Answer: (a) To find the total number of fish that enter the lake over the 5-hour period from midnight to 5 A.M., we need to integrate the rate of fish entering the lake over this time period:
Total number of fish = ∫0^5 E(t) dt
Using the given function for E(t), we get:
Total number of fish = ∫0^5 (20 + 15 sin(πt/6)) dt
Using integration rules, we can solve this:
Total number of fish = 20t - (90/π) cos(πt/6) | from 0 to 5
Total number of fish = (100 - (90/π) cos(5π/6)) - (0 - (90/π) cos(0))
Total number of fish ≈ 121
Therefore, approximately 121 fish enter the lake over the 5-hour period.
(b) To find the average number of fish that leave the lake per hour over the 5-hour period, we need to calculate the total number of fish that leave the lake over this time period and divide by 5:
Total number of fish leaving the lake = L(0) + L(1) + L(2) + L(3) + L(4) + L(5)
Total number of fish leaving the lake = (4 + 20.1(0)^2) + (4 + 20.1(1)^2) + (4 + 20.1(2)^2) + (4 + 20.1(3)^2) + (4 + 20.1(4)^2) + (4 + 20.1(5)^2)
Total number of fish leaving the lake ≈ 257.5
Average number of fish leaving the lake per hour = Total number of fish leaving the lake / 5
Average number of fish leaving the lake per hour ≈ 51.5
Therefore, approximately 51.5 fish leave the lake per hour on average over the 5-hour period.
(c) To find the time when the greatest number of fish are in the lake, we need to find the maximum value of the function N(t) = E(t) - L(t) over the interval 0 ≤ t ≤ 8. We can do this by taking the derivative of N(t) with respect to t and setting it equal to zero:
N'(t) = E'(t) - L'(t)
N'(t) = (15π/6)cos(πt/6) - 40.2t
Setting N'(t) = 0, we get:
(15π/6)cos(πt/6) - 40.2t = 0
Simplifying and solving for t gives:
t ≈ 2.78 or t ≈ 6.22
Since 0 ≤ t ≤ 8, the time when the greatest number of fish are in the lake is t ≈ 2.78 hours after midnight (approximately 2:47 A.M.) or t ≈ 6.22 hours after midnight (approximately 6:13 A.M.).
To justify this, we can use the second derivative test. Taking the second derivative of N(t) gives:
N''(t) = -(15π2/36)sin(πt/6) - 40.2
At t ≈ 2.78, N''(t) is negative, which means that N(t) has a local maximum at this point. Similarly, at t ≈ 6.22, N''(t) is positive, which also means that N(t) has a local maximum at this point. Therefore, these are the times when the greatest number of fish are in the lake.
(d) To determine if the rate of change in the number of fish in the lake is increasing or decreasing at 5 A.M. (t = 5), we need to find the sign of the second derivative of N(t) at t = 5. Taking the second derivative of N(t) gives:
N''(t) = -(15π2/36)sin(πt/6) - 40.2
Plugging in t = 5, we get:
N''(5) = -(15π2/36)sin(5π/6) - 40.2
Simplifying, we get:
N''(5) ≈ -60.5
Since N''(5) is negative, the rate of change in the number of fish in the lake is decreasing at 5 A.M. (t = 5). This means that the number of fish entering the lake is decreasing faster than the number of fish leaving the lake, so the total number of fish in the lake is decreasing.
(a) Approximately 131 fish enter the lake over the 5-hour period from midnight to 5 A.M.
(b) The average number of fish that leave the lake per hour over the same period is approximately 14.8.
(c) The greatest number of fish in the lake occurs at time t = 2.94 hours, or approximately 2 hours and 56 minutes past midnight.
(d) The rate of change in the number of fish in the lake is increasing at 5 A.M.
(a) To find the total number of fish that enter the lake over 5 hours, we need to integrate the function E(t) from t=0 to t=5:
∫[0,5] E(t) dt = ∫[0,5] (20 + 15 sin(πt/6)) dt
This evaluates to approximately 131 fish.
(b) The average number of fish that leave the lake per hour can be found by calculating the total number of fish that leave the lake over 5 hours and dividing by 5:
∫[0,5] L(t) dt = ∫[0,5] (4 + 20.1t^2) dt
This evaluates to approximately 74 fish, so the average number of fish that leave the lake per hour is approximately 14.8.
(c) To find the time at which the greatest number of fish is in the lake, we need to find the maximum of the function N(t) = ∫[0,t] E(x) dx - ∫[0,t] L(x) dx over the interval [0,8]. We can do this by finding the critical points of N(t) and evaluating N(t) at those points. The critical point is at t = 2.94 hours, and N(t) is increasing on either side of this point, so the greatest number of fish is in the lake at time t = 2.94 hours.
(d) The rate of change in the number of fish in the lake at 5 A.M. can be found by calculating the derivative of N(t) at t=5. The derivative is positive, so the rate of change in the number of fish is increasing at 5 A.M.
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the diagonals of a rhombus are 8 and 10cm respectively. find the area of the rhombus
[tex]\sf Let \ d_1 \ and \ d_2 \ be \ the \ lengths \ of \ the \ sides \ of \ diagonals.[/tex]
[tex]\sf Given \ that \ d_1=8 \ cm[/tex]
[tex]\sf And \ d_2=10 \ cm[/tex]
[tex]\therefore\sf Area \ of \ rhombus=\dfrac{1}{2} (d_1)(d_2)=\dfrac{1}{2}(8)(10)=40 \ cm^2[/tex]
[tex]\rightarrow\boxed{\sf Area \ of \ rhombus=40 \ cm^2}[/tex]
Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting Carmen needs 2 cups of her frosting mixture how many cups of strawberry frosting and how many cups of lemon frosting will Carmen need
Carmen needs (6/7) cups of strawberry frosting and (1 1/7) cups of lemon frosting to make 2 cups of the frosting mixture.
To determine the amount of strawberry frosting and lemon frosting that Carmen needs to make 2 cups of the frosting mixture, we need to use a proportion.
Let x be the amount of strawberry frosting needed in cups, and y be the amount of lemon frosting needed in cups.
From the given information, we know that Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting. Thus, the ratio of the amounts of strawberry frosting to lemon frosting is:
x/y = (1/4)/(1/3)
Simplifying this ratio, we get:
x/y = 3/4
We also know that the total amount of frosting needed is 2 cups, so:
x + y = 2
Using substitution, we can solve for x:
x + (4/3)x = 2
(7/3)x = 2
x = (6/7) cups
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Learning Task 2: Let's Illustrate! During the month of February, Dr. Orfega recorded the number of CoViD-19 patients who came in of the hospital each day. The results are as follow: 15, 11, 13, 10, 18, 6, 9, 10, 15, 11, 12. Illustrate the following: 1) Q₁ 5) Pss 2) Q3 D. 3) D4 4) D Assimilation (Time Frame: 30 minutes!
Answer:
6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 18
Q1 (the first quartile) represents the data point that separates the lowest 25% of the data from the rest of the data. To find Q1, we can use the formula:
Q1 = (n + 1) / 4
where n is the total number of data points.
In this case, n = 11, so:
Q1 = (11 + 1) / 4 = 3rd data point
So, Q1 is 10.
Q3 (the third quartile) represents the data point that separates the highest 25% of the data from the rest of the data. To find Q3, we can use the formula:
Q3 = 3(n + 1) / 4
In this case:
Q3 = 3(11 + 1) / 4 = 9th data point
So, Q3 is 15.
D4 represents the fourth decile, which is the data point that separates the lowest 40% of the data from the rest of the data. To find D4, we can use the formula:
D4 = (n + 1) / 10 * 4
In this case:
D4 = (11 + 1) / 10 * 4 = 5th data point
So, D4 is 11.
D Assimilation represents the data point that is closest to the mean (average) of the data. To find D Assimilation, we first need to find the mean of the data:
Mean = (6 + 9 + 10 + 10 + 11 + 11 + 12 + 13 + 15 + 15 + 18) / 11 = 12
The data point closest to the mean is 12, so:
D Assimilation = 12
Pss (the range) represents the difference between the largest and smallest data points. In this case:
Pss = 18 - 6 = 12
6 9 10 10 11 11 12 13 15 15 18
Dss=12
Q1=10 Q3=15
D4=11
Step-by-step explanation:
Use the image below to find x: Show your steps and identify the TRIG RATIO that you used to find x.
The measure of the angle x in the circle is 65 degrees
Solving for x in the circleFrom the question, we have the following parameters that can be used in our computation:
The circle
On the circle, we have the angle at the vertex of the triangle to be
Angle = 100/2
Angle = 50
The sum of angles in a triangle is 180
So, we have
x + x + 50 = 180
Evaluate the like terms,
2x = 130
So, we have
x = 65
Hence, the angle is 65 degrees
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A spring with an m-kg mass and a damping constant 5 (kg/s) can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons. If the spring is stretched 1 meters beyond its natural length and then released with zero velocity, find the mass that would produce critical damping. m = kg
The mass can be any value greater than zero.
To find the mass that would produce critical damping, we first need to find the damping coefficient, which is given by:
c = damping constant * 2 * √m
where m is the mass in kg.
In this case, c = 5 * 2 * √m = 10√m.
Next, we can use the equation for the displacement of a damped harmonic oscillator to find the value of m that produces critical damping:
x = e^(-ct/2m) * (A + Bt)
where x is the displacement from equilibrium, t is time, A and B are constants determined by the initial conditions, and c and m are the damping coefficient and mass, respectively.
For critical damping, we want the system to return to equilibrium as quickly as possible without oscillating, so we set the damping coefficient equal to the critical damping coefficient:
c = 2 * √km
where k is the spring constant.
Since the spring can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons, we know that the spring constant is:
k = F/x = 2/0.5 = 4 N/m
Substituting this value into the equation for critical damping, we get:
10√m = 2 * √(4m)
Squaring both sides and simplifying, we get:
100m = 16m
84m = 0
Since this is a contradiction, there is no value of m that produces critical damping. Therefore, the mass can be any value greater than zero.
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3x − 15y = 11 in slope intercept form
Answer:
To convert the equation 3x - 15y = 11 into slope-intercept form, we need to solve for y.
First, we'll subtract 3x from both sides:
-15y = -3x + 11
Next, we'll divide both sides by -15:
y = (3/15)x - (11/15)
Simplifying the fraction:
y = (1/5)x - (11/15)
This is the slope-intercept form, where the slope is 1/5 and the y-intercept is -11/15.
Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles.
What is the area of the plot?
Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles. The area of the plot is 384 sq meters.
The Pythagorean theorem is a fundamental geometric idea that deals with the connections between the sides of right triangles. The square of the length of the hypotenuse (c) of a right triangle is equal to the sum of the squares of the lengths of the other two sides, according to the theorem (a and b). This may be stated mathematically as follows:
c² = a² + b²
Pythagoras, the ancient Greek mathematician who is credited with inventing the theorem, is named for him. It is employed in domains like physics, astronomy, and surveying and has extensive applications in mathematics, science, and engineering.
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Answer:
Step-by-step explanation:
The plot is in the shape of a trapezium with two sides measuring 32 m and 24 m, and two other sides measuring 25 m each.
To find the area of the plot, we need to first find the height of the trapezium. We can use the Pythagorean theorem to do this.
The side opposite to the right angle is the hypotenuse of the right-angled triangle formed by the two sides measuring 25 m each. So,
h² = 25² - 24²
h² = 625 - 576
h² = 49
h = 7
Therefore, the height of the trapezium is 7 m.
The area of a trapezium is given by the formula:
Area = (sum of parallel sides) x (height) / 2
In this case, the sum of the parallel sides is:
32 + 24 = 56
So, the area of the plot is:
Area = 56 x 7 / 2
Area = 196 m²
Therefore, the area of the plot is 196 square meters.
36 inches in 3 feet
rate=____ unit rate ___
Answer:
Rate: 36:3
Unit Rate: 12:1
Step-by-step explanation:
Triangle ABC has vertices A(-1,1), B(1,3) and C(4,1). The image of ABC after the transformation matrix T=
The coordinates of transforming image of the vertices of the triangle ABC are A' (1, -1) ,B' (3, 1) , and C' (1, 4).
In triangle ABC,
Coordinates of the vertices of triangle ABC are,
A(-1,1), B(1,3) and C(4,1)
The transformation T y=x reflects the points across the line y=x.
The image of each point, we simply swap the x and y coordinates of each point.
So, applying the transformation T y=x to the vertices of triangle ABC, we get,
A' = (-1, 1) → (1, -1)
B' = (1, 3) → (3, 1)
C' = (4, 1) → (1, 4)
This implies,
The image of triangle ABC under the transformation T y=x is triangle A'B'C', where,
A' is located at (1, -1)
B' is located at (3, 1)
C' is located at (1, 4)
Therefore, in triangle ABC labeling the coordinates of the vertices of A'B'C' after transformation are as follows,
A' (1, -1)
B' (3, 1)
C' (1, 4)
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The above question is incomplete, the complete question is:
Triangle ABC has vertices A(-1,1), B(1,3) and C(4,1). The image of ABC after the transformation T y=x is A’ B’ C’. State and label the coordinates of A’ B’ C’.
Find the area of the shaded region. Provide an answer accurate to the
nearest tenth.
18 ft
10 ft
Thus, the area of the shaded part is found to be 50 sq. ft.
Define about area of the shaded region:The shaded region's area is most frequently found in common geometry problems. Such problems always have a minimum of two forms, and you must determine the area for each shape as well as the darkened zone by deducting the smaller shape's area from the larger.
Rectangle's area :
Area has two dimensions: length and width. Square units like square inches, square feet, or square metres are used to measure area.
Multiply its length by the width to determine the area of a rectangle. A is equal to L * W, where * denotes multiplication, L is the length, W is the breadth, and A is the area.Length of shaded part = 5 ft
width of shaded part = 10 ft
Area = 5*10
Area = 50 sq. ft
Thus, the area of the shaded part is found to be 50 sq. ft.
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Correct question:
For the given figure find the area of the shaded region.
Length BC = 18 ft
Length CD = 10 ft
4. what is the difference in the measures of center?
5. what is the variability of grades each week?
6. what conclusions can you draw about the test?
Measures of center are statistical tools used to determine the central tendency of a dataset, including mean, median, and mode.
The difference between these tools is how they capture the central tendency.
Variability of grades refers to how much grades fluctuate from week to week, which can be measured using statistical tools such as range, variance, and standard deviation. Without specific information about the test, it is not possible to draw conclusions.
However, analyzing the measures of center and variability can provide insights into student performance and grading consistency.
Further analysis, such as comparing grades to class averages or identifying patterns over time, may reveal additional information.
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salvador recorded in this list the heights in millimeters of each of his bean plants.
52, 46, 51, 32,50
which 2 inequalities best describe, h, the plant heights in millimeters?
h < 32, h > 52
h> 32, h < 52
h < 46, h > 52
h < 46, h > 52
The two inequalities that best describe the plant heights in millimeters are: h > 32 and h < 52. This is because all the recorded heights fall within this range. The other options do not include all the recorded heights or include heights that are not recorded.
To find the best inequalities that describe the plant heights (h) in millimeters, we need to determine the minimum and maximum heights from the given list.
List of plant heights: 52, 46, 51, 32, 50
Minimum height: 32 mm
Maximum height: 52 mm
Now we can write the inequalities that best describe the plant heights:
h > 32 (heights are greater than 32 mm)
h < 52 (heights are less than 52 mm)
Your answer: h > 32, h < 52
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Find the limit of (7x3)/(4x2-2x+10) as x approaches infinity."
To find the limit of (7x3)/(4x2-2x+10) as x approaches infinity, we need to divide the highest power of x in the numerator and denominator, which is x3, by the highest power of x in the denominator, which is x2. This gives us: (7x3)/(4x2-2x+10) = (7/4)x
As x approaches infinity, the value of (7/4)x also approaches infinity. Therefore, the limit of (7x3)/(4x2-2x+10) as x approaches infinity is infinity.
To find the limit of (7x^3)/(4x^2-2x+10) as x approaches infinity, we'll first look at the highest powers of x in the numerator and denominator.
In this case, the highest power of x in the numerator is x^3, and in the denominator, it's x^2. Since the highest power of x in the numerator is greater than that in the denominator, the limit will go to infinity (or -infinity) depending on the coefficients of the highest powers.
For this function, the coefficients are positive (7 for x^3 and 4 for x^2), so the limit as x approaches infinity will be positive infinity.
Your answer: The limit of (7x^3)/(4x^2-2x+10) as x approaches infinity is positive infinity.
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Ghost riders co. has an eps of $1.65 that is expected to grow at 8.5 percent per year. if the pe ratio is 19.15 times, what is the projected stock price in 4 years?
The projected stock price of Ghost Rider Co. in 4 years is $45.24.
First, we need to calculate the future EPS of Ghost Rider Co. in 4 years. We can do this using the formula for the future value of an annuity:
[tex]FV = PV x (1 + r)^n[/tex]
where FV is the future value, PV is the present value, r is the growth rate, and n is the number of years.
Using this formula, we get:
[tex]FV = $1.65 x (1 + 0.085)^4 = $2.36[/tex]
Next, we can use the following formula to determine the anticipated stock price:
Estimated stock price = EPS x PE ratio
When we enter the values we have, we obtain:
Projected stock price = $2.36 x 19.15 = $45.24
Therefore, the projected stock price of Ghost Rider Co. in 4 years is $45.24.
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Jason borrowed $5000 to go with the money he'd saved to buy a tractor. The finance charge on the loan was $55 and the term on the loan was 360 days. What was the APR for Jason's loan?
O 0. 011%
O 1. 116%
O 4. 015%
O 1. 527%
The answer is option B: 1.116%.
To find the APR(Annual Percentage Rate) for Jason's loan, we first need to calculate the total amount of interest he paid.
The finance charge of $55 is the interest paid for the 360-day term.
To find the total interest, we can use the formula:
Total interest = (finance charge / loan amount) x (days in a year / loan term in days)
Plugging in the values, we get:
Total interest = (55 / 5000) x (365 / 360)
Total interest = 0.011 x 1.01389
Total interest = 0.01116 or 1.116%
Therefore, the APR for Jason's loan is 1.116%.
The answer is option B: 1.116%.
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The line on a coordinate plane makes an angle of depression 32 degrees. What is the slope of the line
The slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.
To find the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees,:
Step 1: Determine the angle of elevation. Since the angle of depression is 32 degrees, the angle of elevation is also 32 degrees, because they are alternate angles.
Step 2: Use the tangent function to find the slope. The tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle (rise) to the side adjacent to the angle (run). In this case, the tangent of the angle of elevation (32 degrees) is equal to the slope of the line.
Step 3: Calculate the tangent of 32 degrees. Using a calculator or a trigonometric table, you can find that tan(32°) ≈ 0.625.
So, the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.
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A builder wishes to fence in 80000 m2 of land in a rectangular shape. for security reasons, the fence along the front part of the land will cost $60 per meter, while the fence for the other three sides will cost $20 per meter.
how much of each type of fence should the builder buy to minimize the cost of the fence?
determine the length of the fence along the front part of the land that will be cost $60 per meter.
(give your answer as a whole or exact number.)
To minimize the cost of the fence, the builder should use the expensive fence along the shorter side of the rectangular shape, as this will require less length of the expensive fence. Let's say the length of the rectangle is x meters and the width is y meters. Then the area of the rectangle is given by:
A = xy = 80000
And the perimeter of the rectangle is:
P = 2x + 2y
We are given that the cost of the fence along the front part of the land will cost $60 per meter, while the fence for the other three sides will cost $20 per meter. So the total cost of the fence is:
C = 60x + 20(2x + 2y)
Simplifying this expression, we get:
C = 100x + 40y
We can now use the area equation to eliminate one of the variables. Solving for y, we get:
y = 80000/x
Substituting this expression for y into the cost equation, we get:
C = 100x + 40(80000/x)
Simplifying this expression, we get:
C = 100x + 3200000/x
To minimize this function, we need to take its derivative and set it equal to zero:
dC/dx = 100 - 3200000/x^2 = 0
Solving for x, we get:
x = sqrt(32000) = 178.89
So the length of the rectangle should be approximately 178.89 meters, and the width should be:
y = 80000/178.89 = 446.68
Therefore, the amount of expensive fence needed is 178.89 meters, and the amount of cheap fence needed is:
2(178.89) + 2(446.68) - 178.89 = 893.36 meters
Finally, the length of the fence along the front part of the land that will be cost $60 per meter is simply the width of the rectangle, which is:
y = 446.68 meters (rounded to two decimal places)
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Answer this question please ( Marking best answer brainiest )
A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:
1/8 = 0.125 = 12.5%.
The experimental probabilities are obtained considering the trials, hence:
100 trials: 20/100 = 0.2 = 20%.400 trials: 44/400 = 0.11 = 11%.The more trials, the closer the experimental probability should be to the theoretical probability.
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Just the answer is fine:)
If C is the parabola y = x? from (1, 1) to (-1,1) then Sc(x - y)dx + (y sin y?)dy equals to: Select one: O a. 12 뮤 Ob O b. 124 7 O c. None of these O d. 5 7 O e. 2 7 Check
The correct answer is e. 2/7.
How to evaluate this line integral?To evaluate this line integral, we need to parameterize the curve given by the parabola y = x from (1, 1) to (-1, 1).
Let's let x = t and y = t, where t goes from 1 to -1. Then we can rewrite the integral as follows:
[tex]\int\ C (x - y)\dx + (y \sin y)\dy[/tex]
[tex]= \int\limits^1_{-1} {[(t - t)dt + (t sin t)}\,dt}[/tex]
[tex]= \int\limits^1_{-1} { (t \sin t)} \, dt[/tex]
We can evaluate this integral using integration by parts:
Let u = t and [tex]dv = sin t\ dt[/tex]. Then [tex]du/dt = 1[/tex] and v = -cos t.
Using the formula for integration by parts, we have:
[tex]\int\limits^1_{-1} { (t \sin t)}\, dt = -t \cos t |_{-1}^{1} + \int\limits^1_{-1} { cos t}\, dt[/tex]
= -cos(-1) + cos(1) + sin(-1) - sin(1)
= 2sin(1) - 2cos(1)
Therefore, the value of the line integral is:
[tex]S_c(x - y)dx + (y \sin y)dy = 2\sin(1) - 2\cos(1)[/tex]
Hence, the correct answer is e. 2/7.
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9. define a relation r on the integers, ∀m, n ∈ z, mean if m n is even. is r a partial order relation? prove or give counterexample.
No, the relation r is not a partial order relation.
To prove this, we need to show that r is not reflexive, not antisymmetric, or not transitive.
r is reflexive if ∀a∈Z, a a holds, which means that any integer is related to itself. This is true for r since a a = 2 × a = even.r is antisymmetric if whenever a b and b a, then a = b. This is not true for r since, for example, 2 6 and 6 2, but 2 ≠ 6.r is transitive if whenever a b and b c, then a c. This is not true for r since, for example, 2 6 and 6 4, but 2 is not related to 4.Since r fails to satisfy the antisymmetric property, it is not a partial order relation.
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What is 2 9 as a percentage? give your answer rounded to one decimal place.
2/9 as a percentage is approximately 22.2%.
To convert the fraction 2/9 to a percentage, you simply need to divide the numerator (2) by the denominator (9) and then multiply the result by 100.
1. Divide the numerator by the denominator: 2 ÷ 9 ≈ 0.2222
2. Multiply the result by 100: 0.2222 × 100 = 22.22%
Now, to round the answer to one decimal place, we consider the second digit after the decimal point. In this case, it's 2. Since it's less than 5, we can round down.
So, 2/9 as a percentage rounded to one decimal place is approximately 22.2%.
In summary, converting a fraction to a percentage involves dividing the numerator by the denominator and then multiplying the result by 100. Rounding to a specific decimal place helps in presenting the result in a more easily understandable form, especially when dealing with non-integer values.
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A school Community had planned to reduce the number of Grade 9 students per classroom by constructing additional classrooms however they constructed 4 Less rooms than they planned. As the result the number of students per class was 10 more than they planned if there are 1200 grade 9 students in the school determine the current number of classrooms and the number of students per class
The current number of classrooms is 24, and the number of students per class is 70 if there were a total of 1200 students.
Let us assume that the number of classes = x
Number of students per class = 1200/x
Number of classrooms planned = x - 4
Number of students planned per class = 1200/ x+10
Total number of students = 1200
By using the above data, the equations will be written as:
(1200 / x-4) = (1200/x) +10
By multiplying the equation 2 we get:
1200x = 1200x + [tex]x^{2}[/tex] - 4800 - 40x
[tex]x^{2}[/tex] - 480- 4x = 0
(x-24) (x+20) = 0
x = 24
Number of rooms built = x =24
Number of students per class = (1200/24-10) = 60 students
Therefore, we can conclude that the current number of classrooms is 24, and the number of students per class is 60 + 10 =70.
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Find the values of a and b, if the function defined by f(x) = x^2 + 3x + a , x <= 1
bx + 2, x >= 1 is differentiable at x = 1
To find the values of a and b, we need to ensure that the function is differentiable at x = 1. Thus, the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
First, we need to check that the function is continuous at x = 1. Since the function has different definitions for x <= 1 and x >= 1, we need to check that the limit of the function as x approaches 1 from both sides is the same.
Limit as x approaches 1 from the left (x <= 1):
f(x) = x^2 + 3x + a
lim x->1- f(x) = lim x->1- (x^2 + 3x + a) = 1^2 + 3(1) + a = 4 + a
Limit as x approaches 1 from the right (x >= 1):
f(x) = bx + 2
lim x->1+ f(x) = lim x->1+ (bx + 2) = b + 2
For the function to be continuous at x = 1, these two limits must be equal.
4 + a = b + 2
a = b - 2
Now we need to check that the derivative of the function at x = 1 exists and is equal from both sides.
Derivative of the function for x <= 1:
f(x) = x^2 + 3x + a
f'(x) = 2x + 3
f'(1) = 2(1) + 3 = 5
Derivative of the function for x >= 1:
f(x) = bx + 2
f'(x) = b
f'(1) = b
For the function to be differentiable at x = 1, these two derivatives must be equal.
5 = b
Substituting b = 5 into the equation we found earlier for a, we get:
a = 5 - 2 = 3
Therefore, the values of a and b that make the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
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Gianna keeps track of the number of people inside a music hall to attend a concert by looking at the number of scanned tickets. She plotted the data on the graph below, where x = 0 x=0 represents the time at 6 p.m., then drew a line of best fit. What does the point ( 1 , 92 ) (1,92) represent?
Note that the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.
How is this so?This is based on the given graph.
Note tha the horizontal axis = x = 0
which corresponds to 6pm
the vertical is the number of scanned tickets.
Also, the pont (1, 92) is the line of best fit so that means that at 7pm which is a n hour after 6pm, there were about 92 persons still in the hall.
Hence we are correct to state that the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.
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Full Question:
See attached image/graph
The double dot plot shows the values in two data sets. express the difference in the measures of center as a multiple of the measure of variation.
no troll comments or i will hack ur device and find out where u live ! okay :)
The difference in measures of center as a multiple of the measure of variation can be expressed using the coefficient of variation.
How to express difference in data?To express the difference in measures of center as a multiple of the measure of variation, you can use the coefficient of variation (CV).
The CV is calculated by dividing the standard deviation (measure of variation) by the mean (measure of center), and then multiplying by 100 to express the result as a percentage.
For example, if the standard deviation of one dataset is 5 and the mean is 10, the CV would be 50%. If the standard deviation of another dataset is 2 and the mean is 8, the CV would be 25%.
To express the difference in measures of center as a multiple of the measure of variation between these two datasets, you would calculate the difference in their means (10-8=2) and divide it by the CV of the combined dataset ((5/10 + 2/8)/2 = 47.5%).
Therefore, the difference in measures of center is approximately 0.042 times the measure of variation (2/47.5%).
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66. Which value of m makes the inequality true?
A. 4
B. 5
3m-4 < 11
C. 6
D. 7
Answer:
The answer to the question provided is choice A, 4.
The value of m which makes the inequality true is, 4
What is Inequality?A relation by which we can compare two or more mathematical expression is called an inequality.
Given that;
The inequality is,
⇒ 3m - 4 < 11
Now,. We can simplify as;
⇒ 3m - 4 < 11
⇒ 3m < 11 + 4
⇒ 3m < 15
⇒ m < 5
Thus, The value of m which makes the inequality true is, 4
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The table shows transactions from a bank account. fill in the missing number for box a.
transaction amount
account balance
transaction 1
150 150
transaction 2
50 100
transaction 3
90 a
transaction 4
-200 b
transaction 5
c 0
btw this is integers
The missing number for box a transaction amount account balance are a = 10, b = 210, c = 210.
Using the information provided in the table, we can fill in the missing numbers as follows:
For transaction 3: The account balance after transaction 2 was $100, and transaction 3 had an amount of $90. Therefore, the account balance after transaction 3 is $190. Hence, the missing number in box a is 190.
For transaction 4: The account balance after transaction 3 was $190, and transaction 4 had an amount of -$200. Therefore, the account balance after transaction 4 is -$10. Hence, the missing number in box b is -10.
For transaction 5: The account balance after transaction 4 was -$10, and transaction 5 had an amount of $c. Therefore, the account balance after transaction 5 is 0. Hence, the missing number in box c is 10.
Therefore, the completed table is:
transaction amount account balance
1 150 150
2 50 100
3 90 190
4 -200-10
5 10 0
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Find the measure of each arc of ⊙ p, where rt is a diameter.
When rt is a diameter of circle p, it divides the circle into two equal halves. Since the sum of angles in a circle is 360 degrees, each half of circle p measures 180 degrees.
Thus, each arc of circle p that is intersected by diameter rt measures half of the circle or 90 degrees.
Therefore, each arc of circle p measures 90 degrees when rt is a diameter.
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Use your mouse or finger to split the
trapezoid into two triangles and a rectangle.
I ready
Trapezoids can be split into two triangles and a rectangle.
Trapezoid is also known as a trapezium which is a closed shape having 4 sides with one pair of parallel sides. Trapezium is quadrilateral with 4 sides The parallel sides of a trapezium are known as the bases, and its non-parallel sides are called legs. A trapezium can also have parallel legs. The parallel sides can be horizontal, vertical, or slanting. Few real-life objects example of trapezium is a lamp, popcorn box etc.
A trapezoid consists of two triangles and one rectangle figure shows how can we cut the trapezium to split the trapezium.
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A circular mirror has a radius of 3. 4 feet rosalinda is decorating the edge of the mirror with Washington tape if she has exactly enough washi tape which measurement is closest to the length of the piece of washi tape in feet
The measurement closest to the length of the piece of washi tape needed is approximately 21.36 feet.
The circumference of the circular mirror can be calculated using the formula C = 2πr, where r is the radius. Plugging in the given radius of 3.4 feet, we get C = 2π(3.4) = 21.36 feet (rounded to two decimal places). Since Rosalinda is decorating the edge of the mirror with washi tape, she needs a piece of tape that is equal in length to the circumference of the mirror. Therefore, the length of the piece of washi tape needed is closest to 21.36 feet.
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