a) The factor form of the polynomial is f(x) = 4 · (x - 3) · (x + 5 / 4).
b) The values of the x-intercepts of the graph of f(x) are represented by the roots found in part a).
c) The function has a minimum and grows in the + y direction.
d) Marking the two x-intercepts, second, generating a line perpendicular to the x-axis that passes the midpoint of the line segment generated by the two intercepts and estimate the location of the vertex and, finally, creating a parabola with a minimum and growing in the + y direction by matching the points.
How to analyze and interpret quadratic functions?
In this problem we have a quadratic function in standard form, that is, an equation of the form f(x) = a · x² + b · x + c, where a, b and c are real coefficients. a) First, we need to complete the square and find the roots to determine the factorized form:
4 · x² - 7 · x - 15 = 0
4 · [x² - (7 / 4) · x - (15 / 4)] = 0
x² - (7 / 4) · x - (15 / 4) = 0
x² - 2 · (7 / 8) · x - (15 / 4) + (289 / 64) = 289 / 64
x² - 2 · (7 / 8) · x + 49 / 64 = 289 / 64
(x - 7 / 8)² = 289 / 64
x - 7 / 8 = ± 17 / 8
x = 7 / 8 ± 17 / 8
Then, the factor form of the polynomial is f(x) = 4 · (x - 3) · (x + 5 / 4).
b) The values of the x-intercepts of the graph of f(x) are represented by the roots found in part a).
c) The end behavior of the graph is inferred from the lead coefficient. A positive coeffcient , which is the case of the function, indicates that vertex of the polynomial is a minimum and parabola grows in the + y direction.
d) In accordance with algebra, we can generate a quadratic equation by knowing three distinct points on Cartesian plane.
First, mark the two x-intercepts, second, generate a line perpendicular to the x-axis that passes the midpoint of the line segment generated by the two intercepts and, finally, generate a parabola with a minimum and growing in the + y direction.
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what is the answer to this?
Answer:
the coordinates of equations
1.4x+y=-7
for x =0
f(0,y)=4(0)+y=-7
y=-7
for y=0
f(x,0)=4x+0=-7
4x=-7
x=-7/4
2.x-y=2
for x=0
f(0,y)=0-y=2
-y=2
y=-2
for y=0
f(x,0)=x-0=2
x=2
graph all of the equations coordinates or you can look the image
so the answers are : {-3,-1}
CMIIW
Answer For X (in degrees)
Answer:
I cant see your question can you show it properly??If Peter pays $84 to buy a watch after discount of 20% find the original price of the watch
Answer: 105 dollars
Step-by-step explanation:
Let's say that the original price x dollars, we can use the equation to find the original price.
80%x = 84
x = 105
Answer:
$105
Step-by-step explanation:
The multiplier is 0.8
To get the price after the discount,the original price was multiplied by 0.8(100 - 20 = 80% = 0.8
Therefore to obtain the original price from the price after the discount, we use reverse percentage:
$84 ÷ 0.8 = $105
Drag the tiles to the correct boxes to complete the reasoning of the proof. Not all tiles will be used.
May this help other people in the near future!!!!!
The missing reasons for each of the given statements are respectively;
1) Substitution
2) Definition of a Straight angle because a straight angle is 180°.
3) Congruent Angles have equal measures
How to prove angles in a triangle?
We are given that;
m∠1 = 90°
m∠1 + m∠2 = m∠5
∠3 ≅ ∠4
Statement 1; 90° + m∠2 + m∠4 = 180°
Reason 1; Substitution
Statement 2; m∠1 + m∠2 + m∠3 = 180°
Reason 2; Definition of a Straight angle because a straight angle is 180°.
Statement 3; m∠3 = m∠4
Reason 3; Congruent Angles have equal measures
Thus, the missing reasons for each of the given statements are respectively;
1) Substitution
2) Definition of a Straight angle because a straight angle is 180°.
3) Congruent Angles have equal measures
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the set of integers less than -20
In mathematical notation, the set of integers less than -20 as follows:
{-21, -22, -23, -24, -25, ...}
The set of integers less than -20 is the set of all whole numbers that are smaller than -20. Integers include both positive and negative whole numbers, as well as zero.
In mathematical notation, we can represent the set of integers less than -20 as follows:
{-21, -22, -23, -24, -25, ...}
The set includes all numbers that are smaller than -20, going indefinitely in the negative direction. Each number in the set is obtained by subtracting 1 from the previous number in the set.
For example:
-21 is the integer immediately smaller than -20, so it is included in the set.
-22 is the next integer, smaller than -20, so it is also included in the set.
And this pattern continues indefinitely, encompassing all the integers less than -20.
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what base could be written in the blank to make the exponential function model 15% decay ? y= (__1/2__) ^t\12
Answer:
0.85
Step-by-step explanation:
1 - 15% = 1 - 0.15 = 0.85
[tex] y = (0.85)^\frac{t}{12} [/tex]
Answer: 0.85
Heyy how do I solve this
Answer:
1200 centimeters
Step-by-step explanation:
Hello!
1meter=100centimeters
12meter = ?
[tex]thus \: \frac{12m \times 100cm}{1m} = 1200cm[/tex]
Answer:
1200cm
Step-by-step explanation:
1m=100cm
12m=xcm
by multiplying cris cross
x=12*100
x=1200cm
12m=1200cm
Open Ended
a. Find the slope of the line that passes through the points (2,-5) and (-2,3).
b. What are the slope and y-intercept of the equation 2x - 5y = -10?
c. Find the x value so that the line through the points (x,-9) and (0,1) has a slope of -4.
d. Write the equation of a line that passes through the point (-1,5) and has a slope of -7. Your answer should be given in the form y = mx + b.
(PLEASE SHOW WORK, IT CAN BE SIMPLE I JUST NEED TO PUT IT IN FOR CREDIT)
a.
Solution Given:
Slope of the line passing through the point is
m= [tex] \frac{y_2-y_1}{x_2-x_1}=\frac{3-(-5)}{-2-2}=\frac{8}{-4}=-2[/tex]
slope=-2
b.
Solution given:
equation is
2x-5y=-10
2x+10=5y
y=2x/5 +10/5
y=2/5 x +3
comparing above equation with y=mx+c
we get
m=2/5
slope=2/5
c.
Slope of the line passing through the point is
m=[tex] \frac{y_2-y_1}{x_2-x_1}[/tex]
-4=[tex] \frac{1-(-9)}{0-x}[/tex]
-4=10/-x
doing criss cross multiplication
-4*-x=10
4x=10
x=10/4
x=5/2
slope=5/2
d.
we have
equation of line passing through the one point is
[tex] y-y_1=m(x-x_1)[/tex]
y-5=-7(x-(-1))
y=-7(x+1)+5
y=-7x-7+5
y=-7x-2 is a required equation:
What is the length of the line segment PQ on the coordinate grid?
A fourquadrant coordinate grid from negative 50 to positive 50 in increments of 10 is drawn. Point P is plotted on the ordered pair negative 20, 10. Point Q is plotted on the ordered pair 10, 10. The points are connected by a line.
20 units
30 units
40 units
50 units
Using the formula for the distance between two points, the length of the line segment PQ is of 30 units.
What is the distance between two points?Suppose that we have two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The distance between them is given by:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
The coordinates of the endpoints for this problem are given by:
P(-20,10).Q(10,10).Hence the distance is given by:
D = sqrt[(-20 -10)² + (10 - 10)] = 30 units.
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Answer: 30 units
Step-by-step explanation:
Dmitri wants to cover the top and sides of this box with glass tiles that are 1 cm square. How many tiles will he need? Dmitri will need [Blank] glass tiles. The dimentions are...26 , 15, 8.
The number of tiles needed to cover the surface area of the box excluding the bottom is: 1,046 tiles.
What is the Surface Area of a Box?The surface area of a box is the area surrounding all its faces. A box has 6 rectangular faces. Therefore, the total surface area of the box equals the sum of all 6 rectangular faces.
What is the Surface Area of a Box?SA = 2(lw + lh + hw), where:
l = lengthw = widthh = height of the boxThe image attached below shows the box Dmitri wants to cover. Since the bottom of the box would be excluded, therefore:
The surface area to be covered = surface area of the box - area of the bottom rectangular face
The surface area to be covered = 2(lw + lh + hw) - (l)(w)
l = 26
w = 15
h = 8
Substitute
The surface area to be covered = 2(l×w + lh + hw) - (l)(w) = 2·(15·26+8·26+8·15) - (26)(15) =
The surface area to be covered = 1436 - 390 = 1,046 cm
Area of one tile = 1 cm square
Number of tiles needed = 1,046/1
Number of tiles needed = 1,046 tiles.
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. A school band sold 432 tickets for their upcoming spring concert. This represents about 85.5% of the number of tickets sold last year. How many tickets were sold last year? (Round to the nearest whole number)
List the angles in order from the largest to the smallest for ΔABC. AB⎯⎯⎯⎯⎯⎯⎯=14, AC⎯⎯⎯⎯⎯⎯⎯⎯=15, BC⎯⎯⎯⎯⎯⎯⎯⎯=16
Answer:
BC 1
AC 2
AABC 3
Step-by-step explanation:
BC=16
AC = 15
AABC= 15
Answer: C, B, A
Step-by-step explanation: It just is
What is the divisor the dividend is 5/9
Answer
5/9 is the dividend
Step-by-step explanation:
How many numbers of possible live card hands (hands in five-card poker) drawn without replacement from a standard deck of 52 playing cards?Please give full details!
The combination shows that the numbers of possible live card hands drawn without replacement from a standard deck of 52 playing cards is 2,598,960.
How to explain the information?A permutation is the act of arranging the objects or numbers in order while combinations are the way of selecting the objects from a group of objects or collection such that the order of the objects does not matter.
Since the order does not matter, it means that each hand is a combination of five cards from a total of 52.
We use the formula for combinations and see that there are a total number of C( 52, 5 ) = 2,598,960 possible hands.
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1. What is the measure of angle x? O 60° O 90° O 45° O 30°
Answer:
Angle x= 60°
Step-by-step explanation:
For the triangle HIL, you're going to add 30+90=120
Angles in a triangle add up to 180.
So, therefore,
180-120=60 so angle L must equal 60.
HJ is a straight line and angles in a straight line add up to 180.
180-90=90
Angle JIL is equal to 90.
To find y you add 90+45 which equals 135.
Again, angles in a triangle add up to 180.
180-135=45
So,
y = 45
Now we are told that IJK is a right angle and that we are given that IJL is 45. 45 is half of 90 so LJK must be 45.
To find angle JLK we must add angle L and angle y.
60+45=105
Angles in a straight line add up to 180. So,
180-105=75
75 = Angle JLK
75+45=120
Angles in a triangle add up to 180 so,
180-120= 60
Angle x= 60°
Hope this helped, if so please award me with the brainliest if possible. If you require further assistance from me comment below! :)
Name the intersection of plane K and plane L.
Answer:
line NM
Step-by-step explanation:
Since planes extend infinitely in all directions, planes intersect at a line, not at a line segment.
A manufacturer of widgets wants to test a new widget-producing machine to determine if it can make an average of 25 widgets per second before deciding to invest in the machine. The standard to reject the new machine is if it makes an average of less than 25 widgets per second. Here are data from a small random sample:
25.6, 26.2, 22.5, 20.5, 26.4, 27.4, 23.6, 26.9, 25.7, 24.9
Assuming the population follows a normal distribution, is there evidence that the new machine should be rejected at the 0.01 significance level? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.
Using the t-distribution, it is found that since the p-value is greater than 0.01, there is no evidence that the new machine should be rejected at the 0.01 significance level.
What are the hypothesis tested?At the null hypothesis, it is tested if the average is not less than 25, that is:
[tex]H_0: \mu \geq 25[/tex]
At the alternative hypothesis, it is tested if the average is less than 25, that is:
[tex]H_1: \mu < 25[/tex]
What is the test statistic?The test statistic is given by:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
The parameters are:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.Considering the situation described, the values of the parameters are given as follows:
[tex]\overline{x} = 24.97, \mu = 25, s = 2.16, n = 10[/tex]
Hence the test statistic is:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{24.97 - 25}{\frac{2.16}{\sqrt{10}}}[/tex]
t = -0.04
What is the p-value and the conclusion?Using a t-distribution calculator, for a left-tailed test, as we are testing if the mean is less than a value, with 10 - 1 = 9 df and t = -0.04, the p-value is of 0.4844.
Since the p-value is greater than 0.01, there is no evidence that the new machine should be rejected at the 0.01 significance level.
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Sound is measured in decibels, using the formula d=10log(p/p0) where p is the intensity of the sound and p0 is the weakest sound the human ear can hear. A horn has a decibel warning of 20. how many times more intense is this horn compared to the weakest sound heard to the human ear?
Solving the given logarithmic equation, it is found that the horn is 100 times more intense compared to the weakest sound heard to the human ear.
What is the equation for the sound in decibels?The equation is given by:
[tex]d = 10\log{\frac{p}{p_0}}[/tex]
In which:
d is the intensity of the sound in decibels.p is the intensity of the sound.[tex]p_0[/tex] is the weakest sound that the human ear can hear.In this problem, we have that d = 20, and we have to solve the logarithmic equation for p to find how many times more intense the sound is:
[tex]d = 10\log{\frac{p}{p_0}}[/tex]
[tex]20 = 10\log{\frac{p}{p_0}}[/tex]
[tex]\log{\frac{p}{p_0}} = 2[/tex]
The logarithm is inverse of the function [tex]10^x[/tex], hence we apply the function to both sides to find the ratio.
[tex]\frac{p}{p_0} = 10^2[/tex]
[tex]\frac{p}{p_0} = 100[/tex]
Hence, the horn is 100 times more intense compared to the weakest sound heard to the human ear.
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Use the graph to determine a. the function's domain; b. the function's range; c. the
x-intercepts, if any; d. the y-intercept, if there is one; e. the following function
values.
f(-3)
f(0)
7-8-5-4-3-2-1
Q
2
Part a
The domain is the set of x-values, which is [tex](-\infty, \infty)[/tex]
Part b
The range is the set of y-values, which is [tex](-\infty, -2][/tex]
Part c
The x-intercepts is when y=0, which there are none of.
Part d
The y-intercept is when x=0, which is at (0, -2).
Do the first three questions with Percise steps on how to do it
Answer:
1. x = 20
2. x = 3
3. RST = 22 degrees
Step-by-step explanation:
1. Since QR bisects PQS, the measure of the angles PQR and PQS should be equal, so we can set their expressions equal to each other, and then solve.
4x-10 = -3x+130
4x = -3x +140
7x = 140
x = 20
2. Since there is a CAB has a right angle, the measure of angles CAD and BAD should add up to 90 degrees. So we can set the sum of their expressions equal to 90 degrees.
(5x+57) + (x+15) = 90
6x + 72 = 90
6x = 18
x = 3
3. I can't see where the R is but if it is on the empty line then we can find RST by subtracting the measure of angle TSU from angle RSU.
TSU - RSU = RST
91 - 69 = 22 degrees
RST = 22 degrees
Answer:
7. m∠PQR =70° m∠PQS = 140°
8. m∠CAD = 18° m∠BAD = 72°
9. m∠RST = 22°
Step-by-step explanation:
Question 7
If QR bisects (divides into two equal parts) ∠PQS then:
⇒ m∠PQR = m∠RQS
⇒ 4x - 10 = -3x + 130
⇒ 4x - 10 + 10 = -3x + 130 + 10
⇒ 4x = -3x + 140
⇒ 4x + 3x = -3x + 140 + 3x
⇒ 7x = 140
⇒ 7x ÷ 7 = 140 ÷ 7
⇒ x = 20
Substitute the found value of x into the expression for m∠PQR:
⇒ m∠PQR = 4(20) - 10 = 70°
As QR bisects ∠PQS:
⇒ m∠PQS = 2m∠PQR = 2 × 70° = 140°
Question 8
From inspection of the given diagram, ∠BAC = 90°.
⇒ m∠CAD + m∠BAD = 90
⇒ x + 15 + 5x + 57 = 90
⇒ 6x + 72 = 90
⇒ 6x + 72 - 72 = 90 - 72
⇒ 6x = 18
⇒ 6x ÷6 = 18 ÷ 6
⇒ x = 3
Substitute the found value of x into the expressions for the two angles:
⇒ m∠CAD = 3 + 15 = 18°
⇒ m∠BAD = 5(3) + 57 = 72°
Question 9
From inspection of the given diagram (and assuming R is on the empty line segment):
m∠RSU = m∠RST + m∠TSU
⇒ 91° = m∠RST + 69°
⇒ 91° - 69° = m∠RST + 69° - 69°
⇒ 22° = m∠RST
⇒ m∠RST = 22°
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Point J is located at -19. Points K and L are each 8 units away from Point J. Where
are K and L located?
K= L=
So, K and L are located at K = -11 or L = -27 or K = -27 or L = -11
How to find the location of K and L?Givent that Point J is located at -19. Points K and L are each 8 units away from Point J. This implies that the modulus of K - J or modulus of L - J = 18.
So, |K - J| = 8
⇒ K - J = 8 or -(K - J) = 8
Substituting the value of J = -19 into the equation, we have
⇒ K - J = 8 or -(K - J) = 8
⇒ K - (-19) = 8 or -(K - (-19)) = 8
⇒ K + 19 = 8 or (K - (-19)) = -8
⇒ K + 19 = 8 or K + 19 = -8
⇒ K = 8 - 19 or K = -8 - 19
⇒ K = - 11 or K = -27
Also, |L - J| = 8
⇒ L - J = 8 or -(L - J) = 8
Substituting the value of J = -19 into the equation, we have
⇒ L - J = 8 or -(L - J) = 8
⇒ L - (-19) = 8 or -(L - (-19)) = 8
⇒ L + 19 = 8 or (L - (-19)) = -8
⇒ L + 19 = 8 or L + 19 = -8
⇒ L = 8 - 19 or L = -8 - 19
⇒ L = - 11 or L = -27
So, K and L are located at K = -11 or L = -27 or K = -27 or L = -11
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A baker uses 3 cups (c) of flour to bake 2 loaves of bread. If 1 pound (lb) of flour is equivalent to 3 c, how many loaves of bread can the baker make with a 10 lb-bag of flour? 3 (Round the answer to the nearest whole number.)
The number of loaves of bread can the baker make with a 10 lb-bag of flour is 20 loaves of bread.
Number of loavesRatio of flour to loaves of bread = 3 cups : 2 loaves
1 pound (lb) of flour = 3 cups of flour10 pounds (lb) of flour = 30 cups of flourNumber of loaves of bread can the baker make with a 10 lb-bag of flour
Equate the ratio of flour to loaves of bread3 : 2 = 30 : x
3/2 = 30/x
3 × x = 2 × 30
3x = 60
x = 60/3
x = 20 loaves of bread
Therefore, the number of loaves of bread can the baker make with a 10 lb-bag of flour is 20 loaves of bread.
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A cereal box filling machine is designed to release an amount of 16 ounces of cereal into each box, and the machine’s manufacturer wants to know of any departure from this setting. The engineers at the factory randomly sample 150 boxes of cereal and find a sample mean of 15.75 ounces. If we know from previous research that the population is normally distributed with a standard deviation of 1.46 ounces, is there evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.
Using the z-distribution, it is found that since the p-value is less than 0.05, there is evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance.
What are the hypothesis tested?At the null hypothesis, it is tested if the mean is not different to 16 ounces, that is:
[tex]H_0: \mu = 16[/tex]
At the alternative hypothesis, it is tested if the mean is different, hence:
[tex]H_1: \mu \neq 16[/tex]
What is the test statistic?The test statistic is:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.[tex]\sigma[/tex] is the standard deviation of the population.n is the sample size.The parameters for this problem are:
[tex]\overline{x} = 15.75, \mu = 16, \sigma = 1.46, n = 150[/tex].
Hence:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{15.75 - 16}{\frac{1.46}{\sqrt{150}}}[/tex]
z = -2.1
What is the p-value and the conclusion?Using a z-distribution calculator, for a two-tailed test, as we are testing if the mean is different of a value, with z = -2.1, the p-value is of 0.0357.
Since the p-value is less than 0.05, there is evidence that the mean amount of cereal in each box is different from 16 ounces at 0.05 significance.
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find the positive square roots by division method of 151,321
The positive square roots of the number 151,321 according to the task content can be determined by means of division as; 389.
What are the square roots of 151,321 by means of division method?It follows from.the task content above that the number given is; 151,321 whose positive square roots is to be determined.
Upon testing different integers as divisor on the number 151,321; it is concluded that the only positive integer by which 151,321 can be divided to result in a whole is; 389.
Hence, the positive square root of the number 151,321 is; 389.
Consequently, it can be concluded that the positive square root of the number, 151,321 as in the task content is; 389 which is itself a prime number as it is only divisible by 1 and itself.
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HELP PLS!!!!!!!!!!!!
Answer:(3+3)(1+3) = 24
Step-by-step explanation: There is a 1 and 3 3's. We know that added up, the sum is not 24. So it can't be all addition. But, 3 is a factor of 6 which is a factor of 24. There are 2 3's, so we add them up to get 6. Then, we have a 1 and a 3 left. we add these up to get 4. 6x4= 24.
Help me with this math question :)
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
To prove two lines parallel, we need to check if the two angles involving the lines acts as some angle pair with defininte property.
In the given figure,
Angle 3 is congruent to Angle 19( by Alternate Exterior angle pair )
Hence, the two lines are parallel.
[tex] \qquad \large \sf {Conclusion} : [/tex]
Option C is correctWhich table represents exponential growth? A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 6, 8. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 8, 16. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 7, 11. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 2, 4, 6, 10.'
The table that represents exponential growth is the second table.
What is exponential growth?Exponential growth simply means a process that increases quantity over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself.
Exponential growth is the pattern of data which shows sharper increases over time. In finance, compounding creates exponential returns and the savings accounts with a compounding interest rate can show exponential growth.
In this case, the second table described represents an exponential function. It has a common ratio of 4, meaning that each y-value is multiplied by 4 to get to the next value. Here, every time the Xs increase by 1, the Ys increase by a multiple of 4.
One of the best examples of exponential growth is observed in bacteria. Here, it takes bacteria roughly an hour to reproduce through prokaryotic fission. This illustrates exponential growth.
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Answer: B
Step-by-step explanation:
need heeeelp please
Answer:9[tex]a^{10}[/tex][tex]b^{-6}[/tex]=[tex]\frac{9a^{10} }{b^6}[/tex]
Step-by-step explanation:
Answer:
9*a^10*(1/(b^6))
Step-by-step explanation:
(-3a^5*b^-3)^2
(-3)^2 * (a^5)^2 * (b^-3)^2
9a^10*b^-6
9*a^10*(1/(b^6))
QUESTION 3 The probability that it will rain on a given day is 63%. A child has a 12% chance of falling in dry weather and is three times as likely to fall in wet weather. Draw a tree diagram to represent all outcomes of the above information. What is the probability that a child will not fall on any given day? What is the probability that a child will fall in dry weather? 3.1 draw a tree diagram to represent all outcomes of the above information
Answer + Step-by-step explanation:
the probability that a child will not fall on any given day :
= (63 × 64 + 27 × 88) ÷ 100
= 64.08 %
the probability that a child will fall in dry weather :
= (27 × 12) ÷ 100
= 3.24 %
Starting from the geometric series, use power series operations to
determine the Maclaurin series. See picture
a. By replacing [tex]x[/tex] with [tex]-x^2[/tex] in the power series, we get
[tex]\displaystyle \frac1{1+x^2} = \sum_{n=0}^\infty (-x^2)^n = \sum_{n=0}^\infty (-1)^n x^{2n}[/tex]
Integrate both sides to recover [tex]\arctan(x)[/tex] on the left.
[tex]\displaystyle \int \frac{dx}{1+x^2} = \int \sum_{n=0}^\infty (-x^2)^n = \sum_{n=0}^\infty (-1)^n x^{2n} \, dx[/tex]
[tex]\displaystyle \arctan(x) = C + \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1} \, dx[/tex]
By letting [tex]x=0[/tex] on both sides, we find [tex]C=0[/tex], so that
[tex]\displaystyle \arctan(x) = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1} \, dx[/tex]
Then dividing both sides by [tex]x[/tex] gives
[tex]\displaystyle \boxed{\frac{\arctan(x)}x = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n} \, dx}[/tex]
b. Let [tex]x=\frac1{\sqrt3}[/tex]. Then
[tex]\displaystyle \frac{\arctan\left(\frac1{\sqrt3}\right)}{\frac1{\sqrt3}} = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \left(\dfrac1\sqrt3\right)^{2n}[/tex]
[tex]\displaystyle \sqrt3 \arctan\left(\frac1{\sqrt3}\right) = \sum_{n=0}^\infty \frac1{2n+1} \left(-\frac13\right)^n[/tex]
Taking the first few terms from the infinite series, we can approximate
[tex]n=0 \implies \sqrt3\arctan\left(\dfrac1{\sqrt3}\right) \approx 1[/tex]
[tex]0\le n\le1 \implies \sqrt3\arctan\left(\dfrac1{\sqrt3}\right) \approx 1+\dfrac13\left(-\dfrac13\right) = \dfrac89[/tex]
which together suggest the value we want is bounded between 8/9 and 9/9 = 1, hence [tex]\boxed{p=8}[/tex].
Since the series is alternating and converges on [tex]-1<x<0\cup0<x<1[/tex],
[tex]\displaystyle \left| \sum_{n=0}^\infty a_n - \sum_{n=0}^0 a_n \right| < |a_1| = \left|\frac13 \left(-\frac13\right)^1\right| = \frac19[/tex]
and
[tex]\displaystyle \left| \sum_{n=0}^\infty a_n - \sum_{n=0}^1 a_n \right| < |a_2| = \left|\frac15 \left(-\frac13\right)^2\right| = \frac1{45}[/tex]
which tells us the first approximation is off by at most 1/9 from the actual value of [tex]\frac\pi{2\sqrt3}[/tex], whereas the second approximation is off by at most 1/45 from the actual value. In other words, the second approximation is closer, so [tex]\frac\pi{2\sqrt3}[/tex] is closer to [tex]\frac p9[/tex] :
[tex]\dfrac89 \approx 0.8889[/tex]
[tex]\dfrac{\pi}{2\sqrt3}\approx0.9069[/tex]
[tex]\dfrac99 = 1[/tex]