The probability of commuter's morning train being canceled precisely six times in a month is closest to 5%.
Given:
A commuter's 6:15 am train to NYC is cancelled three times during a typical month. The commuter uses NJ transit.To find: The probability of the commuter's morning train being canceled precisely six times in a month.
Let X be the number of train cancellations in a month.
As the train cancellations follow a Poisson distribution, the formula for the Poisson distribution is given as:
P(X = x) = (e-λ λx) / x!
where λ is the average number of train cancellations in a month and x is the number of train cancellations in a month.
Now, we need to calculate the probability of a commuter's morning train being canceled precisely six times in a month. Hence, x = 6.
Substitute the given values into the Poisson formula:
P(X = 6) = (e-3 36) / 6!≈ 0.0504 ≈ 0.05
Therefore, the probability of the commuter's morning train being canceled precisely six times in a month is closest to 5%.
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Let f(a) = { x) = S1 0 if 0 < x < 1/2 if 1/2 < x < T. Find the Fourier cosine series and the Fourier sine series. What is the full Fourier series? Explicitly characterize the values of x E R where each converges pointwise.
Given function is { x) = {0, if 0 < x < 1/2, 1, if 1/2 < x < 1}.
Step-by-step explanation: Given function is { x) = {0, if 0 < x < 1/2, 1, if 1/2 < x < 1}.
The function is an even function because the function is symmetric with respect to the y-axis (i.e.) { -x) = {x). So, the Fourier series has only cosine terms. Therefore, the Fourier cosine series of the given function is given by:
f(x) = a0/2 + Σ an cos(nπx/L),
where L is the period of the function.
Since the function is even, the Fourier series reduces to f(x) = a0/2 + Σ an cos(nπx/L) ...(1) , where a0 = 1/L ∫f(x)dx, an = 2/L ∫f(x)cos(nπx/L)dx for n = 1, 2, 3, ..., n. Let L = 1,
then a0 = 1/1 ∫0^1 f(x)dx = 1/2 an = 2/1 ∫0^1 f(x)cos(nπx)dx for n = 1, 2, 3, ..., n.
a1 = 2 ∫1/2^1 cos(nπx)dx = 1/nπ sin(nπx) from 1/2 to 1
= [1/nπ sin(nπ/2) - 1/nπ sin(0)]
= 2/nπ sin(nπ/2)
Hence, the Fourier cosine series is given by f(x) = 1/2 + 2/π ∑[sin(nπ/2)/n] cos(nπx) ...(2)for n = 1, 2, 3, ...Similarly, the Fourier sine series of the given function is given by: f(x) = Σ bn sin(nπx/L)where L is the period of the function. Since the function is even, there are no sine terms in the Fourier series. So, the Fourier sine series is zero, i.e., bn = 0 for n = 1, 2, 3, ....Hence, the full Fourier series is the same as the Fourier cosine series, which is given byf(x) = 1/2 + 2/π ∑[sin(nπ/2)/n] cos(nπx) ...(3)for n = 1, 2, 3, ...The Fourier series converges pointwise to f(x) for x in (0, 1/2) U (1/2, 1).The Fourier series does not converge at x = 0 and x = 1/2 because the function is not continuous at these points.
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Let p be a real number with 0 < p < 1, and n an integer which is greater than or equal to one. Recall that a binomial random variable X is one for which Prob(X = k): = (*) p* (1 k (1 − p)n-k for k = 0,1, n, and Prob(X x) for any x other than one of these n+1 = possible values.
a. In the case n 3 and p = 3/4, compute E(X) and Var(X).
b. Using (a) as a model case, compute E(X) and Var(X) for any value of p and n. (Hint: Write the formula from the binomial theorem and use differentiation.)
c. What is the value of p such that Var(X) is the smallest?
d. For any t > 0, compute E(etx). (Hint: Use the binomial theorem.)
The expected value E(X) of a binomial random variable X can be calculated as n * p, and the variance Var(X) can be calculated as n * p * (1 - p). These formulas can be generalized for any values of p and n, and the value of p that minimizes the variance can be found by setting the derivative of Var(X) with respect to p equal to zero.
a. In part (a), we are given specific values for n (3) and p (3/4). The expected value E(X) of a binomial random variable X can be calculated as n * p, which gives us:
3 * 3/4
= 2.25.
The variance Var(X) can be calculated as n * p * (1 - p), which gives us:
3 * 3/4 * (1 - 3/4)
= 0.5625.
b. In part (b), we generalize the calculation of E(X) and Var(X) for any value of p and n. Using the binomial theorem, we can expand (p + (1 - p))ⁿ and differentiate it to find the coefficients for E(X) and Var(X).
c. To find the value of p that minimizes the variance Var(X), we can take the derivative of Var(X) with respect to p binomial, set it equal to zero, and solve for p. This will give us the value of p that minimizes the variance.
d. For any t > 0, we can calculate E(e^(tx)) using the binomial theorem by substituting e^t for p in the expansion of (p + (1 - p))ⁿ. This will give us the expected value of the exponential of tx.
Therefore, the expected value E(X) of a binomial random variable X can be calculated as n * p, and the variance Var(X) can be calculated as n * p * (1 - p). These formulas can be generalized for any values of p and n, and the value of p that minimizes the variance can be found by setting the derivative of Var(X) with respect to p equal to zero.
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A train travels along a horizontal line according to the function s(t) = –13 + 3t2 – 4t – 4 where t is measured in hours and s is measured in miles. What is the velocity of the train after 4 hours?
The velocity of the train after 4 hours is 20 miles per hour.
To find the velocity of the train after 4 hours, we need to differentiate the given function s(t) with respect to t.
Velocity is the derivative of position with respect to time.
That is,v(t) = ds(t)/dtTo differentiate s(t) = –13 + 3t² – 4t – 4, we differentiate each term separately.v(t) = d/dt(-13) + d/dt(3t²) - d/dt(4t) - d/dt(4)v(t) = 0 + 6t - 4
The velocity of the train after 4 hours is given by substituting t = 4 in the above equation.v(4) = 6(4) - 4 = 20
The velocity of the train after 4 hours is 20 miles per hour.To sum up, the velocity of the train after 4 hours is 20 miles per hour.
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Consider the curve defined by 2x2+3y2−4xy=36 .
(a) Show that ⅆyⅆx=2y−2x3y−2x .
(b) Find the slope of the line tangent to the curve at each point on the curve where x=6
(c) Find the positive value of x at which the curve has a vertical tangent line. Show the work that leads to your answer.
(a) `dy/dx = (2y - 2x)/(3y - 2x)`.
(b) The slope of the tangent line at points where x = 6 is 0.
(c) the curve has a vertical tangent line when x = (3/2)y.
(a) To show that `dy/dx = (2y - 2x)/(3y - 2x)`, we need to find the derivative of `y` with respect to `x`. We can do this by implicitly differentiating the given equation.
Differentiating both sides of the equation with respect to `x`, we get:
4x(dx/dx) + 6y(dy/dx) - 4[(dx/dx)y + x(dy/dx)] = 0
Simplifying the equation, we have:
4x + 6y(dy/dx) - 4xy - 4xy - 4x(dy/dx) = 0
Rearranging the terms and combining like terms, we get:
(6y - 4x)(dy/dx) = 8x - 8xy
Dividing both sides by (6y - 4x), we obtain:
dy/dx = (8x - 8xy)/(6y - 4x)
Simplifying further, we have:
dy/dx = (2x(4 - 4y))/(2(3y - 2x))
Canceling out the common factors, we get:
dy/dx = (2y - 2x)/(3y - 2x)
Therefore, `dy/dx = (2y - 2x)/(3y - 2x)`.
(b) To find the slope of the tangent line at the points where x = 6, substitute x = 6 into the expression we found for `dy/dx` in part (a):
dy/dx = (2(6) - 2(6))/(3y - 2(6))
= 0/(3y - 12)
= 0
The slope of the tangent line at points where x = 6 is 0.
(c) To find the value of x at which the curve has a vertical tangent line, we need to find the point(s) where the slope `dy/dx` is undefined. In other words, we need to find the values of x where the denominator of `dy/dx` becomes zero.
Setting the denominator equal to zero and solving for x:
3y - 2x = 0
2x = 3y
x = (3/2)y
So, the curve has a vertical tangent line when x = (3/2)y.
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In the figure shown, what is the measure of the indicated angle?
Answer:
60 degrees
Step-by-step explanation:
Each triangle needs to add up to 180 total degrees. 70+50=120,
180
-
120
___
60
There are 30 students going on a field trip. Each car can take 4 students. Which inequality would be used to find the least number of cars needed?
Please Help! I'll give Brainliest!
Answer:8 cars
Step-by-step explanation:
to find the least amount of cars dived 30/4 which equilds 7.5
Since there are 2 remaining students, an additional car will be needed bringing the total to 8 cars.
Determine a condition on |x - 4| that will assure that:
(a)∣∣x−2∣∣<21,
(b)∣∣x−2∣∣<10−2.
Given the expression |x - 4|, condition on |x - 4| that will assure that:(a)|x - 2| < 2/1(b)|x - 2| < 0.01
Given expression |x - 4|, the two possible values are: x - 4 if x > 4 -(x - 4) if x < 4Let us solve each part of the question separately:
(a)Part (a) can be expressed as follows:|x - 2| < 2/1Subtracting 2 from both sides of the in equality |x - 2| - 2 < 0Adding 4 to both sides of the inequality. |x - 2| - 2 + 4 < 0|x - 2| - 2 + 4 = |x - 4| < 0Since it is impossible to have an absolute value less than 0, therefore there is no solution.
(b)Part (b) can be expressed as follows:|x - 2| < 0.01 Subtracting 2 from both sides of the inequality |x - 2| - 2 < -0.01Adding 4 to both sides of the inequality. |x - 2| - 2 + 4 < -0.01|x - 2| - 2 + 4 = |x - 4| < -0.01Since it is impossible to have an absolute value less than 0, therefore there is no solution.
Thus, there are no solutions for the given conditions.
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What is the volume of the cylinder above?
A. 168 units^3
B. 96 units^3
C. 84 units^3
D. 112 units^3
The volume of the oblique cylinder is calculated as: B. 96π units³.
What is the Volume of a Cylinder?Volume = πr²h, where h is the height and r is the radius of the given cylinder.
Given the following:
Radius = 4 unitsHeight = 6 unitsVolume = πr²h = π(4²)(6)
Volume = 96π units³ (option B)
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suppose a hand of four cards is drawn from a STANDARD DECK of playing cards with replacement , determine the probability of exactly one card is jack:
Therefore, the probability of exactly one card being jack when a hand of four cards is drawn from a standard deck of playing cards with replacement is 0.073 or 7.3%.
Suppose a hand of four cards is drawn from a standard deck of playing cards with replacement, the probability of exactly one card being jack can be determined using the following steps:Step 1: Determine the total number of possible outcomes when four cards are drawn from a standard deck of 52 cards with replacement. The total number of possible outcomes = 52 × 52 × 52 × 52 = 7,311,616.Step 2: Determine the total number of ways in which exactly one card can be a jack. There are four jacks in a standard deck of 52 cards, so the total number of ways in which exactly one card can be a jack = 4 × 48 × 48 × 48 = 53,333,632.Step 3: Determine the probability of exactly one card being jack. Probability of exactly one card being jack = Total number of ways in which exactly one card can be a jack / Total number of possible outcomes= 53,333,632/ 7,311,616 = 7.28 ≈ 0.073 or 7.3%.Therefore, the probability of exactly one card being jack when a hand of four cards is drawn from a standard deck of playing cards with replacement is 0.073 or 7.3%.
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A particular high school claims that its students have unusually high math SAT scores. A random sample of 50 students from this school was selected, and the mean math SAT score was 544. Is the high school justified in its claim? Explain since it within the range of a usual event, namely within of the mean of the because the score) sample means (Round to two decimal places as needed)
The school is not justified to make this claim because of the reasons defined.
The following is a statement that might be made about the high school to justify its claim No, because the z-score of Z = 1.06 is not uncommon because it does not fall within the range of a typical event, namely within 2 standard deviations of the sample mean.
It has been given to us that:
μ = 511
σ = 119
Sample size (n) = 55
and
s = 119 / √55
= 16.046
As we all know,
Only when z > 2 then, the high school's allegation is valid and warranted.
To locate,
Z's value is
So,
Z = ( X - μ )/σ
by applying the Central Limit Theorem to the values,
z = ( 528 - 511 ) / 16.046
= 1.06
Since, z < 2, as a result, the allegation is unjustified.
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Correct question:
The average math SAT score is 511 with a standard deviation of 119. A particular high school claims that its students have unusually high math SAT scores. A random sample of 55 students from this school was selected, and the mean math SAT score was 528. Is the high school justified in its claim? Explain. ▼ No Yes , because the z-score ( nothing) is ▼ unusual not unusual since it ▼ does not lie lies within the range of a usual event, namely within ▼ 1 standard deviation 2 standard deviations 3 standard deviations of the mean of the sample means. (Round to two decimal places as needed.)
The probability of event A is Pr(A)=1/3 The probability of the union of event A and event B, namely A UB, is Pr(AUB)=5/6 Suppose that event A and event B are disjoint. Pr(B) = [....]
Given that the probability of event A is Pr(A) = 1/3 and the probability of the union of event A and event B, namely AUB, is Pr(AUB) = 5/6. The probability of event B is Pr(B) = 2/3.
Suppose that event A and event B are disjoint.
The probability of event B is Pr(B) = 1/2.
To find the probability of event B.
For disjoint events A and B, we know that A ∩ B = Φ (empty set).
Thus, we can express the union of A and B as: AUB = A + B, where A and B are disjoint.
In general, the probability of the union of two events can be expressed as: P(AUB) = P(A) + P(B) - P(A ∩ B).
For disjoint events, the intersection of the events is always an empty set.
Thus, P(A ∩ B) = 0.
Using this information, we can write:
P(AUB) = P(A) + P(B) - P(A ∩ B)
= P(A) + P(B) - 0
= P(A) + P(B)
Given P(A) = 1/3 and P(AUB) = 5/6, we can solve for P(B) as follows:
5/6 = P(A) + P(B)
=> P(B) = 5/6 - P(A)
=> P(B) = 5/6 - 1/3
=> P(B) = 2/3
Thus, the probability of event B is Pr(B) = 2/3.
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At a certain university, the average cost of books was $330 per
student last semester and the population standard deviation was $75. This
semester a sample of 50 students revealed an average cost of books of $365 per
student. The Dean of Students believes that the costs are greater this semester.
What is the test value for this hypothesis?
The test value for this hypothesis is 3.0.
What is the test value for the hypothesis that the average cost of books is greater this semester at a certain university?The test value for this hypothesis can be calculated using the formula for a one-sample t-test:
test value = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
Population mean (last semester) = $330Sample mean (this semester) = $365Sample size = 50Population standard deviation = $75Calculating the test value:
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10
Write the equation that describes this situation. Use^for exponents.
7000 dollars is placed in an account with an annual interest rate of 6.5%
for 15 years.
Answer:
15(6.5% * 7000)+7000=y
Step-by-step explanation:
15(6.5% * 7000)+7000=y
im not sure tho
What is the difference, in meters, between the length of the longest line and the length of the shortest line?
Answer:
[tex]Range = 3.169m[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
Determine the difference between the shortest and the longest
This question implies that we calculate the range.
[tex]Range = Longest - Shortest[/tex]
From the table, we have:
[tex]Longest = 8.7m[/tex]
[tex]Shortest = 5.531m[/tex]
So, we have:
[tex]Range = 8.7m- 5.531m[/tex]
[tex]Range = 3.169m[/tex]
Danny has a scale drawing of his house. If
3 inches (in) on the scale drawing equals
7 feet on the real house, what is the actual
height of the house?
5.4 in
Answer:
151.2
Step-by-step explanation:
7x12=84
84/3=28
28x5.4=151.2
What is the slope of a line perpendicular to the line y=2/3 x + 3 ( just find the slope)
: 1. Two equilateral triangles are always similar. 2. The diagonals of a rhombus are perpendicular to each other. 3. For any event, 0
Both the given statements are true
1. Two equilateral triangles are always similar: True.
An equilateral triangle is a triangle in which all three sides are equal. Since two equilateral triangles have the same shape and size, they are always similar. Similarity means that the corresponding angles are equal, and the corresponding sides are in proportion.
2. The diagonals of a rhombus are perpendicular to each other: True.
In a rhombus, opposite sides are parallel, and all sides have equal length. The diagonals of a rhombus bisect each other at right angles, which means they are perpendicular to each other. This property holds true for all rhombuses, regardless of their size or orientation.
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Given question is incomplete, the complete question is below
State true or false
1. Two equilateral triangles are always similar.
2. The diagonals of a rhombus are perpendicular to each other.
True or False: All horizontal lines have a y-intercept.
Answer:
If you are looking at a graph then yes it will have a y-axis
Step-by-step explanation:
Answer:
True.
Step-by-step explanation:
A horizontal line goes on infinitely on both ends will eventually cross the y-axis, making a y-intercept.
$10 000 is invested at 3.75% compounded semi-annually. How long would it take for the principal to triple in value.
The time it takes for the principle to triple in value is t = 18.792 years.
To determine how long it would take for a principal of $10,000 to triple in value at an interest rate of 3.75% compounded semi-annually, we can use the compound interest formula. By rearranging the formula and solving for time, we can find the answer.
The compound interest formula can be expressed as A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of compounding periods per year, and t is the time in years.
In this case, we have P = $10,000, r = 3.75% (or 0.0375 as a decimal), and n = 2 since compounding occurs semi-annually.
We want to find the time it takes for the principal to triple, so A = 3P. Substituting the known values into the compound interest formula, we have:
3P = P(1 + r/n)^(nt)
Canceling out the common factor of P on both sides, we get:
3 = (1 + r/n)^(nt)
Taking the natural logarithm (ln) of both sides to isolate the exponent, we have:
ln(3) = nt ln(1 + r/n)
Now, we can solve for t by dividing both sides of the equation by n ln(1 + r/n) and simplifying:
t = ln(3) / (n ln(1 + r/n))
Substituting the given values of r = 0.0375 and n = 2, we can calculate the value of t.
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I take out a 4,000 loan. It's a simple interest loan. Find the interest I get after 4 years at a rate of 6%
Answer:
960
Step-by-step explanation:
Let P, R and T denote principal amount, rate of interest and time period.
Principal amount of loan (P) = 4,000
Time period (T) = 4 years
Rate of interest (R) = 6%
Simple interest is calculated using the following formula:
Simple interest [tex]=\frac{4000(4)(6)}{100} =960[/tex]
So,
Simple interest is equal to 960
Write 3^4 in expanded form. (3^4 means 3 raised to the fourth power.)
A: 3x3
B :3x3x3
C: 3x3x3x3
D: 3x3x3x3x3
Answer:
c because 3.3.3.3 is 3 to the 4th power expanded
Phil has 5 times as many toy race cars as Richard has. Phil has 425 toy race cars. How many race cars does Richard have? *
Answer:
85
Step-by-step explanation:
425 divided by 8= 85
Answer:
He as 85 race cars.
Step-by-step explanation:
Just divide 425 by 5 and you have your answer.
Slove the system of the linear equations by either sus substitution or elimination 8x-12y=20 4x-4y=-4
Answer:
x = -8 and y = -7
Step-by-step explanation:
I will solve your system by substitution.
(You can also solve this system by elimination.)
8x−12y=20;4x−4y=−4
Step: Solve8x−12y=20for x:
8x−12y+12y=20+12y(Add 12y to both sides)
8x=12y+20
8x
8
=
12y+20
8
(Divide both sides by 8)
x=
3
2
y+
5
2
Step: Substitute
3
2
y+
5
2
forxin4x−4y=−4:
4x−4y=−4
4(
3
2
y+
5
2
)−4y=−4
2y+10=−4(Simplify both sides of the equation)
2y+10+−10=−4+−10(Add -10 to both sides)
2y=−14
2y
2
=
−14
2
(Divide both sides by 2)
y=−7
Step: Substitute−7foryinx=
3
2
y+
5
2
:
x=
3
2
y+
5
2
x=
3
2
(−7)+
5
2
x=−8(Simplify both sides of the equation)
Which values of N and p define a random graph ensemble G(N, p) with average degree (k) = 40 and variance of the degree distribution o2 = 50? = Select one: = a. p = 0.25, N = 501 b. p = 1/10, N = 401 p = 1/5, N = 501 = = C. = d. None of the above.
The values of N and p that define the random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50 are N = 201 and p = 0.2.
The values of N and p that define a random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50, we can use the following formulas:
k = (N-1) × p
σ² = (N-1) × p × (1-p)
Plugging in the given values:
k = 40
σ² = 50
We can solve these equations to find the values of N and p:
From the first equation:
40 = (N-1) × p
From the second equation:
50 = (N-1) × p × (1-p)
By substituting the value of (N-1) × p from the first equation into the second equation, we can solve for p.
40 = 50 × (1-p)
1-p = 40/50
1-p = 0.8
p = 1 - 0.8
p = 0.2
Now, we can substitute the value of p back into the first equation to solve for N:
40 = (N-1) × 0.2
200 = N-1
N = 200 + 1
N = 201
Therefore, the correct values of N and p that define the random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50 are N = 201 and p = 0.2.
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Martin recorded the low temperatures at his house for one week. The temperatures are shown below.
-7, -3, 4, 1, 2, 8, 7
Approximately what was the average low temperature for the week?
Α. 7
B. "1
C. 1
D "8
What is the solution to the equation below?
0.5n = 6
It's 12 because if you divide 6 by 0.5 you should get 12, so basically use the opposite operation.
Hope that helps!
Let X and Y be two continuous random variables with joint probability density function Calculate the positive constant b. Show the result with at least two decimal places. 5 -bcx cb - bzycb f(x,y) = 0 otherwise
The positive constant b is 0. This is obtained by setting the coefficient of the xy^2 term to zero in the equation derived from equating the integral of the joint probability density function to 1.
To compute the positive constant b, we need to calculate the integral of the joint probability density function (pdf) over the entire probability space and set it equal to 1 since it represents a valid probability density.
∫∫ f(x, y) dx dy = 1
Since the joint pdf is defined as:
f(x, y) = 5 - bcx * cb - bzycb
And it is zero otherwise, we can set up the integral as follows:
∫∫ (5 - bcx * cb - bzycb) dx dy = 1
To solve this integral, we need to determine the limits of integration. Since the joint pdf is not specified outside of the equation, we assume it is defined for all real values of x and y.
∫∫ (5 - bcx * cb - bzycb) dx dy = ∫∫ 5 - bcx * cb - bzycb dx dy
Integrating with respect to x first:
∫ (5x - bcx^2/2 * cb - bzy * cb) ∣∣ dy = 1
Now integrating with respect to y:
(5xy - bcxy^2/2 * cb - bzy^2/2 * cb) ∣∣ dy = 1
Since this equation holds for all real values of x and y, we can ignore the limits of integration.
Next, we can solve for b by equating the integral to 1 and simplifying:
(5xy - bcxy^2/2 * cb - bzy^2/2 * cb) = 1
Simplifying further:
5xy - bcxy^2/2 - bzy^2/2 = 1
Now, we can compare the coefficients of the terms on both sides of the equation:
- bc/2 = 0 (since there is no xy^2 term on the right-hand side)
Solving for b:
bc = 0
Since we are looking for a positive constant b, we can conclude that b = 0.
Therefore, the positive constant b is 0.
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Joe earns a monthly salary of 250 plus a commission on his total sales. Last month his total sales were $7,289 and he earned a total of $1,275. What is his commission rate?
Answer: Joe earns a monthly salary of 250 plus a commission on his total sales. Last month his total sales were $7,289 and he earned a total of $1,275. What is his commission rate?
Step-by-step explanation:
250 + $7,289 + $1,275 = 8814
The elephants at the Putnam Zoo are fed 9 1/2 barrels of corn each day. The buffalo are fed 1/2 as much corn as the elephants. How many barrels of corn are the buffalo fed each day?
Answer:
7/20
Step-by-step explanation:
What is the slope of the line connecting the pair of points (0,7) (4,12)
Answer: 5/4
Step-by-step explanation: That should be right because I have big brain. Mark brainlist please :)