Answer:
Step-by-step explanation:
We can use trigonometry to solve this problem. Let's call the height of the antenna h and the length of the wire connecting the top of the antenna to the ground d.From the problem statement, we know that d = 7 feet and the angle of elevation θ is 75 degrees. The angle of elevation is the angle between the horizontal and the line of sight to the top of the antenna.We can use the tangent function to find h:tan(θ) = opposite / adjacentIn this case, the opposite side is the height of the antenna h, and the adjacent side is the length of the wire d + 0. This is because the wire touches the ground at a point 7 feet away from the base of the antenna, so the total length of the wire is d + 0.Substituting the values we have:tan(75 degrees) = h / (7 feet + 0)Simplifying:h = (7 feet) × tan(75 degrees)Using a calculator:h ≈ 24.16 feetTherefore, the height of the antenna is approximately 24.16 feet.
Make a number line and mark all the points that represent the following values of x.
x < 2 or x > -1
The number line for x < 2 or x >-1 has been drew.
What is number line?
A horizontal line with evenly spaced numerical increments is referred to as a number line. How the number on the line can be answered depends on the numbers present. The use of the number is determined by the question that it corresponds to, such as when graphing a point.
Here the given inequality is
x < 2 or x > -1.
We need to draw that in number line.
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determine whether the series converges, and if so find its sum. ∑k=1[infinity]4k 37k−1
This limit evaluates to 37/4, which is greater than 1. Therefore, the ratio test tells us that the series diverges.
To determine whether the series ∑k=1[infinity]4k 37k−1 converges, we can use the ratio test. The ratio test states that if the limit as k approaches infinity of the absolute value of the ratio of the (k+1)th term to the kth term is less than 1, then the series converges absolutely.For more such question on limit
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Help me find the area please and can you provide solving steps please!
The total area of the given figure is 55m² respectively.
What is the area?The term "area" describes how much room a two-dimensional figure occupies.
The volume of a one-dimensional figure is zero.
A rectangle's area can be calculated by multiplying the figure's length and width or by counting each individual square unit.
The two words do in fact differ in some ways.
The word "area" denotes "space" on a surface, in a place, or elsewhere.
However, the word "place" communicates the idea of a "spot," or a specific area of space.
The primary distinction between the two words, namely area, and place, is this.
So, to find the area:
First, we will find the area of the triangle and then multiply the answer by 2 as there can be made 2 triangles one on each side.
Then, we will find the area of the rectangle and then add it to get the final answer.
Area of a triangle:
1/2 * b * h
1/2 * 4 * 5
2 * 5
10 * 2 (2 triangles)
20 m²
Area of the rectangle:
l * b
7 * 5
35 m³
Total area: 35 + 20 = 55m²
Therefore, the total area of the given figure is 55m² respectively.
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last year, justin opened an investments account with $6600. at the end of the year, the amount in the account had decreased by 24$ (please help with A and B)
the year-end amount in Justin's account is $6575.44.
How to solve the question?
(a) To write the year-end amount in terms of the original amount, we can use the following formula:
Year-end amount = Original amount - Decrease
Substituting the given values, we get:
Year-end amount = $6600 - $24 = $6576
Now, to express the year-end amount in terms of the original amount, we can divide both sides of the above equation by the original amount:
Year-end amount / Original amount = ($6600 - $24) / $6600
Simplifying this expression, we get:
Year-end amount / Original amount = 0.9964
Therefore, the year-end amount is 0.9964 times the original amount.
(b) Using the answer from part (a), we can determine the year-end amount in Justin's account as follows:
Year-end amount = 0.9964 x $6600
Simplifying this expression, we get:
Year-end amount = $6575.44
Therefore, the year-end amount in Justin's account is $6575.44.
It is worth noting that the decrease of $24 corresponds to a decrease of approximately 0.36% in the original amount. This decrease could be due to various factors, such as market fluctuations or fees associated with the investments account. To better understand the reasons behind the decrease, Justin may want to review the account statement and consult with a financial advisor. Additionally, he may want to reevaluate his investment strategy and consider diversifying his portfolio to mitigate risks and maximize returns.
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The distances (y), in miles, of two cars from their starting points at certain times (x), in hours, are shown by the equations below:
Car A:
y = 52x + 70
Car B:
y = 54x + 56
After how many hours will the two cars be at the same distance from their starting point and what will that distance be? (1 point)
a
6 hours, 420 miles
b
6 hours, 434 miles
c
7 hours, 420 miles
d
7 hours, 434 miles
Answer:
d) 7 hours, 434 miles
Step-by-step explanation:
We need to find the time when the distances of both cars from their starting points will be equal. That is, we need to find the value of x for which the equations for Car A and Car B will give the same value of y. We can set the two equations equal to each other and solve for x:
52x + 70 = 54x + 56
Subtracting 52x and 56 from both sides, we get:
14 = 2x
x = 7
So the two cars will be at the same distance from their starting points after 7 hours. To find the distance at that time, we can substitute x=7 into either of the two equations and solve for y:
y = 52(7) + 70 = 434 (using the equation for Car A)
y = 54(7) + 56 = 386 (using the equation for Car B)
Therefore, the correct answer is:
d) 7 hours, 434 miles
Watch the video on left sided Riemann sumse before answering this question.For the integral ∫³1 x² dx, compute the area of the first (left-most) rectangle in the n=10 left sided Riemannsum. Round your answer to the tenths place.
The area of the first rectangle in the n=10 left sided Riemannsum is 0.2.
To compute the area of the first rectangle in the n=10 left sided Riemann sum for the integral ∫³1 x² dx, we need to divide the interval [1,3] into 10 subintervals of equal length. The width of each rectangle is then given by the length of each subinterval, which is Δx=(3-1)/10=0.2.
The left endpoint of each subinterval is used to determine the height of the rectangle. Since we are looking for the left-most rectangle, we use the left endpoint of the first subinterval, which is x₁=1.
The height of the rectangle is given by f(x₁)=x₁²=1²=1.
Therefore, the area of the first rectangle is A₁=Δx*f(x₁)=0.2*1=0.2.
Rounding to the tenths place, we get the final answer of 0.2.
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Pls help Pls pLS i really need help asap
The locus of all points that are the same distance from A as from B.
Explanation:
Let's observe the steps as,
1. Connect A and B and draw a line segment AB.
2. Construct an arc above and below the line segment AB by taking A as the center and radius greater than AB/2.
3. Construct an arc above and below the line segment AB by taking B as the center and radius greater than AB/2, and the points C and D will be obtained.
4. Connect C and D and draw a line segment CD.
The locus of all points that are the same distance from A as from B.
Explanation:
Let's observe the steps as,
1. Connect A and B and draw a line segment AB.
2. Construct an arc above and below the line segment AB by taking A as the center and radius greater than AB/2.
3. Construct an arc above and below the line segment AB by taking B as the center and radius greater than AB/2, and the points C and D will be obtained.
4. Connect C and D and draw a line segment CD.
Tank A contains 50 gallons of water in which 2 pounds of salt has been dissolved. Tank B contains 30 gallons of water in which 3 pounds of salt has been dissolved. A brine mixture with a concentration of 0.8 pounds of salt per gallon of water is pumped into tank A at the rate of 3 gallons per minute. The well-mixed solution is then pumped from tankA to tankB at the rate of 4 gallons per minute. The solution from tank is also pumped through another pipe into tank A at the rate of 1 gallon per minute, and the solution from tank is also pumped out of the system at the rate of 3 gallons per minute. From the options below, select the correct differential equations with initial conditions for the amounts A(t), and B(t), of salt in tanks A and B, respectively, at time t. O dA/dt=2.4−2A/25+30B,dB/dt=2A/25−152B, with A(0)=2,B(0)=3. O dA /dt=3−A/25+15y,dB/dt=2A/−2B/15, with A(0)=2,B(0)=3. O dA/dt=3−2A/+5B,dB/dt=25A−15B, with A(0)=2,B(0)=3. O dA/dt=2.4−25A+15B,dB/dt=50A−30B, with A(0)=2,B(0)=3.
The correct differential equations with initial conditions for the salt amounts in tanks A and B are 2.4 - 25A + 15B, dB/dt = 50A - 30B, with A(0) = 2, B(0) = 3. Option 4 is correct.
Let A(t) and B(t) be the amounts of salt in tank A and tank B at time t, respectively. Then we can write the differential equations as follows
The rate of change of salt in tank A is given by:
dA/dt = (0.8 * 3) - (3/50)*A + (1/50)*B
The first term on the right-hand side represents the salt that is added to tank A when the brine mixture is pumped into it. The second term represents the salt that is removed from tank A when the mixture is pumped out of it to tank B. The third term represents the salt that is added to tank A when the mixture is pumped from tank B into it.
The rate of change of salt in tank B is given by
dB/dt = (3/50)*A - (3/10)*B
The first term on the right-hand side represents the salt that is pumped from tank A into tank B. The second term represents the salt that is removed from tank B when the mixture is pumped out of it.
The initial conditions are A(0) = 2 and B(0) = 3.
Option 1, dA/dt = 2.4 - 2A/25 + 30B, dB/dt = 2A/25 - 152B
The differential equation for dA/dt in option 1 does not match with the one we derived. Therefore, option 1 is incorrect.
Option 2, dA/dt = 3 - A/25 + 15B, dB/dt = 2A/-2B/15
The differential equation for dB/dt in option 2 is missing a multiplication sign between 2A and -2B/15. This mistake renders the entire option 2 invalid.
Option 3, dA/dt = 3 - 2A/+5B, dB/dt = 25A - 15B
The differential equation for dA/dt in option 3 has a typo. There should be a negative sign between 2A and 5B in the numerator. This mistake renders the entire option 3 invalid.
Option 4, dA/dt = 2.4 - 25A + 15B, dB/dt = 50A - 30B
The differential equations in option 4 match with the ones we derived. Therefore, option 4 is the correct answer.
Hence, the correct differential equations with initial conditions for the amounts A(t) and B(t), of salt in tanks A and B, respectively, at time t are
dA/dt = (0.8 * 3) - (3/50)*A + (1/50)*B, dB/dt = (3/50)*A - (3/10)*B, with A(0) = 2, B(0) = 3.
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Prove the below statement.
If integers x and y where x < y are consecutive, then they have opposite parity.
Consecutive integers x and y with x < y have opposite parity because when one integer is even, the other must be odd, ensuring they are never both even or both odd.
To prove this statement, let's analyze even and odd integers. An even integer is defined as any integer that can be expressed as 2n, where n is an integer. An odd integer is defined as any integer that can be expressed as 2n + 1, where n is an integer.
Let x be an integer. If x is even, it can be expressed as 2n. The next consecutive integer, y, will be x + 1, which can be expressed as 2n + 1, making y odd. Conversely, if x is odd, it can be expressed as 2n + 1. The next consecutive integer, y, will be x + 1, which can be expressed as (2n + 1) + 1 = 2(n + 1), making y even.
Therefore, if x and y are consecutive integers with x < y, they will always have opposite parity.
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should all organizations try to collect and analyze big data? why or why not? what management, organization, and technology issues should be addressed before a company decides to work with big data?
While big data can offer significant insights to businesses, organizations must assess their specific needs and capabilities before deciding to work with it. Once a decision is made, management, organizational, and technology issues must be addressed to ensure that the investment in big data .
Big data has become a buzzword in the world of business. It refers to massive amounts of structured and unstructured data that is generated by businesses on a daily basis. The potential insights that big data can offer is enormous, but it requires the right management, organization, and technology to handle it effectively.
If a company decides to work with big data, it must address several management, organization, and technology issues. For example, a management issue that must be addressed is the identification of key business objectives. It is important to understand the business problem that big data will help to solve.
From an organizational perspective, data governance is a crucial issue. Organizations must establish clear policies and procedures for collecting, storing, and analyzing data. They must also ensure that they have the right people with the necessary skills to handle big data.
From a technology perspective, organizations must invest in the right hardware and software to collect, store, and analyze big data. They must also ensure that they have the necessary security measures in place to protect the data.
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find d2y/dx2 for the curve given by x=1/2t^2 and y=t^2 t
The d²y/dx² for the given curve is (1 - 1/x) / [tex]x^(^3^/^2^)[/tex].
How to calculate the second derivative of given curve?To find d²y/dx² we need to use the chain rule and implicit differentiation.
First, we can express t² in terms of x and y using the equation x = (1/2)t²and solving for t²:
t² = 2x
Next, we can take the derivative of both sides of the equation with respect to x:
d/dx (t²) = d/dx (2x)
Using the chain rule, we have:
d/dx (t²) = d/dt (t²) * dt/dx
To find dt/dx, we can take the derivative of both sides of the equation x = (1/2)t² with respect to t:
d/dt (x) = d/dt (1/2)t²
1 = t * dt/dt
dt/dt = 1/t
dt/dx = 1 / (dt/dt) = t
Substituting these expressions into the previous equation, we have:
2t * dt/dx = 2
t * dt/dx = 1
dt/dx = 1/t
Now, we can use the chain rule and implicit differentiation to find d²y/dx²:
d/dx (2x) = d/dx (t²) * dt/dx
2 = 2t * (1/t)³ * dy/dx + 2t² * d²y/dx²
Simplifying, we get:
d²y/dx² = (2 - 2/t²) / 2t
Substituting t² = 2x, we get:
d²y/dx² = (1 - 1/x) / [tex]x^(^3^/^2^)[/tex]
Therefore, the second derivative of y with respect to x is (1 - 1/x) / [tex]x^(^3^/^2^)[/tex].
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Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h+4 h
Write a polynomial expression, in simplified form, that represents the area of the canvas
Do NOT put any spaces in your answer
If you borrow $120,000 at an APR of 7% for 25 years, you will pay $848.13 per month. If you borrow the same amount at the same APR for 30 years, you will pay $798.36 per month.
a. What is the total interest paid on the 25-year mortgage?
b. What is the total interest paid on the 30-year mortgage?
c. How much more interest is paid on the 30-year loan? Round to the nearest dollar.
d. If you can afford the difference in monthly payments, you can take out the 25-year loan and save all the interest from part c.
What is the difference between the monthly payments of the two different loans? Round to the nearest dollar.
The total interest paid on the 25-year mortgage is given as $134,439.
How to solveIf you borrow $120,000 at an APR of 7% for 25 years, you will pay $848.13 per month. If you borrow the same amount at the same APR for 30 years, you will pay $798.36 per month.
a. What is the total interest paid on the 25-year mortgage? $134,439.
Thus, it can be seen that the total interest paid on the 25-year mortgage is given as $134,439.
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A beam from a lighthouse is visible for a distance of 3 mi.
To the nearest square mile, what is the area covered by the
beam as it sweeps in an arc of 150°?
Answer:
The beam from the lighthouse covers a circular area, and we are given that the maximum distance at which the beam is visible is 3 miles. This means that the radius of the circle is 3 miles.
To find the area of the circle covered by the beam as it sweeps in an arc of 150°, we need to calculate what fraction of the circle's total area corresponds to this arc. To do this, we can use the formula:
fraction of circle's area = (central angle of arc / 360°)
In this case, the central angle of the arc is 150°, so the fraction of the circle's area covered by the arc is:
fraction of circle's area = 150° / 360°
fraction of circle's area = 5/12
Therefore, the area covered by the beam is:
area = fraction of circle's area x total area of circle
area = (5/12) x π x radius^2
area = (5/12) x π x 3^2
area = 3.93 square miles (rounded to the nearest square mile)
Therefore, the area covered by the beam as it sweeps in an arc of 150° is approximately 3.93 square miles.
Simple explanation:
The area covered by the beam from the lighthouse as it sweeps in an arc of 150° is approximately 3.93 square miles (rounded to the nearest square mile).
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Goods bought for $1200 had to be sold off at a discount of 12%. What is the selling price?
To find the selling price after applying a discount of 12% to goods bought for $1200, we can follow these steps:
Step 1: Calculate the discount amount.
Discount Amount = 12% of $1200
Discount Amount = 0.12 * $1200
Discount Amount = $144
Step 2: Subtract the discount amount from the original price to get the selling price.
Selling Price = Original Price - Discount Amount
Selling Price = $1200 - $144
Selling Price = $1056
So, the selling price after applying a discount of 12% to goods bought for $1200 is $1056.
Three randomly chosen Michigan students were asked how many round trips they made to Canada last year. Their replies were 3, 4, 5. The geometric mean is
A. 3.877 B. 4.000 C. 3.915 D. 4.422
The geometric mean of a set of numbers is found by multiplying them all together and then taking the nth root, where n is the number of values. In this case, the three values are 3, 4, and 5. So, the geometric mean is the cube root of (3 x 4 x 5) which is 3.915. Therefore, the answer is C.
To find the geometric mean of the number of round trips made by the three Michigan students, we will use the formula:
Geometric Mean = (Product of the numbers)^(1/n)
Where n is the number of values.
In this case, the numbers are 3, 4, and 5, so we will calculate:
Geometric Mean = (3 * 4 * 5)^(1/3)
Geometric Mean = 60^(1/3)
Geometric Mean ≈ 3.915
Therefore, the correct answer is C. 3.915.
The geometric mean of a set of numbers is found by multiplying them all together and then taking the nth root, where n is the number of values. In this case, the three values are 3, 4, and 5. So, the geometric mean is the cube root of (3 x 4 x 5) which is 3.915. Therefore, the answer is C.
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A regular polygon has 20 vertices, then the number of line segments
whose two ends are two non-consecutive vertices of this polygon is
a) 190
b) 170
c) 380
d) 360
The number of line segments whose two ends are two non-consecutive vertices of this polygon is 170
To find the number of line segments whose two ends are two non-consecutive vertices, we can first find the total number of line segments possible by selecting any two vertices.
For a polygon with n vertices, the number of ways to select two vertices is given by the binomial coefficient nC2, which is n(n-1)/2.
For this polygon with 20 vertices, the number of line segments possible is 20(20-1)/2 = 190.
However, we must subtract the number of line segments that connect consecutive vertices (sides of the polygon) since we only want non-consecutive vertices.
Since there are 20 sides, there are 20 line segments that connect consecutive vertices.
Therefore, the number of line segments whose two ends are two non-consecutive vertices of the polygon is 190 - 20 = 170.
So, the answer is (b) 170.
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The constant of proportionality is m=_
the slope goes by several names
• average rate of change
• rate of change
• deltaY over deltaX
• Δy over Δx
• rise over run
• gradient
• constant of proportionality
however, is the same cat wearing different costumes.
to get the slope of any straight line, we simply need two points off of it, let's use those two in the picture below
[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{48})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{128}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{128}-\stackrel{y1}{48}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{3}}} \implies \cfrac{ 80 }{ 5 } \implies \text{\LARGE 16}[/tex]
Draw a trend line. Write an equation of the linear model. Predict the number of wins of a pitcher with an ERA of 6.
Therefore, the equation is y= -2.6x + 16 and the pitcher has y = 0.4 number of wins
How to solveFirst, we have to draw a trend line. From this, we know that the intercept
Now we can find the slope of the model
We are going to use points (1, 14), (2.5, 10)
m = -2.6
Now we can simplify the equation
y = -2.6x + 16
Now we just substitute 6 in this equation
Therefore, y = 0.4
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Help please. I would appreciate it.
The Solution of the equations is shown below.
To check the equation we have substitute the values of ordered pair.
1. y = -2x + 6
at x= 3
y = -2(3) +6 = -6 + 6= 0
Thus, the solution is (3, 0)
2. y= 6-3x
At x = 3
y =6 -9 = -3
Thus, the solution is (3, -3)
3. y= -5x + 2
At x= -5
y= 15+ 2= 7
Thus, the solution is (-5, 7)
4. y = 10 - 4x
At x = -4
y= 10+ 16 = 26
Thus, the solution is (-4, 26)
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ANSWER THIS QUESTION QUICKLY PLS!
A committee of five people is formed by selecting members from a list of 10 people.
How many different committees can be formed?
Enter your answer in the box.
Answer:
How much is 100×4 please
Muons are unstable subatomic particles with a mean lifetime of 2.2 μs that decay to electrons. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth’s surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth’s surface.
Part A
What is the greatest distance a muon could travel during its 2.2 μs lifetime?
Express your answer with the appropriate units.
Greatest distance a muon could travel during its 2.2 μs lifetime is approximately 660 meters.
How to find the greatest distance a muon could travel during its 2.2 μs lifetime?We'll use the formula:
distance = speed × time
Given that muons travel very close to the speed of light, we can approximate their speed with the speed of light (c), which is approximately 3.0 x 10⁸ meters per second (m/s). The mean lifetime of a muon is 2.2 μs, which is equal to 2.2 x 10⁻⁶ seconds.
Now we can plug the values into the formula:
distance = (3.0 x 10⁸ m/s) × (2.2 x 10⁻⁶ s)
distance = 6.6 x 10² meters
So, the greatest distance a muon could travel during its 2.2 μs lifetime is approximately 660 meters.
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answer the question down below
The height down the wall, in centimeters that the ladder slipped would be 28. 61 cm.
How to find the height of the ladder ?To indicate the ladder's heights before and after slipping as h1 and h2, respectively, is imperative. The ladder maintains a constant length of 3 meters (300 cm) throughout the occurrence.
At first, the ladder's foot stands at a distance of 60 cm from the base of the wall. However, following its slip, the same foot travels an additional 80 cm further away, creating an increased gap between it and the base of the wall, totaling to 140 cm.
The determination of the initial and final heights can be achieved through relying on the Pythagorean theorem:
300 ² = h2² + 140 ²
h2 ² = 300 ² - 140 ²
h2 = 265. 33 cm
Slipped distance would be:
= h1 - h2
= 293. 94 cm - 265. 33 cm
= 28. 61 cm
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By writing each number correct to 1 significant figure, find an estimate for the value of
2.8 × 82.6
27.8-13.9
The value of the phrase to one significant figure is estimated to be 17.
To find an estimate for the value of the expression (2.8 × 82.6)/(27.8-13.9),
we can first perform the calculations using the given values to obtain a more precise answer, and then round the final result to one significant figure.
Using a calculator, we have:
(2.8 × 82.6)/(27.8-13.9) ≈ 16.63
Rounding this to one significant figure, we get:
(2.8 × 82.6)/(27.8-13.9) ≈ 17
Therefore, an estimate for the value of the expression to one significant figure is 17.
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For number 3, how do you get the answer? I don't get how you get those roots
The distance AB is 2√2 units.
The distance ST is 4√2 units.
How to calculate the distance between two points?Distance between two points is the length of the line segment that connects the two points in a plane.
The formula to find the distance between the two points is usually given by d=√((x₂ – x₁)² + (y₂ – y₁)²)
For AB. We have:
A(1, 0) : x₁ = 1 , y₁ = 0
B(3, -2) : x₂ = 3 , y₂ = -2
d = √((3 – 1)² + (-2 – 0)²)
d = √8
d = √(4 * 2)
d = √4 * √2
d = 2 * √2
d = 2√2
Thus, the distance AB is 2√2
For ST. We have:
S(2, 3) : x₁ = 2 , y₁ = 3
T(6, -1) : x₂ = 6 , y₂ = -1
d=√((6 – 2)² + (-1 – 3)²)
d = √32
d = √(16 * 2)
d = √16 * √2
d = 4 * √2
d = 4√2
Thus, the distance ST is 4√2
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find a formula for the exponential function passing through the points ( − 3 , 1250 ) (-3,1250) and ( 1 , 2 ) (1,2)
The formula for the exponential function passing through the points (-3, 1250) and (1, 2) is y = (2/25) * 25^x.
To find the formula for the exponential function passing through the points (-3, 1250) and (1, 2), follow these steps:
1. An exponential function has the form y = ab^x, where a and b are constants.
2. Use the given points to create two equations:
For point (-3, 1250):
1250 = ab^(-3) (Equation 1)
For points (1, 2):
2 = ab^(1) (Equation 2)
3. Solve for one of the constants (e.g., a) using one of the equations (Equation 2):
a = 2/b
4. Substitute this value of a into the other equation (Equation 1):
1250 = (2/b) * b^(-3)
5. Solve for b:
1250 = 2b^2
b^2 = 625
b = 25 (since b must be positive in an exponential function)
6. Substitute the value of b back into the equation for a:
a = 2/25
7. Plug a and b into the general exponential function formula:
y = (2/25) * 25^x
The formula for the exponential function passing through the points (-3, 1250) and (1, 2) is y = (2/25) * 25^x.
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determine whether the series is convergent or divergent. [infinity] k = 1 k2 k2 − 4k 7a) Convergentb) DivergentIf it is convergent, find its sum. (If the quantity diverges, enter DIVERGES
To determine whether the series [infinity] k = 1 k2 k2 − 4k 7 is convergent or divergent, we can use the limit comparison test.
First, we note that k2 − 4k = k(k − 4), so the denominator of the terms in the series can be factored as k2(k − 4).
Next, we compare the series to the p-series ∑1/k2, which we know is convergent.
We take the limit as k approaches infinity of (k2/k2(k − 4)) = 1/(k − 4). This limit approaches 0 as k approaches infinity.
Therefore, by the limit comparison test, the series is convergent.
To find its sum, we can use the partial fraction decomposition:
1/(k2(k − 4)) = A/k + B/k2 + C/(k − 4)
Multiplying both sides by k2(k − 4), we get:
1 = A(k − 4) + Bk + Ck2
Plugging in k = 0 gives:
1 = -4A
So A = -1/4.
Plugging in k = 1 gives:
1 = -3A + B + C
So B + C = 13/12.
Plugging in k = 2 gives:
1/4 = -2A + B + 4C
So -8A + 4B + 16C = 1.
Solving this system of equations gives:
A = -1/4, B = 5/12, C = 1/3
Therefore, the sum of the series is:
∑ k = 1 to infinity 1/(k2(k − 4)) = ∑ k = 1 to infinity (-1/4k + 5/12k2 + 1/3(k − 4))
= (-1/4)∑ k = 1 to infinity 1/k + (5/12)∑ k = 1 to infinity 1/k2 + (1/3)∑ k = 1 to infinity 1/(k − 4)
= (-1/4)∞∑ k = 1 1/k + (5/12)π2/6 + (1/3)∞∑ k = 5 1/k
= (-1/4)ln(∞) + (5/72)π2 + (1/3)ln(∞)
= DIVERGES (since ln(∞) diverges to infinity)
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find the radius of convergence, r, of the series. [infinity] 4(−1)nnxn n = 1 r = 1 find the interval, i, of convergence of the series. (enter your answer using interval notation.) i = (−1,1)
In the series, the radius of convergence is r = 1, and the interval of convergence is (-1,1].
To find the radius of convergence of the series
∞
Σ [tex]4(-1)^n n*x^n[/tex]
n=1
we use the ratio test:
lim |[tex]4(-1)^{(n+1)}*(n+1)*x^{(n+1)}[/tex]| |[tex]4(x)(-1)^n*n*x^n[/tex]|
n->∞ |[tex]4(-1)^n*n*x^n[/tex]| |[tex]4(-1)^n*n*x^n[/tex]|
= lim |x|/n
n->∞
The limit of |x|/n approaches 0 as n approaches infinity, as long as |x| < 1. Therefore, the series converges absolutely for |x| < 1.
On the other hand, if |x| > 1, then the limit of the absolute value of the series terms does not approach zero, and therefore the series diverges.
If |x| = 1, then the series may or may not converge, depending on the value of x. In fact, when x = 1, the series becomes the alternating harmonic series, which converges. When x = -1, the series becomes 4 - 4 + 4 - 4 + ..., which oscillates and does not converge. Therefore, the interval of convergence is (-1,1].
Since the radius of convergence is the same as the distance from the center of the series (which is 0) to the nearest point where the series diverges or fails to converge, we have r = 1.
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A school in new zeland collected data about the employment status of the mother and father in two parent-families. The two-way table of column relative frequencies below shows the data
A family where the family works part time is twice and likely as a family where the father is not working to have the mother work part time is true
We use the following representation:
FF ---> Father works full-time
FP ---> Father works part-time
FW ---> Father not working
MF --->Mother works full-time
MP ---> Mother works part-time
MW ---> Mother not working
Next, we test each of the 4 options, till one of the options is true (see attachment)
Choice A:
The claim in choice A is that:
P(FP and MP) = 2×P(FW and MP)
From the given table, we have:
P(FP and MP)= 2×P(FW and MP)
0.14=2×0.07
0.14=0.14
Hence, a family where the family works part time is twice and likely as a family where the father is not working to have the mother work part time
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Use the definitions of even, odd, prime, and composite to justify each of your answers.
Exercise
Assume that k is a particular integer.
a. Is − 17 an odd integer?
b. Is 0 an even integer?
c. Is 2k − 1 odd?
This is because 2k is always an even integer (by definition) and subtracting 1 from an even integer always results in an odd integer. So, 2k - 1 is odd for any integer value of k.
a. Yes, -17 is an odd integer because it satisfies the definition of an odd integer, which is an integer that can be written in the form 2n + 1 for some integer n. In this case, we can write -17 as 2(-9) + 1, which means it is odd.
b. Yes, 0 is an even integer because it satisfies the definition of an even integer, which is an integer that can be written in the form 2n for some integer n. In this case, we can write 0 as 2(0), which means it is even.
c. No, we cannot determine whether 2k - 1 is odd or even based on the information given. However, we can say that it is always an odd integer when k is an integer. This is because 2k is always an even integer (by definition) and subtracting 1 from an even integer always results in an odd integer. So, 2k - 1 is odd for any integer value of k.
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This is because 2k is always an even integer (by definition) and subtracting 1 from an even integer always results in an odd integer. So, 2k - 1 is odd for any integer value of k.
a. Yes, -17 is an odd integer because it satisfies the definition of an odd integer, which is an integer that can be written in the form 2n + 1 for some integer n. In this case, we can write -17 as 2(-9) + 1, which means it is odd.
b. Yes, 0 is an even integer because it satisfies the definition of an even integer, which is an integer that can be written in the form 2n for some integer n. In this case, we can write 0 as 2(0), which means it is even.
c. No, we cannot determine whether 2k - 1 is odd or even based on the information given. However, we can say that it is always an odd integer when k is an integer. This is because 2k is always an even integer (by definition) and subtracting 1 from an even integer always results in an odd integer. So, 2k - 1 is odd for any integer value of k.
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