The canoe traveled approximately a distance of 21.9ft.
let the distance the canoe travels be "d".
Initial position = 65 ft
After 7s,
The Final position is 40ft away
From the given information the canoe moves =(65-40)=25ft closer
Let x=25ft
Using the tangent function
we know that, tan(50degree)=x/d
therefore, d=25/tan(50 degree)
d=21.9ft
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The measurements of the angles of a triangle have a ratio of 4:5:6. What are the three angle measurements? Classify the triangle by its sides and its angles.
please help with 21-24
The three angles of the triangle are 48 degrees, 60 degrees, and 72 degrees and it is a acute triangle.
The three angles of the triangle 4x, 5x, and 6x, where x is a constant of proportionality.
We know that the sum of the angles of a triangle is always 180 degrees, so we can set up an equation:
4x + 5x + 6x = 180
Simplifying, we get:
15x = 180
Dividing both sides by 15, we get:
x = 12
So the three angles of the triangle are:
4x = 4(12) = 48 degrees
5x = 5(12) = 60 degrees
6x = 6(12) = 72 degrees
all three angles are less than 90 degrees, the triangle is an acute triangle.
Therefore, the three angles of the triangle are 48 degrees, 60 degrees, and 72 degrees and it is a acute triangle.
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Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is to be $500. What is your expected net winnings?
Answer:
Expected Value = -1(999/1000) + 499(1/1000) = -500/1000 = -50 cents
Step-by-step explanation:
The life time of a certain device has CDF
F (x) = 1 − e^−λx2 x > 0; λ > 0
Derive the pdf of X and determine its mean, mode, median and variance.
The probability density function (pdf) of the lifetime of a certain device is expressed as: f (x) = 2λxe^−λx² ; x > 0; λ > 0
Mean: E(x) = -2λ + C
Mode: x = 0
Median: x = ±√(-ln(0.5)/λ)
Variance: Var(x) = 0
What is probability density function?Probability density function (PDF) describes the relative likelihood for a random variable to take on a given value. It is a continuous function that is used to represent the probability distribution of a continuous random variable.
The probability density function (pdf) of the lifetime of a certain device is expressed as:
f (x) = 2λxe^−λx² ; x > 0; λ > 0
Mean: The mean (or expected value) of the probability density function is given by:
E(x) = ∫x.f(x)dx = ∫x.2λxe^−λx²dx
= -2λe^−λx² + C
E(x) = -2λ + C
Mode: The mode of the probability density function is the value at which the maximum density occurs. This occurs at the point where the derivative of the function is equal to 0.
f′(x) = 2λxe^−λx²
= 0
x = 0
Therefore, the mode of the probability density function is 0.
Median: The median is the value of the random variable for which the cumulative distribution function is equal to 0.5.
F(x) = 1 − e^−λx² = 0.5
1 − e^−λx² = 0
e^−λx² = 0.5
−λx² = ln(0.5)
x² = -ln(0.5)/λ
x = ±√(-ln(0.5)/λ)
Therefore, the median of the probability density function is given by:
x = ±√(-ln(0.5)/λ)
Variance: The variance of the probability density function is given by:
Var(x) = E(x²) - (E(x))²
E(x²) = ∫x².f(x)dx
= ∫x^2.2λxe^−λx²dx
= -2λe^−λx² + C
E(x²) = -2λ + C
Var(x) = -2λ + C - (-2λ + C)²
= 0
Therefore, the variance of the probability density function is 0.
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a. ¿Cuál es el pago mensual de la hipoteca a 25 años, de $200,000 al 4% al primer centavo?
Lauren over-filled the homemade pecan pie that she was baking for Thanksgiving, so the pie needed additional cooking time. Lauren decided to place a strip of aluminum foil around the edge of the crust so that it would not burn. If Lauren used a pie pan with a 12-inch diameter, how long, to the nearest inch, should the strip of foil be?
A. 19 inches
B. 24 inches
C. 113 inches
D. 38 inches
Applying the circumference formula, the strip of aluminum foil should be about 38 inches long, which corresponds to answer choice D.
How to Calculate Circumference?To determine the length of the aluminum foil strip, we need to find the circumference of the pie pan, which is the distance around the edge of the pan.
The formula for the circumference of a circle is:
C = 2πr
where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle (which is half the diameter).
Since the pie pan has a diameter of 12 inches, the radius is 6 inches. Therefore:
C = 2π(6)
C ≈ 37.7 inches
This is approximately, D. 38 inches.
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Which graph shows the image of ABC after a rotation about the origin
Answer:
I need the rotation in degrees around the origin (flip around x axis, 90 degrees counter clockwise) or send all of the graph options
Step-by-step explanation:
A set of data items is normally distributed with a mean of 70 and a standard deviation of 8. Convert 70 to a z-score.
z70=
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
When 70 to a z-score, the resulting z-score is 0
Converting 70 to a z-score.From the question, we have the following parameters that can be used in our computation:
Mean = 70
Standard deviation = 8
Score = 70
To convert 70 to a z-score, we make use of the following equation
z = (score - mean)/Standard deviation
Substitute the known values in the above equation, so, we have the following representation
z = (70 - 70)/8
Evaluate the expression
z = 0
Hence, converting 70 to a z-score. we get 0
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Eulerize this graph in the most efficient way possible, considering the weights of the edges.
We can eulerize this graph in the most efficient way possible by making these vertices have even degree through adding an edge between them.
How do we Eulerize a graph?We can Eulerize a graph by ensuring to add edges to the graph in a way that preserves its connectivity while ensuring that all vertices have even degree.
A good method to do this is by adding edges between pairs of vertices that have odd degree until all vertices have even degree.
In the scenario above, all the vertices have odd degree except for the first and last vertices. To make these vertices have even degree, we go ahead and add an edge between them.
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Compare the numbers with
>, <, =.
69
-0.43
- 4
-2.3
Answer:
-4 < -2.3 < -0.43 < 69
Step-by-step explanation:
If you look at a number line,
-4 is the first to come and as we approach 0, it's -2.3 and then -0.43. After 0 is only 69 leftover.
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15.3
21. A squirrel is standing on the branch of a tree. The angle of elevation from a point on the ground to the squirrel
is 48°. The ground distance from the point to the tree is 28ft. How high above the ground is the squirrel?
Round your answer to the nearest foot.
48⁰
28 ft
21
Answer:
The height of the squirrel is 31 feet.
Step-by-step explanation:
You need to know your Right Triangle Trigonometry to do this problem.
Rt Triangle trig is all about ratios. Angles and ratios.
In your question, there is a rt triangle. The side measure given, 28 is next to the angle. The math word for "next to" is "adjacent" You know the adjacent side. The squirrel's height is the opposite side. The ratio that puts together adjacent and opposite is tangent.
tan Angle = opposite/adjacent
tan 48° = x/28
multiply both sides by 28
28•tan48° = x
You have to use a calculator that has trig functions. It will have buttons that say "sin", "cos", and "tan"
Enter 28 × tan48° =
It will return 31.0971504152
Your question asks you to round to the nearest whole.
x = 31
The squirrel's height in the tree is 31ft.
The garment factory plans to make 690 sets of garments. It has been done for 5 days, with an average of 75 sets per day. The rest needs to be completed in 2.5 days, how many sets are done on average per day (step-by-step answer)
The garment factory needs to produce an average of 126 sets per day in the next 2.5 days to complete the order of 690 sets.
We have,
The number of sets produced in the first 5 days.
= 5 x 75
= 375
The number of sets remaining.
= 690 - 375
= 315
The number of sets to be completed in the next 2.5 days.
= 315
The average number of sets to be completed per day.
= 315 / 2.5
= 126
Therefore,
The garment factory needs to produce an average of 126 sets per day in the next 2.5 days to complete the order of 690 sets.
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1. 7/8 + 1 1/4
2. 1 2/6 + 2 3/8
3. 2 3/4 + 1 1/6
4. 2 2/7 - 1
Answer:
1. 17/8 2. 89/24 3. 47/12 4. 9/7
Answer:
1) 29
2) 57
3) 91
4) 15
Step-by-step explanation:
1) 7/8 + 11/4
find lcm
lcm=8
multiply through by 8
8×7/8+8×11/4
Gives: 7 + 22
=29
2) 12/6+ 11/4
lcm=12
multiply through by 12
12× 12/6 + 12× 11/4
24 + 33
=57
3) 23/4 + 11/6
lcm=12
multiply through by 12
12× 23/4 + 12× 11/6
=69 +22
=91
4) 22/7 - 1
lcm=7
multiply through by 7
7 ×22/7 - 7 ×1
22-7
=15
The volume of a cone is 678.24 cubic inches. What is the height of the cone?
The height of the cone is 18 inches.
How to find the height of a cone?The height of the cone can be found as follows:
volume of a cone = 1 / 3 πr²h
where
r = radius of the coneh = height of the coneTherefore,
r = 6 inches
volume of the cone = 678.24
h = ?
Hence,
678.24 = 1 / 3 × 3.14 × 6² × h
678.24 = 1 / 3 × 3.14 × 36 × h
37.68h = 678.24
divide both sides of the equation by 37.68
h = 678.24 / 37.68
h = 18 inches
Therefore,
height of the cone = 18 inches
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4x + 3y = 18
4x - 2y = 18
Answer:
To solve the system of linear equations:
4x + 3y = 18
4x - 2y = 18
we can use the method of elimination or substitution.
Method 1: Elimination
In this method, we eliminate one of the variables by adding or subtracting the equations in a way that eliminates one of the variables. Here's how we can do it:
Multiply the first equation by 2, and the second equation by 3 to eliminate x:
8x + 6y = 36
12x - 6y = 54
Now, subtract the second equation from the first:
8x + 6y - (12x - 6y) = 36 - 54
8x + 6y - 12x + 6y = -18
-4x + 12y = -18
Divide both sides by -4 to isolate x:
-4x/(-4) + 12y/(-4) = -18/(-4)
x - 3y = 4.5
Now, we can substitute this value of x into one of the original equations, let's use the first equation:
4(4.5) + 3y = 18
18 + 3y = 18
3y = 18 - 18
3y = 0
y = 0
So, the solution to the system of equations is x = 4.5 and y = 0.
Method 2: Substitution
In this method, we solve one of the equations for one variable and then substitute that expression into the other equation to solve for the other variable. Here's how we can do it:
Solve the first equation for x:
4x + 3y = 18
4x = 18 - 3y
x = (18 - 3y)/4
Now, substitute this expression for x into the second equation:
4[(18 - 3y)/4] - 2y = 18
18 - 3y - 2y = 18
-5y = 18 - 18
-5y = 0
y = 0
Now, substitute this value of y back into the expression for x:
x = (18 - 3(0))/4
x = 18/4
x = 4.5
So, the solution to the system of equations is x = 4.5 and y = 0, which is consistent with the solution obtained using the elimination method.
Answer:
(4.5,0)
Step-by-step explanation:
The point that makes this system of equations true is (4.5,0.
When x=4.5, and y=0 both of these equations equal each other.
You can find this point either by using your calculator, or graphing the equations and seeing where they intersect.
FIND THE SUM OF GEOMETRIC SERIES 3+2.55+2.1675+1.842375+...
The sum of geometric series 3+2.55+2.1675+1.842375+...
Hence, the correct option is B.
To find the sum of a geometric series, we can use the formula
S = a(1 - [tex]r^{n}[/tex]) / (1 - r)
Where
S = sum of the geometric series
a = first term of the series
r = common ratio of the series
n = number of terms in the series
In this case, we can see that the first term is 3 and the common ratio is 2.55/3 = 0.85. We can also see that there are an infinite number of terms in the series.
Using the formula for an infinite geometric series, we can simplify the formula to
S = a / (1 - r)
By putting these values we have, we get
S = 3 / (1 - 0.85) = 20.
Hence, the sum of the geometric series is 20.
Hence, the correct option is B.
The question is incomplete and the complete question is '' FIND THE SUM OF GEOMETRIC SERIES 3+2.55+2.1675+1.842375+...
A. 8
B. 20
C. 12
D. 4 ''
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If a = 3 and b = -2, what is the value of a2 + 3ab ”“ b2?
Select the correct answer. Which statement best describes the zeros of the function h(x) = (x + 9)(x2 − 10x + 25)? A. The function has three complex zeros. B. The function has three distinct real zeros. C. The function has one real zero and two complex zeros. D. The function has two distinct real zeros.
All the zeroes of the function are,
x = - 9, 5, 5
Given that;
The function is,
h (x) = (x + 9) (x² - 10x + 25)
Now, We can find all the zeroes of the function as;
h (x) = (x + 9) (x² - 10x + 25)
h (x) = (x + 9) (x² - 5x - 5x + 25)
h (x) = (x + 9) ((x - 5) - 5 (x - 5))
h (x) = (x + 9) (x - 5) (x - 5)
Thus, All the zeroes of the function are,
x + 9 = 0
x = - 9
(x - 5)² = 0
x = 5, 5
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Find the accumulated value of an investment of $15,000 for 5 years at an interest rate of 6.5% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously. Round answers to the nearest cent.
Answer:
a. Compounded semiannually: $19,216.67
b. Compounded quarterly: $19,338.56
c. Compounded monthly: $19,416.32
d. Compounded continuously: $19,451.08
Step-by-step explanation:
The formula for the accumulated value (A) of an investment with an initial principal (P), an interest rate (r) and a compounding period (n) for a number of years (t) is:
A = P(1 + r/n)^(n*t)
Using this formula, we can calculate the accumulated value for each of the compounding periods given:
a. Compounded semiannually:
A = $15,000(1 + 0.065/2)^(2*5) = $19,216.67
b. Compounded quarterly:
A = $15,000(1 + 0.065/4)^(4*5) = $19,338.56
c. Compounded monthly:
A = $15,000(1 + 0.065/12)^(12*5) = $19,416.32
d. Compounded continuously:
A = $15,000e^(0.0655) = $19,451.08
Therefore, the accumulated value of the investment varies depending on the compounding period, with more frequent compounding resulting in a higher accumulated value.
different equation dy÷dx+ytanx=secx
Answer:
First, we rearrange the equation to isolate the y-term on one side:
dy/dx + ytanx = secx
Then, we multiply both sides by the integrating factor, which is e^(∫tanx dx) = e^(ln|secx|) = |secx|: | secx| dy/dx + ysecx tanx = 1
Next, we can write this as the derivative of a product using the product rule: d/dx (y |secx|) = 1
Integrating both sides with respect to x, we get: y |secx| = x + C
where C is the constant of integration. Solving for y, we have:
y = (x + C)/|secx|
Note that there is a singularity at x = (2n + 1)π/2, where the denominator |secx| is zero. At these points, the solution is not defined
Which graph shows the solution to the system of linear equations? y equals negative one third times x plus 1 y = −2x − 3 a coordinate grid with one line that passes through the points 0 comma 1 and 4 comma 0 and another line that passes through the points 0 comma negative 1 and 1 comma negative 3 a coordinate grid with one line that passes through the points 0 comma 1 and 3 comma 0 and another line that passes through the points 0 comma negative 3 and 1 comma negative 5 a coordinate grid with one line that passes through the points 0 comma 1 and 3 comma negative 1 and another line that passes through the points 0 comma negative 1 and 2 comma negative 5 a coordinate grid with one line that passes through the points 0 comma 1 and 4 comma negative 2 and another line that passes through the points 0 comma negative 2 and 1 comma negative 5
The graph of the system of linear equations can be seen in the image at the end.
Which is the graph of the system of linear equations?Here we have the following system of equations:
y = -(1/3)x + 1
y = -2x - 3
To graph the system of equations, we just need to graph both of these lines in the same coordinate axis. The point where the lines intercept are the solutions of the system.
In the image at the end we can see the graph of the system of equations, there we also can see that the solution is the point (-2.4, 1.8)
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Calculate the circumference, given the following diameters. Use T[= 22 7 I. 30 cm 2. 15 m 3. 40 ft 4. 20 cm 5. 8 m 6. 12 cm 7. 22 in. 8. 34 m 9. 10m 10. 34 cm
Required circumference of the given diameters are 94.29 cm, 471.43 m, 377.14 ft, 62.86 cm, 251.43 m, 37.71 cm, 69.71 in, 1068.57 m, 314.29 m, 106.86 cm respectively.
What is circumference of circle?
The circumference of a circle is the distance around the edge or boundary of the circle. It is a fundamental geometric property of a circle and is defined as the product of the circle's diameter and pi (π), which is a mathematical constant approximately equal to 3.14.
The formula to calculate the circumference of a circle is C = 2πr or C = πd
where C represents the circumference, r represents the radius, and d represents the diameter of the circle,
1. Diameter = 30 cm
Circumference = π x Diameter
= 22/7 x 30
= 94.29 cm
2. Diameter = 15 m
Circumference = π x Diameter
= 22/7 x 15 x 100
= 471.43 m
3. Diameter = 40 ft
Circumference = π x Diameter
= 22/7 x 40 x 12
= 377.14 ft
4. Diameter = 20 cm
Circumference = π x Diameter
= 22/7 x 20
= 62.86 cm
5. Diameter = 8 m
Circumference = π x Diameter
= 22/7 x 8 x 100
= 251.43 m
6. Diameter = 12 cm
Circumference = π x Diameter
= 22/7 x 12
= 37.71 cm
7. Diameter = 22 in
Circumference = π x Diameter
= 22/7 x 22
= 69.71 in
8. Diameter = 34 m
Circumference = π x Diameter
= 22/7 x 34 x 100
= 1068.57 m
9. Diameter = 10 m
Circumference = π x Diameter
= 22/7 x 10 x 100
= 314.29 m
10. Diameter = 34 cm
Circumference = π x Diameter
= 22/7 x 34
= 106.86 cm
Therefore, the circumference of the given diameters are 94.29 cm, 471.43 m, 377.14 ft, 62.86 cm, 251.43 m, 37.71 cm, 69.71 in, 1068.57 m, 314.29 m, 106.86 cm.
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It's illegal to bet on political races in the United States, but overseas betting on American elections is common. As of this writing, Hillary Clinton is the favorite to win the presidency in 2016, with odds listed as 8/11. Explain what that means, including relating it to probability.
Answer:
57.89%
Step-by-step explanation:
Implied probability = (1 / (odds + 1)) x 100%
Using the odds of 8/11 for Hillary Clinton:
Implied probability = (1 / (8/11 + 1)) x 100% = 57.89%
There is a 57.89% chance that Hillary Clinton will win the presidency.
One can of soda has a capacity of 355mL. How may liters of soda does 12.
12 cans of soda hold 4.26 liters of soda.
The process for determining the total volume of soda would be the same using the specific capacity and number of cans.
We need to determine how many milliliters of soda 12 cans hold. We can do this by multiplying the capacity of one can (355 mL) by the number of cans (12):
355 mL/can x 12 cans = 4,260 mL
Now we need to convert this amount to liters. We know that 1 liter is equal to 1,000 milliliters, so we can divide the total volume in milliliters by 1,000:
4,260 mL ÷ 1,000 mL/L = 4.26 L
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A two-day environmental clean up started at 9AM on the first day. The number of workers fluctuated as shown in the following figure.
graph of number of workers vs. time; between t=0 hours and t=8 hours the graph behaves as a sine curve with centerline 30, amplitude 10 and period 16; between t=12 and t=20, the graph is constant at 20. Then, at t=20 the graph becomes a sine curve again, starting at its minimum and completing one period between t=20 and t=36. The graph is then constant at 20 until t=44, before increasing as a sine again until t=48.
Suppose that the workers were paid 15 dollars per hour for work during the time period 9 am to 5 pm and were paid 22.5 dollars per hour for work during the rest of the day. What would the total personnel costs of the clean up have been under these conditions?
total cost =
dollars
The total personnel costs of the cleanup would be $8,625.
How to determine?Between 9AM and 5PM, a total of 8 hours, the workers are present during the interval t=0 to t=8. During this interval, the number of workers is given by:
f(t) = 30 + 10sin((2π/16)t)
We can calculate the number of workers for each hour using this formula:
f(0) = 30 + 10sin(0) = 30
f(1) = 30 + 10sin(π/8) ≈ 38.66
f(2) = 30 + 10sin(π/4) = 40
f(3) = 30 + 10sin(3π/8) ≈ 38.66
f(4) = 30 + 10sin(π/2) = 40
f(5) = 30 + 10sin(5π/8) ≈ 38.66
f(6) = 30 + 10sin(3π/4) = 30
f(7) = 30 + 10sin(7π/8) ≈ 21.34
f(8) = 30 + 10sin(π) = 20
So, during the first 8 hours, the number of workers fluctuates between 20 and 40.
Between 5PM and 9AM the next day, a total of 16 hours, the workers are present during the intervals t=8 to t=12, t=20 to t=36, and t=44 to t=48. During these intervals, the number of workers is given by:
Between t=8 and t=12, the number of workers is decreasing from 40 to 20. We can approximate this by a linear function:
f(t) = 40 - 5t
Between t=20 and t=36, the number of workers is constant at 20.
Between t=44 and t=48, the number of workers is increasing from 20 to 40. We can use the same linear function as before, but with t shifted by 44:
f(t) = 40 - 5(t-44)
We can now calculate the total personnel costs by multiplying the number of workers at each time interval by the appropriate hourly rate:
From 9AM to 5PM (8 hours): 30 workers on average, at a rate of $15 per hour:
8 × 30 × 15 = $3,600
From 5PM to 9AM the next day (16 hours):
From t=8 to t=12 (4 hours): average of 30 workers, at a rate of $22.5 per hour:
4 × 30 × 22.5 = $2,700
From t=20 to t=36 (16 hours): constant 20 workers, at a rate of $22.5 per hour:
16 × 20 × 22.5 = $7,200
From t=44 to t=48 (4 hours): average of 30 workers, at a rate of $22.5 per hour:
4 × 30 × 22.5 = $2,700
The total cost is the sum of the costs for each time interval:
$3,600 + $2,700 + $7,200 + $2,700 = $16,200
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Solve the right triangle using the information given. Round answers to two decimal places, if necessary. a = 4, b = 6; Find c, α, and β.
Step-by-step explanation:
first, remember Pythagoras :
c² = a² + b²
c is the Hypotenuse (the side opposite of the 90° angle).
a and b are the legs.
so, in our case
c² = 4² + 6² = 16 + 36 = 52
c = sqrt(52) = 7.211102551... ≈ 7.21
and then remember the law of sine :
a/sin(A) = b/sin(B) = c/sin(C)
a, b, c are the sides, A, B, C are the corresponding opposite angles (here also called alpha, beta. gamma would be the 90° angle).
so,
4/sin(alpha) = c/sin(90) = c
4 = c×sin(alpha)
sin(alpha) = 4/c = 4/7.211102551... = 0.554700196...
alpha = 33.69006753...° ≈ 33.69°
we don't need to do that much calculation for the third angle beta, as we know the sum of all angles in a triangle is always 180°.
so,
180 = alpha + beta + 90 = 33.69 + beta + 90
90 = 33.69 + beta
56.30993247... = beta ≈ 56.31°
there are n people seated at a round table.
Step-by-step explanation:
circular permutations of n objects =(n-1)!
Given the demand function D(p) = 400-4p².
Find the Elasticity of Demand at a price of $8.
At this price, we would say the demand is:
-Inelastic
-Elastic
-Unitary
Based on this, to increase revenue we should:
-Keep Prices Unchanged
-Lower Prices
-Raise Prices
The Elasticity of Demand at a price of $8 is 0.056.
Given the demand function,
D(p) = 400 - 4p²
The formula to find the elasticity of demand is,
e = p/D
Here p is the price and D is the quantity demanded.
When p = $8,
D = 400 - (4 × 8²) = 144
Elasticity of demand = 8 / 144 = 0.056
Hence the elasticity of demand is 0.056.
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A company needs to deliver product to each of their 5 stores around the Dallas, TX area. Driving distances between the stores are shown below. Find a route for the driver to follow, returning to the distribution center in Fort Worth:
a. Using Nearest Neighbor starting in Fort Worth
b. Using Repeated Nearest Neighbor
Using Nearest Neighbor starting in Fort Worth, the route will have a total path length of 183.
How to calculate the valueUsing Nearest Neighbor starting in
Foethliceth:
We take the cheapest way to travel,
19 38 41 D42 F
d+ F A 43 M
Total path length = 19 +43 +38 +41 +42 = 183
Using Repeated Nearest. Neighbor:
The repetitive nearest algorithm says. to tey each veetex as starting point I then choose the best answer.
Feam Fort Worth:
F 19 A 43 M 38 p 4D 42 F
Total path length. = 183
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Which of the following equations represents a line that is perpendicular to the line that passes through the points below?
(-3,10) (3,8)
The equation of the line that is perpendicular to the line passing through the points (-3, 10) and (3, 8) is y = 3x + 9.
To find a line that is perpendicular to another line, we need to use the fact that the slopes of the two lines are negative reciprocals of each other.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the slope formula:
slope = (y₂ - y₁)/(x₂ - x₁)
Therefore, the slope of the line passing through the points (-3, 10) and (3, 8) can be found as follows:
slope = (8 - 10)/(3 - (-3)) = -1/3
Now, to find the equation of a line perpendicular to this line, we need to find the negative reciprocal of the slope -1/3, which is 3.
Therefore, the slope of the line perpendicular to the given line is 3.
Now, we can use the point-slope form of a linear equation to find the equation of the perpendicular line. The point-slope form of a linear equation is:
y - y₁ = m(x - x₁)
where m is the slope of the line, and (x₁, y₁) is any point on the line.
We know the slope of the perpendicular line is 3, and we can choose any point on the line. Let's choose the midpoint of the given points (-3, 10) and (3, 8), which is
=> ((-3+3)/2, (10+8)/2) = (0, 9).
So, using the point-slope form of the equation with m = 3 and (x1, y1) = (0, 9), we get:
y - 9 = 3(x - 0)
Simplifying this equation, we get:
y = 3x + 9
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Each of P, Q and R had a certain amount of money. P gave some money to Q and R in such a way that the amounts with them were doubled. Now Q gave some money to P and R such that the amounts with them were doubled. Finally R also gave some money with him to P and Q such that the amounts with them were doubled. At this stage, each of them had 240.What was the initial amount with P?
The initial amount of money with P was 2Q0 = 2 * 3Q0 = 6Q0.
What is amount?Amount is the sum of money or quantity of something. It is usually measured in numerical values such as dollars, pounds, or euros. It is used to describe the total of something, such as how much money is in a bank account or the total cost of a purchase. Amount can also be used to describe a quantity of something such as the amount of time spent on a project or the amount of people present at an event.
Let us denote the initial amount of money with P, Q and R as P0, Q0 and R0 respectively. We know that P gave some money to Q and R such that their amounts were doubled, which implies that P0 = 2(Q0 + R0). Similarly, Q gave some money to P and R such that their amounts were doubled, which implies that Q0 = 2(P0 + R0). Finally, R gave some money to P and Q such that their amounts were doubled, which implies that R0 = 2(P0 + Q0). Substituting the value of P0 from the first equation in the third equation, we get:
2(Q0 + R0) = 2(P0 + Q0)
2Q0 + 2R0 = 2P0 + 2Q0
R0 = P0
Substituting the value of R0 in the first equation, we get:
P0 = 2(Q0 + R0)
P0 = 2(Q0 + P0)
P0 = 2Q0 + P0
P0 = 2Q0
Finally, substituting the value of P0 in the second equation, we get:
Q0 = 2(P0 + R0)
Q0 = 2(2Q0 + R0)
Q0 = 4Q0 + R0
R0 = 3Q0
Therefore, the initial amount of money with P, Q and R can be calculated as follows:
P0 = 2Q0
Q0 = 3Q0
R0 = 3Q0
Therefore, the initial amount of money with P was 2Q0 = 2 * 3Q0 = 6Q0.
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