To approximate the sum with an error of magnitude less than 5×10−6, we can use the alternating series test and the remainder estimate for alternating series. The alternating series test tells us that the sum of an alternating series is between any two consecutive partial sums. Therefore, we can approximate the sum by computing the first few partial sums until the difference between two consecutive partial sums is less than 5×10−6.
Let's start by computing the first few partial sums:
S1 = 1/4!
S2 = 1/4! - 1/8!
S3 = 1/4! - 1/8! + 1/12!
S4 = 1/4! - 1/8! + 1/12! - 1/16!
We can use a calculator to compute these partial sums and get:
S1 ≈ 0.00004166667
S2 ≈ 0.00004114583
S3 ≈ 0.00004166666
S4 ≈ 0.00004166667
We can see that the difference between S3 and S4 is less than 5×10−6, so we can approximate the sum as:
∑n=0[infinity](−1)n 1 (4n)! ≈ S3 = 0.00004166666
To estimate the error of this approximation, we can use the remainder estimate for alternating series:
|Rn| ≤ an+1
where Rn is the error of the nth partial sum, and an+1 is the absolute value of the next term in the series. In this case, an+1 = 1/[(4n+4)!], so we have:
|Rn| ≤ 1/[(4n+4)!]
We can use a calculator to find the smallest n such that |Rn| < 5×10−6:
1/[(4n+4)!] < 5×10−6
n ≥ 9
Therefore, the error of our approximation is less than 1/[(4×9+4)!] ≈ 2.8×10−13, which is smaller than 5×10−6.
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A balloon rises at the rate of 10 ft/sec from a point on the ground 100 feet from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet above the ground.
The rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet above the ground is 1/20 degree/sec.
What is angle of elevation?It is described as a relationship between an oblique line from the observer's eye to an item above his eye and the horizontal plane. This angle eventually develops above the surface. The angle of elevation is made in such a way that it is above the observer's eye, as the name suggests.
Calculation for the rate of change of angle of elevation-
According to the question,
A balloon rises at the rate of 10 ft/sec from a point on the ground 100 feet from an observer.
The balloon is 100 feet above the ground.
Let the height of the balloon be 'h' ft.
Then tanØ = h/100
h = 100tanØ
Differentiate the above equation with respect to time.
dh/dt = 100 sec² Ø (dØ/dt)
We know dh/dt = 10 ft/sec.
when h = 100
tanØ = 100/100 = 1/1
tanØ = Perpendicular/Base
On comparing above two equations,
Perpendicular = 1; Base = 1.
By Pythagoras the hypotenuse is,
H² = P² + B²
= 1² + 1²
= 2
H = √2
Now, calculate the value of cosØ.
cosØ = B/H = 1/√2
Squaring both side
cos² Ø = 1/2
sec² Ø = 1/cos² Ø
sec² Ø = 2
The main equation was,
dh/dt = 100 sec² Ø (dØ/dt)
Substitute the values,
dØ/dt = (dh/dt )/100 sec² Ø
= (10)/(100×2)
dØ/dt = 1/20 degree/sec.
Therefore, the rate of change of the angle of elevation of the balloon from the observer is 1/20 degree/sec.
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4. Given: 21 22
Prove: all b
What is the missing reason in the proof?
1.41 42
2.21 24
3. 2224
1
4. all b
4
2
Statements | Reasons
1. Given
2. Vertical angles are congruent.
3.?
4. If two lines and a transversal form
congruent corresponding angles,
then the lines are parallel.
b
Vertical angles are always equal and congruent.
What we vertical angles?The information is incomplete. An overview will be given. It should be noted that vertical angles simply means the angles that are opposite each other when two lines cross.
Vertical angles are always equal and congruent. When two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
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Chandra invests $75 every month at 4.8%/a compounded monthly for 8 years. What is the total amount of Chandra's investments after 8 years? Show your work.
The total amount of Chandra's investments after 8 years is $110.03
Compound interestThe formula for calculating the compound interest is expressed as;
A = P(1+r/n)^nt
where
P is the amount invested = $75
r is the rate. = 0.048
t is the time =8years
n = 12 (monthly)
Substitute
A = 75(1+0.048/12)^96
A = 75(1.467)
A = 110.03
Hence the total amount of Chandra's investments after 8 years is $110.03
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The chess club started with a membership of only 4 students, but the membership grew at a rate of 2.56% each week. The number of students is modeled by f(x) = 4(1.0256)x.
A: Create a question you could ask that could be answered only by graphing or using a logarithm.
The question that could be answered by using a logarithm is explained below
How to explain the logarithm?The question is: Let your dependent variable in the function be y. Write the function that models the independent variable in terms of y, using logarithms.
The question is asking you to solve y = f(x)
You already did this using a constant for y. Do the same thing with y instead of the constant.
y = 4(1.0256^x)
y/4 = 1.0256^x . . . . . . . divide by 4
log(y/4) = x·log(1.0256) . . . . . take logs
log(y/4)/log(1.0256) = x . . . . . divide by the coefficient of x
Now, you have a model for x in terms of y, which is what the question is asking for.
x = log(y/4)/log(1.0256)
When y=12, this is .x = log(12/4)/log(1.0256) ≈ 43.46 weeks
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(x)/(12)=(y)/(13)=(z)/(15 )3x+2y=35 tìm x , y , z
Step-by-step explanation:
z is the answer of your question 15 is correct answer
Write the function for the graph A. f(x) =2•(4) x B.f(x) = 4 •(2) x C. f(x) = 4•(8) x D. f(x) =2•(8) x
The exponential equation that represent the graph is y = 2(4)^x
Exponential functionsThe standard exponential function is expressed as
y = ab^x
Given the coordinate points (1, 8) and (0,2), the equivalent equations are;
8 = ab^1
2 = ab^0
from equation 2, a = 2
8 = 2b
b = 4
Hence the exponential equation that represent the graph is y = 2(4)^x
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If 15 actuators have failed today, what is the probability that a) at least 10 are repairable? b) from 3 to 8 are repairable? c) exactly 5 are repairable?
The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
According to the statement
we have given that If 15 actuators have failed and we have to find the probabilities on some conditions.
we know that the formula of probabilities is
probability = possible outcomes / total outcomes
So,
at least 10 are repairable = 1 - (10 are not repairable)at least 10 are repairable = 1 - 10/15
at least 10 are repairable = (15 - 10)/15
at least 10 are repairable = (5)/15
at least 10 are repairable = 1/3
from 3 to 8 are repairable = 3/15 *8/15from 3 to 8 are repairable = 1/5 *8/15
exactly 5 are repairable = 5/15exactly 5 are repairable = 1/3
These are the probabilities of the given conditions.
So, The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
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What is the domain of the exponential function shown below?
F(x) = 5.3*
O A. All real numbers except 5
OB. All real numbers
C. x<0
OD. x>0
The function will exist on all real numbers. Hence the domain of the exponential function given is all real numbers
Domain of a functionThe domain are independent values of an expression for which it exist. The standard exponential equation is expressed as y = ab^x
Given the exponential function as shown:
y = 5(3)^x
The function will exist on all real numbers. Hence the domain of the exponential function given is all real numbers
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Hi! It’d be really helpful if someone could explain he two questions below, thanks!
The number (2^22-2^20) (3^24-3^22) (4^26-4^24) is equal to 2^x × 3^y × 5^z. Determine the value of x+y+z.
Answer:
96
Step-by-step explanation:
(2^22-2^20) (3^24-3^22) (4^26-4^24)
=[(2^2-1)2^20][(3^3-1)3^22][(4^2-1)4^24]
=3*2^20*8*3^22*15*4^24
=2^23*3^23*4^24*15
=2^23*3^23*2^48*3*5
=2^71*3^24*5.
∵(2^22-2^20) (3^24-3^22) (4^26-4^24)=2^x × 3^y × 5^z，
∴x=71, y=24, z=1,
∴x+y+z=71+24+1=96.
Solve the equation below by factorising.
2x^2- 10x= 0
Answer:
x = 0 , x = 5
Step-by-step explanation:
2x² - 10x = 0 ← factor out 2x from each term on the left side
2x(x - 5) = 0
equate each factor to zero and solve for x
2x = 0 ⇒ x = 0
x - 5 = 0 ⇒ x = 5
Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f _________.
The differentiation of the stated series expansion of f term-by-term to obtain the corresponding series expansion for the stated derivative of f is [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)Xⁿ⁻¹
What are the steps to arrive at the above solution?f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ
⇒ f'(x) = d/dx [[tex]\sum_{n=0}^{\infty}[/tex](-1)ⁿ 3ⁿXⁿ]
= [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿ d/dx (xⁿ)
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿ(nxⁿ⁻¹)
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿnxⁿ⁻¹
= [tex]\sum_{n=1}^{\infty}[/tex] ((-1)ⁿnxⁿ⁻¹)3ⁿ
= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)xⁿ-1
What is differentiation?In mathematics, differentiation is used to calculate rates of change. In mechanics, for example, velocity is the rate of change of displacement (with regard to time).
The acceleration is the rate of change of velocity (with respect to time).
What is the practical use of Differentiation in real life?They are employed in many fields, including
biologyeconomicsphysicschemistry, and engineering.They can be used to represent:
exponential growth and decay, species population expansion, or changes in investment return over time.Learn more about differentiation at:
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Full Question:
Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f.
If f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ
In trapezoid LOVE, the length of the midsegment is 2x inches. If one
parallel side is x inches, what is the length of the other parallel side?
The length of the other parallel side is 3x
How to determine the other parallel side?The given parameters are:
Midsegment= 2x
One parallel side = x
The midsegment is calculated as:
Midsegment = 0.5 * (sum of parallel sides)
So, we have
2x = 0.5 *(x + Other side)
Divide by 0.5
4x = x + Other side
Evaluate the like terms
Other side = 3x
Hence, the length of the other parallel side is 3x
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PLEASE GUYS THIS IS THE HARDEST MATH PORBLEMS PLEASE ASAP PLEASE I NEED HELP PLEASE T_T
The simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
Simplifying an expressionFrom the question, we are to simplify the expression
From the given information,
[tex]f(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30}[/tex]
and
[tex]g(x) =\frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]
Also,
[tex]R(x) = f(x) \div g(x)[/tex]
∴ [tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \div \frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]
[tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \times \frac{x^{2}-12x+36 }{x^{2}-9x+20}[/tex]
Factoring each of the quadratics
[tex]R(x) =\frac{x^{2}-7x-4x+28 }{x^{2}-6x-5x+30} \times \frac{x^{2}-6x-6x+36 }{x^{2}-4x-5x+20}[/tex]
[tex]R(x) =\frac{x(x-7)-4(x-7) }{x(x-6)-5(x-6)} \times \frac{x(x-6)-6(x-6) }{x(x-4)-5(x-4)}[/tex]
[tex]R(x) =\frac{(x-4)(x-7)}{(x-5)(x-6)} \times \frac{(x-6)(x-6)}{(x-5)(x-4)}[/tex]
Simplifying
[tex]R(x) =\frac{(x-7)}{(x-5)} \times \frac{(x-6)}{(x-5)}[/tex]
[tex]R(x) =\frac{(x-7)(x-6)}{(x-5)(x-5)}[/tex]
[tex]R(x) =\frac{x^{2} -6x-7x+42}{x^{2} -5x-5x+25}[/tex]
[tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
Hence, the simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
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Please help Urgent
1. Explain the steps to use coordinates in calculating the perimeter and area of polygons? Explain your procedure when working with diagonal sides and irregular shapes on the coordinate plane.
2. Explain how the distance formula and slope formula are used to classify various quadrilaterals and triangles.
The way to use the distance formula to classify quadrilateral include:
Use the distance formula to measure the lengths of the sides.Use the slope to check whether the sides are perpendicular and form right angles.Use the slope to check whether the diagonals are perpendicular to eachHow to illustrate the information?The steps to calculate the perimeter and area of a polygon:
Step 1: Plot the given points on the graph.
Step 2: Connect the points and identify the figure.
Step 3: Determine lengths by counting the grid lines.
Step 4: Use a formula to calculate the perimeter and area if one exists.
Step 5: If there are no formulas for a figure, the perimeter can be calculated by summing the lengths of the sides. We might estimate the area without an area formula by counting the number of small squares enclosed.
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Find the area of the polygon formed whe these points are connected a(3,7)b(3,3)c(7,3)d(7,7)
Here, given
sides of polygon= a(3,7)b(3,3)c(7,3)d(7,7)
Let, 3 = x1 & 7= y1 , 3= x2 & 7= y2 , 7= x3 & 3= y3 , 7= x4 & 7= y4
[tex]\frac{1}{2}[/tex] [det.(9-21)+det.(9-21)+det.(49-21)+det.(49-21)] {we took determinants of all the given sides}
[tex]\frac{1}{2}[/tex] (32)
16cm^2
Hence, the area of the polygon is 16cm2.
The formula for calculating the area of a regular polygon is: area = (number of sides x 1 side length x apothem distance) / 2, where the value of the apothem is the formula apothem = [(1) Length side). / {2 × (tan (180 / number of sides))}].
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PLEASE HELP IM STUCK
The slope-intercept equation for the given line is y = 2x + 3
Equation of a straight lineFrom the question, we are to determine the equation for the line
The slope-intercept form of the equation of a straight line is
y = mx + b
Where m is the slope
and b is the y-intercept
First, we will determine the slope
Slope of the line = (3 - 0) / (1.5 - 0)
Slope of the line = 3/1.5
Slope of the line = 2
∴ m = 2
On the graph, we can observe that the y-intercept is 3
∴ b = 3
Thus,
The equation becomes
y = 2x + 3
Hence, the slope-intercept equation for the given line is y = 2x + 3
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10 points please help me
Answer:
(a)
A = 3
B = 17
C = 8
D = 15
(b)
surface area = 240
(using the formula)
Which of the following formulas, when evaluated using rational values for the variables, will result in an irrational number?
The table gives the average age of airplanes per airline at Camila’s local airport. By what percent is the number of airlines with an average airplane age of 14 less than the number of airlines with an average airplane age of 11 ?
The percentage that the number of airlines with an average airplane age of 14 is less than the number of airlines with an average airplane age of 11 is:
B. 20%.
What is a percentage?The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:
[tex]P = \frac{a}{b} \times 100\%[/tex]
The percentage of airlines with an average age of 14 years is:
[tex]P = \frac{2}{15} \times 100\% = 13.33\%[/tex]
The percentage of airlines with an average age of 11 years is:
[tex]P = \frac{5}{15} \times 100\% = 33.33\%[/tex]
The difference is:
33.33% - 13.33% = 20%.
Hence option B is correct.
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Noah has a summer tree-trimming business. Based on experience, Noah knows that his profit, P, in dollars, can be modelled by = −3x^2 + 150 − 1200, where x is the amount he charges per tree.
1. How much does he need to charge if he wants to break even?
2. How much does he need to charge if he wants to make a profit of $600?
He need to charge 24$ per tree if he wants to make a profit of $600.
According to statement
Noah knows that his profit, P, in dollars, are P= −3x^2 + 150 − 1200.
Now, we find the value of x when the profit is $600
Where X represent the Amount he charges per tree.
The given equation is P= −3x^2 + 150 − 1200.
Substitute the value of p is $600 in the above written equation
Then
600 = −3x^2 + 150 − 1200.
After solving the equation by a substitution method the value of X becomes
600+1200-150 = -3(x)^2
x = (1650/3)^1/2
x= 24$
it means the charge per tree is 24$.
So, He need to charge 24$ per tree if he wants to make a profit of $600.
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If d(4)= 68. thus, joe travels 68 mi in 4 hours. then how much mi does joe travel in 45 minutes
Given that, Joe travels 68 mi in 4 hours, the number of miles Joe will travel in 45 minutes is 12.75.
How many miles did joe travel in 45 minutes?Given that;
Joe travels 68 miles in 4 hoursjoe travel x miles in 45 minutesFirst, we convert 4 hours to minutes: 4 × 60 = 240 minutes
Now,
Joe travels 68 miles in 240 minutes
Joe will travel x miles in 45 minutes
We have;
x × 240 = 68 × 45
x × 240 = 3060
x = 3060 / 240
x = 12.75 miles.
Given that, Joe travels 68 mi in 4 hours, the number of miles Joe will travel in 45 minutes is 12.75.
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Describe fully the single transformation which takes shape A to shape B.
The single transformation which takes shape A to shape B as in the task content is a 90° clockwise rotation and a 6 units shift rightward.
What transformation is responsible for taking A to B?By observation, it follows that the orientation of the shape A differs from that of B as it lags by 90° in the clockwise direction.
Additionally, the resulting image from a 90° clockwise rotation about the left vertice of A still reguires that A be shifted 6 units rightwards to yield shape B.
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Mrs. Laura is between 50 and 80. If you divide her age by 9, the remainder is 1. If you divide by 4, the remainder is 1. How old is Mrs. Laura?
Answer: 73
Step-by-step explanation:
I think she's 73? You have to take multiples of nine and see if they work! I did 72 since I knew that 72 goes into 9 equally and if she is 73 there would be a remainder of 1. After that, I stuck with 73 and since I knew that 18 times 4 is also 72, I knew that 73 divided by 4 would give me 18 with a remainder of 1.
The solution is 37 years, that is, Mrs. Laura's age is 37 years old.
Given information in the problem:
1. Mrs. Laura's, L age is between 50 and 80.
2. When you divide age by 9, the remainder is 1.
3. When you divide age by 4, the remainder is also 1.
To find L exact age, start by checking multiples of 9 that leave a remainder of 1 when divided by 4:
9 × 1 + 1 = 10 (remainder of 1 when divided by 4)
9 × 2 + 1 = 19 (remainder of 1 when divided by 4)
9 × 3 + 1 = 28 (remainder of 1 when divided by 4)
9 × 4 + 1 = 37 (remainder of 1 when divided by 4)
9 × 5 + 1 = 46 (remainder of 2 when divided by 4)
9 × 6 + 1 = 55 (remainder of 3 when divided by 4)
9 × 7 + 1 = 64 (remainder of 0 when divided by 4)
Check multiples of 4 that leave a remainder of 1 when divided by 9:
4 × 1 + 1 = 5 (remainder of 1 when divided by 9)
4 × 2 + 1 = 9 (remainder of 1 when divided by 9)
4 × 3 + 1 = 13 (remainder of 4 when divided by 9)
4 × 4 + 1 = 17 (remainder of 8 when divided by 9)
From the above calculations, see that the only common number between the two sets of multiples is 37.
So, L's age is 37 years old.
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The cost of a fan is Rs 1,600. If Mrs. Khadka purchased it with 13% VAT, how muc did she pay for it?
Answer:
Rs 1808
Step-by-step explanation:
Sp=Selling price
Mp=main price
Sp=Mp+Mp*VATSp=Rs 1600+Rs 1600*13%Sp=Rs 1600+Rs 208Sp=Rs 1808match the following vocabulary
1. the distance around a circle
2. a portion of the circumference of a circle
3. the distance from the center to any point on a circle
4. the distance through the center between points on a circle
5.
the degree measurement of an arc (in reference to its central
angle)
6. an arc which measures less than 180°
7. an arc which measures exactly 180°
arc length
circumference
diameter
minor arc
arc measure
radius
semicircle
The properties of a Circle with their definitions are; As highlighted below
How to identify Properties of a Circle?1) The distance around a circle is called; Circumference
2) A portion of the circumference of a circle is called; Arc Length
3) The distance from the center to any point on a circle is called; Radius.
4) The distance through the center between points on a circle is called; Diameter
5) The degree measurement of an arc (in reference to its central
angle) is called; Arc Measure
6) An arc which measures less than 180° is called; Minor arc
7) An arc which measures exactly 180° is called; Semi Circle
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Find the volume of the regular pyramid.
Select the correct answer.
Which expression and quotient does the diagram model?
The expression (quotient) which is modelled by the given diagram as in the task content is; (x²+x-2)/(x-1) = x +2.
Which expression and quotient is depicted by the diagram model as in the task content?It follows from the task content that the diagram model in discuss by observation involves a quotient in which case, the division is; (x-1).
Additionally, the dividend in the quotient by observation is the sum of terms as follows;
x² + x +x -x -1 -1 = x²+x -2.
Ultimately, the result of the division according to the model is; (x +2).
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. If sin(x + y)=sin x.cos y+cos x.sin y, then find the value of sin 75 degrees
Answer:
[tex]\sf \dfrac{\sqrt{6}+\sqrt{2}}{4}[/tex]
Step-by-step explanation:
Trigonometry:75 = 45 + 30
Sin (x + y) = Sin x *Cos y + Cos x * Sin y
Sin (45 + 30) = Sin 45 * Cos 30 + Cos 45 * Sin 30
[tex]\sf =\dfrac{1}{\sqrt{2}}*\dfrac{\sqrt{3}}{2}+\dfrac{1}{\sqrt{2}}* \dfrac{1}{2}\\\\\\=\dfrac{\sqrt{3}}{2\sqrt{2}}+\dfrac{1}{2\sqrt{2}}\\\\\\=\dfrac{\sqrt{3}*\sqrt{2}}{2*\sqrt{2}*\sqrt{2}}+\dfrac{1*\sqrt{2}}{2\sqrt{2}*\sqrt{2}}\\\\\\=\dfrac{\sqrt{6}}{2*2}+\dfrac{\sqrt{2}}{2*2}\\\\\\=\dfrac{\sqrt{6}+\sqrt{2}}{4}[/tex]
A person has a 0.125 probability of getting the number 3 on a roll of an eight-sided die. if a person rolls the die 24 times, and gets a 3 on one-half of the rolls, what is the difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll?
The difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll is 0.375.
In the question, we are given that a person has a 0.125 probability of getting the number 3 on a roll of an eight-sided die.
This means that the theoretical probability of rolling a 3 on an eight-sided die is 0.125.
Also, we are informed that a person rolls the die 24 times, and gets a 3 on one-half of the rolls, that is, 1/2 * 24 = 12 rolls.
Thus, the experimental probability of rolling a 3 on an eight-sided die = 12/24 = 1/2 = 0.5.
In the question, we are asked to find the difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll.
The difference = The experimental probability - The theoretical probability,
or, the difference = 0.5 - 0.125 = 0.375.
Thus, the difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll is 0.375.
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