Answer:
f(x).g(x) = c0+c1x+c2x2+c3x3+⋯.
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
c₀ = 42 , c₁ =97 , c₂ = 95, c₃ =104, ...…...
Step-by-step explanation:
Suppose that f(x) and g(x) are given by the power series
f(x)=6+7x+3x2+5x3+⋯ and
g(x)=7+8x+3x2+4x3+⋯
By multiplying power series ,
f(x).g(x) = (6+7x+3x2+5x3+⋯).(7+8x+3x2+4x3+⋯)
= 6(7+8x+3x2+4x3+⋯)+7x(7+8x+3x2+4x3+⋯)+3x2(7+8x+3x2+4x3+⋯) + 5x3(7+8x+3x2+4x3+⋯)+......
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
f(x).g(x) = c0+c1x+c2x2+c3x3+⋯.is the form
[tex]f(x).g(x) = 42 +97x +95x^{2} + 104x^{3} +77x^{4} +27x^{5} + 20x^{6}+....[/tex]
c₀ = 42 , c₁ =97 , c₂ = 95, c₃ =104, ...……
In this problem, power series f(x) and g(x) are multiplied together term by term to produce the product h(x). The first few terms of the h(x) series are found by sequentially multiplying the corresponding terms of f(x) and g(x). The procedures produced the first four terms as 42, 98, 170, and 356 respectively.
Explanation:In the field of mathematics, specifically calculus, you can multiply two power series together term by term. In this case, function f(x) = 6 + 7x + 3x2 + 5x3 + ⋯ and g(x) = 7 + 8x + 3x2 + 4x3 + ⋯ are provided. When you multiply these two power series, the product h(x) = f(x) ⋅ g(x) is obtained by multiplying each corresponding term in the series of f(x) and g(x) together.
To find the first few terms we consider it like
First term, c0: f0*g0 = 6*7 =42Second term, c1: f1*g0 + g1*f0 = 7*6 + 8*7 = 98Third term, c2: f2*g0 + f1*g1 + g2*f0 = 18*3*6 + 7*8 + 21*6 = 170Fourth term, c3: f3*g0 + f2*g1 + f1*g2 + f0*g3 = 30*7 + 18*8 + 21*7 + 6*0 = 356and so on for the subsequent terms.
Learn more about Power Series Multiplication here:https://brainly.com/question/31404538
#SPJ11
Finish solving the system of equations, y = x – 15 and y = –2x + 3, using the substitution method. 1. Use substitution to create a one-variable linear equation: x – 15 = –2x + 3 2. Solve to determine the unknown variable in the equation: 3x = 18 x = 6 3. Substitute the value of the variable into either original equation to solve for the other variable. 4. Write the solution to the system of equations as an ordered pair. The solution to the system is .
Answer:
THE ANSWER IS (6,-9)
Step-by-step explanation:
The solution to the equation is ( 6 , -9 )
The value of x = 6
The value of y = -9
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the first equation be represented as A
Now , the value of A is
y = x - 15 be equation (1)
Let the second equation be represented as B
Now , the value of B is
y = -2x + 3 be equation (2)
Substituting the value of equation (1) in equation (2) , we get
x - 15 = -2x + 3
On simplifying the equation , we get
Adding 2x on both sides of the equation , we get
x + 2x - 15 = 3
Adding 15 on both sides of the equation , we get
3x = 18
Divide by 3 on both sides of the equation , we get
x = 6
Therefore , the value of x is 6
Substitute the value of x in equation (1) , we get
y = x - 15
y = 6 - 15
y = -9
Therefore , the value of y is -9
Hence , the values of x and y of the equation is ( 6 , -9 )
To learn more about equations click :
https://brainly.com/question/19297665
#SPJ6
n a study of the accuracy of fast food drive-through orders, Restaurant A had 314 accurate orders and 61 that were not accurate. a. Construct a 90% confidence interval estimate of the percentage of orders that are not accurate. b. Compare the results from part (a) to this 90% confidence interval for the percentage of orders that are not accurate at Restaurant B: 0.147less thanpless than0.206. What do you conclude?
Answer:
a. 0.1576<p<0.2310
b. The two restaurants likely have similar order rates which are inaccurate.
Step-by-step explanation:
a. We first calculate the proportion, [tex]\hat p[/tex]:
[tex]\hat p=\frac{61}{314}\\\\=0.1943[/tex]
-We use the z-value alongside the proportion to calculate the margin of error:
[tex]MOE=z\sqrt{\frac{\hat p(1-\hat p)}{n}}\\\\=1.645\times \sqrt{\frac{0.1943(1-0.1943)}{314}}\\\\=0.0367[/tex]
The confidence interval at 90% is then calculated as:
[tex]CI=\hat p\pm MOE\\\\=0.1943\pm 0.0367\\\\=[0.1576,0.2310][/tex]
Hence, the confidence interval at 90% is [0.1576,0.2310]
b. From a above, the calculated confidence interval is 0.1576<p<0.2310
-We compare the calculated CI to the stated CI of 0.147<p<0.206
-The two confidence intervals overlap each other and have the same value for 0.1576<p<0.206
-Hence, we conclude that the two restaurants likely have similar order rates which are inaccurate.
How do I express the following as a fractional part of a year?
3 month
55 days
1 month
7 months
120 days
Answer:
See below.
Step-by-step explanation:
A year has 12 months and 365 days.
3 months: 3/12 year = 1/4 year
55 days: 55/365 year = 11/73 year
1 month: 1/12 year
7 months = 7/12 year
120 days = 120/365 year = 24/73 year
Answer:
3 months = 3/12 | 55 days = 55/365 | 1 month = 1/12 | 7 months = 7/12 | 120 days = 120/365
Step-by-step explanation:
All you have to do is write the numbers with the total underneath. With months, it is out of 12 because there are 12 months in a year. With days, it is out of 365 days because there are 365 days in total in a year.
If needed, simplify the answer to its simplest form
Inquiries arrive at a record message device according to a Poisson process of rate 15 inquiries per minute. The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________.
Answer:
0.0498 = 4.98%
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
Inquiries arrive at a record message device according to a Poisson process of rate 15 inquiries per minute.
Each minute has 60 seconds.
So a rate of 1 inquire each 4 seconds.
The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately
Mean of 1 inquire each 4 seconds, so for 12 seconds [tex]\mu = \frac{12}{4} = 3[/tex]
This probability is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498[/tex]
The probability that it takes more than 12 seconds for the first inquiry to arrive in a Poisson process at a rate of 15 inquiries per minute is calculated using the exponential distribution formula: e^{-15*0.2}.
The student is asking for the probability that it takes more than 12 seconds for the first inquiry to arrive at a record message device, with inquiries arriving according to a Poisson process with a rate of 15 inquiries per minute. Since the time between arrivals in a Poisson process follows an exponential distribution, we can calculate this probability using the exponential distribution's formula.
The mean interval between inquiries is the inverse of the rate, which for 15 inquiries per minute is 1/15 minute per inquiry, or 4 seconds per inquiry. To convert minutes to seconds, multiply by 60. Therefore, the average interval is 4 seconds.
The exponential distribution gives us the probability that the time until the first event (inquiry) exceeds a certain amount, t, which is P(T > t) = e-λ*t, where λ is the rate and T is the time. In this case, t is 12 seconds or 0.2 minutes. Therefore, the probability (P) that the time is more than 12 seconds is: P(T > 0.2) = e-15*0.2.
Simplify.
Remove all perfect squares from inside the square root.
V72 =
Final answer:
To simplify √72 by removing all perfect squares, you factor 72 into 2³ × 3², then take out the square root of 4 and 9 to get 6√(2) as the simplified result.
Explanation:
To simplify the square root of 72 and remove all perfect squares from inside the square root, you first need to factor 72 into its prime factors. The prime factorization of 72 is 2³ × 3². This can be rewritten as (2² × 3²) × 2, which simplifies to (4 × 9) × 2.
Since the square root of 4 and 9 are perfect squares, you can take them out of the square root, resulting in √(4) × √(9) × √(2) = 2 × 3 × √(2) = 6√(2).
Therefore, the simplified form of √72 is 6√(2), which removes all perfect squares from inside the square root.
Ms.Foster built a hexagon by combining two trapezoids that were exactly the same size and shape. What fraction of the area of the whole shape is each trapezoid?
Answer:
[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
1. Let's draw the trapezoids, then combine them. The first trapezoid has larger Base measuring 4.67 cm, parallel and minor base =2, an area of 4.98
2. Since the other one is a copy, same area, same base. The junction of both trapezoids generates a hexagon. We have another trapezoid with an area of 4.98. The hexagon has a total area of 9.96
3. So each trapezoid has exactly 1/2 of the area of the hexagon.
What can we conclude for the following linear homogeneous equation? t2y''+3ty'+y=0, t>0. y1=t is a solution. By the method of reduction of order, we can ALWAYS find another independent solution y2 satisfying W(y1,y2)≠ 0 y1=t−1 is a solution. By the method of reduction of order, we can ALWAYS find another independent solution y2 satisfying W(y1,y2)≠ 0 None of these y1=t is a solution. By the method of reduction of order, we can SOMETIMES find another independent solution y2 satisfying W(y1,y2)≠ 0 y1=t−1 is a solution. By the method of reduction of order, we can SOMETIMES find another independent solution y2 satisfying W(y1,y2)≠ 0
Answer:
Required conclusion is that if [tex]y_1, y_2[/tex] satisfies given differential equation and wronskean is zero then they are considered as solution of that differential equation.
Step-by-step explanation:
Given differential equation,
[tex]t^2y''+3ty'+y=0[/tex] [tex] t>0\hfill (1)[/tex]
(i) To verify [tex]y_1(t)=t[/tex] is a solution or not we have to show,
[tex]t^2y_{1}^{''}+3ty_{1}^{'}+y_1=0[/tex]
But,
[tex]t^2y_{1}^{''}+3ty_{1}^{'}+y_1=(t^2\times 0)=(3t\times 1)+t=4t\neq 0[/tex]
hence [tex]y_1[/tex] is not a solution of (1).
Now if [tex]y_2=t-1[/tex] is another solution where [tex]y_2(t)=t-1[/tex] then,
[tex]t^2y_{2}^{''}+3ty_{2}^{'}+y_2=0[/tex]
But,
[tex]t^2y_{2}^{''}+3ty_{2}^{'}+y_2=(t^2\times 0)+(3t\times 1)+t-1=4t-1\neq 0[/tex]
so [tex]y_2[/tex] is not a solution of (1).
(ii) Rather the wronskean,
[tex]W(y_1,y_2)=y_{1}y_{2}^{'}-y_{2}y_{1}^{'}=(t\times 1)-((t-1)\times 1)=t-t+1=1\neq 0[/tex]
Hence it is conclude that if [tex]y_1, y_2[/tex] satisfies (i) along with condition (ii) that is wronskean zero, only then [tex]y_1, y_2[/tex] will consider as solution of (1).
At crescent high school, 108 students plan on going to an in-state college and 63 students plan on going out of state college what is the ratio of students planning on going to an in state college to students planning on going to an out state college
Answer:
12/7
Step-by-step explanation:
To set up a ratio of "thing 1" to "thing 2", build a fraction with "thing 1" on top (numerator) and "thing 2" on the bottom (denominator).
The ratio of in-state students to out-of-state students is [tex]\frac{108}{63}[/tex] and this can be simplified ("reduced") by dividing both numbers by 9 to get the fraction 12/7.
Final answer:
Explanation on finding the ratio of students planning in-state college to out-of-state college.
Explanation:
The ratio of students planning on going to an in-state college to students planning on going to an out-of-state college:
To find the ratio, you need to divide the number of students planning in-state college by the number planning out-of-state college.
Number of students planning in-state college: 108
Number of students planning out-of-state college: 63
Ratio: 108:63 which simplifies to 36:21 or 12:7
The expression ?/g+1 + (2g+1/g - g+1/2g) simplifies to 3g^2+10g+1/2g^2+2g. The unknown value is...
A.) 1
B.) 2
C.) 3
D.) 4
Answer: 3
Step-by-step explanation:
E2020 answer
Answer:
3
Step-by-step explanation:
answer on edge
as part of a fundraiser, a local organization collected 418 returnable bottles and cans, some worth 5 cents each and the rest worth 10 cents each. if the total value of the cans and bottles was $25.00, how many 5 cent bottles or cans and how many 10 cent bottles or cans were collected
Using a system of equations, it was determined there were 336 bottles worth 5 cents each and 82 bottles worth 10 cents each, summing up to a total of $25.
The local organization collected 418 returnable bottles and cans, some worth 5 cents each and the rest worth 10 cents each. The total value was $25.00. We need to find out how many bottles were worth 5 cents and how many were worth 10 cents.
We can set up two equations to solve this problem using algebra. Let's assume that the number of 5-cent bottles is x and the number of 10-cent bottles is y.
The first equation will represent the total number of bottles: x + y = 418.
The second equation will represent the total value of these bottles: 0.05x + 0.10y = 25.
To solve this system of equations, we can multiply the second equation by 100 to eliminate the decimals and then apply the method of substitution or elimination to find the values of x and y.
Rewrite the second equation without decimals: 5x + 10y = 2500.
Multiply the first equation by 5: 5x + 5y = 2090.
Subtract the modified first equation from the second equation: 5y = 410.
Divide both sides by 5 to find y: y = 82.
Substitute y = 82 into the first equation and solve for x: x = 418 - 82 = 336.
Therefore, there were 336 bottles or cans worth 5 cents each and 82 bottles or cans worth 10 cents each.
A bag has 2 blue marbles, 3 red marbles, and 5 white marbles. Which events have a probability greater than 5 ? Select three
options
choosing 1 blue marble
choosing 1 red marble
ll choosing 1 red marble, not replacing it, and then choosing a blue marble
choosing 1 white marble, replacing it, and choosing another white marble
choosing 1 white marble
Save and Exit
Submit
Mark this and retum
Answer:
choosing 1 red marble ; choosing 1 white marble, replacing it, and choosing another white marble ; and choosing 1 white marble
Step-by-step explanation:
There are 2+3+5 = 10 marbles. The probability of choosing 1 blue marble is 2/10 = 1/5; this is not greater than 1/5.
The probability of choosing 1 red marble is 3/10; this is greater than 2/10, which is the same as 1/5.
The probability of choosing 1 red marble is 3/10; not replacing it and choosing a blue marble would then be a probability of 2/9. Together this is a probability of 3/10(2/9) = 6/90 = 3/45; this is smaller than 9/45, which is the same as 1/5.
The probability of choosing 1 white marble is 5/10 = 1/2; replacing it and choosing another white marble would be 1/2. Together this is a probability of 1/2(1/2) = 1/4; this is greater than 4/20, which is the same as 1/5.
The probability of choosing 1 white marble is 5/10 = 1/2. This is greater than 2/10, which is the same as 1/5.
Answer:
b,d,e are the awnsers
Step-by-step explanation:
The function rule of a certain function is y = -3 x + 1. If the input is 7, what is the output?
Here are the choices,
1. -22
2. -21
3. -20
4. -24
Rover’s dog bowl is pictured below. Approximately how much water does it hold? Use 3.14 for π and round your answer to the nearest cubic inch.
Answer:
57 in³
Step-by-step explanation:
Area of cylinder= [tex]\pi {r}^{2} h[/tex]
Let's find the radius.
Radius= Diameter ÷2
Radius= 6 ÷2
Radius= 3 in.
Area of Rover's dog bowl
= 3.14(3)²(2)
= 56.52
= 57 in³ (nearest cubic inch)
Answer:
57 in3
Step-by-step explanation:
Solve for x.
2(4x− 9) = 5(x − 4)
Answer:
2(4x− 9) = 5(x − 4)
=>8x- 18= 5x-20
=>8x-5x= -20+18
=>3x = -2
=>x= -2/3
Short=7 Long=
30 60 90 triangle
Final answer:
To find the "long" side opposite the 60° angle in a 30-60-90 triangle with a "short" side length of 7, we multiply 7 by √3 to get 7√3.
Explanation:
The student appears to be asking about the relationships between the sides of a 30-60-90 triangle, which is a special type of right triangle. In such a triangle, the sides are in the ratio 1:√3:2. So, if the "short" side opposite the 30° angle is given as 7, then the "long" side opposite the 60° angle can be found by multiplying the short side by √3. To find the hypotenuse (opposite the 90° angle), we would multiply the short side by 2.
Step-by-step to find the long side:
Identify the given "short" side, which is opposite the 30° angle, as 7.Use the special triangle ratio for a 30-60-90 triangle to find the "long" side by calculating 7 × √3.Therefore, the length of the "long" side opposite the 60° angle is 7√3.
A rectangle has a length of 11 meters less than 10 times its width. If the area of the rectangle is 9888 square meters, find the length of the rectangle.
Step-by-step explanation:
The Length ,
L
=
284
f
t
.
Explanation:
Given:
Rectangle
Area,
A
=
8804
f
t
2
let W bet he width of the rectangle
L be the length of the rectangle
L
=
10
W
−
26
E
q
u
a
t
i
o
n
1
substitute to
e
q
u
a
t
i
o
n
2
A
=
(
L
)
(
W
)
e
q
u
a
t
i
o
n
2
A
=
(
10
W
−
26
)
(
W
)
8804
=
(
10
W
−
26
)
(
W
)
factor
8804
=
2
(
5
W
−
13
)
(
W
)
divide both sides by 2
4402
=
(
5
W
−
13
)
(
W
)
4402
=
5
W
2
−
13
W
transposing 4402 to the right side of the equation
0
=
5
W
2
−
13
W
−
4402
by quadratic formula
W
=
−
(
−
13
)
+
√
(
−
13
)
2
−
4
(
5
)
(
−
4402
)
2
(
5
)
W
=
[
13
+
√
169
+
88040
]
10
W
=
13
+
(
√
88209
)
10
W
=
13
+
297
10
W
=
310
10
W
=
31
ft
Thus ,
L
=
10
W
−
26
=
10
(
31
)
−
26
L
=
284
f
t
.
answer
W
=
−
(
−
13
)
−
√
−
(
−
13
2
)
−
4
(
5
)
(
−
4402
)
2
(
5
)
this is discarded since this will yield a negative
To find the length of the rectangle, set up an equation using the given information. Solve the quadratic equation to find the width and substitute it back to find the length.
Explanation:To find the length of the rectangle, we can set up an equation using the given information. Let's assume the width of the rectangle is 'w' meters. According to the problem, the length of the rectangle is 10 times its width minus 11 meters, so it can be represented as 10w - 11 meters. The area of the rectangle is given as 9888 square meters. We know that the formula for the area of a rectangle is length times width, so we can write the equation as:
w (10w - 11) = 9888
Expanding the equation and rearranging terms, we get:
10w^2 - 11w - 9888 = 0
Now, we can solve this quadratic equation for 'w' and find the width of the rectangle. Once we have the width, we can substitute it back into the expression for the length (10w - 11) to find the length of the rectangle.
Learn more about Finding the length of a rectangle here:https://brainly.com/question/34224051
#SPJ2
A buyer owes a supplier $820. The terms were 3/10, n/60. Within 10 days, the buyer sent in a payment of $140. How much is the new balance? (Round your answers to the nearest hundredth)
Answer:
The new balance is $675.8
Step-by-step explanation:
Solution
Given that
The total amount of loan = $820.
The terms were = 3/10, n/60
Within 10 days, the buyer sent in a payment of =$140
What is the new balance =?
Now,
3/10 = if the amount is paid in 10 days, a discount of is included
n/60 = This means that all amount should be paid within 60 days
Thus,
$140 for 3/10 as this is paid within 10 days
140 *3% = 140 * 3/100
we get
=$4.2 of discount
Then,
The balance amount becomes $ 820 - 140 -4.2
=$675.8
What is the volume of the prism?
Answer: the volume is: 280.8
Step-by-step explanation:
volume= length x width x height
Answer:
140.4 cm^3
Step-by-step explanation:
The formula for calculating volume of a triangular prism : 1/2 × h × l × b
1/2 × 4 × 9 × 7.2 = 140.4 cm^3
What is another name used for permanent cement?
Answer:
Zinc Phosphate cement
Answer:
luting agent
Step-by-step explanation:
For which of the following procedures would you include a temporary
luting agent is basicslly the same thing
Of the 4,700 students at Medium Suburban College (MSC), 50 play collegiate soccer, 60 play collegiate lacrosse, and 96 play collegiate football. Only 4 students play both collegiate soccer and lacrosse, 5 play collegiate soccer and football, and 17 play collegiate lacrosse and football. No students play all three sports. ____ % of the college soccer players also play one of the other two sports at the collegiate level.
Answer:
18%
Step-by-step explanation:
Of the 50 soccer players, 4 play soccer and lacrosse, and 5 play soccer and football. That is, 9 of the 50 players also play one of the other sports.
9/50 × 100% = 18%
18% of soccer players also play another sport.
The University of Montana ski team has nine entrants in a men's downhill ski event. The coach would like the first, second, and third places to go to the team members. In how many ways can the nine team entrants achieve first, second, and third places
Answer:
504 ways
Step-by-step explanation:
The first position can go in 9 ways
The second position can go to the team in 8 ways
The third position can go to the team in 7 ways
Therefore, the first,second and third position can go to the team in:
9 X 8 X 7=504 ways
or Simply, we can use Permutation.
[tex]^9P_3=504[/tex] ways
Write the ratio of hands to cellphones in simplest form. it was 8 hands and 4 phones
Answer:
The ratio is 2:1
Step-by-step explanation:
This is since there are 2 times the amount of hands as there are phones
What is 1/6 divide by 1/3
0.555555556 so that's it hope this helped
Answer:
0.5
Step-by-step explanation:
Because I said so
Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.42cm and a standard deviation of 0.36cm. Using the empirical rule, what percentage of the apples have diameters that are between 7.06cm and 7.78cm
Answer:
68% of the diameters are between 7.06 cm and 7.78 cm
Step-by-step explanation:
Mean diameter = μ = 7.42
Standard Deviation = σ = 0.36
We have to find what percentage of diameters will be between 7.06 cm and 7.78 cm. According to the empirical rule, for a bell-shaped data:
68% of the values are within 1 standard deviation of the mean. i.e. between μ - 1σ and μ + 1σ95% of the values are within 2 standard deviations of the mean. i.e. between μ - 2σ and μ + 2σ99.7% of the values are within 3 standard deviation of the mean. i.e. between μ - 3σ and μ + 3σSo, we first need to find how many standard deviations away are the given two data points. This can be done by converting them to z-score. A z score tells us that how far is a data value from the mean. The formula to calculate the z-score is:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
x = 7.06 converted to z score will be:
[tex]z=\frac{7.06-7.42}{0.36}=-1[/tex]
x = 7.78 converted to z score will be:
[tex]z=\frac{7.78-7.42}{0.36}=1[/tex]
This means the two given values are 1 standard deviation away from the mean and we have to find what percentage of values are within 1 standard deviation of the mean.
From the first listed point of empirical formula, we can say that 68% of the data values lie within 1 standard deviation of the mean. Therefore, 68% of the diameters are between 7.06 cm and 7.78 cm
Approximately 68% of the apples have diameters between 7.06cm and 7.78cm.
Explanation:To determine the percentage of apples with diameters between 7.06cm and 7.78cm, we can use the empirical rule which is based on the standard deviation. According to the empirical rule, approximately 68% of the apples will fall within one standard deviation of the mean, which in this case is between 7.42 - 0.36 and 7.42 + 0.36. In other words, between 7.06cm and 7.78cm. Therefore, approximately 68% of the apples have diameters between 7.06cm and 7.78cm.
Learn more about empirical rule here:https://brainly.com/question/35669892
#SPJ3
Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of the sphere x2 + y2 + z2 = 9. (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk x2 + y2 ≤ 9, oriented downward, and S2 = S1 ∪ S.)
Close off the hemisphere [tex]S[/tex] by attaching to it the disk [tex]D[/tex] of radius 3 centered at the origin in the plane [tex]z=0[/tex]. By the divergence theorem, we have
[tex]\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV[/tex]
where [tex]R[/tex] is the interior of the joined surfaces [tex]S\cup D[/tex].
Compute the divergence of [tex]\vec F[/tex]:
[tex]\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2[/tex]
Compute the integral of the divergence over [tex]R[/tex]. Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:
[tex]\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}[/tex]
So the volume integral is
[tex]\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5[/tex]
From this we need to subtract the contribution of
[tex]\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S[/tex]
that is, the integral of [tex]\vec F[/tex] over the disk, oriented downward. Since [tex]z=0[/tex] in [tex]D[/tex], we have
[tex]\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k[/tex]
Parameterize [tex]D[/tex] by
[tex]\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath[/tex]
where [tex]0\le u\le 3[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to be
[tex]\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k[/tex]
Then taking the dot product of [tex]\vec F[/tex] with the normal vector gives
[tex]\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v[/tex]
So the contribution of integrating [tex]\vec F[/tex] over [tex]D[/tex] is
[tex]\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4[/tex]
and the value of the integral we want is
(integral of divergence of F) - (integral over D) = integral over S
==> 486π/5 - (-81π/4) = 2349π/20
The process involves applying the Divergence Theorem over a closed surface formed by adding a bottom disk to the sphere and converting the given surface integral into a volume integral which is easier to calculate. F(x, y, z) must be used accurately in all calculations.
Explanation:To solve this problem, we need to apply the Divergence Theorem to evaluate the surface integral of the given vector field F(x, y, z) over the top half of the sphere. Before we do that, we must first compute the integrals over S1 and S2, where S1 is the disk x² + y² ≤ 9, oriented downward, and S2 = S1 ∪ S.
On applying the Divergence Theorem over a closed surface S₁ + S₂ obtained by adding a bottom disk to the sphere, we can convert the given surface integral into a volume integral over the region inside the closed surface.
Once we obtain this volume integral, this should simplify our calculations, as volume integrals are typically easier to evaluate than surface integrals. This strategy utilises the power of the Divergence Theorem, which connects the flow of a vector field across a surface to the behavior of the field inside the volume enclosed by the surface.
Remember to use the correct vector field formulas for F when calculating the integrals over S1 and S2. Ensure each step is carefully followed so errors are not made.
Learn more about Divergence Theorem here:https://brainly.com/question/33180298
#SPJ3
Sally invests $10,500 in an account that earns 6% annual simple interest. Assuming she makes no additional deposits or withdrawals, how much interest will Sally earn after 4 years? Will mark brainliest PLSSSSS HELLLLLP
Answer:
$2,520
Step-by-step explanation:
The simple Interest earned on an deposit, P at a rate of r% for a period of t years is calculated using the formula:
[tex]\text{ Simple Interest}=\dfrac{Principal*Time*Rate}{100}[/tex]
P=$10,500
R=6%
T=4 years
Therefore:
[tex]\text{ Simple Interest}=\dfrac{10500*4*6}{100}\\=\$2,520[/tex]
Sally will earn $2520 interest after 4 years.
solve for x: logx(16)=2
Answer:
4, -4
Step-by-step explanation:
Take the (+2)th root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
x = 4 , − 4
y=2x−1
5x−4y=1
Is (1,1)(1,1)left parenthesis, 1, comma, 1, right parenthesis a solution of the system?
Answer:
(1,1) is a solution of the system.
Step-by-step explanation:
Let's solve the system.
y = 2x - 1
5x - 4y = 1
In the first equation, y is already separated as a function of x. So we replace in the second equation;
5x - 4(2x - 1) = 1
5x - 8x + 4 = 1
4 - 1 = 8x - 5x
x = 1
y = 2x - 1 = 2(1) - 1 = 1
(1,1) is a solution of the system.
Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his time between pitches are normally distributed with a mean of seconds and a standard deviation of seconds. What percentage of his times between pitches are longer than seconds
The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his time between pitches are normally distributed with a mean of 29 seconds and a standard deviation of 2.1 seconds. What percentage of his times between pitches are longer than 31 seconds ?
Given Information:
Mean pitching time = μ = 29 seconds
Standard deviation of pitching time = σ = 2.1 seconds
Required Information:
P(X > 31) = ?
Answer:
[tex]P(X > 31) = 17.11 \%[/tex]
Step-by-step explanation:
What is Normal Distribution?
We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.
We want to find out the probability that what percentage of his times between pitches are longer than 31 seconds.
[tex]P(X > 31) = 1 - P(X < 31)\\P(X > 31) = 1 - P(Z < \frac{x - \mu}{\sigma} )\\P(X > 31) = 1 - P(Z < \frac{31 - 29}{2.1} )\\P(X > 31) = 1 - P(Z < \frac{2}{2.1} )\\P(X > 31) = 1 - P(Z < 0.95)\\[/tex]
The z-score corresponding to 0.95 is 0.8289
[tex]P(X > 31) = 1 - 0.8289\\P(X > 31) = 0.1711\\P(X > 31) = 17.11 \%[/tex]
Therefore, 17.11% of his times between pitches are longer than 31 seconds.
How to use z-table?
Step 1:
In the z-table, find the two-digit number on the left side corresponding to your z-score. (e.g 0.9 1.4, 2.2, 0.5 etc.)
Step 2:
Then look up at the top of z-table to find the remaining decimal point in the range of 0.00 to 0.09. (e.g. if you are looking for 0.95 then go for 0.05 column)
Step 3:
Finally, find the corresponding probability from the z-table at the intersection of step 1 and step 2.
Select the correct answer.
What is the solution for p in the equation?
1/3p + 1/2p = 7/6p +5 + 4
A.
p = -6
B.
p = -1
C.
p = 1
D.
p = 6
To solve for p in the equation 1/3p + 1/2p = 7/6p + 5 + 4, you need to combine like terms and isolate p. The solution is p = 27.
Explanation:To solve for p in the equation 1/3p + 1/2p = 7/6p + 5 + 4, you need to combine like terms and isolate p on one side of the equation.
Multiply the fractions by their respective denominators to eliminate the fractions. This gives us 2/6p + 3/6p = 7/6p + 9.
Combine like terms. On the left side, 2/6p + 3/6p = 5/6p. On the right side, 7/6p + 9 remains unchanged.
Subtract 5/6p from both sides to isolate p. This gives us 7/6p - 5/6p = 9.
Combine like terms on the left side. 7/6p - 5/6p = 2/6p.
Simplify the equation further. 2/6p = 9 can be reduced to 1/3p = 9.
Multiply both sides of the equation by the reciprocal of 1/3, which is 3/1. This gives us p = 9 * (3/1) = 27.
Therefore, the solution for p in the equation is p = 27.