The work done by [tex]\vec F[/tex] is
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r[/tex]
where [tex]C[/tex] is the given curve and [tex]\vec r(t)[/tex] is the given parameterization of [tex]C[/tex]. We have
[tex]\mathrm d\vec r=\dfrac{\mathrm d\vec r}{\mathrm dt}\mathrm dt=k(1-2t)\,\vec\imath+\vec\jmath[/tex]
Then the work done by [tex]\vec F[/tex] is
[tex]\displaystyle\int_0^1((t-kt(1-t))\,\vec\imath+kt^2(1-t)\,\vec\jmath)\cdot(k(1-2t)\,\vec\imath+\vec\jmath)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1((k-k^2)t-(k-3k^2)t^2-(k+2k^2)t^3)\,\mathrm dt=-\frac k{12}[/tex]
In order for the work to be 1, we need to have [tex]\boxed{k=-12}[/tex].
Pretend you're playing a carnival game and you've won the lottery, sort of. You have the opportunity to select five bills from a money bag, while blindfolded. The bill values are $1, $2, $5, $10, $20, $50, and $100. How many different possible ways can you choose the five bills? (Order doesn't matter, and there are at least five of each type of bill.) A. 56 B. 120 C. 288 D. 462
Answer: 462
Step-by-step explanation:
The general theorem of combination says that there are [tex]C(n+r-1, r)[/tex], with r-combinations from a set having n elements when repetition of elements is allowed.
Here the number of denomination: [tex]n = 7[/tex] , r =5
Also order doesn't matters.
Then the number of different possible ways can you choose the five bills is given by :-[tex]C(7+5-1, 5)= C(11,5)\\\\=\dfrac{11!}{5!(11-5)!}\\\\=462[/tex]
Hence, the number of different possible ways can you choose the five bills is 462.
A computer password system requires 6 DIFFERENT letters (case sensitive) from the 26 alphabets. For example, AAAAAA and aaaaaa are two different passwords. Is this a PERMUTATION or a COMBINATION problem? How many passwords can be created?
Answer: 325
Step-by-step explanation:
Combination is a way to calculate the total outcomes of an event where order of the outcomes does not matter where as a Permutation is a way of arranging the elements of a set into a order or a sequence . Here order matters.
If we want to create password with 6 different letters then order matters.
Hence, we use permutations.
The number of passwords created is given by :-
[tex]^{26}P_6=\dfrac{26!}{2!(26-2)!}\\\\=\dfrac{26\times25\times24!}{2\times24!}=325[/tex]
Hence, the number of passwords created = 325
On a very hot summer day and a few months later on a very cold winter day, you go outside and take your temperature. Each time your body temperature is 37 degrees Celsius. This example illustrates ____.
Answer:
This example illustrates Homeostasis.
Step-by-step explanation:
Consider the provided information.
On a very hot summer day and a few months later on a very cold winter day, you go outside and take your temperature. Each time your body temperature is 37 degrees Celsius that means your body's tissues, and cells helps you to attain the stability and constancy required for proper functioning.
The medical definition of Homeostasis is:
Homeostasis is a property of tissues, and cells that helps the stability and constancy required for proper functioning to be maintained and regulated. it is a state maintained through the continuous adjustment of biochemical and physiological processes.
The provided example explain the process of Homeostasis.
Thus, this example illustrates Homeostasis.
Final answer:
The unchanging body temperature of 37 degrees Celsius in both hot and cold external conditions exemplifies thermoregulation, the body's ability to maintain a constant internal temperature through homeostasis.
Explanation:
The scenario of your body temperature remaining constant at 37 degrees Celsius, regardless of whether it is a hot summer day or a cold winter day, illustrates the concept of thermoregulation. Thermoregulation is the ability of an organism to keep its body temperature within certain boundaries, even when the surrounding temperature is very different.
Your body does this through negative feedback mechanisms similar to a thermostat in a house. For example, on a hot day, if your body temperature rises, your skin produces sweat and the blood vessels near your skin's surface dilate to release heat and cool you down. Conversely, in cold weather, the blood vessels constrict, and shivering generates heat to maintain your body temperature. This constant adjustment keeps your body's core temperature steady, enabling the efficient functioning of enzymes and bodily processes that are optimized for a temperature of around 37 degrees Celsius.
This biological thermostat works continuously to keep internal conditions stable, a state known as homeostasis. In physiological terms, about 60 percent of the energy generated from the production of ATP (adenosine triphosphate) by your cells is released in the form of heat, contributing to the maintenance of your body temperature.
What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible?
Answer:
The answer is 0 and no.
Step-by-step explanation:
Consider the provided information.
If the event is impossible then the probability of that event will be 0.
Now, consider the statement 'Suppose that a probability is approximated to be zero-based on empirical results.'
Empirical probability is based on observations and if you do more observation then the probability of an event might be increased.
Thus, this means that the event is not impossible.
Answer:
THE ANSWER IS WHAT THE GUY ABOVE ME SAID....0
Step-by-step explanation:
Two points on L1 and two points on L2 are given. Use the slope formula to determine if lines L1 and L2 are parallel, perpendicular, or neither.
L1: (1, 10) and (-1, 7)
L2: (0, 3) and (1, 5 )
Answer:
The lines L1 and L2 neither parallel nor perpendicular
Step-by-step explanation:
* Lets revise how to find a slope of a line
- If a line passes through points (x1 , y1) and (x2 , y2), then the slope
of the line is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
- Parallel lines have same slopes
- Perpendicular lines have additive, multiplicative slopes
( the product of their slopes is -1)
* Lets solve the problem
∵ L1 passes through the point (1 , 10) and (-1 , 7)
- Let (1 , 10) is (x1 , y1) and (-1 , 7) is (x2 , y2)
∴ x1 = 1 , x2 = -1 and y1 = 10 , y2 = 7
∴ The slope of L1 is [tex]m1 = \frac{7-10}{-1-1}=\frac{-3}{-2}=\frac{3}{2}[/tex]
∵ L2 passes through the point (0 , 3) and (1 , 5)
- Let (0 , 3) is (x1 , y1) and (1 , 5) is (x2 , y2)
∴ x1 = 0 , x2 = 1 and y1 = 3 , y2 = 5
∴ The slope of L2 is [tex]m2=\frac{5-3}{1-0}=\frac{2}{1}=2[/tex]
∵ m1 = 3/2 and m2 = 2
- The two lines have different slopes and their product not equal -1
∴ The lines L1 and L2 neither parallel nor perpendicular
By calculating the slopes of L1 and L2, we find that they are 1.5 and 2 respectively. Since they are neither the same nor negative reciprocals, L1 and L2 are neither parallel nor perpendicular.
To determine if lines L1 and L2 are parallel, perpendicular, or neither, we need to calculate the slopes of both lines using the slope formula:
Slope formula: (y2 - y1) / (x2 - x1)
Calculating the slope of L1:
Points on L1: (1, 10) and (-1, 7)
Slope of L1 = (7 - 10) / (-1 - 1) = (-3) / (-2) = 1.5
Calculating the slope of L2:
Points on L2: (0, 3) and (1, 5)
Slope of L2 = (5 - 3) / (1 - 0) = 2 / 1 = 2
Since the slopes of L1 (1.5) and L2 (2) are neither the same nor negative reciprocals of each other, the lines L1 and L2 are neither parallel nor perpendicular.
Work out the Wronskian for the following pairs of functions and state whether the are linearly dependent or linearly independent in the interval x €(01). (Please Show All Work!!! Y; (x)=sin(x); y2(x)= xsin(x) Please Show All Your Work, Or Risk Losing All Points x+(0, ') Votre Pom
The Wronskian determinant is
[tex]\begin{vmatrix}\sin x&x\sin x\\\cos x&x\cos x+\sin x\end{vmatrix}=\sin x(x\cos x+\sin x) - x\sin x\cos x=\sin^2x[/tex]
which is non-zero for all [tex]x\in(0,1)[/tex], so the solutions are linearly independent.
Evaluate the sum or explain why it diverges: Sigma^infinity_k = 3(-3/2)^k
Answer:
Step-by-step explanation:
Remember that in the geometric serie if | r | < 1 the serie converges and if | r | ≥1 the serie diverges.
I suppose that the serie starts at 0, so using the geometric serie with r = | [tex]\frac{-3}{2}[/tex] | > 1 the serie diverges.
2. In a barnyard, there is an assortment of chickens and cows. Counting heads, one gets 80; counting legs, one gets 184. How many of each are there? Note: To receive credit for this problem you need to define the variables, write a system of equations and solve the system to answer the question.
Answer:
68 chickens and 12 cows.
Step-by-step explanation:
Let x represents the number of chicken and y represents the number of cows in the barnyard,
Given,
Total heads = 80
⇒ x + y = 80 ------(1),
Also, total legs = 184,
Since, a chicken has two legs and cow has 4 legs,
⇒ 2x + 4y = 184 -----(2),
Equation (2) - 2 × equation (1),
We get,
4y - 2y = 184 - 160
2y = 24
y = 12
From equation (1),
x + 12 = 80 ⇒ x = 80 - 12 = 68
Hence, the number of chicken = 68,
And, the number of cows = 12
The area under the normal curve between the mean and a score for which z= - 1.20 is
a. .2563
b. .7563
c. .1138
d. .3862
e. .3849
Answer: e. 0.3849
Step-by-step explanation:
We know that the mean lies exactly at the middle of the normal curve .
The z-score of the mean value is 0.
Also According to the standard normal probability table, the probability value of z=0 is P(z<0)=0.5.
And the probability value of z=-1.20 is P(z<-1.2) =0.1150697.
Now, the area under the normal curve between the mean and a score for which z= - 1.20 is given by :-
[tex]P(z<0)-P(z<-1.2)=0.5-0.1150697=0.3849303\approx0.3849[/tex]
Hence, the area under the normal curve between the mean and a score for which z= - 1.20 is 0.3849 .
candy box is made from a piece of cardboard that measures 45 by 24 inches. Squares of equal size will be cut out of each comer. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume? inches should be cut away from each corner to obtain the maximum volume. A square with a side of length (Round to the nearest hundredth as needed.)
Answer:
Each square should have 5 inches of side and area = 25 square inches.
Step-by-step explanation:
Candy box is made that measures 45 by 24 inches.
Let the squares of equal size x inches has been cut out of each corner.
The sides will then be folded up to form a rectangular box.
Now we have to find the size of square that should be cut from each corner to obtain maximum volume of the box.
Now the box is with length = (45 - 2x) inches
and width = (24 - 2x) inches
and height = x inches
Volume of the candy box = Length × width × height
V = (45 - 2x)(24 - 2x)(x)
V = x(1080 - 48x -90x + 4x²)
= x(1080 - 138x + 4x²)
= 4x³ - 138x² + 1080x
Now we will find the derivative of volume and equate it to zero.
[tex]\frac{dV}{dx}=12x^{2}-276x+1080=0[/tex]
12(x² - 23x + 90) = 0
x² - 23x + 90 = 0
x² - 18x - 5x + 90 = 0
x(x - 18) - 5(x - 18) = 0
(x - 5)(x - 18)=0
x = 5, 18
Now for x = 18 Width of the box will be = (24 - 2×18) = 24 - 36 = -12
Which is not possible.
Therefore, x = 5 will be the possible value.
Therefore, square having area 25 square inches should be cut out from each corner to get the maximum volume of candy box.
The size of the square that should be cut away from each corner to obtain the maximum volume for a box made from a cardboard measuring 45 by 24 inches is 3 inches.
Explanation:To find the size of the square that should be cut from each corner to obtain the maximum volume, we should first make an equation for the volume of the box. If x is the length of the side of the square, then the dimensions of the box are (45-2x) by (24-2x) by x, thus the volume of the box V is (45-2x)(24-2x)x.
By using calculus, we can find the derivative of this function, set it to zero and solve, this will give the critical points where the maximum and minimum volumes will be.
The derivative is found to be -4x^2 + 138x - 1080. Setting this to zero and solving, we find that x = 3 and x = 90 are the critical points for the maximum and minimum volumes. Since we cannot cut corners more than 24 inches (this would make the width negative), x = 3 inches is the only feasible solution.
So, 3 inches should be cut away from each corner to obtain the maximum volume.
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1. Let a; b; c; d; n belong to Z with n > 0. Suppose a congruent b (mod n) and c congruent d (mod n). Use the definition
of congruence to
(a) prove that a + c congruent b + d (mod n).
(b) prove that ac congruent bd (mod n).
Expert Answer
Answer:
Proofs are in the explantion.
Step-by-step explanation:
We are given the following:
1) [tex]a \equi b (mod n) \rightarrow a-b=kn[/tex] for integer [tex]k[/tex].
1) [tex] c \equi d (mod n) \rightarrow c-d=mn[/tex] for integer [tex]m[/tex].
a)
Proof:
We want to show [tex]a+c \equiv b+d (mod n)[/tex].
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(a+c)-(b+d)=rn. If we do that we would have shown that [tex]a+c \equiv b+d (mod n)[/tex].
kn+mn = (a-b)+(c-d)
(k+m)n = a-b+ c-d
(k+m)n = (a+c)+(-b-d)
(k+m)n = (a+c)-(b+d)
k+m is is just an integer
So we found integer r such that (a+c)-(b+d)=rn.
Therefore, [tex]a+c \equiv b+d (mod n)[/tex].
//
b) Proof:
We want to show [tex]ac \equiv bd (mod n)[/tex].
So we have the two equations:
a-b=kn and c-d=mn and we want to show for some integer r that we have
(ac)-(bd)=tn. If we do that we would have shown that [tex]ac \equiv bd (mod n)[/tex].
If a-b=kn, then a=b+kn.
If c-d=mn, then c=d+mn.
ac-bd = (b+kn)(d+mn)-bd
= bd+bmn+dkn+kmn^2-bd
= bmn+dkn+kmn^2
= n(bm+dk+kmn)
So the integer t such that (ac)-(bd)=tn is bm+dk+kmn.
Therefore, [tex]ac \equiv bd (mod n)[/tex].
//
The population of bacteria in a Petri Dish is growing at a rate of 0.8t^3 + 3.5 thousand per hour. Find the total increase in bacteria during the first 2 hours.
Answer:
[tex]p=9900\\[/tex] bacterias in the initial two hours
Step-by-step explanation:
the growing rate is given by the ecuation
[tex]p(t)=0.8(t)^{3} +3.5 [/tex] thousand per hour
for t=2 we have
[tex]p(2)=0.8(2)^{3} +3.5 = 9.9[/tex] thousand
[tex]p=9900\\[/tex] bacterias
In two hours we have 9900 bacterias
Which is relatively better: a score of 87 on a psychology test or a score of 52 on an economics test? Scores on the psychology test have a mean of 92 and a standard deviation of 5. Scores on the economics test have a mean of 62 and a standard deviation of 6.
Answer: Psychology test score is relatively better than economics test score.
Step-by-step explanation:
For this question, we are using Z-score to compare the psychology test score and economics test score.
Z-scores are an approach to compare results from a test with a "normal" population.
Z = [tex]\frac{X - u}{S.D}[/tex]
where,
X - Test score
u - Mean
S.D - Standard deviation
Psychology Test:
Z = [tex]\frac{87 - 92}{5}[/tex]
= [tex]\frac{-5}{5}[/tex]
= -1
Economics Test:
Z = [tex]\frac{52 - 62}{6}[/tex]
= [tex]\frac{-10}{6}[/tex]
= -1.6
Hence, above calculation shows that z- score for pshychology test is greater than the z- score for economics test. so, psychology test score is better than economics test score.
To determine which score is relatively better, we need to use the concept of z-scores, which measure how many standard deviations a particular score is from the mean.
Explanation:To determine which score is relatively better, we need to use the concept of z-scores, which measure how many standard deviations a particular score is from the mean. The formula for calculating the z-score is:
z = (X - μ) / σ
where X is the score, μ is the mean, and σ is the standard deviation.
For the 87 on the psychology test:
z = (87 - 92) / 5 = -1
For the 52 on the economics test:
z = (52 - 62) / 6 = -1.67
Since a higher z-score indicates a score that is relatively better, we can conclude that the score of 87 on the psychology test is relatively better than the score of 52 on the economics test.
To help plan its nursing staff schedule, a large hospital uses simple exponential smoothing to forecast the daily number of hospital beds that will be occupied on each of the next few days. Using a smoothing parameter of 0.56 , the forecast for today's number of occupied beds was 385, although at day's end the actual number of occupied beds was reported to be 386. Using this information, calculate a forecast of the daily bed count for each of the next few days. Round your answer to the nearest integer.
Answer:386
Step-by-step explanation:
We have given
Smoothing parameter [tex]\left ( \alpha \right )=0.56[/tex]
Forecasted demand[tex]\left ( F_t\right )=385[/tex]
Actual demand[tex]\left ( D_t\right )=386[/tex]
And Forecast is given by
[tex]F_{t+1}=\alpha D_t+\left ( 1-\alpha \right )F_t[/tex]
[tex]F_{t+1}=0.56\cdot 386+\left ( 1-0.56\right )385=385.56\approx 386[/tex]
[tex]F_{t+2}=0.56\cdot 386+\left ( 1-0.56\right )385.56=385.806\approx 386[/tex]
[tex]F_{t+3}=0.56\cdot 386+\left ( 1-0.56\right )385.806=385.914\approx 386[/tex]
[tex]F_{t+4}=0.56\cdot 386+\left ( 1-0.56\right )385.914=385.962\approx 386[/tex]
(a) Find the point at which the given lines intersect. r = 2, 3, 0 + t 3, −3, 3 r = 5, 0, 3 + s −3, 3, 0 (x, y, z) = (b) Find an equation of the plane that contains these lines.
Answer:
(x,y,z)=(5,0,3)
[tex]((x,y,z)-(2,3,0))*(-1,1,0)=0[/tex]
Step-by-step explanation:
a)
The problem requires to find the intersection point of the lines, at that point the position 'r' of the lines is the same:
[tex]r_{1} =r_{2} \\(2,3,0)+(3,-3,3)t=(5,0,3)+(-3,3,0)s\\[/tex]
First, built the parametric equation system; this is just a simplification coordinate to coordinate of the vector equation:
[tex]2+3t=5-3s\\3-3t=3s\\3t=3[/tex]
From the last equation,
[tex]t=1[/tex]
And for whatever of the other two,
[tex]s=0[/tex]
You can check that replacing t=1 and s=0 the point gotten is (5,0,3), which is the intersection point (the point that belongs to both lines).
b) The plane is defined by an orthogonal direction. The equation of the plane uses the fact that the dot product between two orthogonal vectors is always zero.
The general equation of a plane is:
[tex]((x,y,z)-(x_{0},y_{0},z_{0}))*(n)=0[/tex]
Where (x,y,z) are the variables that may be part of the plane or not, [tex](x_{0},y_{0},z_{0})[/tex] is a point that belongs to the plane and n is a vector which is orthogonal to the plane.
Due that both lines belong to the plane, the cross product between their direction vectors will give us the orthogonal vector.
[tex]n=(3,-3,3)X(-3,3,0)=(-9,-9,0)[/tex]
We can divide (-9,-9,0) by nine, because we only need the direction and the division does not affect it.
[tex]n=(-1,-1,0)[/tex]
Finally, we know that both lines are inside the plane, so any point that belong to a line, belong to the plane. For this reason, let's select any point, for example: (2,3,0) (It could be another). So, the equation of the plane is:
[tex]((x,y,z)-(2,3,0))*(-1,-1,0)=0[/tex]
The intersection point of the lines can be obtained by equating the parametric forms of the lines and finding the values of parameters. The equation of the plane containing these lines can be obtained using the directional vectors of these lines, which essentially define the plane.
Explanation:First, we need to find the common point at which the given lines intersect. We can do this by setting r = 2, 3, 0 + t 3, −3, 3 and r = 5, 0, 3 + s −3, 3, 0 to be equal, and finding the values of t and s that make this true. This gives us the (x, y, z) coordinates of the intersection point.
To find the equation of the plane that contains these lines, we know that any point on this plane can be expressed as a linear combination of the directional vectors of these lines, which are (3, -3, 3) and (-3, 3, 0). Therefore, the equation of the plane can be written in the form of Ax + By + Cz = D, where (A, B, C) is a normal vector to the plane, and D is a constant that can be determined by substituting the coordinates of any point on the plane.
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The following hypotheses are given. H0 : π ≤ 0.70 H1 : π > 0.70 A sample of 100 observations revealed that p = 0.75. At the 0.05 significance level, can the null hypothesis be rejected?
Answer:
The null hypothesis can't be rejected.
Step-by-step explanation:
Given information:
Null hypothesis:
H₀ : π ≤ 0.70
Alternative hypothesis:
H₁ : π > 0.70
We need to check whether the null hypothesis is rejected or accepted.
If P-value < α, then we reject the null hypothesis H₀.
If P-value ≥ α, then we accept the null hypothesis H₀.
A sample of 100 observations revealed that p = 0.75 at the 0.05 significance level.
Here 0.75>0.05, it means p > α, therefore we can not reject the null hypothesis.
Final answer:
Explaining the rejection of a null hypothesis at a 0.05 significance level based on a sample proportion of 0.75.
Explanation:
The question:
The hypotheses given are H0: π ≤ 0.70 and H1: π > 0.70. A sample of 100 observations resulted in p = 0.75. At the 0.05 significance level, can the null hypothesis be rejected?
Step 1: Calculate the z-score for the given sample proportion.
Step 2: Find the p-value associated with the z-score.
Step 3: Compare the p-value to the significance level α (0.05) to decide whether to reject the null hypothesis or not.
Conclusion:
At the 0.05 significance level, the null hypothesis can be rejected because the p-value is less than 0.05, indicating sufficient evidence to conclude that the proportion is indeed greater than 0.70.
g Use the properties of the natural logarithm to rewrite each logarithmic expression as an expression with a single logarithm. a . ln ( x + 5 ) − ln ( x − 5 ) = b . ln ( x + 5 ) + ln ( x − 5 )
Answer:
Given:
(a.) ㏑( x + 5 ) − ㏑( x − 5 )
(b.) ㏑( x + 5 ) + ㏑( x − 5 )
To compute the above expression, we'll use the properties of natural logarithm. i.e.
㏑(a) − ㏑(b) = ㏑[tex]\frac{a}{b}[/tex]
∴ ㏑( x + 5 ) − ㏑( x − 5 ) = ㏑[tex]\frac{x+5}{x-5}[/tex]
Similarly
㏑(a) + ㏑(b) = ㏑[tex](a\times b)[/tex]
∴ ㏑( x + 5 ) + ㏑( x − 5 ) = ㏑([tex]x^{2}[/tex]-25)
Kevin and Randy Muiso have a jar containing 61 coins, all of which are either quarters or nickels. The total value of the coins in the jar is $12.05. How many of oach type of coin do they have? The jar contains iguarters
Answer:
They have 45 quarters and 16 nickels.
Step-by-step explanation:
Let x be the number of quarters and y be the number of nickels in the jar,
Since, the jar contains 61 coins,
⇒ x + y = 61 ------(1)
Also, 1 quart = $ 0.25 and 1 nickel = $ 0.05,
So, the total cost = ( 0.25x + 0.05y ) dollars,
According to the question,
0.25x + 0.05y = 12.05,
⇒ 25x + 5y = 1205 -----(2),
Equation (2) - 5 × Equation (1),
20x = 1205 - 305
20x = 900
⇒ x = 45,
From equation (1)m,
y = 61 - 45 = 16,
Hence, they have 45 quarters and 16 nickels.
Let's solve this problem step by step.
We have two types of coins: quarters and nickels. Let's use two variables to represent the number of each type of coin in the jar.
Let \( Q \) represent the number of quarters and \( N \) represent the number of nickels. Since we have two unknowns, we'll need two equations to solve for them.
1. The total number of coins is 61:
\[ Q + N = 61 \] (Equation 1)
2. The total value of the coins is $12.05. Since quarters are worth 25 cents each and nickels are worth 5 cents each, we can convert this total value into cents to avoid dealing with dollars and make the calculation easier.
\[ 12.05 dollars = 1205 cents \]
Now we set up an equation based on the value of the coins:
\[ 25Q + 5N = 1205 \] (Equation 2)
These are our two equations:
\[ Q + N = 61 \]
\[ 25Q + 5N = 1205 \]
Let's solve this system of linear equations.
First, we can simplify the second equation by dividing by 5 to make the numbers smaller and easier to work with:
\[ 5Q + N = 241 \] (Simplified Equation 2)
Now, let's subtract Equation 1 from the Simplified Equation 2 to eliminate \( N \):
\[ (5Q + N) - (Q + N) = 241 - 61 \]
\[ 5Q + N - Q - N = 241 - 61 \]
\[ 4Q = 180 \]
Divide both sides by 4 to solve for \( Q \):
\[ Q = \frac{180}{4} \]
\[ Q = 45 \]
Now we know there are 45 quarters. To find the number of nickels, we plug the value of \( Q \) back into Equation 1:
\[ Q + N = 61 \]
\[ 45 + N = 61 \]
\[ N = 61 - 45 \]
\[ N = 16 \]
Therefore, there are 45 quarters and 16 nickels in the jar.
The brain volumes (cm cubed ) of 20 brains have a mean of 1053.5 cm cubed and a standard deviation of 122.3 cm cubed . Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1248.1 cm cubed be significantly high?
Answer:
2.3775
Step-by-step explanation:
find the orthogonal projection of v= [19,12,14,-17] onto the subspace W spanned by [ [ -4,-1,-1,3] ,[ 1,-4,4,3] ] proj w (v) = [answer,answer,answer,answer]
Hence, we have:
[tex]proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}][/tex]
Step-by-step explanation:By the orthogonal decomposition theorem we have:
The orthogonal projection of a vector v onto the subspace W=span{w,w'} is given by:
[tex]proj_W(v)=(\dfrac{v\cdot w}{w\cdot w})w+(\dfrac{v\cdot w'}{w'\cdot w'})w'[/tex]
Here we have:
[tex]v=[19,12,14,-17]\\\\w=[-4,-1,-1,3]\\\\w'=[1,-4,4,3][/tex]
Now,
[tex]v\cdot w=[19,12,14,-17]\cdot [-4,-1,-1,3]\\\\i.e.\\\\v\cdot w=19\times -4+12\times -1+14\times -1+-17\times 3\\\\i.e.\\\\v\cdot w=-76-12-14-51=-153[/tex]
[tex]w\cdot w=[-4,-1,-1,3]\cdot [-4,-1,-1,3]\\\\i.e.\\\\w\cdot w=(-4)^2+(-1)^2+(-1)^2+3^2\\\\i.e.\\\\w\cdot w=16+1+1+9\\\\i.e.\\\\w\cdot w=27[/tex]
and
[tex]v\cdot w'=[19,12,14,-17]\cdot [1,-4,4,3]\\\\i.e.\\\\v\cdot w'=19\times 1+12\times (-4)+14\times 4+(-17)\times 3\\\\i.e.\\\\v\cdot w'=19-48+56-51\\\\i.e.\\\\v\cdot w'=-24[/tex]
[tex]w'\cdot w'=[1,-4,4,3]\cdot [1,-4,4,3]\\\\i.e.\\\\w'\cdot w'=(1)^2+(-4)^2+(4)^2+(3)^2\\\\i.e.\\\\w'\cdot w'=1+16+16+9\\\\i.e.\\\\w'\cdot w'=42[/tex]
Hence, we have:
[tex]proj_W(v)=(\dfrac{-153}{27})[-4,-1,-1,3]+(\dfrac{-24}{42})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=\dfrac{-17}{3}[-4,-1,-1,3]+(\dfrac{-4}{7})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=[\dfrac{68}{3},\dfrac{17}{3},\dfrac{17}{3},-17]+[\dfrac{-4}{7},\dfrac{16}{7},\dfrac{-16}{7},\dfrac{-12}{7}]\\\\i.e.\\\\proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}][/tex]
A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.
To find the outward flow through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] integrate the dot product of velocity and surface normal. The outward flux is [tex]\(810 \times 32 \pi\).[/tex]
To calculate the rate of flow outward through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] let's compute the flux of the given vector field through the cylindrical surface. The outward flux through a closed surface is given by:
[tex]\[\Phi = \int_S \mathb{v} \cdot \mathb{dS}.\][/tex]
Surface Parameterization and Surface Normal.
The cylinder has radius r = 2, so the general parameterization is:
- [tex]\(x = 2 \cos(\theta)\),[/tex]
- [tex]\(y = 2 \sin(\theta)\),[/tex]
- z = z,
where [tex]\(0 \leq \theta \leq 2\pi\) and \(0 \leq z \leq 4\).[/tex]
The outward normal for the cylindrical surface is [tex]\(\mathb{n} = \cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}\).[/tex]
The differential surface element is:
[tex]\[\mathb{dS} = 2 \, d\theta \, dz \, (\cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}).\][/tex]
Dot Product of Velocity and Normal :
The given velocity field is:
[tex]\[\mathb{v} = z \mathb{i} + y^2 \mathb{j} + x^2 \mathb{k}.\][/tex]
The dot product of the velocity with the surface normal is:
[tex]\[\mathb{v} \cdot \mathb{dS} = 2 \, dz \, d\theta \, (z \cos(\theta) + (4 \sin^2(\theta)) \sin(\theta)).\][/tex]
Integrate to Find the Flux :
The flux through the cylindrical surface is given by:
[tex]\[\int_0^4 \int_0^{2\pi} 2 \, (z \cos(\theta) + 4 \sin^2(\theta)) \, dz \, d\theta,[/tex]
Separate and compute the integral:
- The integral of [tex]\(z \cos(\theta)\)[/tex] over[tex]\((0, 2\pi)\)[/tex] is zero (because [tex]\(\cos(\theta)\)[/tex] has symmetric oscillations).
- The integral of [tex]\(4 \sin^2(\theta)\) over \((0, 2\pi)\) is \(2\pi \cdot 4\),[/tex] since [tex]\(\sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta))\).[/tex]
This results in [tex]\(8\pi\).[/tex]
To compute the flux, multiply by 4 : [tex]\[8\pi \times 4 = 32\pi.\][/tex]
Since the density of the fluid is 810 kg/m³, the outward flux of the fluid through the cylinder, considering the density, is : [tex]\[810 \times 32 \pi.[/tex]
This would be the answer, with the expression [tex]\(810 \times 32 \pi\)[/tex] giving the rate of flow outward through the cylinder.
Complete question : A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 4, 0 ≤ z ≤ 4.
WEEK 8 DISCUSSION 250 words in word file so i can copy
Closing â What class, knowledge, or lesson do feel would prepare you for this course? Should there be a prerequisite course or a prep course before this class? If so, what should it include? What did you feel was the most helpful part of this course in learning the material? What topic was the most interesting to you? It was a pleasure having you in class!
Give me robux on robloz
Step-by-step explanation:
.........sd.....
The solutions to the inequality y > −3x + 2 are shaded on the graph. Which point is a solution? (0, 2) (2, 0) (1, −2) (−2, 1)
Answer:
(2,0)
Step-by-step explanation:
The solution of the inequality [tex]y>-3x+2[/tex] is shown in attached diagram.
The boundary line is dotted line, because the sign of inequality is without notion "or equal to". The dotted line means that points lying on this line are not solutions of the inequality. The solutions are those points lying in the shaded region.
From the points (0,2), (2,0), (1,-2), (-2,1) only point (2,0) lies on the shaded region, so only point (2,0) is a solution to the inequality
Answer:
on ed it says its B
Step-by-step expla
3. (6 points) Determine whether the relation R on the set A is an equivalence relation a. (3 pts) A = {1,2,3,4, 5) R={(1,1), (1,2), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,4), (5,5) OPC b. (3 pts) A = {a, b, c R={(a, a), (a, c), (b, b), (c, a), (c, c)}
Answer:
a is not an equivalence relation.
b is an equivalence relation.
Step-by-step explanation:
a.
A = {1,2,3,4, 5) R={(1,1), (1,2), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,4), (5,5)
To see if is an equivalence relation you need to see if you have these 3 things:
Part 1: xRx for all x in A. This is the reflexive property.
Do we? Yes we have all these points in R: (1,1), (2,2) ,(3,3) ,(4,4), and (5,5).
Part 2: If xRy then yRx. This is the symmetic property.
Do we? We have (1,2) but not (2,1). So it isn't symmetric.
Part 3: If xRy and yRz then xRz.
Do we? We are not going to check this because there is no point. We have to have all 3 parts fot it be an equivalence relation.
b.
A = {a, b, c} R={(a, a), (a, c), (b, b), (c, a), (c, c)}
To see if is an equivalence relation you need to see if you have these 3 things:
Part 1: xRx for all x in A. This is the reflexive property.
Do we? Yes we have all these points in R: (a,a),(b,b), and (c,c).
Part 2: If xRy then yRx. This is the symmetric property.
Do we? We have (a,c) and (c,a). We don't need to worry about any other (x,y) since there are no more with x and y being different. This is symmetric.
Part 3: If xRy and yRz then xRz.
Do we? We do have (a,c), (c,a), and (a,a).
We do have (c,a), (a,c), and (c,c).
So it is transitive.
Question b has all 3 parts so it is an equivalence relation.
ESTIMATE THE ANSWER BY ROUNDING EACH NUMBER TO THE NEAREST 10. 961 + 27.8 + 693.0 + 573 +76.4 =
Answer:
2330
Step-by-step explanation:
In rounding nearest 10,
If the ones place digit is 5 or more then 5 then we round the number to upper 10, while if the ones place digit is less than 5 then we round the number to lower 10,
For example : 34 ≈ 30 while 36 ≈ 40,
Now, given expression,
961 + 27.8 + 693.0 + 573 +76.4
By the above statement 961≈ 960, 27.8 ≈ 30, 693.0 ≈ 690, 573 ≈ 570 and 76.4 ≈ 80,
Hence, 961 + 27.8 + 693.0 + 573 +76.4 = 960 + 30 + 690 + 570 + 80 = 2330.
Answer:
2330
Step-by-step explanation:
961 + 27.8 + 693.0 + 573 +76.4 =
Rounding to the nearest 10
960 + 30 + 690 + 570 + 80
2330
Consider the set of differences, denoted with d, between two dependent sets: 84, 85, 83, 63, 61, 100, 98. Find the sample standard deviation sd and round it to the nearest tenth.
Answer:
The sample standard deviation is 15.3.
Step-by-step explanation:
Given data items,
84, 85, 83, 63, 61, 100, 98,
Number of data items, N = 7,
Let x represents the data item,
Mean of the data points,
[tex]\bar{x}=\frac{84+85+83+63+61+100+98}{7}[/tex]
[tex]=82[/tex]
Hence, sample standard deviation would be,
[tex]\sigma= \sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (x_i-\bar{x})^2}[/tex]
[tex]=\sqrt{\frac{1}{6}\sum_{i=1}^{7} (x_i-82)^2}[/tex]
[tex]=\sqrt{\frac{1}{6}\times 1396}[/tex]
[tex]=\sqrt{232.666666667}[/tex]
[tex]=15.2534149182[/tex]
[tex]\approx 15.3[/tex]
The sample standard deviation of the dataset: 84, 85, 83, 63, 61, 100, 98 is approximately 15.3 when rounded to the nearest tenth.
Explanation:To find the sample standard deviation of the given set, we first need to calculate the mean of the data set. Then, each number in the data set should be subtracted from the mean, and the results squared. These squared differences should be summed and divided by the number of data values minus one, which gives the variance. Taking the square root of the variance gives the sample standard deviation.
Let's do this step by step for the given dataset: 84, 85, 83, 63, 61, 100, 98.
Find the mean (average): (84+85+83+63+61+100+98) / 7 = 82 Subtract the mean from each number and square the result: (84-82)^2, (85-82)^2, ..., (98-82)^2. Sum these squared differences: 4+9+1+361+441+324+256 = 1396. Divide this sum by the number of data values minus one (n - 1): 1396 / (7 - 1) = 232.67 (This is the variance). Take the square root of the variance to get the sample standard deviation: √232.67 = 15.3 (rounded to the nearest tenth). Learn more about Sample Standard Deviation here:https://brainly.com/question/30952578
#SPJ3
M1Q5.) Enter your answer as a fully reduced fraction.
Answer:
9/17
Step-by-step explanation:
Chance of being over 40:
[tex] \frac{20 + 30 + 35}{255} = \frac{1}{3} [/tex]
Chance of drinking
root beer:
[tex] \frac{25 + 20 + 30}{255} = \frac{75}{255} [/tex]
Chance of drinking root beer and being over 40
[tex] \frac{1}{3} \times \frac{75}{255} = \frac{25}{255} [/tex]
Chance of drinking root beer OR being over 40
[tex] \frac{1}{3} + \frac{75}{255} - \frac{25}{255} = \frac{135}{255} = \frac{9}{17} [/tex]
20+30+35/255
1/3 chance of being 40+ years old
20+25+30/255
75/255 chance of drinking root beer
75/255 * 1/3
25/255 chance of drinking root beer being 40+ years old
75/255 - 25/255 * 1/3
135/255
9/17 chance of drinking root beer under the age of 40 years old.
Best of Luck!
The number N(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. Initially, N(0) = 500, and it is observed that N(1) = 1000. Solve for N(t) if it is predicted that the limiting number of people in the community who will see the advertisement is 50,000.
Answer:
[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]
Step-by-step explanation:
The logistic equation is
[tex]N(x)=\frac{c}{1+ae^{-rx}}[/tex]
where:
c/(1+a) is the initial value.
c is the limiting value
r is constant determined by growth rate
So we are given that:
N(0)=500 or that c/(1+a)=500
If your not sure about his initial value of c/(1+a) then replace x with 0 in the function N:
[tex]N(0)=\frac{c}{1+ae^{-r \cdot 0}}[/tex]
Simplify:
[tex]N(0)=\frac{c}{1+ae^{0}}[/tex]
[tex]N(0)=\frac{c}{1+a(1)}[/tex]
[tex]N(0)=\frac{c}{1+a}[/tex]
Anyways we are given:
[tex]\frac{c}{1+a}=500[/tex].
Cross multiplying gives you [tex]c=500(1+a)[/tex].
We are also giving that N(1)=1000 so plug this in:
[tex]N(1)=\frac{c}{1+ae^{-r \cdot 1}}[/tex]
Simplify:
[tex]N(1)=\frac{c}{1+ae^{-r}}[/tex]
So this means
[tex]1000=\frac{c}{1+ae^{-r}}[/tex]
Cross multiplying gives you [tex]c=1000(1+ae^{-r})[/tex]
We are giving that c=50000 so we have these two equations to solve:
[tex]50000=500(1+a)[/tex]
and
[tex]50000=1000(1+ae^{-r})[/tex]
I'm going to solve [tex]50000=500(1+a)[/tex] first because there is only one constant variable here,[tex]a[/tex].
[tex]50000=500(1+a)[/tex]
Divide both sides by 500:
[tex]100=1+a[/tex]
Subtract 1 on both sides:
[tex]99=a[/tex]
Now since we have [tex]a[/tex] we can find [tex]r[/tex] in the second equation:
[tex]50000=1000(1+ae^{-r})[/tex] with [tex]a=99[/tex]
[tex]50000=1000(1+99e^{-r})[/tex]
Divide both sides by 1000
[tex]50=1+99e^{-r}[/tex]
Subtract 1 on both sides:
[tex]49=99e^{-r}[/tex]
Divide both sides by 99:
[tex]\frac{49}{99}=e^{-r}[/tex]
Take natural log of both sides:
[tex]\ln(\frac{49}{99})=-r[/tex]
Multiply both sides by -1:
[tex]-\ln(\frac{49}{99})=r[/tex]
So the function N with all the write values plugged into the constant variables is:
[tex]N(x)=\frac{50000}{1+99e^{\ln(\frac{49}{99})x}}[/tex]
Final answer:
The question involves applying the logistic growth equation to determine the number of people who will see an advertisement over time, given initial conditions and the carrying capacity. The process includes finding the growth rate from the provided data and using it to solve the logistic growth formula for any time t.
Explanation:
The number of people in a community who are exposed to a particular advertisement is described by the logistic growth equation. Given that initially N(0) = 500, and after one unit of time N(1) = 1000, and the carrying capacity is 50,000, we want to solve for N(t), the number of people who will see the advertisement at any time t.
The logistic growth model can be written as:
N(t) = K / (1 + (K - N_0) / N_0 ×[tex]e^{(-rt)}[/tex]
Where:
N(t) is the number of individuals at time t
K is the carrying capacity of the environment
N_0 is the initial number of individuals
r is the growth rate
e is the base of the natural logarithms
We are given that K = 50,000, N_0 = 500, and N(1) = 1000. From N(1), we can find the growth rate r. Re-arranging the logistic equation and substituting the values for N(1), t = 1, K, and N_0, we get an equation that we can solve for r. Once we have found r, we can substitute all known values back into the logistic equation to solve for N(t) for any given value of t.
To find the solution for this kind of problem it might require numerical methods or algebraic manipulation which is beyond this explanation, but once the value of r is found, the N(t) formula can be applied to predict the number of people who will see the advertisement at any given time.
Write an equation of the circle with center (6, 2) and radius 4.
Answer:
[tex](x-6)^2+(y-2)^2=16[/tex].
Step-by-step explanation:
[tex](x-h)^2+(y-k)^2=r^2[/tex] is the equation for a circle with center (h,k) and radius r.
You are given center (6,2) and radius 4.
So we will replace h with 6 and k with 2 and r with 4.
This gives us:
[tex](x-6)^2+(y-2)^2=4^2[/tex]
Simplify:
[tex](x-6)^2+(y-2)^2=16[/tex].
For this case we have that by definition, the equation of a circle is given by:
[tex](x-h) ^ 2 + (y-k) ^ 2 = r ^ 2[/tex]
Where:
[tex](h, k):[/tex]It is the center of the circle
r: It is the radius of the circle
According to the data we have to:
[tex](h, k) :( 6.2)\\r = 4[/tex]
Substituting:
[tex](x-6) ^ 2 + (y-2) ^ 2 = 4 ^ 2\\(x-6) ^ 2 + (y-2) ^ 2 = 16[/tex]
ANswer:
[tex](x-6) ^ 2 + (y-2) ^ 2 = 16[/tex]
HELP! Need quick - Determine whether the data distribution appears to be positively skewed, negatively skewed, or symmetric.
The table below shows the average employee salary (in thousands) at S&S, Inc for the last five years.
A) negatively skewed
B) symmetric
C) positively skewed
Answer:
A. negatively skewed
Step-by-step explanation:
Plot the values of salary against years
You will notice that;
The graph has a long left tailThe tail is longer towards the negative direction of the number lineThis mean that this graph is a negative skewed graph/left-skewed distribution
In the attached sketch of the plot, join the points with a smooth curve and observe the above mentioned properties.
The x-axis scale ranges from 2000 to 2015 where 0=2000, 1=2001,2=2002......,15=2015
Hope this will give you a visual picture of the negatively- skewed distribution