The number of times the inner loop be iterated when the algorithm is implemented and run is 10.
We are given that;
i := 1 to 4, for j := 1
Now,
a. The inner loop will be iterated 10 times when the algorithm is implemented and run.
| i | j | Iterations |
|---|---|------------|
| 1 | 1 | 1 |
| 2 | 1 | 2 |
| 2 | 2 | 3 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 3 | 3 | 6 |
| 4 | 1 | 7 |
| 4 | 2 | 8 |
| 4 | 3 | 9 |
| 4 | 4 | 10 |
b. The number of iterations of the inner loop depends on the value of i. For each i, the inner loop runs from j = 1 to j = i.
So, the total number of iterations is the sum of i from i = 1 to i = n. This is a well-known arithmetic series, which can be written as:
[tex]$\sum_{i=1}^n i = \frac{n(n+1)}{2}$[/tex]
This is the formula for the number of iterations of the inner loop.
Therefore, by algorithm the answer will be 10.
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a. The inner loop will be iterated 10 times when the algorithm is implemented and run.
b. The total number of iterations for the inner loop when the algorithm is implemented and run will be (n^2 + n)/2.
a. In the given algorithm segment, we have two nested loops. The outer loop runs from 1 to 4, and the inner loop runs from 1 to i. Let's analyze how many times the inner loop will be iterated.
For i = 1, the inner loop will run once.
For i = 2, the inner loop will run twice.
For i = 3, the inner loop will run three times.
For i = 4, the inner loop will run four times.
Therefore, the total number of iterations for the inner loop can be calculated by summing the iterations for each value of i:
1 + 2 + 3 + 4 = 10
b. In this algorithm segment, we have the same nested loops as in part a, but the range of the outer loop is from 1 to n, where n is a positive integer.
The inner loop iterates from 1 to i, where i takes the values from 1 to n. So, for each value of i, the inner loop will run i times.
To determine the total number of iterations for the inner loop, we need to sum the iterations for each value of i from 1 to n:
1 + 2 + 3 + ... + n
This is an arithmetic series, and the sum of an arithmetic series can be calculated using the formula:
Sum = (n/2)(first term + last term)
In this case, the first term is 1, and the last term is n. Substituting these values into the formula, we get:
Sum = (n/2)(1 + n) = (n^2 + n)/2
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Numbers from zero to nine are individually selected at random and combined to make a code that contains a six-digit number. Numbers can be repeated. If you were given ten tries to guess the code what would be the probability of guessing the correct code? Give you answer as a fraction. Do not include commas in your answer, for example, 31,000 would be written as 31000.
The probability of guessing correctly within ten tries is 10/1,000,000, simplifying to 1/100,000.
Explanation:This problem is related to probability. The total number of ways to form a six-digit code with numbers from 0 to 9, where numbers can be repeated, is 10^6 because there are 10 possible choices for each of the 6 places. Thus there are 1,000,000 possible codes.
The probability of you correctly guessing the code on any one try would then be 1/1,000,000. If you try ten times, each attempt independent of the others, you still have a 1/1,000,000 chance each try. Combining these ten independent events, the total probability of guessing the correct code in ten tries would be 10/1,000,000.
Therefore, your probability is 10/1,000,000, which simplifies to 1/100,000.
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Show your work to prove that the inverse of f(x) is g(x).
[tex]f(x) = \frac{x+9}{4}\\g(x)=4x-9[/tex]
Answer:
Below.
Step-by-step explanation:
If g(x) is the inverse of g(x) then g(f(x)) = x.
g(f(x)) = 4 (x + 9)/ 4 - 9
= x + 9 - 9
= x.
So it is the inverse.
Also if you find f((g(x)) it is also = to x.
7x-9-11=3x+4+2x
how did u get 12?
Answer:
x = 12
Step-by-step explanation:
Given expression is \[7x-9-11=3x+4+2x\]
Simplifying: \[7x -(9+11) = (3x+2x)+4\]
Or, \[7x - 20 = 5x + 4\]
Bringing all terms containing x to the left side of the equation and all the numeric terms to the right side:
\[7x-5x = 20 + 4\]
=> \[(7-5)x = 24\]
=> \[2x = 24 \]
=> x = \[\frac{24}{2}\]
=> x= 12
Hence the value of x which satisfies the given equation \[7x-9-11=3x+4+2x\] is 12
Write an equation for the line that is parallel to the glven line and that passes
through the given point.
y = x-10;(-6, -29)
Answer:
The answer to your question is y = x - 23
Step-by-step explanation:
Process
1.- Get the slope of the line
If two lines are parallels, it means that they have the same slope.
y = 1x - 10
Slope = m = 1
2.- Get the equation of the line
y - y1 = m(x - x1)
y + 29 = 1(x + 6) Substitution
y + 29 = x + 6 Expanding
y = x + 6 - 29 Simplifying
y = x - 23
Answer:
y = x - 23
Step-by-step explanation:
All lines parallel to the given line y = x - 10 have the same slope (1), and the same form of equaiton: y = x + C, where C is a constant.
We know that the new line passes through (-6, -29). Replacing x with -6, y with -29, we get:
-29 = -6 + C, and thus we find that C = -23.
Thus, the desired equation is
y = x - 23
Which of the following variables for data about a track team is a discrete variable?a) The height of a team memberb) The weight of a team memberc) The number of times that a team member finished first in a raced) The time recorded for the last race that was run by a team membere) The time recorded for a one-mile race by a team member
Answer:
C
Step-by-step explanation:
Discrete data includes numbers that are exclusively integers, i.e, ... -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 7 ..... and so on. It do not include the other real numbers that are not integers. You can recognize discrete data, for example, in a grah, as there will be only isolated points and not a continuous line.
The height of a member is continuous as we can have every number between 0 and the top height. There can be 1.50 m, 1.51m, 1.5000001m, 1.65m, 1,644444449, and so on with every number (obviously, we will not have heights of 5.78 m because of simply nature). So, we discard option a.
Exactly the same as heights happens with weight. We can ahve any weight you may imagine from the less weight to the top. 100.45555555 pounds is a posible weight, for example. With this, we discard option b.
The time is also continuous. Lets think in minutes. A runner can register 7 minutes, 7.2 minutes, 7.098686 minutes, and so on for every number. We can use every fraction you imagine. So we can discard options d and e.
However the number of times is discrete, because the number of races are discrete. There are 1, 2, 3, 4,... races. We can not have 5.5 races, it is impossible. So, the number of times a runner finished ahead is discrete. There is no member that finished 7.2 times first, we can find either 7 times or 8 times, but not 7.2. So, option c is the correct.
Sally was shopping for a turkey tree for Thanksgiving she looked at 48 trees. Of those she found that 7/8 of them were too small. How many of the trees were too small?
Final answer:
Sally looked at 48 turkey trees and found 7/8 were too small. By multiplying 48 by the fraction 7/8, we find that 42 trees were too small.
Explanation:
The question asks us to calculate the number of turkey trees that were too small, based on the total number of trees Sally looked at and the fraction that were too small.
Sally looked at 48 trees in total and found that 7/8 of them were too small. To find the number of too small trees, we multiply the total number of trees by the fraction that were too small:
Number of too small trees = Total number of trees × Fraction too small
Number of too small trees = 48 × 7/8
Calculating this gives us:
Number of too small trees = 48 × 0.875
Number of too small trees = 42
Therefore, out of the 48 turkey trees Sally looked at, 42 of them were too small for her Thanksgiving needs.
A rectangular field is 0.4 kilometers long and 0.35 kilometers wide. What is the area of the field in square meters? Do not round your answer. Be sure to include the correct unit in your answer.
To find the area of the rectangular field in square meters, we first convert the length and width from kilometers to meters. Then, we multiply the length by the width to find the area. This results in an area of 140,000 m².
Explanation:To solve this problem, we must first understand what the question is asking. The question is asking for the area of a rectangular field, and the dimensions are given in kilometers. The area is found by multiplying the length times the width of a shape (in this case, a rectangle).
Then, we need to convert the kilometers to meters because the question asks for the answer in square meters. We know there are 1,000 meters in 1 kilometer. Therefore, the length of the field is 0.4 km * 1,000 = 400 meters, and the width of the field is 0.35 km * 1,000 = 350 meters.
The area is found by multiplying the length by the width, which is 400m * 350m = 140,000 m².
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Jane andre and maria pick apples .Andre picks three times as many pounds as maris .Jane picks two times as .A u pounds as andre .Tbw total weight of tbe apples is 840 pounds .How maby pounds of apples dose andre pick?
Answer: Andre picked 252 pounds of apples
Step-by-step explanation:
Let x = number of pounds of apple picked by Jane.
Let y = number of pounds of apple picked by Andre
Let z = number of pounds of apple picked by Maria
Andre picks three times as many pounds as maria. It means that
y = 3z
Jane picks two times as many pounds as Andre. It means that
x = 2y
The total weight of the apples is 840 pounds. It means that
x + y + z = 840 - - - - - - - - - 1
We will substitute z = y/3 and x = 2y into equation 1
2y + y + y/3 = 840
Cross multiplying with 3
6y + 3y + y = 2520
10y = 2520
y = 2520/10 = 252
x = 2y = 252× 2 = 504
z = y/3 = 252/3 = 84
Joaquín invirtió su dinero a 12% y a 15% obteniendo unos intereses de $3000. Si las cantidades que invirtió hubieran sido intercambiadas, habría tenido un retorno de $2940. ¿Cuánto dinero invirtió a 15%?
Answer:
12,000
Step-by-step explanation:
espero que he ayudado
The second and forth pic is the answers to the first and third pic.. second pic is the answers to first pic and the forth pic is the answer for the third pic..
Answer:
pic # 1 answer is B. 81 x pi
pic#3 answer is C. QC
Step-by-step explanation:
pic #1
the formula to find the area of a circle is pi x the radius to the second power
they give you the diameter. (18)
the radius is half of a diameter (d/2)
so 18/3 = 9
9 to the second power (9 x 9) = 81
so your answer is 81 x pi
pic#2
I don't much, but i do know they asking for the radius and since the radius is half the diameter of a circle then QC makes sense in my book.
hope this is helpful.
The area of the circle is B. 81 π in²
the line segment representing radii are PC and QC
How to find the area of the circleArea of circle is given by the formula
= π d² / 4
where d = 18 in
= π * 18² / 4
= 81 π in²
When C is the center of the circle. The line segment representing radii are lines from the circumference to the center. These lines includes
PC and QC
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A rectangular field will have one side made of a brick wall and the other three sides made of wooden fence. Brick wall costs 10 dollars per meter and wooden fence costs 20 dollars for 4 meters. the area of the field is to be 2400m^2. What length should the brick wall be to give the lowest total cost of wall plus fence?
Answer:
1,152
Step-by-step explanation:
The rectangular field have four sides, where the opposite sides of the field are equal
The length of the brick wall that gives the lowest total cost of the fence is 40 meters
Let the length of the rectangular field be x, and the width be y.
Where: y represents the side to be made of brick wall,
So, the perimeter of the field is calculated using:
[tex]\mathbf{P =2x + 2y}[/tex]
And the area is
[tex]\mathbf{A =xy}[/tex]
The area is given as 2400.
So, we have:
[tex]\mathbf{xy = 2400}[/tex]
Make x the subject in
[tex]\mathbf{x = \frac{2400}y}[/tex]
Rewrite the perimeter as:
[tex]\mathbf{P =2x + y + y}[/tex]
The brick wall is $10 per meter, while the wooden wall is $20 per 4 meters
So, the cost function becomes
[tex]\mathbf{C =\frac {20}4 \times (2x + y) + 10 \times y}[/tex]
[tex]\mathbf{C =5 \times (2x + y) + 10 \times y}[/tex]
Open brackets
[tex]\mathbf{C =10x + 5y + 10y}[/tex]
[tex]\mathbf{C =10x +15y}[/tex]
Substitute [tex]\mathbf{x = \frac{2400}y}[/tex] in the cost function
[tex]\mathbf{C =10 \times \frac{2400}{y} +15y}[/tex]
[tex]\mathbf{C = \frac{24000}{y} +15y}[/tex]
Differentiate
[tex]\mathbf{C' = -\frac{24000}{y^2} +15}[/tex]
Set to 0, to minimize
[tex]\mathbf{-\frac{24000}{y^2} +15 = 0}[/tex]
Rewrite as
[tex]\mathbf{\frac{24000}{y^2} =15}[/tex]
Divide through by 15
[tex]\mathbf{\frac{1600}{y^2} =1}[/tex]
Multiply both sides by y^2
[tex]\mathbf{y^2 =1600}[/tex]
Take square roots of both sides
[tex]\mathbf{y^2 =40}[/tex]
Hence, the length of the brick wall should be 40 meters
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Can someone solve with a system of equations and show work?
Answer:
1
Step-by-step explanation:
Label the points as in the attachment. Then we have ...
P = (a+b)/2Q = (b+c)/2R = (c+d)/2S = (d+e)/2T = (e+a)/2We can form the sum P + R + T and we get ...
P +R +T = (a+b)/2 +(c+d)/2 +(e+a)/2 = a +(b +c +d +e)/2
We can form the sum Q + S and we get ...
Q + S = (b+c)/2 +(d+e)/2 = (b +c +d +e)/2
Subtracting the latter sum from the former one gives ...
P +R +T -(Q +S) = a +(b +c +d +e)/2 -(b +c +d +e)/2 = a
__
So, the value picked by the person with the average "6" was ...
(7 +1 +5) -(9 +3) = 13 -12 = 1
The person with average "6" picked 1.
_____
The system of equations written in matrix form is shown in the second attachment. The inverse of the coefficient matrix is shown in the third attachment. That is where the sum shown above came from.
__
The rest of the picked numbers are ...
P = 2, b = 13, Q = 14, c = 5, R = 6, d = -3, S = -2, e = 9, T = 10
To practice for a competition, Luis swam 0.73 kilometer in the pool each day for 4 weeks. How many meters did Luis swim in those 4 weeks? 1 km = 1,000 m
Luis swam 20440m in those 4 weeks.
Step-by-step explanation:
Distance swam per day = 0.73 km
Time period = 4 weeks
1 week = 7 days
4 weeks = 7*4 = 28 days
Total distance swam = Distance per day * Total days
[tex]Total\ distance\ swam=0.73*28\\Total\ distance\ swam=20.44\ km[/tex]
1 km = 1000m
20.44 km = 20.44*1000
Total distance in meters = 20440 m
Luis swam 20440m in those 4 weeks.
Keywords: multiplication, conversion
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There are 8 88 employees on The Game Shop's sales team. Last month, they sold a total of g gg games. One of the sales team members, Chris, sold 17 1717 fewer games than what the team averaged per employee. How many games did Chris sell? Write your answer as an expression
Answer:
The number of games sold by Chris = [tex](\frac{g}{8} - 17)[/tex]
Step-by-step explanation:
The total number of employees in the team = 8
The total number of games sold by the whole team = g
The number of games sold by Chris = Average Games sold by each member - 17
Now, [tex]\textrm{Average number of games sold by each} = \frac{\textrm{Total number of games sold by team}}{\textrm{Total number of people in team}}\\[/tex]
=[tex]\frac{g}{8}[/tex]
⇒The average number of games sold by each of the team member = g/8
Hence, the number of games sold by Chris = [tex](\frac{g}{8} - 17)[/tex]
It’s not 3 but still need help with it
Answer:
13
Step-by-step explanation:
5-(-8)=13
Answer:
13
Step-by-step explanation:
The number line is really helpful in this case. All you have to do is count the spaces between -8, where A is, and 5, where C is. There's 13 spaces between them, therefore the length of AC is 13.
The ratio of toddlers to infants at a day care center is 7 to 3. If twelve more infants join the day care to change the ratio to 7 to 5, how many toddlers are there at this day care center?
A. 24
B. 36
C. 42
D. 72
E. 120
The number of toddlers at this daycare center is C. 42.
What are ratio and proportion?A ratio is a comparison between two similar quantities in simplest form.
Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.
In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.
Given, The ratio of toddlers to infants at a daycare center is 7 to 3.
let, There be 'T' number of toddlers and 'I' number of infants.
So, T : I = 7 : 3.
T/I = 7/3.
3T = 7I Or I = 3T/7...(i)
Now, Twelve more infants join the daycare to change the ratio to 7 to 5.
Therefore,
T/(I + 12) = 7/5.
5T = 7I + 84..(ii)
5T = 7(3T/7) + 84.
5T = 3T + 84.
2T = 84.
T = 42.
So, The number of toddlers in this day center is 42.
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A simple random sample of size nequals=8181 is obtained from a population with mu equals 77μ=77 and sigma equals 27σ=27. (a) Describe the sampling distribution of x overbarx. (b) What is Upper P (x overbar greater than 81.5 )P x>81.5? (c) What is Upper P (x overbar less than or equals 69.5 )P x≤69.5? (d) What is Upper P (73.4 less than x overbar less than 84.05 )P 73.4
Answer:
a) [tex]P(\bar X>81.5)=1-0.933=0.067[/tex]
b) [tex]P(\bar X<69.5)=0.0062[/tex]
c) [tex]P(73.4<\bar X<84.05)=0.8755[/tex]
Step-by-step explanation:
1) Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu=77,\sigma=27)[/tex]
And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
On this case [tex]\bar X \sim N(77,\frac{27}{\sqrt{81}})[/tex]
Part a
We want this probability:
[tex]P(\bar X>81.5)=1-P(\bar X<81.5)[/tex]
The best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
If we apply this formula to our probability we got this:
[tex]P(\bar X >81.5)=1-P(Z<\frac{81.5-77}{\frac{27}{\sqrt{81}}})=1-P(Z<1.5)[/tex]
[tex]P(\bar X>81.5)=1-0.933=0.067[/tex]
Part b
We want this probability:
[tex]P(\bar X\leq 69.5)[/tex]
If we apply the formula for the z score to our probability we got this:
[tex]P(\bar X \leq 69.5)=P(Z\leq \frac{69.5-77}{\frac{27}{\sqrt{81}}})=P(Z<-2.5)[/tex]
[tex]P(\bar X\leq 69.5)=0.0062[/tex]
Part c
We are interested on this probability
[tex]P(73.4<\bar X<84.05)[/tex]
If we apply the Z score formula to our probability we got this:
[tex]P(73.4<\bar X<84.05)=P(\frac{73.4-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{84.05-\mu}{\frac{\sigma}{\sqrt{n}}})[/tex]
[tex]=P(\frac{73.4-77}{\frac{27}{\sqrt{81}}}<Z<\frac{84.05-77}{\frac{27}{\sqrt{81}}})=P(-1.2<z<2.35)[/tex]
And we can find this probability on this way:
[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)[/tex]
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)=0.9906-0.1151=0.8755[/tex]
There are 10 questions on a discrete mathematics final exam. How many ways are there to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points?
Answer:
There are 12,565,671,261 ways.
Step-by-step explanation:
Here we have to use the combination and repetition formula.
C(n + r-1, r) = [tex]\frac{(n + r-1)!}{r!(n-1)!}[/tex]
Given: n = 10 (The number of questions)
Each question is worth at least 5 points.
10 questions = 10 *5 = 50
The total = 100
r = 100 - 50
r = 50
Now we can apply the formula.
C(10 + 50 -1, 50) = [tex]\frac{(10 + 50 -1)!}{50!(10 -1)!}[/tex]
C(59, 50) = [tex]\frac{59!}{50!9!}[/tex]
Simplifying the above factorial using the calculator, we get
C(59, 50) = 12,565,671,261
There are 12,565,671,261 ways.
There are 14,441,654 ways to assign scores to the problems on the final exam.
Explanation:To find the number of ways to assign scores to the problems, we can use the concept of stars and bars. Let's consider each question as a bar and the points as stars. Since each question is worth at least 5 points, we can subtract 5 from each question's score to make sure it is at least 0. Now, we have a total of 100-5*10 = 50 points to distribute among the questions. Using stars and bars, we can find the number of ways to distribute these points.
The total number of ways to distribute 50 points among 10 questions is given by the formula (n+r-1) choose (n-1), where n is the number of questions (10) and r is the total number of points (50). Plugging in these values, we get (10+50-1) choose (10-1) = 59 choose 9 = 14,441,654 ways to assign scores to the problems.
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About how many cubes were used to make this figure? A. about 40 B. about 70 C. about 100 D. about 140
Answer:
Around 100,i.e. 105
Step-by-step explanation:
In the given cube
Length of the cube = 7 unit cells
Breadth of the cube = 3 unit cells
Height of the cube = 5 unit cells
Therefore the number of unit cubes required to make such big cube is nothing but the volume of the big cube = length*breadth*height
⇒Number of cubes used to make that big cube= 7*5*3
= 105
Hence, option D (around 100) is the correct answer
Form a polynomial function(x) with zeros: -2, multiplicity 1; 1, multiplicity 2; 5, multiplicity 3; and degree 6. Use 1 as the leading coefficient and leave the function in factored form.
Answer:
Remember, a number a is a zero of the polynomial p(x) if p(a)=0. And a has multiplicity n if the factor (x-a) appear n times in the factorization of p(x).
1. Since -2 is a zero with multiplicity 1, then (x+2) is a factor of the polynomial.
2. Since 1 is a zero with multiplicity 2, then (x-1) is a factor of the polynomial and appear 2 times.
3. Since 5 is a zero with multiplicity 3, then (x-5) is a factor of the polynomial and appear 3 times.
Then, the polynomial function with the zeros described above is
[tex]p(x)=(x+2)(x-1)^2(x-5)^3= x^6-15x^5+72x^4-78x^3-255x^2+525x-250[/tex]
Final answer:
The polynomial function with the given zeros -2, 1, and 5, with their respective multiplicities 1, 2, and 3, and leading coefficient 1 is [tex]f(x) = (x + 2)(x - 1)^2(x - 5)^3.[/tex]
Explanation:
To form a polynomial function f(x) with the given zeros and multiplicities, we use the fact that a zero x = a with multiplicity m corresponds to a factor (x - a)^m in the polynomial. Since the leading coefficient should be 1, we simply multiply these factors together. Based on this, the polynomial with zeros -2 (multiplicity 1), 1 (multiplicity 2), and 5 (multiplicity 3) is:
[tex]f(x) = (x + 2)(x - 1)^2(x - 5)^3[/tex]
This polynomial is of degree 6, as the sum of the multiplicities of the zeros (1+2+3) equals the degree.
The radius of a cylindrical water tank is 5.5 ft, and its height is 13 ft. What is the volume of the tank?
Use the value 3.14 for at, and round your answer to the nearest whole number.
Be sure to include the correct unit in your answer.
Final answer:
The volume of the cylindrical water tank is approximately 490 ft³.
Explanation:
To find the volume of a cylindrical water tank, we can use the formula V = πr²h, where r is the radius of the tank and h is the height of the tank. Plugging in the given values, we have V = 3.14 × (5.5 ft)² × 13 ft. Simplifying, we get V ≈ 490 ft³. Rounding to the nearest whole number, the volume of the tank is approximately 490 ft³.
Use a triple integral to Önd the volume of the solid enclosed by the cylinder x 2 + z 2 = 4 and the planes y = 1 and y + z = 4.
Answer:
The volume should not be bigger than the volume if the cross section was a square and thus the volume enclosed would be V=12π.
Step-by-step explanation:
using cylindrical coordinates
x= rsin θ
z= rcos θ
y=y
therefore
y+z=4 → y= 4-z = 4-r cos θ
also from x²+z²=4 → -2≤x≤2 , -2≤z≤2
therefore since y= 4-z → 6≤y≤2 → it does not overlap with the plane y=1
V=∫∫∫dV = ∫∫∫dxdydz = ∫∫∫rdθdrdy = ∫∫rdθdr [(y=4-r cos θ,y=1) ∫ dy] =
∫∫[(4-rcosθ) - 1]rdθdr = ∫∫(3-rcosθ) rdθdr = ∫dθ [r=2,r=0] ∫(3r-r²cosθ) dr
∫ (3/2* 2²- 2³/3 cosθ) dθ =[θ=2π, θ=0] ∫ (6-8/3 cosθ) dθ = 2π*6 - 8/3 sin0 = 12π
thus
V= 12π
to verify it, the volume should not be bigger than the volume if the cross section was a square and thus the volume enclosed would be:
V = [(2-(-2)]² * (6-2) /2 + [(2-(-2)]² * (2-1) = 4³/2 + 4²*2 = 64 > 12π
The top and bottom margins of a poster are 4 cm and the side margins are each 5 cm. If the area of printed material on the poster is fixed at 388 square centimeters, find the dimensions of the poster with the smallest area.
Answer:
The smallest poster has dimension 25.6 cm by 32.05 cm.
Step-by-step explanation:
Let "x" and "y" be the length and the width of the poster.
The margin of a poster are 4 cm and the side margins are 5 cm.
The length of the print = x - 2(4) = x - 8
The width of the print = y - 2(5) = y - 10
The area of the print = (x- 8)(y -10)
The area of the print is given as 388 square inches.
(x-8)(y -10) = 388
From this let's find y.
y -10 = [tex]\frac{388}{(x - 8)}[/tex]
y = [tex]\frac{388}{x - 8} + 10[/tex] -------------------(1)
The area of the poster = xy
Now replace y by [tex]\frac{388}{x - 8} + 10[/tex], we get
The area of the poster = x ([tex]\frac{388}{x - 8} + 10[/tex])
= [tex]10x + \frac{388x}{x - 8}[/tex]
To minimizing the area of the poster, take the derivative.
A'(x) = [tex]10 + 388(\frac{-8}{(x-8)^{2} } )[/tex]
A'(x) = [tex]10 - \frac{3104}{(x-8)^2}[/tex]
Now set the derivative equal to zero and find the critical point.
A'(x) = 0
[tex]10 - \frac{3104}{(x-8)^2}[/tex] = 0
[tex]10 = \frac{3104}{(x-8)^2}[/tex]
[tex](x - 8)^2 = \frac{3104}{10}[/tex]
[tex](x - 8)^2 = 310.4[/tex]
Taking square root on both sides, we get
x - 8 = 17.6
x = 17.6 + 8
x = 25.6
So, x = 25.6 cm takes the minimum.
Now let's find y.
Plug in x = 25.6 cm in equation (1)
y = [tex]\frac{388}{25.6 - 8} + 10[/tex]
y = 22.05 + 10
y = 32.05
Therefore, the smallest poster has dimension 25.6 cm by 32.05 cm.
The minimum area of the poster given that the printed area is 388 sq.cm and the margins are 4cm top/bottom and 5cm on the sides, is found by using calculus and the area constraints. The problem can be solved by expressing width and height in terms of one another and then optimizing the area equation.
To solve the problem, we use calculus and the area constraint to find the dimensions with a minimum area. Let's denote that width of the poster is w and height is h. The entire area of the poster would then be expressed as:
(w+2*5cm)*(h+2*4cm) = poster area
On the other hand, we have a fixed constraint that states the printed area is 388cm², so we have:
w*h = 388 cm²
We can express h as 388/w and substitute this in the poster's area equation, and then use calculus to find the minimum area given these constraints.
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Three machines, individually, can do a certain job in 4, 5, and 6 hours, respectively. What is the greatest part of the job that can be done in one hour by two of the machines working together at their respective rates?(A) 11/30
(B) 9/20
(C) 3/5
(D) 11/15
(E) 5/6
Answer:
(B) 9/20
Step-by-step explanation:
The fastest machine can do 1/4 of the job in 1 hour. The second-fastest machine can do 1/5 of the job in 1 hour. Together, these two machines can do ...
(1/4) +(1/5) = (5+4)/(5·4) = 9/20
of the job in 1 hour.
9/20 of the job can be done in 1 hour by two of the machines.
Raj has 40% of his weekly paycheck automatically dispositive into his savings account this week $160 is dispositive into the account Raj wants to know the total amount of his paycheck this week
Answer:
Raj can look on his pay stub to find the total is $400
Step-by-step explanation:
The relation between the deposit and the total pay is ...
deposit = 0.40 × total pay
total pay = deposit / 0.40 = 160/0.40 = 400
Raj was paid $400 this week.
In the 6/55 lottery game, a player picks six numbers from 1 to 55. How many different choices does the player have if repetition is not allowed? Note that the order of the numbers is not important.
Answer: 28989675
Step-by-step explanation:
The number of ways to choose r things out of n things ( if order doesn't matter) is given by :_
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]
Given : In the 6/55 lottery game, a player picks six numbers from 1 to 55.
Then , the number of ways to choose 6 numbers out of 55 is if repetition is not allowed :
[tex]^{55}C_6=\dfrac{55!}{6!(55-6)!}\\\\=\dfrac{55\times54\times53\times52\times51\times49!}{6\times5\times4\times3\times2\times1\times49!}\\\\=\dfrac{55\times54\times53\times52\times51}{6\times5\times4\times3\times2\times1}\\\\=28989675[/tex]
Hence, the player have 28989675 choices.
When repetition is not allowed, a player in the 6/55 lottery game can make 32,468,436 different choices.
Explanation:When repetition is not allowed, the number of different choices a player has in the 6/55 lottery game can be determined using the concept of combinations. A combination is a selection where the order of the elements does not matter.
To calculate the number of combinations, we can use the formula:
C(n, r) = n! / (r! * (n-r)!)
In this case, n = 55 (total number of choices) and r = 6 (number of choices to be made). Substituting these values into the formula:
C(55, 6) = 55! / (6! * (55-6)!)
Simplifying further:
C(55, 6) = 55 * 54 * 53 * 52 * 51 * 50 / (6 * 5 * 4 * 3 * 2 * 1)
This simplifies to:
C(55, 6) = 32,468,436
Therefore, a player has 32,468,436 different choices in the 6/55 lottery game when repetition is not allowed.
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Ahab needs to mix 3.5 cups of liquid iced tea concentrate with 3 cups of water to make iced tea. Ahab has 21 cups of iced tea concentrate. How much iced tea can he make?
Answer:
39 cups
Step-by-step explanation:
If we assume that the 3.5 cups of concentrate make 3.5+3 = 6.5 cups of tea, we can use the proportion ...
6.5/3.5 = x/21
to find the x cups of tea Ahab can make with 21 cups of concentrate.
Multiplying by 21, we get ...
x = 21(6.5/3.5) = 39
Ahab can make 39 cups of tea.
A two dimensional shape is formed by the intersection of the cone and a plane perpendicular to its base and passing through its vertex. What is the two dimensional shape?
Answer:
D
Step-by-step explanation:
it is a triangle.
Answer:
D. Triangle
Step-by-step explanation:
D. Triangle
A student showed the steps below while solving the equation by graphing. Step 1: Write a system of equations: Step 2: Use the change of base formula to rewrite the equations: Step 3: Graph the two equations: Step 4: Identify the x-value at the point of intersection: In which step did the student make the first mistake? step 1 step 2 step 3 step 4
Answer:
Step Three is the error
Step-by-step explanation:
right on edge
The student made the first mistake in step 2. Use the change of base formula to rewrite the equations is incorrect.
The reason is that the change of base formula is typically used in logarithms, not when graphing systems of equations.
When solving a system of linear equations by graphing, you would typically write the equations in the form y=mx+b and then graph them on the coordinate plane to find the point of intersection.
The change of base formula is used to convert logarithms from one base to another, not to rewrite equations in general.
In this case, the student simply rewrote the equations using different notation without changing their meaning.
The correct method would be to graph the original equations and identify the x-value at the point of intersection.
A study showed that the ratio of the number of people who get their news from social media to the number of people who their news elsewhere is 3:7 Based on the ratio, how many people in a town of 800 people get their news from social media
Answer: 240 people get the news from social media
Step-by-step explanation:
The ratio of the number of people who get their news from social media to the number of people who their news elsewhere is 3:7
Total ratio is the sum of the proportion of those that get their news from social media and those that got their news elsewhere.
It becomes 3+7 = 10
Total number of people in the town is 800
To determine the number of people that got their news from social media, we will divide the proportion of those that get their news from social media by the total ratio and multiply by the total number of people in the town. It becomes
3/10 × 800 = 240