Simplified form of expression is,
⇒ - 3√3
We have to given that,
An expression is,
⇒ 2√27 + √12 - 3√3 - 2√12
We can simplify it as,
⇒ 2√27 + √12 - 3√3 - 2√12
⇒ 2 √9×3 + √3×4 - 3√3 - 2√3×4
⇒ 2 × 3 √3 - 2√3 - 3√3 - 2 × 2√3
⇒ 6√3 - 2√3 - 3√3 - 4√3
⇒ - 3√3
Therefore, Simplified form of expression is,
⇒ - 3√3
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Final answer:
The expression simplifies to 4sqrt(3) after breaking down the square roots of 27 and 12 into 3√(3) and 2√(3), respectively, and combining like terms.
Explanation:
To simplify the expression 2√(27) + √(12) - 3√(3) - 2√(12), we should first simplify √(27) and √(12) individually.
Square root of 27 can be written as √(9*3). Since 9 is a perfect square, we get 3√(3). Similarly, √(12) can be simplified to √(4*3). As 4 is a perfect square, this simplifies to 2√(3).
Now, substituting back into the original expression, we get 2(3√(3)) + 2√(3) - 3√(3) - 2(2√(3)), which simplifies to 6√(3) + 2√(3) - 3√(3) - 4√(3).
Combining like terms, we obtain 6√(3) - 2√(3), which simplifies further to 4√(3).
Therefore, the simplified form of the original expression is 4√(3).
PLEASE HURRY
WILL GIVE BRAINLIEST
What is the equation for the hyperbola shown?
Find the answer in the attachment.
The hyperbola's equation is x² / 3600 - y² / 121 = 1, centered at the origin (0,0). Its vertices are at (60,0), (-60,0) on the x-axis, and (0,11), (0,-11) on the y-axis.
To find the equation of the hyperbola, we need to determine its center and the distances from the center to the vertices along the x and y axes. The general equation of a hyperbola centered at (h, k) is given by:
(x - h)² / a² - (y - k)² / b² = 1
Where (h, k) is the center of the hyperbola, and 'a' and 'b' are the distances from the center to the vertices along the x and y axes, respectively.
In this case, since the hyperbola is symmetric along the x and y axes, the center is at the origin (0, 0). Also, we know the distance from the center to the vertices along the x-axis is 60 units (60 and -60) and along the y-axis is 11 units (11 and -11).
So, a = 60 and b = 11.
Now we can plug these values into the equation:
x² / (60)² - y² / (11)² = 1
Simplifying further:
x² / 3600 - y² / 121 = 1
And that's the equation of the hyperbola.
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which of the following recursive formulas represent the same arithmetic sequence as the explicit formula an=5+(n-1)2
a. a1=5
an=an-1+2
b. a1=5
an=(an-1+2)5
c. a1=2
an=an-1+5
d. a1=2
an=an-1*5
Answer:
Choice A:
[tex]a_1=5[/tex]
[tex]a_{n}=a_{n-1}+2[/tex]
Step-by-step explanation:
[tex]a_n=5+(n-1)2[/tex]
means we looking for first term 5 and the sequence is going up by 2.
In general,
[tex]a_n=a_1+(n-1)d[/tex]
means you have first term [tex]a_1[/tex] and the sequence has a common difference of d.
So it is between the first two choices.
The explicit form of an arithmetic sequence is: [tex]a_n=a_1+(n-1)d[/tex]
An equivalent recursive form is [tex]a_n=a_{n-1}+d \text{ where } a_1 \text{ is the first term}[/tex]
So d again here is 2.
So choice a is correct.
[tex]a_1=5[/tex]
[tex]a_{n}=a_{n-1}+2[/tex]
Answer: Option a
[tex]\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.[/tex]
Step-by-step explanation:
The arithmetic sequences have the following explicit formula
[tex]a_n=a_1 +(n-1)*d[/tex]
Where d is the common difference between the consecutive terms and [tex]a_1[/tex] is the first term of the sequence:
The recursive formula for an arithmetic sequence is as follows
[tex]\left \{ {{a_1} \atop {a_n=a_{(n-1)}+d}} \right.[/tex]
Where d is the common difference between the consecutive terms and [tex]a_1[/tex] is the first term of the sequence:
In this case we have the explicit formula [tex]a_n=5+(n-1)*2[/tex]
Notice that in this case
[tex]a_1 = 5\\d = 2[/tex]
Then the recursive formula is:
[tex]\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.[/tex]
The answer is the option a.
PLEASE HELP AS FAST AS POSSIBLE PLEASE HELP PLEASE
Evaluate 0.00008 ÷ 640,000,000.
A) 1.25 × 10 -15
B) 1.25 × 10 -14
C) 1.25 × 10 -13
D) 1.25 × 10 -12
Answer:
0.00008 ÷ 640,000,000 means
8*10^-5 ÷ 6.4*10^8
so let's collect to simplify the operation
(8÷6.4)*(10^-12) -5-7=-12
then the answer becomes 1.25×10^-14 that is B
Answer:
option C
Step-by-step explanation:
Evaluate 0.00008 ÷ 640,000,000.
0.00008 can be written in standard notation
Move the decimal point to the end
so it becomes [tex]8 \cdot 10^{-5}[/tex]
for 640,000,000 , remove all the zeros and write it in standard form
[tex]64 \cdot 10^7[/tex]
Now we divide both
[tex]\frac{8 \cdot 10^{-5}}{64 \cdot 10^7}[/tex]
Apply exponential property
a^m divide by a^n is a^ m-n
[tex]\frac{8}{64} =0.125[/tex]
[tex]\frac{10^{-5}}{10^7}=10^{-12}[/tex]
[tex]0.125 \cdot 10^{-12}= 1.25 \cdot 10^{-13}[/tex]
Steps for solving 3x + 18 = 54 are shown.
Explain how Step 1 helps solve the equation.
3x+18 = 54
3x + 18-18 = 54 -18
3x = 36
3x_36
33
X = 12
Original equation
Step 1
Step 2
Step 3
Step 4
) A. Adding 18 to both sides undoes the subtraction.
O
B. Adding 18 to both sides combines like terms.
O
C. Subtracting 18 from both sides isolates the variable term.
O
D. Subtracting 18 from both sides isolates the variable.
C
By subtracting 18 to both sides you are separating the 3x because you are not ready to isolate the variable (x) in step 1
Answer:
WELP i got little different answers and for me it was A
A.
Adding 18 to both sides isolates the variable term.
B.
Adding 18 to both sides isolates the variable.
C.
Subtracting 18 from both sides undoes the subtraction.
D.
Subtracting 18 from both sides combines like terms.
The Montanez family is a family of four people. They have used 3,485.78 gallons of water so far this month. They cannot exceed 7,250.50 gallons per month during the drought season. Write an inequality to show how much water just one member of the family can use for the remainder of the month, assuming each family member uses the same amount of water every month.
Answer:
x ≤ 3764.72
Step-by-step expl anation:
The Montanez family cannot use more than 7250.50 gallons, this means that they can use less than or equal to 7250.50 gallons, this tells you which sign to use. The variable x can be used to describe how much water they have left to use and then you add 3,485.78 gallons to x.
x + 3485.78 ≤ 7250.50 This inequality means that the amount of water the family has yet to use added to 3485.78 gallons cannot exceed 7250.50 gallons.
Next, you simplify the inequality using the property of inequalities.
x ≤ 7250.50 - 3485.78
x ≤ 3764.72
The required inequality is x<941.18 for the Montanez family if they have used 3485.78 gallons of water.
What is inequality?It is also a relationship between variables but this is in greater than , less than, greater than or equal to , less than or equal to is used.
How to form an inequality?Let the water used by a member be x
so
x*4=7250.50-3485.78
4x=3764.72
x=941.18
Hence one member can only use 941.18 gallons of water in the month.
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one x-intercept for a parabola is at the point (2, 0). use the quadratic formula to find the other x-intercept for the parabola defined by y=x^2-3x+2
Answer:
Step-by-step explanation:
There are 3 ways to find the other x intercept.
1) Polynomial Long Division.
Divide x^2 - 3x + 2 by the binomial x - 2, because by the Factor Theorem if a is a root of a polynomial then x - a is a factor of said polynomial.
2) Just solving for x when y = 0, by using the quadratic formula.
[tex]x^2 - 3x + 2 = 0\\x_{12} = \frac{3 \pm \sqrt{9 - 4(1)(2)}}{2} = \frac{3 \pm 1}{2} = 2, 1[/tex].
So the other x - intercept is at (1, 0)
3) Using Vietta's Theorem regarding the solutions of a quadratic
Namely, the sum of the solutions of a quadratic equation is equal to the quotient between the negative coefficient of the linear term divided by the coefficient of the quadratic term.
[tex]x_1 + x_2 = \frac{-b}{a}[/tex]
And the product between the solutions of a quadratic equation is just the quotient between the constant term and the coefficient of the quadratic term.
[tex]x_1 \cdot x_2 = \frac{c}{a}[/tex]
These relations between the solutions give us a brief idea of what the solutions should be like.
What is the solution of 4.5x-100>125
Answer:
x > 50Step-by-step explanation:
[tex]4.5x-100>125\qquad\text{add 100 to both sides}\\\\4.5x-100+100>125+100\\\\4.5x>225\qquad\text{divide both sides by 4.5}\\\\\dfrac{4.5x}{4.5}>\dfrac{225}{4.5}\\\\x>50[/tex]
The solution to the inequality 4.5x - 100 > 125 is x > 50.
The given inequality is 4.5x - 100 > 125.
To solve it, first, add 100 to both sides to get 4.5x > 225. Then, divide by 4.5 to find x > 50. Therefore, the solution to the inequality is x > 50.
how to divide (x^2+5x-6)/(x-1)
Answer:
x+6
Step-by-step explanation:
Let's see if the numerator is factorable.
Since the coefficient of x^2 is 1 (a=1), all you have to do is find two numbers that multiply to be -6 (c) and add up to be 5 (b).
Those numbers are 6 and -1.
So the factored form of the numerator is (x+6)(x-1)
So when you divide (x+6)(x-1) by (x-1) you get (x+6) because (x-1)/(x-1)=1 for number x except x=1 (since that would lead to division by 0).
Anyways, this is what I'm saying:
[tex]\frac{(x+6)(x-1)}{(x-1)}=\frac{(x+6)\xout{(x-1)}}{\xout{(x-1)}}[/tex]
[tex]x+6[/tex]
PLEASE HELP!!! Given the functions, f(x) = 6x + 2 and g(x) = x - 7, perform the indicated operation. When applicable, state the domain restriction. (f/g)(x)
To find (f/g)(x) with f(x) = 6x + 2 and g(x) = x - 7, one must divide f(x) by g(x). The domain restriction occurs because division by zero is not defined, so we exclude the x value that makes g(x) zero, which is x = 7.
Explanation:To perform the indicated operation (f/g)(x) with the given functions f(x) = 6x + 2 and g(x) = x - 7, we need to divide the function f(x) by the function g(x). This operation is equivalent to finding the quotient of the two functions, which is expressed as:
(f/g)(x) = f(x)/g(x) = (6x + 2)/(x - 7)
The domain restriction occurs when the denominator, g(x), is equal to zero since division by zero is undefined. So we must find the value of x for which g(x) = 0. Since g(x) = x - 7, setting this equal to zero gives us:
x - 7 = 0 → x = 7
Therefore, the domain of the function (f/g)(x) is all real numbers except for x = 7, because at x = 7 the function is undefined. The domain of (f/g)(x) can be expressed as ℜ - {7}, where ℜ represents the set of all real numbers.
What is the slope of st.line xcosa+ysina=p? ( Find by using derivative)
Answer:
Assume that [tex]a[/tex] and [tex]p[/tex] are constants. The slope of the line will be equal to
[tex]\displaystyle -\frac{\cos{(a)}}{\sin{(a)}} = \cot{(a)}[/tex] if [tex]\sin{a} \ne 0[/tex];Infinity if [tex]\sin{a} = 0[/tex].Step-by-step explanation:
Rewrite the expression of the line to express [tex]y[/tex] in terms of [tex]x[/tex] and the constants.
Substract [tex]x\cdot \cos{(a)}[/tex] from both sides of the equation:
[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].
In case [tex]\sin{a} \ne 0[/tex], divide both sides with [tex]\sin{a}[/tex]:
[tex]\displaystyle y = - \frac{\cos{(a)}}{\sin{(a)}}\cdot x+ \frac{p}{\sin{(a)}}[/tex].
Take the first derivative of both sides with respect to [tex]x[/tex]. [tex]\frac{p}{\sin{(a)}}[/tex] is a constant, so its first derivative will be zero.
[tex]\displaystyle \frac{dy}{dx} = - \frac{\cos{(a)}}{\sin{(a)}}[/tex].
[tex]\displaystyle \frac{dy}{dx}[/tex] is the slope of this line. The slope of this line is therefore
[tex]\displaystyle - \frac{\cos{(a)}}{\sin{(a)}} = -\cot{(a)}[/tex].
In case [tex]\sin{a} = 0[/tex], the equation of this line becomes:
[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].
[tex]x\cos{(a)} = p[/tex].
[tex]\displaystyle x = \frac{p}{\cos{(a)}}[/tex],
which is the equation of a vertical line that goes through the point [tex]\displaystyle \left(0, \frac{p}{\cos{(a)}}\right)[/tex]. The slope of this line will be infinity.
Kevin’s car requires 5 liters of diesel to cover 25 kilometers. How many liters of diesel will Kevin need if he has to cover 125 kilometers?
Answer: 25 litres
Step-by-step explanation: If it takes 5 litres of diesel to cover 25km, it means it will take more litres to cover 125km. So you have to divide 125km by 25km then you will get 5 which you will then multiply by 5 litres to get 25 litres of diesel.
Which graph shows the line y-2 = 2(x + 2)?
A/B/co
A. Graph A
B. Graph B
C. Graph D
D. Graph c
Answer:
The answer is Graph A.
Step-by-step explanation:
i got the answer from the teachers answer key
Graph A shows the line y-2 = 2(x + 2).
The answer is option A
What is a straight line graph?
The graph follows a straight line equation shows a straight line graph.
equation of a straight line is y=mx+cy represents vertical line y-axis.x represents the horizontal line x-axis. m is the slope of the lineslope(m)=tan∅=y axis/x axis.
c represents y-intercepts (it is the point at which the line cuts on the y-axis)Straight line graphs show a linear relationship between the x and y values.solving the equation:-
y-2 = 2(x + 2)
y=2x+4+2
Y=2x+6
y-intercepts = 6 (Graph A represents this).
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Solve for x: 5 over x equals 4 over quantity x plus 3
5
3
−3
−15
Answer:
x = - 15
Step-by-step explanation:
The equation is [tex]\frac{5}{x}=\frac{4}{x+3}[/tex]
We now cross mulitply and do algebra to figure the value of x (shown below):
[tex]\frac{5}{x}=\frac{4}{x+3}\\5(x+3)=4(x)\\5x+15=4x\\5x-4x=-15\\x=-15[/tex]
Hence x = -15
Answer:
D
Step-by-step explanation:
the number of three-digit numbers with distinct digits that be formed using the digits 1,2,3,5,8 and 9 is . The probability that both the first digit and the last digit of the three-digit number are even numbers .
Answer:
a)120
b)6.67%
Step-by-step explanation:
Given:
No. of digits given= 6
Digits given= 1,2,3,5,8,9
Number to be formed should be 3-digits, as we have to choose 3 digits from given 6-digits so the no. of combinations will be
6P3= 6!/3!
= 6*5*4*3*2*1/3*2*1
=6*5*4
=120
Now finding the probability that both the first digit and the last digit of the three-digit number are even numbers:
As the first and last digits can only be even
then the form of number can be
a)2n8 or
b)8n2
where n can be 1,3,5 or 9
4*2=8
so there can be 8 three-digit numbers with both the first digit and the last digit even numbers
And probability = 8/120
= 0.0667
=6.67%
The probability that both the first digit and the last digit of the three-digit number are even numbers is 6.67% !
1.
[tex]6\cdot5\cdot4=120[/tex]
2.
[tex]|\Omega|=120\\|A|=2\cdot4\cdot1=8\\\\P(A)=\dfrac{8}{120}=\dfrac{1}{15}\approx6.7\%[/tex]
square root of 3 x^2 times square root of 4x
[tex]\bf \sqrt{3x^2}\cdot \sqrt{4x}\implies \sqrt{3x^2\cdot 4x}\implies \sqrt{12x^2x}\implies \sqrt{4\cdot 3\cdot x^2x} \\\\\\ \sqrt{2^2\cdot 3\cdot x^2x}\implies 2x\sqrt{3x}[/tex]
What type of angles are 1 and 5?
vertical
supplementary
corresponding
complementary
Answer:
corresponding
Step-by-step explanation:
Answer:
Corresponding
Step-by-step explanation:
I like to call corresponding angles, the copy and paste angles because you can copy and paste the top intersection over the bottom intersection; the angles that lay down on top of each other are the corresponding angles. 1 and 5 do this.
Use the intercepts from the graph below to determine the equation of the function.
A) 4x-3y=12
B) -4x-3y=12
C) 4x-3y=-12
D) -4x+3y=-12
ANSWER
C) 4x-3y=-12
EXPLANATION
The intercept form of a straight line is given by:
[tex] \frac{x}{x - intercept} + \frac{y}{y - intercept} = 1[/tex]
From the the x-intercept is -3 and the y-intercept is 4.
This is because each box is one unit each.
We substitute the intercepts to get:
[tex] \frac{x}{ - 3} + \frac{y}{4} = 1[/tex]
We now multiply through by -12 to get
[tex] - 12 \times \frac{x}{ - 3} + - 12 \times \frac{y}{4} = 1 \times - 12[/tex]
[tex]4x - 3y = -12[/tex]
The correct choice is C.
write a compound inequality that represents each situations all real numbers that are greater than -8 but less than 8
Answer:
[tex]-8 < x < 8[/tex]
Step-by-step explanation:
Your compound inequality will include two inequalities.
These are:
x > -8
x < 8
Put your lowest number first, ensuring that your sign is pointed in the correct direction.
[tex]-8 < x[/tex]
Next, enter your higher number, again making sure that your sign is pointing in the correct direction.
[tex]-8 < x < 8[/tex]
Answer:
-8 < r < 8
Step-by-step explanation:
Let r = real number
Greater than >
r>-8
less than <
r <8
We want a compound inequality so we combine these
-8 < r < 8
Determine if the two figures are congruent and explain your answer.
To determine if two figures are congruent, we need to check if one can be obtained from the other through a combination of rigid transformations. The specific transformations that can be applied to one figure to make it coincide with the other need to be described to explain if the figures are congruent or not.
If two figures are congruent, we need to check if one can be obtained from the other through a combination of rigid transformations. Rigid transformations include translations, rotations, and reflections. If we can apply a series of these transformations to one figure to make it coincide exactly with the other, then the figures are congruent.
Without specific information about the figures, I can provide a general explanation of how you might approach this:
Translation (Slide): Check if one figure can be translated (slide) to coincide with the other. If you can move one figure to overlap with the other without rotating or reflecting it, they might be congruent.Rotation (Turn): Check if one figure can be rotated to match the orientation of the other. If a rotation can make the two figures coincide, they may be congruent.Reflection (Flip): Check if one figure can be reflected (flipped) to match the other. If a reflection can be applied to one figure to make it coincide with the other, they may be congruent.To explain your answer, describe the specific transformations (translations, rotations, reflections) that can be applied to one figure to make it coincide with the other. If you can perform a sequence of these transformations to superimpose one figure onto the other, then the figures are congruent. If not, they are not congruent.
The equation of a circle in general form is x2+y2+20x+12y+15=0 . What is the equation of the circle in standard form?
ANSWER
[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]
EXPLANATION
The equation of the circle in general form is given as:
[tex] {x}^{2} + {y}^{2} + 20x + 12y + 15 = 0[/tex]
To obtain the standard form, we need to complete the squares.
We rearrange the terms to obtain:
[tex] {x}^{2} + 20x + {y}^{2} + 12y = - 15 [/tex]
Add the square of half the coefficient of the linear terms to both sides to get:
[tex]{x}^{2} + 20x +100 + {y}^{2} + 12y + 36 = - 15 + 100 + 36[/tex]
Factor the perfect square trinomial and simplify the RHS.
[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]
This is the equation of the circle in standard form.
15.5
tons
155 tons =how many
pounds
An ant can run an inch per second. How long will it take the same ant to run a foot?
Answer:
12 seconds.
Step-by-step explanation:
There are 12 inches in a foot so the ant can run a foot in 1 *12 = 12 seconds.
If a scalene triangle has its measures 4 m, 11 m and 8 m, find the largest angle.
A. 129.8
B. 90.0
C. 34.0
D. 16.2
Answer:
129.8 approximately
Step-by-step explanation:
So this sounds like a problem for the Law of Cosines. The largest angle is always opposite the largest side in a triangle.
So 11 is the largest side so the angle opposite to it is what we are trying to find. Let's call that angle, X.
My math is case sensitive.
X is the angle opposite to the side x.
Law of cosines formula is:
[tex]x^2=a^2+b^2-2ab \cos(X)[/tex]
So we are looking for X.
We know x=11, a=4, and b=8 (it didn't matter if you called b=4 and a=8).
[tex]11^2=4^2+8^2-2(4)(8)\cos(X)[/tex]
[tex]121=16+64-64\cos(X)[/tex]
[tex]121=80-64\cos(X)[/tex]
Subtract 80 on both sides:
[tex]121-80=-64\cos(X)[/tex]
[tex]41=-64\cos(X)[/tex]
Divide both sides by -64:
[tex]\frac{41}{-64}=\cos(X)[/tex]
Now do the inverse of cosine of both sides or just arccos( )
[these are same thing]
[tex]\arccos(\frac{-41}{64})=X[/tex]
Time for the calculator:
X=129.8 approximately
if you are paid $5.50/hour for mowing yards, and you take 3 1/3 hours to mow a yard, how much money are you owed?
[tex]\bf \begin{array}{ccll} \$&hour\\ \cline{1-2} 5.5&1\\ x&3\frac{1}{3} \end{array}\implies \cfrac{5.5}{x}=\cfrac{1}{3\frac{1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{3\cdot 3+1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{10}{3}}\implies \cfrac{5.5}{x}=\cfrac{\frac{1}{1}}{\frac{10}{3}} \\\\\\ \cfrac{5.5}{x}=\cfrac{1}{1}\cdot \cfrac{3}{10}\implies \cfrac{5.5}{x}=\cfrac{3}{10}\implies 55=3x\implies \stackrel{\textit{about 18 bucks and 33 cents}}{\cfrac{55}{3}=x\implies 18\frac{1}{3}=x}[/tex]
Answer:
$18.3
Step-by-step explanation:
If you are paid $5.50/hour for mowing yards, and you take 3 1/3 hours to mow a yard, you should earn $18.3.
3 1/3 hours
$5.50 and hour
$5.50 x 3 = $16.5
$5.50 / 3 = $1.8
$16.5 + $1.8 = $18.3
Therefore, you are owed $18.3.
What is the sum of the complex numbers below?
(5+7i)+(-2+6i)
A. -3 +13i
B. 3+13i
C. -3-13i
D. 3-13i
Answer:
B
Step-by-step explanation:
Given
(5 + 7i) + (- 2 + 6i ) ← remove parenthesis and collect like terms
= 5 + 7i - 2 + 6i
= 3 + 13i → B
The sum of the complex number is 3 + 13i.
Option B is the correct answer.
We have,
To find the sum of the complex numbers (5+7i) and (-2+6i), you can simply add the real parts together and add the imaginary parts together separately.
Real part: 5 + (-2) = 3
Imaginary part: 7i + 6i = 13i
Combining the real and imaginary parts, we get:
Sum = 3 + 13i
Therefore,
The sum of the complex number is 3 + 13i.
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If a fair coin is tossed 11 times, in how many different ways can the sequence of heads and tails appear?
2048 different ways can the sequence of heads and tails appear.
What is possibility?The definition of a possibility is something that may be true or might occur, or something that can be chosen from among a series of choices.
According to the question
Fair coin is tossed 11 times
There are two possibilities for each flip (Heads or Tails). You multiply those together to get the total number of unique sequences.
Here’s an example for 2 flips: HH - HT - TT - TH. (That’s 2 × 2, or [tex]2^{2}[/tex])
Here’s 3 flips: HHH - HHT - HTH - HTT - THH - THT - TTH - TTT. (That’s
2× 2 × 2, or [tex]2^{3}[/tex]).
For ten flips, it’s [tex]2^{10}[/tex]… which is 1024.
For 11 flips, it's [tex]2^{11}[/tex].......Which is 2048
2048 different ways can the sequence of heads and tails appear.
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Final answer:
There are 2048 different ways that a sequence of heads and tails can appear after tossing a fair coin 11 times, because each coin toss has two possible outcomes and the events are independent.
Explanation:
If a fair coin is tossed 11 times, the number of different sequences of heads and tails that can appear is calculated using the formula for the number of outcomes of binomial events. Since each toss of the coin has two possible outcomes (either a head or a tail), and we are tossing the coin 11 times, we use the power of 2 raised to the 11th power, which gives us 211 = 2048. Therefore, there are 2048 different ways a sequence of heads and tails can appear after tossing a coin 11 times.
The reason why there are so many combinations is that each coin toss is independent of the previous one, with a 50 percent chance of landing on either side. This principle is used in probability theory to calculate the possible outcomes of repeated binary events. As an interesting note, if we were to look at possibilities of a specific number of heads and tails, such as 10 heads and 1 tail or vice versa, this would be represented in Pascal's triangle, which reflects the coefficients in the binomial expansion.
The polynomial P(x) = 2x^3 + mx^2-5 leaves the same remainder when divided by (x-1) or (2x + 3). Find the value of m and the remainder.
The polynomial also leaves the same remainder also leaves the same remainder when divided by (qx+r), find
the values of q and r.
Answer:
m=7
Remainder =4
If q=1 then r=3 or r=-1.
If q=2 then r=3.
They are probably looking for q=1 and r=3 because the other combinations were used earlier in the problem.
Step-by-step explanation:
Let's assume the remainders left when doing P divided by (x-1) and P divided by (2x+3) is R.
By remainder theorem we have that:
P(1)=R
P(-3/2)=R
[tex]P(1)=2(1)^3+m(1)^2-5[/tex]
[tex]=2+m-5=m-3[/tex]
[tex]P(\frac{-3}{2})=2(\frac{-3}{2})^3+m(\frac{-3}{2})^2-5[/tex]
[tex]=2(\frac{-27}{8})+m(\frac{9}{4})-5[/tex]
[tex]=-\frac{27}{4}+\frac{9m}{4}-5[/tex]
[tex]=\frac{-27+9m-20}{4}[/tex]
[tex]=\frac{9m-47}{4}[/tex]
Both of these are equal to R.
[tex]m-3=R[/tex]
[tex]\frac{9m-47}{4}=R[/tex]
I'm going to substitute second R which is (9m-47)/4 in place of first R.
[tex]m-3=\frac{9m-47}{4}[/tex]
Multiply both sides by 4:
[tex]4(m-3)=9m-47[/tex]
Distribute:
[tex]4m-12=9m-47[/tex]
Subtract 4m on both sides:
[tex]-12=5m-47[/tex]
Add 47 on both sides:
[tex]-12+47=5m[/tex]
Simplify left hand side:
[tex]35=5m[/tex]
Divide both sides by 5:
[tex]\frac{35}{5}=m[/tex]
[tex]7=m[/tex]
So the value for m is 7.
[tex]P(x)=2x^3+7x^2-5[/tex]
What is the remainder when dividing P by (x-1) or (2x+3)?
Well recall that we said m-3=R which means r=m-3=7-3=4.
So the remainder is 4 when dividing P by (x-1) or (2x+3).
Now P divided by (qx+r) will also give the same remainder R=4.
So by remainder theorem we have that P(-r/q)=4.
Let's plug this in:
[tex]P(\frac{-r}{q})=2(\frac{-r}{q})^3+m(\frac{-r}{q})^2-5[/tex]
Let x=-r/q
This is equal to 4 so we have this equation:
[tex]2u^3+7u^2-5=4[/tex]
Subtract 4 on both sides:
[tex]2u^3+7u^2-9=0[/tex]
I see one obvious solution of 1.
I seen this because I see 2+7-9 is 0.
u=1 would do that.
Let's see if we can find any other real solutions.
Dividing:
1 | 2 7 0 -9
| 2 9 9
-----------------------
2 9 9 0
This gives us the quadratic equation to solve:
[tex]2x^2+9x+9=0[/tex]
Compare this to [tex]ax^2+bx+c=0[/tex]
[tex]a=2[/tex]
[tex]b=9[/tex]
[tex]c=9[/tex]
Since the coefficient of [tex]x^2[/tex] is not 1, we have to find two numbers that multiply to be [tex]ac[/tex] and add up to be [tex]b[/tex].
Those numbers are 6 and 3 because [tex]6(3)=18=ac[/tex] while [tex]6+3=9=b[/tex].
So we are going to replace [tex]bx[/tex] or [tex]9x[/tex] with [tex]6x+3x[/tex] then factor by grouping:
[tex]2x^2+6x+3x+9=0[/tex]
[tex](2x^2+6x)+(3x+9)=0[/tex]
[tex]2x(x+3)+3(x+3)=0[/tex]
[tex](x+3)(2x+3)=0[/tex]
This means x+3=0 or 2x+3=0.
We need to solve both of these:
x+3=0
Subtract 3 on both sides:
x=-3
----
2x+3=0
Subtract 3 on both sides:
2x=-3
Divide both sides by 2:
x=-3/2
So the solutions to P(x)=4:
[tex]x \in \{-3,\frac{-3}{2},1\}[/tex]
If x=-3 is a solution then (x+3) is a factor that you can divide P by to get remainder 4.
If x=-3/2 is a solution then (2x+3) is a factor that you can divide P by to get remainder 4.
If x=1 is a solution then (x-1) is a factor that you can divide P by to get remainder 4.
Compare (qx+r) to (x+3); we see one possibility for (q,r)=(1,3).
Compare (qx+r) to (2x+3); we see another possibility is (q,r)=(2,3).
Compare (qx+r) to (x-1); we see another possibility is (q,r)=(1,-1).
Given the function f(x) = 2x – 1 and the linear function g(x), which function has a greater value when x = 3?
A.f(x) is greater.
B.g(x) is greater.
C.f(x) and g(x) are the same when x=3
D.g(x) is undefined when x=3
Answer:
Option B. g(x) is greater
Step-by-step explanation:
step 1
Find the value of f(x) when the value of x is equal to 3
we have
f(x)=2x-1
substitute the value of x=3
f(3)=2(3)-1=5
step 2
Find the value of g(x) when the value of x is equal to 3
Observing the graph
when x=3
g(3)=7
step 3
Compare the values
f(x)=5
g(x)=7
so
g(x) > f(x)
g(x) is greater
Answer:
Correct option is:
B. g(x) is greater
Step-by-step explanation:
Firstly, we find the value of f(x) when x=3
f(x)=2x-1
substitute the value of x=3
f(3)=2×3-1=5
On observing the graph, we see that g(x)=7 when x=3
Now, on Comparing the values of f(x) and g(x) when x=3
f(3)=5
g(3)=7
so, g(x) > f(x) when x=3
So, Correct option is:
B. g(x) is greater
Need help with this problem h+-3=4 please
Answer:
h=7
Step-by-step explanation:
[tex]h+(-3)=4[/tex]
may be rewritten as
[tex]h-3=4[/tex]
as adding a negative is the same as subtracting a positive.
To solve, add 3 to both sides.
[tex]h-3=4\\h=7[/tex]
Answer:
h=7
Step-by-step explanation:
1) Add three to both sides
2) You should get h=7
A jet plane travels 2 times the speed of a commercial airplane. The distance between Vancouver
and Regina is 1730 km. If the flight from Vancouver to Regina on a commercial airplane takes
140 minutes longer than a jet plane, what is the time of a commercial plane ride of this route? (please show steps:))
Answer:
The time of a commercial airplane is 280 minutes
Step-by-step explanation:
Let
x -----> the speed of a commercial airplane
y ----> the speed of a jet plane
t -----> the time that a jet airplane takes from Vancouver to Regina
we know that
The speed is equal to divide the distance by the time
y=2x ----> equation A
The speed of a commercial airplane is equal to
x=1,730/(t+140) ----> equation B
The speed of a jet airplane is equal to
y=1,730/t -----> equation C
substitute equation B and equation C in equation A
1,730/t=2(1,730/(t+140))
Solve for t
1/t=(2/(t+140))
t+140=2t
2t-t=140
t=140 minutes
The time of a commercial airplane is
t+140=140+140=280 minutes