We have
[tex]2+\dfrac54+\dfrac{11}{16}+\dfrac{23}{64}+\cdots=\displaystyle\sum_{n=0}^\infty\frac{3\cdot2^n-1}{4^n}[/tex]
(notice that each denominator is a power of 4, and each numerator is one less than some multiple of 3, in particular 3 times some power of 2)
Recall for [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]
So we have
[tex]\displaystyle\sum_{n=0}^\infty\frac{3\cdot2^n-1}4=3\sum_{n=0}^\infty\left(\frac12\right)^n-\sum_{n=0}^\infty\left(\frac14\right)^n=\frac3{1-\frac12}-\frac1{1-\frac14}=\boxed{\frac{14}3}[/tex]
Following is a portion of the regression output for an application relating maintenance expense (dollars per month) to usage (hours per week) for a particular brand of computer terminal.
ANOVA
df SS MS F Significance F
Regression 1 1575.76
Residual 8 349.14
Total 9 1924.90
Coefficient Standard Error t Stat P-value
Intercept 6.1092 0.9361
Usage 0.8931 0.149
A) Write the estimated regression equation (to 4 decimals).
B) Use a t test to determine whether monthly maintenance expense is related to usage at the .05 level of significance (to 2 decimals, if necessary).
1. Reject the null hypothesis
2. Do not reject the null hypothesis
C) Monthly maintenance expense______to usage.
1. Is related
2. Is not related
D) Did the estimated regression equation provide a good fit?
1. yes
2. no
E) Explain.
Answer:
Explained below.
Step-by-step explanation:
The ANOVA and Regression output for an application relating maintenance expense (dollars per month) to usage (hours per week) for a particular brand of computer terminal is provided.
(A)
The estimated regression equation equation is:
[tex]y=6.1092+0.8931x[/tex]
Here,
y = maintenance expense (dollars per month)
x = usage (hours per week) for a particular brand of computer terminal
(B)
Consider the Regression output.
The hypothesis to test whether monthly maintenance expense is related to usage is:
H₀: The monthly maintenance expense is not related to usage, i.e. β = 0.
Hₐ: The monthly maintenance expense is related to usage, i.e. β ≠ 0.
Compute the test statistic as follows:
[tex]t=\frac{b}{S.E._{b}}=\frac{0.8931}{0.149}=5.99[/tex]
Compute the p-value as follows:
[tex]p-value=2\times P (t_{8}<5.99}=0.00033[/tex]
The null hypothesis will be rejected if the p-value is less than the significance level.
p-value = 0.00033 < α = 0.05
Reject the null hypothesis.
(C)
Monthly maintenance expense is related to usage.
(D)
Yes, the estimated regression equation provide a good fit.
Since the regression coefficient is significant it can be concluded that the regression equation estimated is a good fit.
From the regression output given, the solution to the questions given are outlined thus ;
[tex] Null \: hypothesis : H_{0} : β = 0 [/tex] [tex] Alternative \: hypothesis : H_{1} : β ≠ 0 [/tex]1.)
Regression equation :
y = bx + c b = slope ; c = interceptHence, the estimated regression equation is;
y = 0.8931x + 6.10922.)
We can calculate the T-statistic value thus ;
[tex] T-statistic = \frac{b}{SE_{b}}[/tex] [tex]SE_{b} = Standard \: error \: of \: slope[/tex] df = 8Hence, the T-statistic is given as ;
[tex] T-statistic = \frac{0.8931}{0.149} = 5.99[/tex]
Pvalue (2 tailed) = 0.00033
Decison Region :
[tex] Reject \: H_{0} \: if \: Pvalue \: < \: α [/tex]Since 0.00033 < 0.05 ; we reject the Null hypothesis.
3.)
Hence, we conclude that monthly expense is related to usage.
4.)
Since, the correlation Coefficient, β ≠ 0 ; Yes, the correlation provides a good fit as it is significant.
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b) Use Greens theorem to find∫x^2 ydx-xy^2 dy where ‘C’ is the circle x2 + y2 = 4 going counter clock wise.
It looks like the integral you want to find is
[tex]\displaystyle \int_C x^2y\,\mathrm dx - xy^2\,\mathrm dy[/tex]
where C is the circle x ² + y ² = 4. By Green's theorem, the line integral is equivalent to a double integral over the disk x ² + y ² ≤ 4, namely
[tex]\displaystyle \iint\limits_{x^2+y^2\le4}\frac{\partial(-xy^2)}{\partial x}-\frac{\partial(x^2y)}{\partial y}\,\mathrm dx\,\mathrm dy = -\iint\limits_{x^2+y^2\le4}(x^2+y^2)\,\mathrm dx\,\mathrm dy[/tex]
To compute the remaining integral, convert to polar coordinates. We take
x = r cos(t )
y = r sin(t )
x ² + y ² = r ²
dx dy = r dr dt
Then
[tex]\displaystyle \int_C x^2y\,\mathrm dx - xy^2\,\mathrm dy = -\int_0^{2\pi}\int_0^2 r^3\,\mathrm dr\,\mathrm dt \\\\ = -2\pi\int_0^2 r^3\,\mathrm dr \\\\ = -\frac\pi2 r^4\bigg|_{r=0}^{r=2} \\\\ = \boxed{-8\pi}[/tex]
What is the approximate diameter of a sphere whose surface area is 83.96 square inches? Use π = 3.14.
Answer:
5.17
Step-by-step explanation:
The surface area of a sphere is 4[tex]\pi[/tex]r².
83.96=4[tex]\pi[/tex]r²
Divide by 4
20.99=3.14r²
divide by 3.14
6.6847=r²
take the square root
2.585=r
mulitply by 2 (diameter is twice the radius)
5.17
The diameter of the sphere is 5.17 inches.
What is surface area?The space occupied by any two-dimensional figure in a plane is called the area. The area of the outer surface of any object is called as the surface area.
The surface area of a sphere is 4πr².
83.96=4πr²
Divide by 4
20.99=3.14r²
Divide by 3.14
6.6847=r²
Take the square root.
2.585=r
Multiply by 2 (diameter is twice the radius).
r= 2 x 2.585
r = 5.17 inches
Therefore, the diameter of the sphere is 5.17 inches.
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The probability of drawing a red candy at random from a bag of 25 candies is 2/5.
After 5 red candies are removed from the bag, what is the probability of randomly drawing a red candy from the bag?
Please include an explanation!
Answer:
1/4
Step-by-step explanation:
Probability of red * number of candies
2/5 * 25 = 10
There are 10 red candies
25-10 = 15
There are 15 other candies
Remove 5 red
10-5 = 5 red candies
Now we have 5 red candies and 15 other candies= 20 candies
P(red) = red/total = 5/20 = 1/4
The length of rectangle is double of it's breadth .if its perimeter is 48 cm then find it's length.
Step-by-step explanation:
Let breadth be x,
then length is 2x cm,
perimeter of rectangle=2(l+b),
48=2(2x+x),
48÷2=3x
24=3x
24÷3=x
8cm =x,
B=8cm
L=2×8=16cm
Answer:
length =2x
breadth= let x be breadth
Perimeter = 48cm
now
Perimeter = 2 (l+b)
or, 48 cm = 2 (2x+x)
or, 48cm= 4x + 2x
or, 48 cm = 6x
or, 48cm÷6 =x
x = 8 cm
breadth= 8cm
length = 8 ×2 =16cm
The pattern 96, 92, 88, 84, ________ follows the "subtract 4" rule. Study the pattern to find traits that are not obvious in the rule. Explain.
pleas in one or 2 sentences
6 people consists of 3 married couples. Each couple wants to sit with older partner on the left.
Required:
a. How many ways can they be seated together in the row?
b. Suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat?
c. Suppose the six people consist of three married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?
a. The first part asks for how many ways they can be seated together in a row. Therefore we want the permutations of the set of 6 people, or 6 factorial,
6! = 6 [tex]*[/tex] 5
= 30 [tex]*[/tex] 4
= 360 [tex]*[/tex] 2 = 720 possible ways to order 6 people in a row
b. There are two cases to consider here. If the doctor were to sit in the left - most seat, or the right - most seat. In either case there would be 5 people remaining, and hence 5! possible ways to arrange themselves.
5! = 5 [tex]*[/tex] 4
= 20 [tex]*[/tex] 3
= 120 [tex]*[/tex] 1 = 120 possible ways to arrange themselves if the doctor were to sit in either the left - most or right - most seat.
In either case there are 120 ways, so 120 + 120 = Total of 240 arrangements among the 6 people if the doctor sits in the aisle seat ( leftmost or rightmost seat )
c. With each husband on the left, there are 3 people left, all women, that we have to consider here.
3! = 3 [tex]*[/tex] 2 6 ways to arrange 3 couples in a row, the husband always to the left
Which equation is in standard form?
a. X +3= -5y
b. 5-y=x
c. y =3x + 6
d. -8x+ 3y = 12
选项 (d)
option (d)
!!!!!!!!!
Dylan paid $3.50 for 12 folders. How much would be pay for 30 folders?
Answer:
$8.75 (approximately)
BRAINLIEST, PLEASE!
Step-by-step explanation:
3.5/12 = 0.29...
0.29... x 30 = 8.75...
Adam’s house is 2 centimeters from Juan’s house on a map. If each centimeter on the map represents 6 kilometers, how far apart are the two houses?
Answer:
Since 1 cm represents 6 km
2 cm will represent 12 km
Hence Adam's house is 12 km away
Answer:
The answer is C. 3
Step-by-step explanation:
If u divide 6 divided by 2 it will get you 3
i also got it right bc i took the quiz.
Discuss the validity of the following statement. If the statement is always true, explain why. If not, give a counterexample. If the odds for E equal the odds against E', then P(E)P(F)=P(E∩F)
Correction:
Because F is not present in the statement, instead of working onP(E)P(F) = P(E∩F), I worked on
P(E∩E') = P(E)P(E').
Answer:
The case is not always true.
Step-by-step explanation:
Given that the odds for E equals the odds against E', then it is correct to say that the E and E' do not intersect.
And for any two mutually exclusive events, E and E',
P(E∩E') = 0
Suppose P(E) is not equal to zero, and P(E') is not equal to zero, then
P(E)P(E') cannot be equal to zero.
So
P(E)P(E') ≠ 0
This makes P(E∩E') different from P(E)P(E')
Therefore,
P(E∩E') ≠ P(E)P(E') in this case.
Amy is a software saleswoman. Let Y represent her total pay (in dollars). Let X represent the number of copies of "English is Fun" she sells. Suppose that X and Y are related by the equation 110X +2300 = Y.
Answer the questions below. Note that a change can be an increase or decrease.
What is the change in Amy's total pay for each copy of "English is Fun"?
What is Amy's total pay if she doesn't sell any copies of "English is Fun"?
Answer:
1) For every copy she sells, her pay increases by $110
2) Her total pay is 2300
Step-by-step explanation:
1) X is the number of copies she sells. In the equation 110X+ 2300 = Y, X will determine how many times 110 is multiplied. So, for every increase by one in X, Y will also go up by 110
eg.
110(50) + 2300 = 7800 -- if she sells 50 copies
110(51) + 2300 = 7910 -- if she sells 51 copies,
2) If she doesn't sell any copies, the equation becomes 110 * 0 + 2300. Anything multiplied by 0 equals 0, so the equation equals 0 + 2300 = 2300 = Y
Therefore, if she doesn't sell any copies, she will get a pay of $2300
You have two lines. Line A measures 12/16, and line B measures 3/4.
a) Line A is longer
b) The lines are equal.
c) Line B is longer.
Answer:
B
Step-by-step explanation:
Let's compare the two values of 12/16 and 3/4.
Notice that 12/16 isn't a simplified fraction; both the numerator and denominator are divisible by 4. So divide the top and bottom by 4:
12 / 4 = 3
16 / 4 = 4
We are left with:
12/16 = 3/4
Now, we see that Line A and Line B are equal in length, so the answer is B.
~ an aesthetics lover
State whether the given measurements determine zero, one, or two triangles. B = 84°, b = 28, c = 25 (5 points)
Answer:
1
Step-by-step explanation:
The given angle is opposite the longest given side, so the parameters define exactly 1 triangle.
__
Additional comment
When the given angle is opposite the shortest given side, there may be 0, 1, or 2 triangles, depending on the angle and the ratio of sides. For the case here, there is only one possibility.
Identify the triangle, ABC, which has a 72∘ angle and a 36∘ angle.
Answer:
isosceles acute
Step-by-step explanation:
sum of angles in a triangle = 180
to find third angle, subtract 72 & 36 from 180 and you get 72
72, 36, and 72 are all less than 90 so it will be an acute triangle
It will also be isosceles bc there are 2 angles of the same measure
The expression is 3 (x+4)-(2x+7) what is that equivalent too
Answer:
x+19
Step-by-step explanation:
3x+12 - 2x+7 = x+19
Answer:
x-19
Step-by-step explanation:
expand the brackets
3x+12-2x-7
3x-2x-12-7
x-19
You run a souvenir store that sells key rings. You can get 50 key rings from your first supplier for $.50 cents each. You can get the same 50 key rings from your second supplier for $30 total, or you can get them from your third supplier for $27.50. How much will you pay if you get the best deal?
Answer:
$25
Step-by-step explanation:
.5 * 50 = 25
25<27.5<30
The cheapest supplier is the first one.
is x^2 + 1 a polynomial
Answer: Answer: x^2 - 1 is not a polynomial as after simplifying it there will be no variable.
Step-by-step explanation:
i need some help with this!
Answer:
A. 4.19 yd³
Step-by-step explanation:
Volume of a sphere = 4/3πr³
V = 4/3 ( 3.14) (1³)
= 4/3 ( 3.14) (1)
= 4.19 yd³
Answer:
[tex]A\approx 4.19\ yd^2[/tex]
Step-by-step explanation:
The following formula can be used to find the volume of a sphere,
[tex]A=\frac{4}{3}(\pi)r^3[/tex]
Where (r) represents the radius of the sphere or the distance from the center of the sphere to its outer edge. The term ([tex]\pi[/tex]) represents the numerical constant ([tex]3.1415...[/tex]). In this problem, one is given that the radius of the sphere is (1). Substitute this value into the formula and simplify to solve,
[tex]A=\frac{4}{3}(\pi)r^3[/tex]
[tex]A=\frac{4}{3}(\pi)(1)^3[/tex]
[tex]A=\frac{4}{3}(\pi)1[/tex]
[tex]A=\frac{4}{3}(\pi)[/tex]
[tex]A\approx \frac{4}{3}*3.14[/tex]
[tex]A\approx \frac{12.56}{3}[/tex]
[tex]A\approx 4.19\ yd^2[/tex]
The graph of F(x), shown below, has the same shape as the graph of
G(x) = x, but it is shifted up 2 units. What is its equation?
Greetings from Brasil...
We know that the translations are established as follows:
→ Horizontal
F(X + k) ⇒ k units to the left
F(X - k) ⇒ k units to the right
→ Vertical
F(X) + k ⇒ k units up
F(X) - k ⇒ k units down
G(X) = X and F(X) is G(X) shifted up 2 units.
In the statement it is said that there was a translation of 2 units upwards, so
F(X) = G(X) + k where k = 2 units up
F(X) = X + 2find the missing side round to the nearest tenth
Answer:
[tex]cos49 = \frac{26}{x} \\ x = 39.6305[/tex]
Find the radius of a circle with a diameter of 14 cm
Answer:
r = 7 cm
Step-by-step explanation:
The radius is 1/2 of the diameter
r = d/2
r =14/2
r = 7 cm
What are two solutions of x
Answer:
Answer is attached below :)
A box contain 12 balls in which 4 are white 3 are blue and 5 are red.3 balls are drawn at random from the box.find the chance that all three are selected
Answer:
3/11
Step-by-step explanation:
From the above question, we have the following information
Total number of balls = 12
Number of white balls = 4
Number of blue balls = 3
Number of red balls = 5
We solve this question using combination formula
C(n, r) = nCr = n!/r!(n - r)!
We are told that 3 balls are drawn out at random.
The chance/probability of drawing out 3 balls = 12C3 = 12!/3! × (12 - 3)! = 12!/3! × 9!
= 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1/(3 × 2 × 1) × (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)
= 220 ways
The chance of selecting 3 balls at random = 220
To find the chance that all the three balls are selected,
= [Chance of selecting (white ball) × Chance of selecting(blue ball) × Chance of selecting(red balls)]/ The chance/probability of drawing out 3 balls
Chance of selecting (white ball)= 4C1
Chance of selecting(blue ball) = 3C1 Chance of selecting(red balls) = 5C1
Hence,
= [4C1 × 3C1 × 5C1]/ 220
= 60/220
= 6/22
= 3/11
The chance that all three are selected is = 3/11
Find the length of segment YZ in the diagram below.
Answer:
2√2
Step-by-step explanation:
hope you understand.
Vhat is the volume of the right rectangular prism?
Will mark brainliest
Answer:
432 mm³
Step-by-step explanation:
Volume of a Rectangular Prism: V = lwh
Step 1: Define variables
l = 8
w = 6
h = 9
Step 2: Plug into formula
V = 8(6)(9)
Step 3: Evaluate
V = 48(9)
V = 432
And we have our answer!
PLEASE HELP! (3/4) - 50 POINTS -
Answer:
C
Step-by-step explanation:
The set of data will only become more narrow when the standard deviation is decreased, so D isn't correct. The data isn't going to shift directions unless there's a translation, so A and B are both out. That leaves us with C. The opposite of answer D.
Answer:
C. It produces a wider range of probable values
Step-by-step explanation:
The set of data that we have cannot shift in directions unless there is a translation, so therefore, A and B are both out. The set of data would become smaller when the standard deviation is decreases so therefore, D isn't correct. So, that leaves us with only one answer.
C. It produces a wider range of probably values.
The sentence "the length of a rectangle is 6 feet more than the width" can be translated as / = 6 + w.
O False
0 True
Answer:
true
Step-by-step explanation:
if the width is w and the length is l then 6 more than the width is w+6 and that is equal to the length so l = w+6
Suppose the bacteria population in a specimen increases at a rate proportional to the population at each moment. There were 100 bacteria 4 days ago and 100,000 bacteria 2 days ago. How many bacteria will there be by tomorrow
9514 1404 393
Answer:
about 3,160,000,000
Step-by-step explanation:
"Increases at a rate proportional to population" means the growth is exponential. It can be modeled by the equation ...
p = ab^t
We can find 'a' and 'b' using the given data points.
100 = ab^(-4) . . . . . . . population 4 days ago
100,000 = ab^(-2) . . . population 2 days ago
Dividing the second equation by the first, we find ...
1000 = b^2
b = 1000^(1/2)
Substituting for b in the first equation, we have ...
100 = a(1000^(1/2))^(-4) = a(1000^-2)
100,000,000 = a
Then the population model is ...
p = 100,000,000×1000^(t/2)
__
Tomorrow (t=1), the population will be ...
p = 100,000,000 × 1000^(1/2) ≈ 31.6 × 100,000,000
p ≈ 3,160,000,000 . . . . . bacteria by tomorrow
_____
Additional comment
We could write this as ...
p = 10^(8+1.5t)
Then for t=1, this is p = 10^(8+1.5) = 10^0.5 × 10^9 = 3.16×10^9
Prove that if a and b are integers, then for any integer k one has (a,b) = (a + kb,b). (Hint: Show that they are mutually divisible.)
Answer:
The operation:
(a,b) is equal to the rest of the division of a by b.
Now, if we have:
(a + kb,b) = (a,b) + (k*b,b)
But if we have that k and b are integers, then:
(k*b)/b = k
So b divides k*b into a whole number, this means that (k*b,b) = 0
then:
(a + kb,b) = (a,b) + (k*b,b) = (a,b) + 0 = (a,b)