When multiplying a whole number with a fraction, the first factor will be reduced to a fraction or a decimal, depending on the problem.
In both of the given problems, we are multiplying a whole number with a fraction. When we multiply a whole number with a fraction, we can simply multiply the whole number with the numerator of the fraction and keep the denominator as it is. So, in the first problem, we have 10 multiplied by 1/2. Multiplying 10 with 1 gives us 10, and then we divide it by 2, which gives us 5. Therefore, the first factor, which is 10, will be reduced to 5 in this problem.
In the second problem, we have 15 multiplied by 7/2. Multiplying 15 with 7 gives us 105, and then we divide it by 2, which gives us 52.5. Therefore, the first factor, which is 15, will be reduced to 52.5 in this problem.
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Write 5.3*10^-5 in standard notation.
The scientific notation 5.3*10‐⁵ in standard notation gives 0.000053
Writing the number 5.3*10‐⁵ in standard notation.From the question, we have the following parameters that can be used in our computation:
Write 5.3*10‐⁵ in standard notation.
The above number is written in scientific notation
The standard notation of a number a * 10ⁿ is represented as
a000 where the 0's are in n places
Using the above as a guide, we have the following:
5.3 * 10‐⁵ = 0.000053
Hence, 5.3*10‐⁵ in standard notation is 0.000053
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A group of students collected old newspapers for a recycling project. The data shows the mass, in kilograms, of old newspapers collected by each student.
23, 35, 87, 64, 101, 90, 45, 76, 105, 60, 55
98, 122, 49, 15, 57, 75, 120, 56, 88, 45, 100.
What percent of students collected between 49 kilograms and 98 kilograms of newspapers? Explain how you got to your solution
Therefore, approximately 45.45% of students collected between 49 and 98 kilograms of newspapers.
Total number of students is 22.
To find the percentage of students who collected between 49 and 98 kilograms of newspapers, we first need to count the number of students who collected within this range. From the given data, we can see that the following students collected between 49 and 98 kilograms of newspapers
87, 64, 90, 76, 60, 55, 57, 75, 56, 88
Percentage of students = (number of students in range / total number of students) x 100
= (10 / 22) x 100
= 45.45%
Therefore, approximately 45.45% of students collected between 49 and 98 kilograms of newspapers.
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Americans consume on average 32. 3 lbs of cheese per year with a standard deviation of 8. 7 lbs. Assume that the amount of cheese consumed each year by an American is normally distributed. An American in the middle 70% of cheese consumption consumes per year how much cheese?
An American in the middle 70% of cheese consumption consumes between 23.1 lbs and 41.5 lbs of cheese per year.
To find the amount of cheese consumed by an American in the middle 70% of cheese consumption, we need to find the z-scores that correspond to the lower and upper bounds of the middle 70% and then convert those z-scores back to the original scale of measurement.
First, we need to find the z-score that corresponds to the 15th percentile (lower bound) and the z-score that corresponds to the 85th percentile (upper bound) of the normal distribution. We can use a standard normal table or a calculator to find these values. Using a calculator, we get:
z_15 = invNorm(0.15) = -1.036
z_85 = invNorm(0.85) = 1.036
Next, we can use the formula:
z = (x - mu) / sigma
where x is the amount of cheese consumed by an American, mu is the mean amount of cheese consumed (32.3 lbs), and sigma is the standard deviation (8.7 lbs), to convert the z-scores back to the original scale of measurement:
For the lower bound:
-1.036 = (x - 32.3) / 8.7
x = -1.036 * 8.7 + 32.3 = 23.1 lbs
For the upper bound:
1.036 = (x - 32.3) / 8.7
x = 1.036 * 8.7 + 32.3 = 41.5 lbs
Therefore, an American in the middle 70% of cheese consumption consumes between 23.1 lbs and 41.5 lbs of cheese per year.
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PLS HELP FAST I ONLY HAVE A LIL TIME LEFT! WILL MARK BRAINLIEST
Answer:
DStep-by-step explanation:
In this set,
the maximum is 35
the minimum is 6
So, we can cross out options A, B
the median is 14
The only option that works for this set is option D
Pls, answer this, 5 points and brainliest for the one who answers first!
Answer: C
Step-by-step explanation:
f moved left 4 spaces in x direction to get to g
so take opposite sign
f(x+4)
Over what interval is the function shown in the table increasing? Decreasing? Y=6x2
without a calculator find out
√28 ÷ √7
Step-by-step explanation:
= sqrt ( 28 ÷7) = sqrt (4) = 2
Answer:
2
Step-by-step explanation:
First we put the two equations together so it would be like this
[tex]\sqrt \frac{28}{7}[/tex]
the square root of that is 4 because 7 goes into 28 four times
so now we have this [tex]\sqrt{4}[/tex]
and the square root of 4 is 2
Cindy weighed the hydrogen peroxide in two containers. the hydrogen peroxide in one container weighed 6.4 ounces. the hydrogen peroxide in the second container weighed 4.07 ounces. find the total number of ounces of hydrogen peroxide using the rules of significant digits.
The total number of ounces of hydrogen peroxide, rounded to up to 1 significant digit after the decimal point, is 10.5 ounces.
According to the question the hydrogen peroxide in the first container weighed 6.4 ounces, and the hydrogen peroxide in the second container weighed 4.07 ounces.
According to the rules of significant digits, the numbers after the decimal point in a sum of difference is the least of that of the numbers to be added or subtracted.
This means that when 6.4 and 4.07 is added, then we will see numbers of decimal places in each number. 6.4 has 1 number after decimal point and 10.47 has 2.
Hence the result of 6.4 + 10.47 will have a minimum of 2 which is 1 decimal place.
Now, 6.4 + 4.07
= 10.47
Rounding it off from the above-mentioned criteria gives us 10.5 ounces.
Therefore, the total number of ounces of hydrogen peroxide, rounded to up to 1 significant digit after the decimal point, is 10.5 ounces.
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What is the equation of the parabola?y = −one eighthx2 + 5 y = one eighthx2 + 5 y = one eighthx2 − 5 y = −one eighthx2 − 5
The equations represent four different parabolas with different shapes and orientations, but all of them have the same axis of symmetry, which is the y-axis (because there is no x term).
What is equation of parabola?The collection of all points in a plane that are equally spaced from a fixed line and another fixed point in the plane that is not on the line is known as a parabola. The focus of the parabola is the fixed point (F) and the fixed point (D) is known as the directrix.
The equation of a parabola in standard form is:
y = a x² + b x + c
where "a" is the coefficient of the quadratic term (x^2), "b" is the coefficient of the linear term (x), and "c" is the constant term.
Looking at the given equations:
y = -1/8 x² + 5, has a negative coefficient for the quadratic term (a = -1/8) and a positive constant term (c = 5).y = 1/8 x² + 5, has a positive coefficient for the quadratic term (a = 1/8) and a positive constant term (c = 5).y = 1/8 x² - 5, has a positive coefficient for the quadratic term (a = 1/8) and a negative constant term (c = -5).y = -1/8 x² - 5, has a negative coefficient for the quadratic term (a = -1/8) and a negative constant term (c = -5).So, the equations represent four different parabolas with different shapes and orientations, but all of them have the same axis of symmetry, which is the y-axis (because there is no x term).
To graph each parabola, you can use the vertex form of the equation:
y = a (x - h)² + k
where (h, k) is the vertex of the parabola.
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Approximately how many inches of ground does kameron cover in
3
full rotations of the unicycle wheel? (use
3.14
as an approximation of pi.)
The total count of inches that a Kameron cover is 301.44 inches, under the condition that we have to use 3.14 as an approximation of π.
The circumference formula of a circle is
C=πd
Here
C = refers to circumference
d = refers to diameter.
Since we know that the spoke length is 16 inches, we can evaluate the diameter by multiplying it by 2. Hence, the diameter of the wheel is 32 inches.
Now that we know the diameter of the wheel, we calculate its circumference using the formula
C=πd.
Staging in this formula
d=32 inches
π=3.14
C = πd
= 3.14 x 32
= 100.48 inches
So Kameron covers 100.48 inches of ground in one full rotation of the unicycle wheel.
Now in order to evaluate how many inches of ground Kameron covers in 3 full rotations of the unicycle wheel, we simply need to multiply this value by 3
100.48 x 3
= 301.44 inches
Thus, Kameron covers approximately 301.44 inches of ground in 3 full rotations of the unicycle wheel.
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Solve the following equation:
8 x five sixths
To solve the equation 8 x five sixths, we first convert the fraction to a decimal by dividing the numerator (5) by the denominator (6), which gives us 0.83.
We then multiply 8 by 0.83 to get the final answer of 6.64. Therefore, 8 x five sixths = 6.64.
In general, to multiply a whole number by a fraction, we can convert the fraction to a decimal and then multiply the whole number by the decimal.
Alternatively, we can convert the whole number to a fraction with a denominator of 1 and then multiply the two fractions by cross-multiplying and simplifying.
In this case, we could also write 8 as 8/1 and multiply it by 5/6 to get (8 x 5)/(1 x 6) = 40/6, which simplifies to 6 and 4/6 or 6.67 (rounded to two decimal places). However, converting the fraction to a decimal is often simpler and more practical.
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A dilation always produces a similar figure. Similar figures have the same ______ but different ______.
Answer:
A dilation always produces a similar figure. Similar figures have the same shape but different sizes.
In a dilation, each point of the original figure is transformed by multiplying its coordinates by a scale factor, which determines the change in size. However, the shape and proportions of the figure remain unchanged. Therefore, the figures obtained through dilation are similar, meaning they have the same shape but different sizes.
help. 100 points guaranteed.
a) Find the general solution of the differential equation dy 2.cy dar 22 +1 3 b) Find the particular solution that satisfies y(0) 2
The particular solution is [tex]y(t) = (1/2c) (t^2/2 + t/2 + 1/2c) + (2 - 1/(4c))e^(-2ct)[/tex].
[tex]dy/dt + 2cy = t^2 + 1[/tex]
To find the general solution of this differential equation, we can start by finding the integrating factor, which is given by:
I(t) = e^(∫2c dt) = [tex]e^(2ct)[/tex]
Next, we can multiply both sides of the differential equation by the integrating factor I(t):
[tex]e^(2ct) dy/dt + 2ce^(2ct) y = (t^2 + 1) e^(2ct)[/tex]
We can now recognize the left-hand side as the product rule of the derivative of the product of y and I(t):
[tex](d/dt)(y e^(2ct)) = (t^2 + 1) e^(2ct)[/tex]
Integrating both sides with respect to t gives:
[tex]y e^(2ct) = ∫(t^2 + 1) e^(2ct) dt + C[/tex]
The integral on the right-hand side can be solved using integration by parts, and we get:
∫([tex]t^2[/tex] + 1) [tex]e^(2ct) dt = (1/2c) e^(2ct) (t^2/2 + t/2 + 1/2c) + K[/tex]
where K is an arbitrary constant of integration.
Substituting this expression back into the previous equation, we get:
[tex]y e^(2ct) = (1/2c) e^(2ct) (t^2/2 + t/2 + 1/2c) + K[/tex]
Dividing both sides by e^(2ct), we obtain the general solution:
[tex]y(t) = (1/2c) (t^2/2 + t/2 + 1/2c) + Ke^(-2ct)[/tex]
where K is an arbitrary constant.
To find the particular solution that satisfies y(0) = 2, we can substitute t = 0 and y(0) = 2 into the general solution and solve for K:
[tex]y(0) = (1/2c) (0^2/2 + 0/2 + 1/2c) + Ke^(0)[/tex]
2 = 1/(4c) + K
Solving for K, we get:
K = 2 - 1/(4c)
Substituting this value of K back into the general solution, we get the particular solution:
[tex]y(t) = (1/2c) (t^2/2 + t/2 + 1/2c) + (2 - 1/(4c))e^(-2ct)[/tex]
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Amelia is saving up to buy a new phone. She already has $100 and can save an
additional $9 per week using money from her after school job. How much total
money would Amelia have after 6 weeks of saving? Also, write an expression that
represents the amount of money Amelia would have saved in w weeks.
The expression that represents the amount of money Amelia would have saved in w weeks is: $9w + $100
Amelia starts with $100 and saves an additional $9 per week for 6 weeks. To find the total amount of money she has after 6 weeks, you can use this formula:
Total money = Initial amount + (Weekly savings × Number of weeks)
Total money = $100 + ($9 × 6)
Total money = $100 + $54
Total money = $154
So, Amelia would have $154 after 6 weeks of saving.
For an expression representing the amount of money Amelia would have saved in w weeks:
Total money (w) = $100 + ($9 × w)
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PLEASEEEEEE HELP ME!!!!!
VERY IMPORTANT!!!!
Amelie spins the following spinner, which has 10
equally sized spaces numbered 1
through 10. The numbers 1 and 7 are colored blue; the numbers 2, 4, and 6 are red; and the numbers 3, 5, 8, 9, and 10 are green. What is the probability that Amelie spins either an odd number or a red number?
Enter your answer as a reduced fraction, like this: 3/14
Answer:
4/5
Step-by-step explanation:
In probability, if there are two events and "or" is used, we will add the probabilities of each event. Since the event that an odd number is spun does not affect whether a red number is spun, we can calculate the probability of each event separately:
Probability of spinning an odd number:
5/10 because there are 5 odd numbers possible & 10 outcomes
Probability of spinning a red number:
3/10 because there are 3 red numbers & 10 outcomes
Now add the probability of each event:
5/10 + 3/10 = 8/10 = 4/5
Buying three movie tickets and a popcorn, which costs $5.50, is the same price as buying two tickets snacks worth a total of 16.50. How much does one movie ticket cost
If buying three movie tickets and a popcorn, which costs $5.50, is the same price as buying two tickets snacks worth a total of 16.50, one movie ticket costs $2.20.
Let the cost of one movie ticket be represented by x.
According to the problem, buying three movie tickets and a popcorn costs $5.50, so we can set up the equation:
3x + $5.50 = 2y
where y is the cost of snacks.
Similarly, buying two tickets and snacks worth a total of $16.50 can be represented by the equation:
2x + y = $16.50
We can solve this system of equations by substituting the first equation into the second equation for y:
2x + (3x + $5.50) = $16.50
5x + $5.50 = $16.50
5x = $11
x = $2.20
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The value of a stock in 1940 is $1. 25. Its value grows
by 7% each year after 1940.
A. ) Write an equation representing the value of the
stock, V(t), in dollars, t years after 1940.
The equation representing the value of the stock, V(t), in dollars, t years after 1940 is V(t) = $1.25 * [tex]1.07^t[/tex].
Let V(0) be the value of the stock in 1940, which is given as $1.25. Then, the value of the stock after t years (t > 0) can be found by multiplying the initial value with the growth factor of 1.07 raised to the power of the number of years of growth. Thus, the equation representing the value of the stock, V(t), in dollars, t years after 1940 is:
V(t) = V(0) * [tex](1 + 0.07)^t[/tex]
Substituting the given value of V(0) = $1.25, we get:
V(t) = $1.25 * [tex](1 + 0.07)^t[/tex]
Simplifying this expression, we get:
V(t) = $1.25 * [tex]1.07^t[/tex]
Therefore, the equation representing the value of the stock, V(t), in dollars, t years after 1940 is V(t) = $1.25 * [tex]1.07^t[/tex].
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WILL GIVE BRAINLIEST!! ANSWER FAST!!!
Given a graph for the transformation of f(x) in the format g(x) = f(x) + k, determine the k value.
k = −3
k = 1
k = 4
k = 5
Answer:
k = -3.
Step-by-step explanation:
To answer this question, we need to use our own knowledge and information. Adding a constant k to a function f(x) shifts the graph of f(x) vertically by k units. If k is positive, the graph moves up. If k is negative, the graph moves down. The value of k can be found by comparing the y-coordinates of corresponding points on the graphs of f(x) and g(x). For example, if g(x) = f(x) + 2, then the graph of g(x) is 2 units above the graph of f(x), and any point (x, y) on f(x) corresponds to a point (x, y + 2) on g(x). Therefore, the answer is: k is the vertical shift of the graph of f(x) to get the graph of g(x). It can be found by subtracting the y-coordinate of a point on f(x) from the y-coordinate of the corresponding point on g(x).
Looking at the graph given, we can see that the graph of g(x) is below the graph of f(x), which means that k is negative. We can also see that one point on f(x) is (0, 3), and the corresponding point on g(x) is (0, 0). Using the formula above, we get:
k = y_g - y_f
k = 0 - 3
k = -3
Therefore, the correct option is k = -3.
The population of a certain bacteria is known to double every 10 hours. Assuming exponential growth, determine the time that it would take for the bacteria to triple in number
It would take approximately 20 hours for the bacteria to triple in number.
Given that the bacteria population doubles every 10 hours, we can use exponential growth to determine the time it would take for the population to triple.
Let's represent the initial population as P0 and the time it takes for the population to triple as t.
Using the concept of exponential growth, we can express the population at time t as P(t) = P0 * 2^(t/10).
Since we want the population to triple, we set P(t) = 3 * P0:
3 * P0 = P0 * 2^(t/10).
We can cancel out P0 from both sides of the equation:
3 = 2^(t/10).
To solve for t, we can take the logarithm of both sides. Using the base-2 logarithm (log2) gives us:
log2(3) = t/10.
Using a calculator, we find that log2(3) is approximately 1.585.
Now, we can solve for t:
1.585 = t/10.
Multiplying both sides of the equation by 10 gives us:
15.85 = t.
Rounding to the nearest hour, the time it would take for the bacteria population to triple is approximately 16 hours.
Therefore, the bacteria population would take approximately 20 hours to triple in number.
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Which triangle has an obtuse angle?
Answer:
Step-by-step explanation:
An obtuse angle has a measure between 90 and 180 degrees. Looks like S and Q have obtuse angles, Its impossible to be sure unless you measure them with a protractor.
A sterling silver platter is made up of a mixture of silver and copper. The ratio of silver to copper is 37:3 by mass. If the platter has a mass of 600 grams, what is the mass, in grams, of the copper in the platter?
A) 18
B) 45
C) 222
D) 555
The mass of copper in the platter is 45 grams, which corresponds to option (B).
What is ratio ?
In mathematics, a ratio is a comparison of two quantities, often expressed as a fraction. Ratios can be used to describe how two quantities relate to each other, and they can be used to make predictions and solve problems in a variety of contexts.
The ratio of silver to copper in the platter is 37:3 by mass, which means that for every 37 grams of silver, there are 3 grams of copper.
Let's call the mass of silver in the platter "s" and the mass of copper "c". We know that the total mass of the platter is 600 grams, so:
s + c = 600
We also know that the ratio of silver to copper is 37:3, which means that:
s÷c = 37÷3
We can use this second equation to solve for s in terms of c:
s:c = 37:3
s = (37÷3)c
Now we can substitute this expression for s into the first equation:
s + c = 600
(37÷3)c + c = 600
(40÷3)c = 600
c = (3÷40) * 600
c = 45
Therefore, the mass of copper in the platter is 45 grams, which corresponds to option (B).
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You are going to run at a constant speed of 7.5
miles per hour for 45
minutes. You calculate the distance you will run. What mistake did you make in your calculation? [Use the formula S=dt
.]
The value of the distance you will run is 5.62500 miles
Calculating the value of the distance you will runFrom the question, we have the following parameters that can be used in our computation:
You are going to run at a constant speed of 7.5 miles per hour For 45 minutesThis means that
Speed = 7.5 miles per hour
Time = 45 minutes
The distance you will run is calculated as
Distance = Speed * Time
Substitute the known values in the above equation, so, we have the following representation
Distance = 7.5 miles per hour * 45 minutes
Evaluate the product
Distance = 5.62500 miles
Hence, the distance is 5.62500 miles
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Which equation does not have infinitely many solutions?
o =
6x + 4 = 2(3x + 2)
+
2x + 5 - 5x + 2 + 3x = 7
-
3x + 13 + 4x – 5 = 7x + 8
-
0 4x - 8 = 2(2x + 3)
-
None of the given equations have infinitely many solutions.
To identify which equation does not have infinitely many solutions among the given options.
1) 0 = 6x + 4 = 2(3x + 2)
2) 2x + 5 - 5x + 2 + 3x = 7
3) -3x + 13 + 4x – 5 = 7x + 8
4) 4x - 8 = 2(2x + 3)
Let's analyze each equation:
1) The equation can be simplified to 0 = 6x + 4, which is not true for all x, so it does not have infinitely many solutions.
2) Simplifying the equation, we get 0 = 7, which is false for any x, so it does not have infinitely many solutions.
3) Simplifying the equation, we get 1x + 8 = 7x + 8, which can be further simplified to -6x = 0, or x = 0. Since it has only one solution, it does not have infinitely many solutions.
4) Expanding the equation, we get 4x - 8 = 4x + 6. It is false for any x, so it does not have infinitely many solutions.
Therefore, none of the given equations have infinitely many solutions.
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If you need 2 1/4 cups of water for 1 cup of rice how much water do you need if you have 1/3 cup of rice?
The amount of water needed for 1/3 cup of rice, is 3/4 cups of water.
How much water is needed for 1/2 cup of rice?The problem asks us to find out how much water is needed for 1/3 cup of rice, given that 2 1/4 cups of water are needed for 1 cup of rice. To solve this problem, we can use a proportion.
A proportion is an equation that says two ratios are equal. In this case, we want to set up a proportion that relates the amount of water needed to the amount of rice.
Let's start by writing down what we know. We know that for 1 cup of rice, we need 2 1/4 cups of water. We can write this as a ratio:
2 1/4 cups water : 1 cup rice
Now we want to figure out how much water we need for 1/3 cup of rice. Let's call the amount of water we need "x" (we don't know what it is yet), and set up another ratio:
x cups water : 1/3 cup rice
We can now set up our proportion by equating these two ratios:
2 1/4 cups water : 1 cup rice = x cups water : 1/3 cup rice
To solve for x, we can cross-multiply and simplify. Cross-multiplying means we multiply the numerator of one ratio by the denominator of the other ratio, like this:
(2 1/4 cups water) * (1/3 cup rice) = (x cups water) * (1 cup rice)
To simplify this, we can convert the mixed number 2 1/4 to an improper fraction:
2 1/4 = 9/4
Now we can substitute these values and multiply:
(9/4 cups water) * (1/3 cup rice) = (x cups water) * (1/1 cup rice)
Multiplying the fractions on the left-hand side gives:
9/12 cups water = (x cups water) * (1/1 cup rice)
Simplifying the fraction on the left-hand side gives:
3/4 cups water = x cups water
So we have found that x, the amount of water needed for 1/3 cup of rice, is 3/4 cups of water. Therefore, if you have 1/3 cup of rice, you would need to use 3/4 cups of water to cook it.
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Given the differential equation dy/dx = x+3/2y, find the particular solution, y = f(x), with the initial condition f(-4)= 5
The particular solution with the given initial condition is:
[tex]y = (5/ e^(16/3)) * e^(x^2/3)[/tex]
To find the particular solution, we need to first separate the variables in the differential equation:
[tex]dy/dx = x + (3/2)y[/tex]
[tex]dy/y = (2/3)x dx[/tex]
Next, we integrate both sides:
[tex]ln|y| = (1/3)x^2 + C[/tex]
where C is the constant of integration.
To find the value of C, we use the initial condition f(-4) = 5:
[tex]ln|5| = (1/3)(-4)^2 + C[/tex]
[tex]ln|5| = (16/3) + C[/tex]
[tex]C = ln|5| - (16/3)[/tex]
Therefore, the particular solution is:
[tex]ln|y| = (1/3)x^2 + ln|5| - (16/3)[/tex]
[tex]ln|y| = (1/3)x^2 + ln|5/ e^(16/3) |[/tex]
[tex]y = ± (5/ e^(16/3)) * e^(x^2/3)[/tex]
However, since we know that f(-4) = 5, we can eliminate the negative solution and obtain:
[tex]y = (5/ e^(16/3)) * e^(x^2/3)[/tex]
So the particular solution with the given initial condition is:
[tex]y = (5/ e^(16/3)) * e^(x^2/3)[/tex]
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please show all work so i can understand! thanks!!
Classify each series as absolutely convergent conditionally convergent, or divergent. DO «Σ a (-1)k+1 k! k=1 b. Σ ka sin 2
For series Σ a(-1)^(k+1)k!, convergence depends on the limit of |a(k+1)/a(k)|. For series Σ ka sin(2), it diverges.
Consider the series Σ a(-1)^(k+1)k!, where a is a sequence of real numbers.
To determine the convergence of this series, we can use the ratio test
lim┬(k→∞)〖|a(k+1)(-1)^(k+2)(k+1)!|/|ak(-1)^(k+1)k!| = lim┬(k→∞)〖(k+1)|a(k+1)|/|a(k)||〗
If this limit is less than 1, then the series converges absolutely. If the limit is greater than 1, the series diverges. If the limit is equal to 1, then the test is inconclusive.
Let's evaluate the limit
lim┬(k→∞)〖(k+1)|a(k+1)|/|a(k)||〗 = lim┬(k→∞)〖(k+1)!/(k!k)|a(k+1)/a(k)||〗 = lim┬(k→∞)〖(k+1)/(k)|a(k+1)/a(k)||〗
Since lim┬(k→∞)〖|a(k+1)/a(k)||〗 exists, we can apply the ratio test again:
if the limit is less than 1, the series converges absolutely.
if the limit is greater than 1, the series diverges.
if the limit is equal to 1, the test is inconclusive.
Therefore, we can classify the series Σ a(-1)^(k+1)k! as either absolutely convergent, conditionally convergent, or divergent depending on the value of the limit.
Consider the series Σ ka sin(2), where a is a sequence of real numbers.
To determine the convergence of this series, we can use the alternating series test, which states that if a series Σ (-1)^(k+1)b(k) is alternating and |b(k+1)| <= |b(k)| for all k, and if lim┬(k→∞)〖b(k) = 0〗, then the series converges.
In this case, we have b(k) = ka sin(2), which is alternating since (-1)^(k+1) changes sign for each term. We also have
|b(k+1)|/|b(k)| = (k+1)|a|/k < k|a|/k = |b(k)|/|b(k-1)|
Therefore, |b(k+1)| <= |b(k)| for all k. Finally, we have
lim┬(k→∞)〖b(k) = lim┬(k→∞)〖ka sin(2)〗 = ∞〗
Since the limit does not exist, the series diverges.
Therefore, we can classify the series Σ ka sin(2) as divergent.
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10. When travelling along King Street or
Queen Street, the distance between any two
parallel streets is always about 1. 42 km.
Queen St.
King St.
Water St.
1. 42 km
,1. 42 km
1 km
Albert St.
1 km
Park St.
How much greater is the distance along
Park Street from King Street to Queen
Street than the distance along Albert Street
from King Street to Queen Street?
Answer:
The distance along Park Street from King Street to Queen Street is 0.42 km greater than the distance along Albert Street from King Street to Queen Street.
Step-by-step explanation:
Since the distance between any two parallel streets along King Street or Queen Street is always about 1.42 km, the distance along Park Street from King Street to Queen Street is:
1.42 km + 1 km + 1.42 km = 3.84 km
Similarly, the distance along Albert Street from King Street to Queen Street is:
1 km + 1.42 km + 1 km = 3.42 km
Therefore, the difference in distance is:
3.84 km - 3.42 km = 0.42 km
So, the distance along Park Street from King Street to Queen Street is 0.42 km greater than the distance along Albert Street from King Street to Queen Street.
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Olympiads School Calculus Class 9 Test 1 -. Find the equation(s) of the tangent line(s) to the curve defined by x² + x²y2 + y = 1 when = -1. (4 marks) . Find the intervals of concavity and any point(s) of inflection for f(x) = x? In x. (4 marks)
The equation of the tangent line to the curve x² + x²y² + y = 1 at the point where x=-1 is (-dx/dy + 2)/(1 - √5)(x + 1). The interval of concavity for f(x) = xlnx is (0, ∞) and there are no points of inflection.
To find the equation(s) of the tangent line(s) to the curve x² + x²y² + y = 1 at x = -1, we need to find the derivative of the curve with respect to x, i.e.,
2x + 2xy²(dx/dy) + dy/dx = 0
At x = -1, we get
-2 + 2y²(dy/dx) + dx/dy = 0
dy/dx = (-dx/dy + 2)/(2y²)
Now, substituting x = -1 in the curve, we get
1 - y + y² = 0
Solving for y, we get
y = (1 ± √5)/2
Substituting y = (1 + √5)/2 in the equation for dy/dx, we get
dy/dx = (-dx/dy + 2)/(2(1 + √5)/4) = (-dx/dy + 2)/(√5 + 1)
Therefore, the equation of the tangent line to the curve at x = -1, y = (1 + √5)/2 is
y - (1 + √5)/2 = (-dx/dy + 2)/(√5 + 1)(x + 1)
Similarly, substituting y = (1 - √5)/2 in the equation for dy/dx, we get
dy/dx = (-dx/dy + 2)/(1 - √5)
Therefore, the equation of the tangent line to the curve at x = -1, y = (1 - √5)/2 is
y - (1 - √5)/2 = (-dx/dy + 2)/(1 - √5)(x + 1)
To find the intervals of concavity and any point(s) of inflection for f(x) = xlnx, we need to find the second derivative of the function with respect to x, i.e.,
f''(x) = (d²/dx²)(xlnx) = d/dx(lnx + 1) = 1/x
Now, to find the intervals of concavity, we need to find the values of x for which f''(x) > 0 and f''(x) < 0. We have
f''(x) > 0 when x > 0, which means the function is concave up on (0, ∞).
f''(x) < 0 when x < 0, which means the function is concave down on (0, ∞).
To find any point(s) of inflection, we need to find the values of x for which f''(x) = 0. However, in this case, f''(x) is never equal to zero. Therefore, there are no points of inflection for the function f(x) = xlnx.
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here are 15 big dogs at the dog park. The ratio of big dogs to small dogs is 5 to 4. How many small dogs are at the dog park? There are 15 big dogs at the dog park. The ratio of big dogs to small dogs is 5 to 4. How many small dogs are at the dog park? A) 27 B) 12 C) 19 D) 9
Answer:
Answer: C) 19
We can solve this problem using the following chain of thought reasoning:
Step 1: We know that the ratio of Big Dogs to Small Dogs is 5 to 4. Therefore, if there are 15 Big Dogs in total, then the total number of Dogs in the park must be the sum of the Big Dogs and the Small Dogs.
Step 2: Since we know the ratio of Big Dogs to Small Dogs is 5 to 4, we can solve for the number of Small Dogs in the park: 15 (Big Dogs) / 5 = 3. Therefore, the total number of Dogs in the park is 15 + 3 = 18.
Step 3: Lastly, since we know that the total number of Dogs in the park is 18, the number of Small Dogs in the park can be found by subtracting the number of Big Dogs from the total: 18 - 15 = 3.
Therefore, the answer is C) 19 Small Dogs at the Dog Park.
Answer:
option B
Step-by-step explanation:
big : small
5 : 4
5 units= 15
1 unit= 15÷5
= 3
4 units= 3×4
= 12
there are 12 small dogs at the dog park
