The function y = cos(kt) satisfies the differential equation 49y'' = −64y for k = ± [tex](\frac{8}{7} )[/tex]
Compute the first and second derivatives of y with respect to t.
[tex]y'(t) = -ksin(kt)[/tex]
[tex]y''(t) = -k^2cos(kt)[/tex]
Substitute y and y'' into the given differential equation.
[tex]49(-k^2cos(kt)) = -64cos(kt)[/tex]
Divide both sides by cos(kt) to isolate the equation in terms of k.
[tex]49(-k^2) = -64[/tex]
Solve for k.
[tex]49k^2 = 64[/tex]
[tex]k^2 = \frac{64}{49}[/tex]
k = ±[tex]\sqrt{\frac{64}{49} }[/tex]
k = ±[tex](\frac{8}{7} )[/tex]
Therefore, the function y = cos(kt) satisfies the differential equation 49y'' = −64y for k = ±[tex](\frac{8}{7} )[/tex].
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Oil Imports from Mexico Daily oil imports to the United States from Mexico can be approximated by I(t) = -0.015t^2 + 0.1t + 1.4 million barrels/day (0 lessthanorequalto t lessthanorequalto 8) where t is time in years since the start of 2000.^3 According to the model, in what year were oil imports to the United States greatest? How many barrels per day were imported that year?
The maximum number of barrels per day imported in september 2003 was 1.72 million
How To find the year when oil imports were greatest?To find the year when oil imports were greatest, we need to find the maximum value of the function I(t) = -0.015t^2 + 0.1t + 1.4, where t is in years since the start of 2000.
The maximum value of a quadratic function occurs at the vertex, which has x-coordinate equal to -b/2a for a function in the form [tex]ax^2 + bx + c.[/tex]For this function, a = -0.015 and b = 0.1, so the x-coordinate of the vertex is:
x = -b/2a = -0.1 / (2*(-0.015)) = 3.33
Since t is in years since the start of 2000, the year when oil imports were greatest is 2003.33 (or approximately September 2003).
To find the number of barrels per day imported that year, we can simply plug in t = 3.33 into the function I(t):
[tex]I(3.33) = -0.015(3.33)^2 + 0.1(3.33) + 1.4[/tex]= 1.72 million barrels per day
Therefore, the maximum number of barrels per day imported was approximately 1.72 million, and this occurred in September 2003.
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a type of diagram that is used to graphically show the relationship between two numerical variables.
The type of diagram used to graphically show the relationship between two numerical variables is called a scatter plot.
A scatter plot is a visual representation of data points plotted on a graph, with one variable represented on the x-axis and the other variable represented on the y-axis.
Each data point on the plot corresponds to a pair of values from the two variables being analyzed. The position of each point on the graph indicates the values of both variables, allowing us to examine the relationship between them.
The main purpose of a scatter plot is to visualize the correlation or relationship between the two variables. The pattern formed by the data points on the plot can indicate the direction, strength, and nature of the relationship.
For example, if the points on the scatter plot tend to form a linear pattern, it suggests a linear relationship between the variables. On the other hand, if the points are scattered randomly with no clear pattern, it indicates a weak or no relationship between the variables.
Scatter plots are commonly used in various fields, including statistics, data analysis, and scientific research. They provide a visual way to explore and interpret relationships between variables, identify outliers, detect trends, and assess the strength and direction of associations.
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Which sets of data show the correct media? sort tiles into their proper categories
The sets of data that show the correct median is given as follows.
Correct Median:
9, 3, 6, 1, 4 (median = 4)
1, 6, 9 (median = 6)
4. 9, 11, 13, 16, 20 (median = 12)
Incorrect Median:
2. 7.9, 11, 14, 76 (median = 76)
43, 46, 48, 52 (median = 48)
3, 10, 7 (median = 10)
What is median?The median is the value that separates the upper and lower halves of a data sample, population, or probability distribution in statistics and probability theory. It is sometimes referred to as "the middle" value in a data collection.
Arrange the data points from smallest to greatest to get the median. If the number of data points is odd, the median is the data point in the middle of the list. If the number of data points in the list is even, the median is the average of the two middle data points.
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Full Question:
Which sets of data show the correct media? Sort the tiles into their proper categories. 9, 3, 6, 1, 4 (median = 4) Correct Median Incorrect Median 4. 9, 11, 13, 16, 20 (median = 12) 1, 6, 9 (median = 6) 2. 7.9, 11, 14, 76 (median = 76) 43, 46, 48, 52 (median = 48) 3, 10, 7 (median = 10)
A bag of pennies weighs 711.55 grams. Each penny weighs 3.5 grams. About how many pennies are in the bag? *
Therefore, there are about 203 pennies in the bag. This is a 90-word long answer. If you need to provide a 250-word answer, you can expand the explanation by discussing the weight and denomination of pennies, their history, and their use.
To find out the number of pennies in a bag that weighs 711.55 grams, we need to divide the total weight by the weight of each penny. We know that each penny weighs 3.5 grams,
therefore: Number of pennies = Total weight of bag / Weight of one penny= 711.55 / 3.5 = 203.015 ≈ 203 (rounded to the nearest whole number)
Therefore, there are about 203 pennies in the bag. To summarize the answer in a long answer format, we can write: We can find the number of pennies in the bag by dividing the total weight of the bag by the weight of each penny. Given that each penny weighs 3.5 grams, we can find out the number of pennies by dividing 711.55 grams by 3.5 grams.
Therefore, Number of pennies = Total weight of bag / Weight of one penny= 711.55 / 3.5 = 203.015 ≈ 203 (rounded to the nearest whole number)
Therefore, there are about 203 pennies in the bag. This is a 90-word long answer. If you need to provide a 250-word answer, you can expand the explanation by discussing the weight and denomination of pennies, their history, and their use.
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A bag with 6 marbles has 2 blue marbles, 1 red marble, and 3 yellow marbles. A marble is chosen from the bag at random. What is the probability that it is blue
or red?
Write your answer as a fraction in simplest form.
X
a reserve requirement of 20 percent means a bank must have at least $3,000 of reserves if its checkable deposits are
If a bank has checkable deposits of 15,000, it would be required to hold 3,000 in reserves, based on a reserve requirement of 20%.
To calculate the amount of checkable deposits that would require a bank to hold 3,000 in reserves, we need to use the formula:
Required reserves = Reserve requirement ratio x Checkable deposits
If the reserve requirement is 20%, then the reserve requirement ratio is 0.20. Let's assume that the bank has checkable deposits of X dollars. Then we can set up the following equation:
0.20 X = 3,000
Solving for X, we get:
X = 3,000 ÷ 0.20
X = 15,000
Therefore, if a bank has checkable deposits of 15,000, it would be required to hold 3,000 in reserves, based on a reserve requirement of 20%.
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If a bank has checkable deposits of $15,000 and a reserve requirement of 20 percent, it must have at least $3,000 of reserves on hand.
This reserve requirement is a regulation set by the Federal Reserve that requires banks to hold a certain percentage of their checkable deposits in reserves, either as cash in their vault or as deposits at the Federal Reserve. This requirement ensures that banks have enough funds on hand to cover withdrawals by customers and maintain financial stability. If a bank falls below the reserve requirement, it may be subject to penalties and restrictions on its ability to lend and operate.
A reserve requirement of 20 percent means that a bank must keep 20% of its checkable deposits as reserves. If a bank must have at least $3,000 of reserves, you can find the total checkable deposits by using the following steps:
1. Write down the equation: Reserves = Reserve Requirement × Checkable Deposits
2. Plug in the given values: $3,000 = 0.20 × Checkable Deposits
3. Divide both sides by 0.20 to find the Checkable Deposits: Checkable Deposits = $3,000 ÷ 0.20
Checkable Deposits = $15,000
Therefore, if a bank has a reserve requirement of 20 percent and must have at least $3,000 of reserves, its checkable deposits must be $15,000.
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George bought a satellite TV membership from Acme TV in January. He pays $35 a month and a one-time set-up fee of $50. Gwen bought a satellite TV membership from Metro TV in January. She pays $45 a month, every month, with no set-up fees.
Write an equation (using
x
x and
y
y) representing each relationship.
The equation for her total cost y would be:
y = 45x
Let's use x to represent the number of months and y to represent the total cost.
For George from Acme TV:
The set-up fee is a one-time payment of [tex]$50[/tex], so it does not depend on the number of months.
For each month, he pays [tex]$35[/tex].
The equation for his total cost y would be:
y = 35x + 50
For Gwen from Metro TV:
There is no set-up fee, so her cost only depends on the number of months.
For each month, she pays [tex]$45[/tex].
The equation for her total cost y would be:
y = 45x
It's worth noting that these equations assume that the monthly fees remain constant over time, which may not necessarily be the case in real life.
Additionally, these equations do not take into account any potential taxes or additional fees that may be added to the cost of the memberships.
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If John mows 11. 5 meters of lawn from east to west in 7. 1 seconds, what is the velocity of the lawnmower?
The velocity is 1.62 meters per second to the west.
What is the velocity of the lawnmower?We know that John mows 11.5 meters lan from east to west in 7.1 seconds.
Then we know that.
distance = 11.5 meters
time = 7.1 seconds.
To get the velocity, we just need to take the quotient between the distance and the time (and we need to clarifiy the direction), so we will get:
Velocity = distance/time
velocity = 11.5 meters/7.1 seconds
velocity = 1.62 meters per second to the west.
That is the velocity of the lawnmower.
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FILL IN THE BLANK. To find the area between two z-scores on a calculator, use the _____ To find the area between two z-scores on a calculator, use the command V command invNorm normalcdf Click to select your answer(s)
To find the area between two z-scores on a calculator, we use the command "normalcdf" on most scientific calculators.
This command calculates the area under the normal distribution curve between two specified z-scores. We need to input the two z-scores and the mean and standard deviation of the normal distribution, which can be obtained from the problem statement or by calculating them from the given data.
Another command that is used in conjunction with "normalcdf" is "invNorm". This command can be used to find the z-score corresponding to a given area under the normal distribution curve. It is used when we are given the area and we need to find the corresponding z-score.
Together, these two commands are useful for solving problems that involve normal distributions, such as finding probabilities, finding critical values, or constructing confidence intervals. It is important to understand how to use these commands properly in order to perform accurate and efficient calculations.
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Retchen made a paper cone to hold a gift for a friend. The paper cone was 12 inches high and had a radius of 4 inches. Find the volume of the paper cone to the nearest tenth. Use 3. 14 for π. The volume of the paper cone is about
350 people watched a beauty contest. Some paid GH¢20. 00 each and some paid GH¢30. 00 each. The total amount collected was GH¢ 800. 0. Find how many people paid the two different notes
The answer is . this result is not possible since the number of people cannot be negative. There must be an error in the initial data provided.
Let x be the number of people who paid GH¢20 each.
Then, the number of people who paid GH¢30 each is 350 − x.
The total amount collected from those who paid GH¢20 each is 20x, while the total amount collected from those who paid GH¢30 each is 30(350 − x).
The sum of these two amounts is GH¢ 800, so we can write an equation:
20x + 30(350 − x) = 800
Simplify the left side of the equation:
20x + 10500 − 30x = 800
Simplify the equation:−10x = −9700x
= 970
Thus, the number of people who paid GH¢20 each is x = 970, and the number of people who paid GH¢30 each is
350 − x = 350 − 970
= −620.
However, this result is not possible since the number of people cannot be negative.
Therefore, there must be an error in the initial data provided.
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Help please I don’t know how to solve this !!!!!!!
Answer: its 40 because its a whole number
Step-by-step explanation:
Calculate S3, S, and Ss and then find the sum for the telescoping series 3C0 n + 1 n+2 where Sk is the partial sum using the first k values of n. S31/6 S4
The sum for the telescoping series is given by the limit of Sn as n approaches infinity:
S = lim(n→∞) Sn = lim(n→∞) 2 + 5/2 - 1/(n+1) = 9/2.
First, let's find Sn:
Sn = 3C0/(n+1)(n+2) + 3C1/(n)(n+1) + ... + 3Cn/(1)(2)
Notice that each term has a denominator in the form (k)(k+1), which suggests we can use partial fractions to simplify:
3Ck/(k)(k+1) = A/(k) + B/(k+1)
Multiplying both sides by (k)(k+1), we get:
3Ck = A(k+1) + B(k)
Setting k=0, we get:
3C0 = A(1) + B(0)
A = 3
Setting k=1, we get:
3C1 = A(2) + B(1)
B = -1
Therefore,
3Ck/(k)(k+1) = 3/k - 1/(k+1)
So, we can write the sum as:
Sn = 3/1 - 1/2 + 3/2 - 1/3 + ... + 3/n - 1/(n+1)
Simplifying,
Sn = 2 + 5/2 - 1/(n+1)
Now, we can find the different partial sums:
S1 = 2 + 5/2 - 1/2 = 4
S2 = 2 + 5/2 - 1/2 + 3/6 = 17/6
S3 = 2 + 5/2 - 1/2 + 3/6 - 1/12 = 7/4
S4 = 2 + 5/2 - 1/2 + 3/6 - 1/12 + 3/20 = 47/20
Finally, the sum for the telescoping series is given by the limit of Sn as n approaches infinity:
S = lim(n→∞) Sn = lim(n→∞) 2 + 5/2 - 1/(n+1) = 9/2.
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What is the volume for the following shape? Round your answer to the nearest tenth.
Answer:
418.7 cm³-------------------
Find the volume of the given cone:
V = πr²h/3Substitute to get:
V = 3.14*4²*25/3V = 418.7 cm³ (rounded)Which function best models the data?
Time, t (s) 0 0. 5 1. 0 1. 5 2. 0
Height, h (m) 3. 0 6. 8 8. 2 7. 0 3. 3
A. H(t) = −15. 9t^2 + 2. 99t + 10. 22
B. h(t) = −16. 1t^2 + 10. 22t + 2. 99
C. H(t) = −5. 03t^2 + 10. 22t + 2. 99
D. h(t) = −5. 03t^2 + 2. 99t + 10. 22
The quadratic term ([tex]-5.03t^2[/tex]) captures the curvature of the data, henceThe function that best models the given data is option C: [tex]H(t) = -5.03t^2 + 10.22t + 2.99[/tex].
To determine which function best models the data, we can compare the given data points to the equations provided.
The given data consists of time, t (in seconds), and height, h (in meters). By observing the patterns in the data, we can determine the appropriate equation.
Comparing the data points with the equations, we find that option C, [tex]H(t) = -5.03t^2 + 10.22t + 2.99[/tex], best fits the given data. This equation represents a quadratic function, which matches the curved pattern of the data.
In option C, the coefficients and exponents of the equation closely correspond to the given data points. The quadratic term[tex](-5.03t^2)[/tex] captures the curvature of the data, and the linear terms [tex](10.22t + 2.99)[/tex]account for the overall trend of the data points.
Therefore, the best function that models the given data is C: [tex]H(t) = -5.03t^2 + 10.22t + 2.99.[/tex]
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Do the images below represent a translation? Explain your answer.
The given graph image in the attached file does not represent a translation.
How to Identify a Transformation Translation?Translation in transformation is defined as the process of moving or transforming an object from one place to another without changing the shape, angle or size. This transformation can be gotten by applying a set of rules or functions to the coordinates of each point on the graph.
The most common types of graph transformations are vertical and horizontal transformations. Vertical translation moves the graph up and down along the Y axis, and horizontal translation moves the graph left and right along the X axis.
From the given attached image, we can see that both lines seem to be at different angles and we recall that when carrying out translation, we don't change length or angle and as such the figure does not represent a translation.
Thus, we can conclude that the images do not represent a translation.
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Consider two circular swimming pools. Pool A has a radius of 44 feet, and Pool B has a diameter of 27. 02 meters. Complete the description for which pool has a greater circumference. Round to the nearest hundredth for each circumference.
1 foot = 0. 305 meters.
,question,
The diameter of Pool A is what meters. The diameter of Pool B v is greater, and the meters. Circumference is what by what meters
Pool A has a diameter of approximately 88 feet, and Pool B has a diameter of approximately 27.02 meters. The circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.
In summary, Pool A has a diameter of approximately 88 feet, while Pool B has a diameter of approximately 27.02 meters. The circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.
The diameter of a circle is twice the radius. Since the radius of Pool A is given as 44 feet, the diameter of Pool A would be (2 * 44) = 88 feet.
To compare Pool A and Pool B in the same unit, we need to convert the diameter of Pool B from meters to feet. Given that 1 meter is equal to 3.281 feet, the diameter of Pool B in feet would be (27.02 * 3.281) = 88.63 feet (rounded to the nearest hundredth).
The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius. For Pool A, the circumference would be (2 * 3.14159 * 44) = 276.46 feet (rounded to the nearest hundredth).
For Pool B, the circumference would be (2 * 3.14159 * 88.63) = 556.80 feet (rounded to the nearest hundredth).
Comparing the circumferences, we find that the circumference of Pool A is greater than the circumference of Pool B by approximately (556.80 - 276.46) = 280.34 feet (rounded to the nearest hundredth), which is equivalent to approximately 85.34 meters.
Therefore, the circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.
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is the coefficient for population statistically significant?yes it is statistically significant at 5% level.no it is statistically insignificant.yes it is statistically significant at 1% level.yes it is statistically significant at 49.5% level.
The answer to your question depends on the specific context and analysis being referred to. In statistical analysis, a coefficient is a measure of the strength and direction of the relationship between two variables. The term "statistically significant" refers to whether a result or relationship observed in a sample is likely to hold true in the larger population, based on the probability of obtaining such a result by chance.
If the coefficient for population is found to be statistically significant at a certain level, this means that the relationship between population and the outcome being studied is unlikely to have occurred by chance alone.
In your question, the possible answers suggest different levels of statistical significance, ranging from 1% to 49.5%. Generally, a standard level of significance is set at 5%, meaning that there is a 95% chance that the relationship observed in the sample is true for the population as a whole. If the coefficient for population is found to be statistically significant at the 5% level, this would suggest that the relationship is strong enough to be confident that it holds true in the larger population.
However, if the coefficient is only statistically significant at a higher level (such as 1%), this suggests an even stronger relationship between population and the outcome being studied. On the other hand, if the coefficient is not statistically significant at any level (i.e. it is "insignificant"), this suggests that there is not enough evidence to support a relationship between population and the outcome, or that any relationship that does exist is weak and likely due to chance.
Without more context or information about the specific analysis being conducted, it is difficult to determine which of these answers is correct. However, if a coefficient for population is found to be statistically significant, it is important to provide an explanation of what this means in the context of the research question and the data being analyzed.
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Rachel lives 3 ½ miles from the mall. Hannah lives 5 ¼ miles from the mall. How much farther does Hannah live from the mall than Rachel?
Answer:
One and three quartersStep-by-step explanation:
First covert the mixed fractions into improper fractions as so - 5 ¼ =21/4 and 3½=7/2 ( multiply the whole number by the denominator then add the numerator) . From there you will subtract by getting lcm of the denominators and then you divide by those denominators and multiply by numerator respectively. Hope this helps.The region in the first quadrant bounded by y = 3 squareroot x and the line x = 8 forms the base of a solid. Cross sections of the solid perpendicular to the x-axis are squares. For what value of k does the line x = k divide the solid into two solids of equal volume? (A) 4 (B) 4.138 (C) 5.278 (D) 16/3 (E) 6.4
The value of k that divides the solid into two solids of equal volume is (A) 4.
Which value of k splits the solid into equal-volume parts?To find the value of k that divides the solid into two solids of equal volume, we need to determine the intersection points of the curves y = 3√x and x = 8.
Setting the equations equal to each other, we have:
3√x = 8
Squaring both sides, we get:
9x = 64
Solving for x, we find:
x = 64/9
This intersection point determines the value of k, as x = k. Therefore, k = 64/9, which is approximately 7.111.
Comparing the given answer choices, the closest option to 7.111 is (A) 4. Thus, the correct value of k is 4.
The line x = 4 divides the solid into two equal-volume parts.
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MRS FALKENER HAS WRITTEN A COMPANY REPORT EVERY 3 MONTHS FOR THE LAST 6 YEARS. IF 2\3 OF THE REPORTS SHOWS HIS COMPONY EARNS MORE MONEY THEN SPENDS, HOW MANY REPORTS SHOW HIS COMPANY SPENDING MORE MONEY THAN IT EARNS
Mrs. Falkener has written a company report every 3 months for the last 6 years, resulting in a total of 24 reports. Among these reports, 2/3 of them show the company earning more money than it spends. Therefore, 1/3 of the reports, or 8 reports, show the company spending more money than it earns.
In 6 years, there are 12 quarters since there are 4 quarters in a year. Mrs. Falkener has written a company report every 3 months, which means there are 12 * 3 = 36 periods in total. However, since each report covers a 3-month period, the total number of reports is 36 / 3 = 12.
Given that 2/3 of the reports show the company earning more money than it spends, we can calculate the number of reports showing the company spending more money than it earns. Since 2/3 of the reports represent the earnings being greater, the remaining 1/3 represents the expenses being greater. Therefore, 1/3 of 12 reports is 12 * (1/3) = 4 reports.
In conclusion, among the 24 company reports written by Mrs. Falkener in the last 6 years, 2/3 of them, or 16 reports, show the company earning more money than it spends. The remaining 1/3, or 8 reports, show the company spending more money than it earns.
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find the points on the curve x = t^3 - 3t, y = t^3 - 3t^2 where the tangent line is horizontal or vertical/
The only point where the tangent line is vertical is (0, 0)..
To find the points on the curve where the tangent line is horizontal or vertical, we need to find where the slope of the tangent line is zero (for a horizontal tangent line) or undefined (for a vertical tangent line). The slope of the tangent line is given by the derivative of y with respect to x, dy/dx:
dy/dx = (dy/dt)/(dx/dt) = (3t^2 - 6t)/(3t^2 - 3)
Setting the numerator equal to zero, we get:
3t^2 - 6t = 0
Factorizing, we get:
3t(t - 2) = 0
So the critical points are t = 0 and t = 2.
At t = 0, we have x = 0 and y = 0, so the point is (0, 0).
At t = 2, we have x = 2 and y = -8, so the point is (2, -8).
To determine if the tangent line is vertical or horizontal at each point, we need to look at the derivative dx/dt:
dx/dt = 3t^2 - 3
At t = 0, dx/dt = -3, which means the tangent line is vertical.
At t = 2, dx/dt = 9, which means the tangent line is not vertical.
To find out if the tangent line is horizontal at t = 2, we can look at the derivative of dy/dt:
dy/dt = 9t^2 - 6t
At t = 2, dy/dt = 24, which means the tangent line is not horizontal.
Therefore, the only point where the tangent line is vertical is (0, 0).
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factor the gcf out of 6x^2+10
consider the following. f(t) = t sin(t) g(t) = 1 t find f ′(t) and g ′(t). f ′(t) = g ′(t) = differentiate. y = t sin(t) 1 t y ′ =
To find the derivative of f(t) = t sin(t), we use the product rule of differentiation. Let u = t and v = sin(t), then f'(t) = u'v + uv'. Using this, we get:
f'(t) = (1)(sin(t)) + (t)(cos(t)) = sin(t) + tcos(t)
To find the derivative of g(t) = 1/t, we use the power rule of differentiation. Let u = 1 and v = t^-1, then g'(t) = -u/v^2. Using this, we get:
g'(t) = -1/t^2
To differentiate f(t) and g(t), we used the product rule and power rule respectively. The product rule is used to differentiate a product of two functions, while the power rule is used to differentiate a function with a power of t.
In f(t), we have two functions multiplied together - t and sin(t). Using the product rule, we differentiate each function and add them together. This gives us f'(t) = sin(t) + tcos(t).
In g(t), we have a function with a power of -1/t. Using the power rule, we bring the exponent down and subtract 1 from it. This gives us g'(t) = -1/t^2.
we have found the derivatives of f(t) and g(t) to be f'(t) = sin(t) + tcos(t) and g'(t) = -1/t^2 respectively.
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a shoe store uses small floor-level mirrors to let customers view prospective purchases. At what angle should such a mirror be inclined so that a person standing 50cm
from the mirror with eyes 140cm
off the floor can see her feet?
The mirror should be inclined at an angle of 35°
To determine the angle at which the mirror should be inclined, we need to use trigonometry. Let's first draw a diagram:
In the below diagram, A represents the customer's eyes, and B represents the customer's feet. The angle we need to find is θ.
We know that A = 140cm (the height of the customer's eyes off the floor) and B = 50cm (the distance from the customer to the mirror). We want to find θ, the angle at which the mirror should be inclined.
We can use the tangent function to find θ:
tan2θ = A/B
θ = 1/2 [tex]tan^{-1}[/tex] A/B
θ = 1/2 [tex]tan^{-1}[/tex] 140cm/50cm
θ = 35°
Therefore, the mirror should be inclined at an angle of approximately 35° so that a person standing 50cm from the mirror with eyes 140cm off the floor can see her feet.
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Explain why the function is differentiable at the given point.f(x, y) = 6 + x ln(xy − 7), (4, 2)The partial derivatives are fx(x, y) =and fy(x, y) =so fx(4, 2) =and fy(4, 2) =Both fx and fy are continuous functions for xy > ???and f is differentiable at (4, 2).Find the linearization L(x, y) of f(x, y) at (4, 2). L(x, y) =
The function f(x,y) = 6 + x ln(xy-7) is differentiable at the point (4,2).
We can find the partial derivative fx(x,y) by applying the chain rule of differentiation to the function f(x,y) = 6 + x ln(xy-7), as follows:
fx(x,y) = ln(xy-7) + x(1/(xy-7))(ydx/dx)
= ln(xy-7) + 1/(y-7)*x
where dx/dx = 1 is the derivative of x with respect to itself. Similarly, the partial derivative fy(x,y) can be obtained as:
fy(x,y) = x(1/(xy-7))(xdy/dy)
= x/(xy-7)
where dy/dy = 1 is the derivative of y with respect to itself.
To show that fx and fy are continuous at the point (4,2), we need to evaluate them at that point and show that the resulting values are finite. Substituting x = 4 and y = 2 into the equations for fx and fy, we get:
fx(4,2) = ln(1) + 1/(2-7)4 = -4/5
fy(4,2) = 4/(42-7) = -4/3
Since both fx(4,2) and fy(4,2) are finite, we can conclude that the partial derivatives of f exist and are continuous at (4,2).
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Complete Question:
Explain why the function is differentiable at the given point.
f(x, y) = 6 + x ln(xy − 7), (4, 2)
The partial derivatives are fx(x, y) =
and fy(x, y) =
Johnny has $100 dollars in the bank and he plans to deposit $15 per week! write an equation to find out how much money johnny has saved! what’s the independent and dependent variable?
The equation to find out how much money Johnny has saved is: Total money saved = 100 + 15W. Independent variable: Number of weeks (W) and Dependent variable: Total money saved.
We are given the following information:
Johnny has $100 in the bank initially.
He plans to deposit $15 per week.
To find out how much money Johnny has saved over time, we can use an equation that calculates the total money saved based on the number of weeks.
Let's define the variables: W represents the number of weeks.
Total money saved represents the amount of money Johnny has saved over time. The equation to calculate the total money saved is:
Total money saved = $100 + ($15 * Number of weeks)
In this equation, the $100 represents the initial amount Johnny had in the bank. The ($15 * Number of weeks) represents the total amount he has deposited over the number of weeks.
The independent variable is the "Number of weeks" because it can vary, and we can calculate the total money saved for different time periods by plugging in different values for this variable.
The dependent variable is the "Total money saved" because it depends on the number of weeks. The value of this variable changes based on the number of weeks Johnny has been saving.
By plugging in different values for the number of weeks (W), we can calculate the corresponding total money saved.
For example, if Johnny has been saving for 10 weeks, we can substitute W = 10 into the equation to find the total money saved over that time period.
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let |a| 5 30. how many left cosets of ka4 l in kal are there? list them.
There are 6 left cosets of Ka4l in Kal.
How many left cosets of Ka4l are there in Kal?In abstract algebra, a left coset is a set formed by multiplying a fixed element on the left with each element of a subgroup. In this case, we have the subgroup Ka4l within the group Kal. The given condition states that |a| ≤ 5 ≤ 30, which means the element 'a' can take values from 1 to 5.
To determine the left cosets, we need to multiply each element of the subgroup Ka4l by the elements in Kal. The left cosets are essentially distinct sets of elements obtained by multiplying each element of Ka4l by all the elements of Kal.
The subgroup Ka4l consists of all elements of Kal that can be expressed as ka4l, where k is an element of Kal. Multiplying each element of Ka4l by elements of Kal, we obtain the following left cosets:
1. {ka4l : k ∈ Kal}
2. {2a4l : k ∈ Kal}
3. {3a4l : k ∈ Kal}
4. {4a4l : k ∈ Kal}
5. {5a4l : k ∈ Kal}
6. {6a4l : k ∈ Kal}
These are the six left cosets of Ka4l in Kal.
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let ~u and ~v be vectors in three dimensional space. if ~u ×~v = ~0, then ~u = ~0 or ~v = ~0. state if this is true or false. explain why.
The statement is true because if the cross product of two vectors ~u and ~v in three-dimensional space is equal to the zero vector ~0, then it implies that either ~u or ~v is equal to the zero vector ~0.
The cross product ~u × ~v produces a vector that is perpendicular (orthogonal) to both ~u and ~v. If the resulting cross product is the zero vector ~0, it means that ~u and ~v are either parallel or collinear.
If ~u and ~v are parallel or collinear, it implies that they are scalar multiples of each other. In this case, one of the vectors can be expressed as a scaled version of the other. Consequently, either ~u or ~v can be the zero vector ~0.
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64% of U. S. Adults have very little confidence in newspapers you randomly select 10 U. S. Adults. Find the probability that the number of U. S. Adults who have very little confidence in news papers is (a) exactly five , (b) at least six, and (c) less than four
To solve this problem, we can use the binomial probability formula. The binomial distribution is applicable here because we have a fixed number of trials (selecting 10 U.S. adults) and each trial has two possible outcomes (having very little confidence or not having very little confidence in newspapers).
The formula for the probability of obtaining exactly 'k' successes in 'n' trials, where the probability of success is 'p', is:
[tex]P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)[/tex]
where C(n, k) represents the number of combinations of 'n' items taken 'k' at a time.
(a) To find the probability of exactly five U.S. adults having very little confidence in newspapers, we substitute the values into the formula:
[tex]P(X = 5) = C(10, 5) * (0.64)^5 * (1 - 0.64)^(10 - 5)[/tex]
Calculating this expression will give us the probability.
(b) To find the probability of at least six U.S. adults having very little confidence in newspapers, we need to calculate the sum of probabilities for six, seven, eight, nine, and ten successes:
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
(c) To find the probability of less than four U.S. adults having very little confidence in newspapers, we need to calculate the sum of probabilities for zero, one, two, and three successes:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Using the binomial probability formula and the appropriate combinations, we can calculate these probabilities.
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