = 75
Step-by-step explanation:
Lets visualize the given data in a table to show the frequency distribution:
Daily expenditure on milk (in Rs) Number of households
0-30 5
30-60 6
60-90 9
90-120 6
120-150 4
Here the maximum frequency is 9.
So, modal class is 60-90.
As the formula to calculate the mode:
[tex]Mode = l_{1} + h (\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} )[/tex]
Here, the maximum
[tex]l_{1} =60, f_{1} =9, f_{0}=6, f_{0}=6, h=30[/tex]
[tex]l=[/tex] is the lower limit of the class
[tex]f_{1} =[/tex] is the frequency of the modal class
[tex]f_{0} =[/tex] is the frequency of the previous modal class
[tex]f_{2} =[/tex] is the frequency of the next previous modal class
[tex]l=[/tex] is the class size
So,
[tex]Mode = l_{1} + h (\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} )[/tex]
[tex]Mode = 60 + 30 (\frac{9-6}{2(9)-6-6} )[/tex]
[tex]Mode = 60 + \frac{(30)(3)}{6}[/tex]
[tex]Mode = 60 + 15=75[/tex]
∴ The mode of the data = 75
Keywords: mode, frequency distribution
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Solve -3.4 - x = -2 1/2
Please include explanation!!
Answer:
x = -0.9
Step-by-step explanation:
You need to isolate x in order to find x so,
-3.4 - x = -2 1/2 or -2.5
-3.4 - x = -2.5
Add 3.4 to -2.5
-3.4 - x = -2.5
+3.4 +3.4
-x = 0.9
x CANNOT be negative so divide -1 to both sides
-x/-1 = 0.9/-1
x = -0.9
Checking answer
-3.4 - (-0.9) = -2.5
-3.4 + 0.9 = -2.5
-2.5 = -2.5
Which table represents the graph of a logarithmic function in the form y-log, when : > 1?
Nex
-1.9
-2.096
-1.75
-1.262
The Table represents the graph of a logarithmic function in the form is first table.
What is Logarithm?A logarithm is defined as the number of powers to which a number must be increased in order to obtain some other numbers. It is the simplest way to express enormous numbers. A logarithm has several key features that demonstrate that logarithm multiplication and division can also be represented in the form of logarithm addition and subtraction.
We have,
y= log (base b) x
Take the value y= -3 and x= 1/8.
So, -3 = log (base b) 1/8
[tex]b^{-3[/tex] = 1/8
[tex]b^{-3[/tex] = 1/2³
[tex]b^{-3[/tex] = [tex]2^{-3[/tex]
On comparing b=2.
Now, if y=1 and x= 2
1= log(base b)2
b= 2
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help me and i’ll mark you brainliest plus you get 20 points !!!!
Answer:
Angle bisector
Step-by-step explanation:
Find the measure of the angle COA. By angle addition postulate,
[tex]m\angle COX=m\angle AOX+m\angle COA[/tex]
From the diagram,
[tex]m\angle COX=80^{\circ}\\ \\m\angle AOX=40^{\circ},[/tex]
then
[tex]80^{\circ}=m\angle COA+40^{\circ}\\ \\m\angle COA=80^{\circ}-40^{\circ}=40^{\circ}[/tex]
Find the measure of the angle BOA. By angle addition postulate,
[tex]m\angle BOX=m\angle AOX+m\angle BOA[/tex]
From the diagram,
[tex]m\angle BOX=60^{\circ}\\ \\m\angle AOX=40^{\circ},[/tex]
then
[tex]60^{\circ}=m\angle BOA+40^{\circ}\\ \\m\angle BOA=60^{\circ}-40^{\circ}=20^{\circ}[/tex]
Find the measure of the angle COB. By angle addition postulate,
[tex]m\angle COX=m\angle BOX+m\angle COB[/tex]
From the diagram,
[tex]m\angle BOX=60^{\circ}\\ \\m\angle COX=80^{\circ},[/tex]
then
[tex]80^{\circ}=m\angle COB+60^{\circ}\\ \\m\angle COB=80^{\circ}-60^{\circ}=20^{\circ}[/tex]
This means, the measures of angles COB and BOA are the same and are equal half the measure of angle COA, so angles COB and BOA are congruent. This means, the ray OB is the angle bisector of angle COA
A taxi covers a certain distance in 5 hours and the speed of 60 k/ml how much time will the taxi take to cover the same distance at a speed of 70 km/h
It will take [tex]4\frac{2}{7}\ hours[/tex] to cover same distance.
Step-by-step explanation:
Given,
Time period = 5 hours
Speed = 60 km/h
Distance = Speed * Time
Distance = 60*5 = 300 km
Now,
Distance = 300km
Speed = 70 km/h
Distance = Speed * Time
Time = [tex]\frac{Distance}{Speed}[/tex]
[tex]Time=\frac{300}{70}\\\\Time=\frac{30}{7}\\\\Time= 4\frac{2}{7}\ hours[/tex]
It will take [tex]4\frac{2}{7}\ hours[/tex] to cover same distance.
Keywords: speed, distance
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what is the best estimate for 52% of 320?
Answer:166.4 or 166
Step-by-step explanation:
Tiffany sketched a picture of a car she used the scale 2 inches : 12 feet the car in her sketch is 8 inches long what is the length in feet of the actual car
Answer:
The actual length of car is 48 feet.
Step-by-step explanation:
Given:
Tiffany sketched a picture of a car she used the scale 2 inches : 12 feet.
In her sketch the car is 8 inches long.
Now, to find the length in feet of the actual car.
Let the actual length of car in feet be [tex]x\ feet.[/tex]
And the length of car in her sketch is 8 inches.
So, the ratio of the scale used by Tiffany as given is 2 inches : 12 feet.
Now, to get the actual length of car by using cross multiplication method:
[tex]\frac{2\ inches}{12\ feet} =\frac{8\ inches}{x\ feet}[/tex]
⇒ [tex]\frac{2}{12} =\frac{8}{x}[/tex]
By cross multiplying we get:
⇒ [tex]2x=96[/tex]
Dividing both sides by 2 we get:
⇒ [tex]48=x[/tex]
⇒ [tex]x=48\ feet.[/tex]
Therefore, the actual length of car is 48 feet.
What is an equation of the line that passes through (2,2) and is parellel to the line y=7x
Answer:
Therefore, equation of the line that passes through (2,2) and is parellel to the line [tex]y=7x[/tex] is [tex]y=7x-12[/tex]
Step-by-step explanation:
Given:
a line [tex]y=7x[/tex]
To Find:
Equation of line passing through ( 2, 2) and is parellel to the line y=7x
Solution:
[tex]y=7x[/tex] ...........Given
Comparing with,
[tex]y=mx[/tex]
Where m =slope
We get
[tex]Slope = m = 7[/tex]
We know that parallel lines have Equal slopes.
Therefore the slope of the required line passing through (2 , 2) will also have the slope = m = 7.
Now the equation of line in slope point form given by
[tex](y-y_{1})=m(x-x_{1})[/tex]
Substituting the points and so we will get the required equation of the line,
[tex](y-2)=7(x-2)=7x-14\\\\y=7x-12......Equation\ of\ line[/tex]
Therefore, equation of the line that passes through (2,2) and is parellel to the line [tex]y=7x[/tex] is [tex]y=7x-12[/tex]
HELP ASAP. need this done.
Answer:
D. 35
Step-by-step explanation:
[tex]20 \times \frac{7}{4} = 35[/tex]
Good morning,
Answer:
D. 35
Step-by-step explanation:
it Has the ratio of 4 to 7 then
it Has the ratio of 4×5 to 7×5 then
it Has the ratio of 20 to 35.
:)
You sell small and large candles at a craft fair. You collect $144 selling a total of 28 candles. How many of each type of candle did you sell?
Final answer:
Without knowing the individual prices of small and large candles, we cannot solve for the exact numbers of each type sold. A system of equations would normally be used, but in this case, essential price information is missing.
Explanation:
You sell small and large candles at a craft fair and collect $144 selling a total of 28 candles. To solve how many of each type of candle you sold, let's set up a system of equations with two variables. Let x represent the number of small candles and y represent the number of large candles.
The first equation comes from the total number of candles:
x + y = 28
The second equation involves the total amount of money collected:
ax + by = 144
where a and b are the prices of small and large candles respectively.
Unfortunately, the question does not provide the individual prices of the small and large candles, so we cannot continue without that information. To solve this system correctly, you would need to know the price of at least one type of candle.
With the missing price information, we cannot define a and b and therefore cannot provide a numerical solution to this question.
What is 4 7/8 x 28in simplest form
Answer:
[tex]4\frac{7}{8} * 28 = \frac{39}{8}*28\\= \frac{39*28}{8} \\=\frac{39*7}{2} \\=\frac{273}{2}\\ =136\frac{1}{2} \\[/tex]
Step-by-step explanation:
Which graph represents the solution to the given system? Y = -6x-2Y+2=-6x
Answer: The graph is attached.
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope and "b" is the y-intercept.
So, having the first equation:
[tex]y=- 6x - 2[/tex]
You can identify that:
[tex]m=-6\\b=-2[/tex]
By definition, the line intersects the x-axis when [tex]y=0[/tex]. Subsituting this value into the equation and solving for "x", you get that the x-intercept is the following:
[tex]0=- 6x - 2\\\\2=-6x\\\\x=-\frac{1}{3}\\\\x=-0.333[/tex]
Now you can graph the line.
Now you must solve for "y" from the second equation:
[tex]y +2=- 6x\\\\y=-6x-2[/tex]
You can identify that:
[tex]m=-6\\b=-2[/tex]
Notice that the slopes and the y-intercepts of the first line and the second line are equal; this means that they are exactly the same line and the System of equations has Infinitely many solutions.
See the graph attached.
Answer:
a
Step-by-step explanation:
Determine the corresponding y-value for x = 35 using y = -x + 175 (slope-intercept form of the equation of a line
Answer:
Step-by-step explanation:
when x=35
y=-35+175=140
What’s the answer please?
ans is 9/4 .................
Find the domain of -1/x^2-4
Answer:
[tex]\mathbb{R}-\{0\}[/tex]
[tex](-\infty,0)\ U\ (0,+\infty)[/tex]
Step-by-step explanation:
The domain of a Function
Given a real function f(x), the domain of f is made of all the values x can take, such that f exists. The function given in the question is
[tex]\displaystyle f(x)=-\frac{1}{x^2}-4[/tex]
Finding the domain of a function is not possible by giving x every possible value and check if f exists in all of them. It's better to find the values where f does NOT exist and exclude those values from the real numbers.
Since f is a rational function, we know the denominator cannot be 0 because the division by 0 is not defined, so we use the denominator to find the values of x to exclude from the domain.
We set
[tex]x^2=0[/tex]
Or equivalently
x=0
The domain of f can be written as
[tex]\mathbb{R}-\{0\}[/tex]
Or also
[tex](-\infty,0)\ U\ (0,+\infty)[/tex]
how many bricks can my truck carry in a full load if each brick weighs 4 pounds 14 ounces and my truck can carry a 3/4 ton load
Answer:
The number of bricks a truck can carry in full load is 308 .
Step-by-step explanation:
Given as :
The weight of each brick = 4 pounds and 14 ounce
The total load a truck can carry = [tex]\dfrac{3}{4}[/tex] tons
Let The number of bricks truck can carry = n bricks
Now, According to question
∵ 1 ounce = 0.0625 pounds
∴ 14 ounce = 0.0625 × 14 = 0.875 pounds
So, Total weight of each brick = 4 pounds + 0.875 pounds = 4.875 pounds
Again
∵ 1 pound = 0.0005 tons
∴ 4.875 pounds = 0.0005 × 4.875 = 0.0024375 tons
Now, Again
The total load a truck can carry = [tex]\dfrac{3}{4}[/tex] tons = 0.75 tons
And The weight of each brick = 0.0024375 tons
So, The number of bricks = [tex]\dfrac{\textrm total load a truck can carry}{\textrm Total weight of each brick}[/tex]
I.e n = [tex]\dfrac{0.75}{0.0024375}[/tex]
∴ n = 307.69 ≈ 308
So, The number of bricks can truck carry = n = 308
Hence, The number of bricks a truck can carry in full load is 308 . Answer
Complete the equation of the line through (-8,-2) and (-4,6). Use exact numbers.
Y=
Answer:
y = 2x + 14
Step-by-step explanation:
First, you have to find the slope by using
[tex] \frac{y2 - y1}{x2 - x1} [/tex]
In other words,
[tex] \frac{ - 2 - 6}{ - 8 + 4} [/tex]
The slope is 2.
Then you plug the rest into point-slope form (you can use either of the points, I used the first one)
[tex]y - y1 = m(x - x1)[/tex]
Remember that m is the slope.
[tex]y + 2 = 2(x + 8)[/tex]
Distribute the slope to the parenthesis
[tex]y + 2 = 2x + 16[/tex]
Isolate the y variable
[tex]y = 2x + 14[/tex]
Answer: y=2x+14
Step-by-step explanation:
Yes
write the equation of the line that is perpendicular to the given line and that passes through the given point. y = -1/3x+5;(4,3)
Answer:
The equation of line perpendicular to given line equation and passing through point (4,3) is y = 3 x - 9 .
Step-by-step explanation:
Given as :
The given equation of one line = y = [tex]\dfrac{-1}{3}[/tex]x + 5
∵ Equation of line in slope-intercept form is written as
y = m x + c
where m is the slope of line
And c is the intercept of y
Now, Comparing given line equation with standard slope intercept line equation
∴ m = [tex]\dfrac{-1}{3}[/tex]
Slope of this line = m = [tex]\dfrac{-1}{3}[/tex]
Now, another line is perpendicular to the given line
For perpendicular lines , the products of slope of lines = - 1
Let the slope of another line = M
So, from perpendicular lines condition
m × M = - 1
∴ M = [tex]\dfrac{-1}{m}[/tex]
I.e M = [tex]\frac{-1}{\frac{-1}{3}}[/tex]
So, M = 3
∴ The slope of other line = M = 3 , and the line passing through point (4,3)
Now, Again
The equation of line in slope-intercept form
I.e y = M x + c
Now, satisfying the points on line
So, 3 = 3 × 4 + c
Or, 3 = 12 + c
∴ c = 3 - 12
i.e c = - 9
or, The other line equation = y = 3 x - 9
Hence, The equation of line perpendicular to given line equation and passing through point (4,3) is y = 3 x - 9 . Answer
verify rolles theorem for f(x) =x^3-x^2-6x+2 in [0,3]
Answer:
See the explanation.
Step-by-step explanation:
Given function [tex]f(x)=x^3-x^2-6x+2[/tex]
And the interval [tex][0,3][/tex]
According to Rolle's Theorem
Let [tex]f(x)[/tex] be differentiable on the open interval [tex](a,b)[/tex] and continuous on the closed interval [tex][a,b][/tex]. Then if [tex]f(a)=f(b)[/tex], then there is at least one point [tex]x\ in\ (a,b)[/tex] where [tex]f'(x)=0[/tex].
So,
[tex]f(0)=0^3-0^2-6\times0+2\\\\f(0)=2\\\\Similarly\\\\f(3)=3^3-3^2-6\times3+2\\\\f(3)=27-9-18+2\\\\f(3)=2\\\\We\ can\ see\ f(0)=f(3)\\\\Now,\\\\f'(x)=\frac{d}{dx}(x^3-x^2-6x+2)\\\\f'(x)=3x^2-2x-6\\\\put\ f'(x)=0\\\\3x^2-2x-6=0\\[/tex]
We will find the value of [tex]x[/tex] for which [tex]f'(x)[/tex] became zero.
[tex]If\ ax^2+bx+c=0\\\\Then,\ x_{1}=\frac{-b+\sqrt{b^2-4ac}}{2a}\\ \\And\ x_{2}=\frac{-b-\sqrt{b^2-4ac}}{2a}\\\\f'(x)=3x^2-2x+6=0\\a=3,\ b=-2,\ c=6\\\\\ x_{1}=\frac{-(-2)+\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2+\sqrt{76}}{6}=\frac{2+2\sqrt{19}}{6}\\\\x_{1}=\frac{1+\sqrt{19}}{3}\\\\x_{2}=\frac{-(-2)-\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2-\sqrt{76}}{6}=\frac{2-2\sqrt{19}}{6}\\\\x_{2}=\frac{1-\sqrt{19}}{3}[/tex]
We can see
[tex]x_{1}=\frac{1+\sqrt{19}}{3}=1.786\\\\and\ 1.786\ is\ in\ [0,3][/tex]
There is at least one point [tex]1.786\ in\ (0,3)[/tex] where [tex]f'(x)=0[/tex].
Rolle's theorem requires a function to be continuous on [a, b], differentiable on (a, b), and have f(a) = f(b). After verifying these conditions are met for f(x) = x^3 - x^2 - 6x + 2, we look for a point in (0, 3) where f'(x) = 0. We cannot find such a point in the interval, suggesting an error in the application of the theorem.
To verify Rolle's theorem for the function f(x) = x^3 - x^2 - 6x + 2 in the interval [0,3], we need to ensure that the function meets the criteria stated by the theorem:
The function f must be continuous on the closed interval [a, b].
The function f must be differentiable on the open interval (a, b).
The function f must satisfy f(a) = f(b).
f(x) = x^3 - x^2 - 6x + 2 is a polynomial, which is continuous and differentiable everywhere. Therefore, it satisfies the first two conditions of Rolle's theorem on any interval, including [0,3].
Now we need to check the third condition:
f(0) = (0)^3 - (0)^2 - 6(0) + 2 = 2
f(3) = (3)^3 - (3)^2 - 6(3) + 2 = 2
Since f(0) = f(3), the third condition is also met. Hence, Rolle's theorem applies, and there must be at least one c in (0, 3) such that f'(c) = 0.
To find c, we take the derivative of f(x):
f'(x) = 3x^2 - 2x - 6
Set f'(x) equal to zero and solve for x:
3x^2 - 2x - 6 = 0
Factors into (3x + 2)(x - 3) = 0
So x can be either x = -2/3 or x = 3. Since -2/3 is not in our interval, the only possibility in the interval (0, 3) is at x = 3. However, in this case, the point x = 3 is an endpoint, hence it's not within the open interval (0, 3).
Thus, we need to conclude that there has been a mistake, as x = 3 does not satisfy the condition for Rolle's theorem within the open interval (0, 3).
Can any one solve 10 I’ll give Brainiest
Answer:
circumference of a circle = 2πr
= 2×22/7×7
= 44/7×7
= 308/7
= 44 inches
3 An equation is shown below.
7x = -56
What value of x makes the equation true?
A X=-63
B x=-8
C x = -0.125
D x = -392
Answer:
Option B is correct.
[tex]x=-8[/tex]
Step-by-step explanation:
Given:
The given equation is.
[tex]7x=-56[/tex]
Solve above equation for value of x.
[tex]7x=-56[/tex]
[tex]x=-\frac{56}{7}[/tex]
[tex]x=-8[/tex]
Therefore. the value of [tex]x=-8[/tex]
Todd wants to find 352 + 116. Break
apart 116 by place value and use the
open number line and the adding on
strategy to find the sum.
Answer
given,
352 + 116
we have to break 116 and use it on open number line
breaking of number 116
116 = 100 + 10 + 6
And addition will be shown in the number line attached below
now,
352 + 100 = 452
452 + 10 = 462
462 + 6 = 468
we know
sum of
352 + 116 = 468
y = -1/4x + 2 and 2y = 1/2x - 4
how do these lines compare?
1. Slopes:
- The first line has a negative slope of -1/4.
- The second line has a positive slope of 1/4.
2. Y-intercepts:
- The first line intersects the y-axis at y = 2.
- The second line intersects the y-axis at y = -2.
To compare these lines, let's first rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
1. For the first equation, [tex]\( Y = -\frac{1}{4}x + 2 \)[/tex]:
- Slope (m₁) = -1/4
- Y-intercept (b₁) = 2
2. For the second equation, [tex]\( 2y = \frac{1}{2}x - 4 \)[/tex]:
- We divide both sides by 2 to isolate y:
[tex]\( y = \frac{1}{4}x - 2 \)[/tex]
- Slope (m₂) = 1/4
- Y-intercept (b₂) = -2
Now, let's compare:
1. Slopes:
- The first line has a negative slope of -1/4.
- The second line has a positive slope of 1/4.
2. Y-intercepts:
- The first line intersects the y-axis at y = 2.
- The second line intersects the y-axis at y = -2.