Answer: The two consecutive integers are 36 and 37.
Step-by-step explanation: Let's call the smaller integer "x".
According to the problem, the larger integer is the next consecutive integer, so we can call it "x + 1".
The problem tells us that the sum of the larger integer and 23 less than the smaller integer is 50. So we can set up the equation:
(x + 1) + (x - 23) = 50
Simplifying the equation, we get:
2x - 22 = 50
Adding 22 to both sides, we get:
2x = 72
Dividing both sides by 2, we get:
x = 36
So the smaller integer is 36. The next consecutive integer is 37.
Therefore, the two consecutive integers are 36 and 37.
statistics show that there is a weak relationship between education and income. please select the best answer from the choices provided t
There is a weak relationship between education and income. Education not only raises the level of income is TRUE.
Effect of educationEducation is the process of facilitating learning, or gaining of knowledge, skills and personal development.
Education effect on also your future. It makes bright future. if you are study in continuation it develops inner skills of human beings.
The income effect describes that the income effect evaluates consumer spending habits based on a change in their income. This is reflected in microeconomics via an upward shift in the downward-sloping demand curve.
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The given question is incomplete, complete question is:
statistics show that there is a weak relationship between education and income. please select the best answer from the choices provided
1. True
2. False
Which of the following represents the solution set to the system of inequalities?
y-5≥-(x + 1)
y ≥ 3x-2
The graph of the solution set for the system of inequalities is on the image at the end.
How to find the solution set?Here we have a system of inequalities, it can be written as:
y - 5 ≥ -(x + 1)
y ≥ 3x - 2
To graph this, we can write both inequalities in linear form:
y ≥ -(x + 1) + 5
y ≥ 3x - 2
We can see that in both cases we have the symbol "≥", so we need to graph both of these lines with solid lines, and then shade the region above of these lines, then the graph of the system is the one you can see at the end.
The region where the two shades intercept (dark blue one) is the set of solutions.
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prove or disprove the following: (a) if f(x) is o(g(x)) then 2f(x) is o(2g(x) ). (b) if f(x) is o(g(x)) then (f(x))2 is o (g(x))2
(a) To prove or disprove the statement "if f(x) is o(g(x)), then 2f(x) is o(2g(x))", we can use the definition of little-o notation.
Recall that f(x) is o(g(x)) if and only if, for any positive constant ε, there exists a positive constant M such that |f(x)| ≤ ε|g(x)| for all x > M.
Using this definition, let's consider the statement in question.
Suppose f(x) is o(g(x)), which means that |f(x)| ≤ ε|g(x)| for some positive constant ε and all x > M.
Now let's consider 2f(x). We can write this as 2f(x) = 2 * f(x), and since f(x) is o(g(x)), we know that |f(x)| ≤ ε|g(x)|. Therefore,
|2f(x)| = |2 * f(x)| ≤ 2 * |f(x)| ≤ 2ε|g(x)|
So we can see that |2f(x)| ≤ 2ε|g(x)|, which means that 2f(x) is also o(g(x)). Therefore, the statement is true.
(b) Now let's consider the statement "if f(x) is o(g(x)), then (f(x))2 is o(g(x))2". Again, we can use the definition of little-o notation.
Suppose f(x) is o(g(x)), which means that |f(x)| ≤ ε|g(x)| for some positive constant ε and all x > M.
Now let's consider (f(x))2. We can write this as (f(x))2 = f(x) * f(x), and since f(x) is o(g(x)), we know that |f(x)| ≤ ε|g(x)|. Therefore,
|(f(x))2| = |f(x) * f(x)| = |f(x)| * |f(x)| ≤ ε|g(x)| * ε|g(x)| = ε2|g(x)|2
So we can see that |(f(x))2| ≤ ε2|g(x)|2, which means that (f(x))2 is also o(g(x))2. Therefore, the statement is true.
In conclusion, we have proven both (a) and (b) to be true.
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In a group of 50 executives, 27 have a type A personality. If one executive is selected at random from this group, what is the probability that this executive has a type A personality?
The probability of selecting an executive with a type A personality can be calculated by dividing the number of executives with a type A personality by the total number of executives in the group. the probability of selecting an executive with a type A personality from this group is 0.54, or 54%.
P(type A) = number of executives with type A personality / total number of executives
P(type A) = 27 / 50
P(type A) = 0.54 or 54%
Therefore, the probability of selecting an executive with a type A personality from this group is 54%. Probability = (Number of desired outcomes) / (Total number of possible outcomes)
Probability = 27 / 50
Now, we can simplify the fraction:
Probability = 0.54
So, the probability of selecting an executive with a type A personality from this group is 0.54, or 54%.
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find the mass of the ball of radius 3 centered at the origin with a density f(rho,φ,θ)=5e−rho3.
The resulting value will give you the mass of the ball of radius 3 centered at the origin with the given density function.
To find the mass of the ball with a radius of 3 centered at the origin, we need to integrate the density function over the volume of the ball.
The density function is given as f(ρ, φ, θ) = 5e^(-ρ^3), where ρ represents the radial distance, φ represents the azimuthal angle, and θ represents the polar angle.
In spherical coordinates, the volume element is given by ρ^2 sin(φ) dρ dφ dθ.
To integrate over the ball, we need to set the limits of integration as follows:
ρ: 0 to 3
φ: 0 to π
θ: 0 to 2π
The mass of the ball can be calculated using the integral:
Mass = ∫∫∫ f(ρ, φ, θ) ρ^2 sin(φ) dρ dφ dθ
Mass = ∫[0 to 2π] ∫[0 to π] ∫[0 to 3] 5e^(-ρ^3) ρ^2 sin(φ) dρ dφ dθ
This integral needs to be evaluated numerically using appropriate software or numerical techniques.
The resulting value will give you the mass of the ball of radius 3 centered at the origin with the given density function.
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The radius r of a circle is increasing at a rate of 5 centimeters per minute. (a) Find the rate of change of the area when r = 12 centimeters. (b) Find the rate of change of the area when r = 32 centimeters.
(a) To find the rate of change of the area when r = 12 centimeters, we need to use the formula for the area of a circle, which is A = πr^2. We can then take the derivative of both sides with respect to time t, which gives us:
dA/dt = 2πr(dr/dt)
Substituting in the given values, we get:
dA/dt = 2π(12)(5) = 120π
Therefore, the rate of change of the area when r = 12 centimeters is 120π square centimeters per minute.
(b) To find the rate of change of the area when r = 32 centimeters, we can use the same formula and approach as in part (a), but with r = 32:
dA/dt = 2πr(dr/dt)
dA/dt = 2π(32)(5) = 320π
Therefore, the rate of change of the area when r = 32 centimeters is 320π square centimeters per minute.
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It’s almost the end of highschool and I need this done by tomorrow. I only need the right side done (the circled even numbers) please help and I’ll grant you brainly.
Answer:
Step-by-step explanation:
What is the greatest common factor of 66,94,and87?
The greatest common factor of 66, 94, and 87 is 1.
To find the greatest common factor (GCF) of these three numbers, we need to find the common factors that they share. First, we can factor each number into its prime factors:
66 = 2 * 3 * 11
94 = 2 * 47
87 = 3 * 29
The only factor that all three numbers share is the number 1. Therefore, the GCF of 66, 94, and 87 is 1.
It's important to note that the GCF is the largest factor that two or more numbers have in common. In this case, the three numbers do not have any factors larger than 1 in common, so the GCF is 1.
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A seal swims on a bearing of 055 degrees for 4km. How far north is the seal from its starting point?
Using the concept of bearing and displacement, the seal is 3.276km from it's starting point.
How far north is the seal from its starting point?To determine how far north the seal is from its starting point, we need to find the northward component of its displacement.
Given that the seal swims on a bearing of 055 degrees, we can consider this as the direction with respect to true north. To find the northward component, we need to calculate the sine of the angle.
The northward component can be found using the formula:
Northward component = Displacement * sin(Bearing)
In this case, the displacement is 4 km, and the bearing is 055 degrees.
Northward component = 4 km * sin(55 degrees)
Using a calculator, we find that sin(55 degrees) ≈ 0.8192.
Northward component ≈ 4 km * 0.8192
Northward component ≈ 3.276 km
Therefore, the seal is approximately 3.2768 km north of its starting point.
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Please help fast I’ll mark brainly
Answer:
[C] 30
Step-by-step explanation:
------------------------------------------------------------------------------------------------------------
Preferred Food
Pizza Burger Other Total
Girl 5 6 4
Gender Boy 7 5 3
Total
------------------------------------------------------------------------------------------------------------
We first have to add them up to find the total.------------------------------------------------------------------------------------------------------------
Preferred Food
Pizza Burger Other Total
Girl 5 6 4 15
Gender Boy 7 5 3 15
Total 12 11 7 30
------------------------------------------------------------------------------------------------------------
Based on what we record in the table. We can see that;
30 people took part in the survey.
RevyBreeze
Answer:
30 people took part in the survey.
find and classify the critical points of f(x, y) = x 2 − 2x y 2 − 4y 5.
The critical points of the function f(x, y) = x^2 - 2xy^2 - 4y^5 can be found by setting the partial derivatives with respect to x and y equal to zero and solving for x and y. Taking the partial derivative with respect to x, we get 2x - 2y^2 = 0. Taking the partial derivative with respect to y, we get -4xy - 20y^4 = 0. Solving these equations simultaneously, we get two critical points: (0, 0) and (2, -1/2).
To classify the critical points, we need to use the second partial derivative test. Taking the second partial derivative with respect to x, we get 2. Taking the second partial derivative with respect to y, we get -8xy - 100y^3. At (0, 0), the second partial derivative with respect to y is zero, so we cannot use the second partial derivative test. At (2, -1/2), the second partial derivative with respect to y is negative, so the critical point is a local maximum. Therefore, the critical points of f(x, y) = x^2 - 2xy^2 - 4y^5 are (0, 0) and (2, -1/2), with the critical point (2, -1/2) being a local maximum.
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Find the surface area PLSSS TYY
The surface area of the prisms found from the sum of the areas of the the individual surfaces of the prism are;
23) 60 cm²
24) 124 ft²
25) 48 mi²
26) 132 yd²
27)50.2 mi²
28) 72 km²
29) 112 yd²
30) 154 mi²
31) 150 mi²
32) 288 km²
33) 156 in²
34) 96.2 in²
What is the surface area of a solid?The surface area of a prism is the sum of the surface areas of the faces on the prism surface.
23) Surface area = 3.5 × 2 × (2 + 6)/2 + 2 × 4 × 2 + 2 × 6 + 2 × 2 = 60
The surface area = 60 cm²
24) Surface area = 3.5 × 2 × (2 + 6)/2 + 2 × 4 × 6 + 6 × 6 + 2 × 6 = 124
The surface area of the prism = 124 ft²
25) Surface area = 2 × (1/2) × 4 × 3 + 3 × 5 + 3 × 3 + 4 × 3 = 48
The surface area = 48 mi²
26) Surface area = 2 × (8 × 3 + 3 × 6 + 3 × 8) = 132
The surface area = 132 yd²
27) Surface area = 1.7 × 2 × (2 + 4)/2 + 4 × 4 + 3 × 2 × 4 = 50.2
The surface area = 50.2 mi²
28) Surface area = 2 × (6 × 3 + 3 × 2 + 2 × 6) = 72
The surface area = 72 km²
29) Surface area = 4 × 4 × 4 + 2 × 6 × 4 = 112
The surface area = 112 yd²
30) Surface area = 4 × 2 × 7 + 2 × 7 × 7 = 154
The surface area = 154 mi²
31) Surface area = 6 × 5 × 5 = 150
The surface area = 150 mi²
32) Surface area = 4 × 5 × 8 + 2 × 8 × 8 = 288
The surface area = 288 km²
33) Surface area = 2.8 × 2 × (6 + 4)/2 + 2 × 8 × 3 + 4 × 8 + 8 × 6 = 156
The surface area = 156 in²
34) Surface area = 2.6 × 2 × (2 + 5)/2 + 2 × 6 × 3 + 2 × 6 + 5 × 6 = 96.2
The surface area = 96.2 mi²
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Find the solution of the following initial value problem. (x)-8x3 + 10x 3 v(8)-48, x0 The solution of the initial value problem is v(x)=
To solve the initial value problem, we first need to find the general solution of the given differential equation, which is a second-order linear differential equation with constant coefficients. The characteristic equation is r^2 - 8x^3 = 0, which has roots r = ±(2x)^(3/2). Therefore, the general solution of the differential equation is v(x) = c1(2x)^(3/2) + c2(-2x)^(3/2).
To find the values of the constants c1 and c2, we use the initial conditions v(8) = -48 and x0 = 2. Substituting x = 8 and v(x) = -48 into the general solution, we get -48 = c1(16)^(3/2) + c2(-16)^(3/2). Simplifying, we get c1 - c2 = -3.
To find c1 and c2, we differentiate the general solution with respect to x and substitute x = 2 and v'(2) = -8 into the resulting expression. We get v'(x) = 3x^(1/2)c1 - 3x^(1/2)c2, and substituting x = 2 and v'(2) = -8, we get -8 = 6c1 - 6c2. Simplifying, we get c1 + c2 = 4/3.
Solving the system of equations c1 - c2 = -3 and c1 + c2 = 4/3, we get c1 = 5/6 and c2 = -17/6. Therefore, the solution of the initial value problem is v(x) = (5/6)(2x)^(3/2) - (17/6)(-2x)^(3/2).
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For x in the isosceles triangle below.
Answer: x=6
Step-by-step explanation:
Since isosceles triangles have two sides of equal length, the side of 24 in is equal to the side of 6x-12. So, with the equation 24=6x-12, you should find x to be 6.
The first three terms of a sequence are given. Round to the nearest thousandth (if necessary). 3 , 9 , 27 , Find the 7th term
The 7th term of the sequence is 729. To find the 7th term, we notice that the sequence is formed by multiplying each term by 3. Therefore, the 4th term is 327=81, the 5th term is 381=243, the 6th term is 3243=729, and the 7th term is 3729=2187.
In this problem, we are given the first three terms of a sequence and asked to find the 7th term. A sequence is a list of numbers in a specific order, where each number is called a term. To find the next term in a sequence, we need to identify the pattern or rule that generates the sequence.
In this case, we notice that each term is obtained by multiplying the previous term by 3. That is, if a_1=3, a_2=9, a_3=27, then a_4=3a_3=81, a_5=3a_4=243, a_6=3a_5=729, and a_7=3a_6=2187.
Therefore, the 7th term is 2187. It is important to round to the nearest thousandth only when we are dealing with decimal numbers. Since the terms of this sequence are integers, we do not need to round.
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Patricio measured the length of the line at Muffles' truffle shop on Tuesday and found
that it was 11 yards long. How many inches long was the line on Tuesday?
The line at Muffles' truffle shop on Tuesday was 396 inches long.
Given that the Tuesday, Patricio counted the line at Muffles' truffle business and discovered that it was 11 yards long.
To convert yards to inches, we need to know that 1 yard is equal to 36 inches.
Therefore, to find the length of the line in inches, we can multiply the length in yards by the conversion factor:
Length in inches = Length in yards × Conversion factor
Length in inches = 11 yards × 36 inches/yard
Length in inches = 396 inches
So, the line at Muffles' truffle shop on Tuesday was 396 inches long.
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f(x)=x^3+1000 i need help
Answer:x=-10
Step-by-step explanation:
i used a app i could be wrong :/
Help FAST PLEASE hich of the following ordered pairs represents the point plotted and labeled A?
The x-axis starts at negative 4, with tick marks every unit up to positive 4. The y-axis starts at negative 4, with tick marks every unit up to positive 4. Point D is two units left of the origin. Point A is one unit left and two units up from the origin. Point C is two units down from the origin. Point F is one unit right and two units up from the origin. Point B is three units right and one unit up from the origin. Point E is three units right and one unit down from the origin.
(−1, 2)
(−2, 0)
(0, −2)
(1, 2)
Answer:
(a) (-1, 2)
Step-by-step explanation:
You want the ordered pair that represents the coordinates of a point 1 unit left and 2 units up from the origin.
CoordinatesThe (x, y) coordinates of a point on the Cartesian plane represent (units right, units up) relative to the origin. When the direction is left or down, the sign of the corresponding coordinate is made negative.
(1 left, 2 up) ⇒ (-1, 2), matching choice A
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What is the unknown fraction in the equation?
Answer:
47/100
Step-by-step explanation:
2/10*10/10= 20/100
67/100-20/100= 47/100.
I need help with this?
The graph of the system of inequalities is attached to the solution.
Given is a system of inequalities, y > -x/3+5 and y ≥ 3,
So, we will simply find the coordinates of both the inequalities, and plot them,
We know that the solution of a system of inequalities is all the part which is common in both the inequalities.
So, here the first inequality,
y > -x/3+5
Finding the coordinates,
y = -x/3+5
Put x = 0
y = 5
(0, 5)
Put y = 0,
x = 15
(15, 0)
Therefore, the inequality will pass from these two lines, and since the sing is > so the shaded part will be above the line and the line will be dotted.
And y ≥ 3,
In this inequality the graph will simply pass by y = 3 and since the sing is ≥ so the shaded part will be above the line and the line will solid line.
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pls help! Find the coordinates of the midpoint of line segment CN with endpoints C(-7, -6) and N(-3,5).
A. (-6,-1)
B. (-4,0)
C. (-10,-1)
D. (-5,-0.5)
The coordinates of the midpoint of the line segment CN with endpoints C(-7, -6) and N(-3,5) are (-5, -0.5). So, the correct answer is:
D. (-5,-0.5)
if c is a circle of radius 5 centered at the point ( 3, -5 ), then evaluate ∮ c ( 5 y − e sin ( x ) ) d x ( 8 x − sin ( y 3 y ) ) d y ∮c(5y-esin(x))dx (8x-sin(y3 y))dy . value = π ⋅
To evaluate this line integral, we will use Green's theorem, which states that for a closed curve C, oriented counterclockwise, and a region R bounded by C, the line integral of the vector field F along C is equal to the double integral of the curl of F over R:
∮c F ⋅ dr = ∬R (curl F) ⋅ dA
In this case, our vector field F is:
F = (5y - e×sin(x)) i + (8x - sin(y³)) j
And the curl of F is:
curl F = ∂(8x - sin(y³))/∂x - ∂(5y - e*sin(x))/∂y = 5e×cos(x) + 3y²×cos(y³)
Now, we need to find the region R bounded by C, which is the circle of radius 5 centered at (3,-5). This is simply the disk with center (3,-5) and radius 5.
Using polar coordinates, we can write the double integral as:
∬R (curl F) ⋅ dA = ∫θ=0..2π ∫r=0..5 (5e×cos(θ) + 3r²×cos(r³×sin(θ)³)) r dr dθ
Evaluating this integral, we get:
∬R (curl F) ⋅ dA = π×(625e + 375)
Therefore, by Green's theorem:
∮c F ⋅ dr = π×(625e + 375)
Substituting F and evaluating the integral, we get:
∮c (5y - e×sin(x)) dx + (8x - sin(y³)) dy = π×(625e + 375)
This is the value of the line integral.
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Other things being equal, an increase in relative U.S. Real GDP growth compared to the rest of the world is likely to cause an increase in the: 1. Quantity of exports. 2. Quantity of imports. 3. Demand for U.S. dollars. 4. International value of the U.S. dollar.
An increase in relative U.S. Real GDP growth compared to the rest of the world is likely to cause an increase in the demand for U.S. dollars and the international value of the U.S. dollar. This is because investors and traders will see the U.S. as a more attractive place to invest, which will increase the demand for U.S. currency. As a result, the international value of the U.S. dollar will increase, making exports more expensive and imports cheaper. This means that the quantity of exports will decrease while the quantity of imports will increase.
The U.S. dollar is a reserve currency, which means that it is widely used in international transactions and is held by foreign governments and central banks. An increase in relative U.S. Real GDP growth makes the U.S. economy more attractive to investors and traders, which increases the demand for U.S. dollars. This higher demand leads to an increase in the international value of the U.S. dollar, which in turn makes exports more expensive and imports cheaper.
In conclusion, an increase in relative U.S. Real GDP growth compared to the rest of the world is likely to lead to an increase in the demand for U.S. dollars and the international value of the U.S. dollar. This will result in a decrease in the quantity of exports and an increase in the quantity of imports.
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50 Points Math Image
Determine the degree of overlap (high, moderate, low, or none).
Answer:
its none
Step-by-step explanation:
can i get brainliest please
The total length of a road trip was 19.2 hours. If highway signs are posted every 0.8 hours, including one at the end of the road trip, how many highway signs will there be on the road trip?
According to the Question predicts that there are 24 highway signs throughout the journey.
How are lengths determined?Length can indeed be measured in a variety of ways, including handspan, foot span, meters, inches, and millimeters. There are two categories of length measurement units: There are conventional units for measuring length and nonstandard ones.
Division will help us resolve this issue. We may calculate of highway signs by dividing the overall driving time, 19.2 hours, by the 0.8-hour includes an aspect interval.
[tex]\sf \dfrac{19.2 \ hours}{0.8 \ hours/sign} =24[/tex]
Therefore, there will be 24 highway signs on the road trip.
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what is the cost of installing a fence around a rectangular shaped lot if the cost of the fence is $3.25 per linear foot and the lot is 80 ft. wide and 120 ft. deep?
The cost of installing a fence around an 80 ft. wide and 120 ft. deep rectangular lot, with the fence priced at $3.25 per linear foot, will be $1,300.
To determine the cost of installing a fence around a rectangular lot, you need to calculate the total length of the fence required and then multiply that by the cost per linear foot. The given dimensions of the lot are 80 feet wide and 120 feet deep.
First, calculate the perimeter of the rectangular lot. The perimeter of a rectangle is given by the formula P = 2L + 2W, where L is the length (or depth) and W is the width. In this case, the perimeter is P = 2(120) + 2(80) = 240 + 160 = 400 feet.
Next, multiply the total length of the fence by the cost per linear foot, which is $3.25. So, the cost of installing the fence is 400 feet × $3.25 per linear foot = $1,300.
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PIZ HEIP ITS GEOMTREY AND I DONT GET IT
The number of cups of cat food the feeder can hold is 12 cups.
What is the total volume of the feeder?The total volume of the feeder is calculated as follows;
total volume = volume of cylinder + volume of cone
The volume of the cylindrical part of the feeder is calculated as;
V = πr²h
where;
r is the radiush is the heightV = π(2.5 in)²(7.5 in)
V = 147.3 in³
The volume of the cone part of the feeder is calculated as;
V = ¹/₃ πr²h
V = ¹/₃ π(2.5 in)²( 4 in )
V = 26.2 in³
Total volume = 147.3 in³ + 26.2 in³
Total volume = 173.5 in³
The number of cups of cat food the feeder can hold is calculated as follows;
n = (173.5 in³) / 14.4 in³
n = 12 cups
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solvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvve
Step-by-step explanation:
We know that
In the isolcels triangle base angles are equal.The sum of triangle is 180°<L = <N so
2(4x + 23)° + (3x + 46)° = 180°.. b/c all sum is 180°
8x + 46 + 3x + 46 = 180°
11x +92 = 180
11x = 180 - 92
11x = 88
11x /11 = 88 / 11
x = 8 so we can solve angle of each <M = 3x+ 46
<M =3(8) +46
<M = 24 + 46
<M =70 so angle M measure 70°
the other 2 sides are congrunt so their base angles are equal.
< L = < N =4x + 23
< L = < N =4(8) + 23
< L = < N =32 + 23
< L = < N =55°
So angle L and N is 55° .
find the general solution of the given differential equation. u'' 02u = cos t, 2 ≠ 02
The given differential equation is a second-order homogeneous linear differential equation with constant coefficients. The characteristic equation is r^2 + 2 = 0, which has roots r = ±sqrt(2)i. Since the roots are complex conjugates, the general solution can be written as u(t) = c1*cos(sqrt(2)t) + c2*sin(sqrt(2)t), where c1 and c2 are constants determined by the initial conditions.
To find the particular solution for the non-homogeneous term cos(t), we can use the method of undetermined coefficients. Since the characteristic equation has imaginary roots, the guess for the particular solution must be of the form u(t) = Acos(t) + Bsin(t), where A and B are constants to be determined. Taking the derivatives of this guess and substituting them into the differential equation, we can solve for A and B.
The final solution is the sum of the homogeneous and particular solutions: u(t) = c1*cos(sqrt(2)t) + c2*sin(sqrt(2)t) + (1/2)*(cos(t) - sqrt(2)*sin(t)). This is the general solution to the given differential equation.
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find two other pairs of polar coordinates of the given polar coordinate, one with r > 0 and one with r < 0. then plot the point. (a) (4, 7/4) (r, ) = (r > 0) (r, ) =
One pair of polar coordinates with r < 0 would be (-4, 7/4 + π), which corresponds to the same point but on the opposite side of the origin.
The given polar coordinate is (4, 7/4). To find two other pairs of polar coordinates, we can use the fact that the point lies on the terminal ray of multiple angles that differ by integer multiples of 2π.
Specifically, we can add or subtract 2π from the given angle to get two other angles that correspond to points with the same distance from the origin.
So, one pair of polar coordinates with r > 0 would be (4, 7/4 + 2π), which corresponds to the same point on the terminal ray but with a larger angle.
One pair of polar coordinates with r < 0 would be (-4, 7/4 + π), which corresponds to the same point but on the opposite side of the origin.
To plot the point (4, 7/4), we first move out 4 units from the origin in the direction of the angle 7/4 (measured counterclockwise from the positive x-axis).
This gives us a point on the terminal ray of the angle with distance 4 from the origin. We then label this point with the given polar coordinates.
To plot the other two points, we repeat the process with the angles we found above, using a dashed line to differentiate them from the original point.
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