The general solution to the given differential equation is y = c₁e⁻t + c₂te⁻t + t².
The given differential equation can be written as y'' - y' - 2y = -2t + 4t², which is a second-order linear differential equation with constant coefficients. To solve this equation, the first step is to solve the associated homogeneous equation, which is y'' - y' - 2y = 0. This homogeneous equation can be solved by using the characteristic equation, r² - r - 2 = 0, which has two roots, r = 2 and r = -1.
The general solution of the homogeneous equation is then y = c₁e²t + c₂e⁻t. To find the particular solution of the non-homogeneous equation, the method of undetermined coefficients is used. Since the non-homogeneous term is 4t², the particular solution of the non-homogeneous equation is yp = t².
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what is the difference in the volumes, in cubic inches, of two spheres, one with a radius of 3 inches and the other with a radius of 5 inches.
The difference in the volumes of the sphere is 410.29 inches³.
What is the difference in the volumes?A sphere is a three dimensional circle in which each point on the circumference is equidistant from its center. The volume of a three-dimensional object is the amount of space in the object.
The formula for the volume of the sphere = 4/3πr³
Where:
π = 3.14r = radiusDifference in the volumes = volume of the larger sphere - volume of the smaller sphere
(4/3 x 5³ x 3.14) - (4/3 x 3³ x 3.14)
523.33 - 113.04 = 410.29 inches³
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Jordann is painting her kitchen. To get the color she wants, she mixes 6 parts red paint and 2 parts yellow paint.
1) Write the ratio of parts red paint to parts yellow paint as a fraction.
2) Then, write the unit rate of red paint to yellow paint. (How many parts of red paint does she need for every 1 yellow part?)
What color do you imagine her kitchen will be?!
Answer:
1. 6:2
2.for every 1 yellow paint, she needs 3 red paints
3.an orange
Step-by-step explanation:
1. It would be 6:2 because for every 2 parts yellow she needs 6 parts red
2. Since the ratio is first 6:2, just divide it by 2 since there is 2 yellow and it's asking how much red you need for 1 yellow
Find the sum of all natural numbers such that they are 5 times bigger than their last digit.
A. 25
B. 75
C. 125
D. 276
E. 389
The sum of the natural numbers is (a) 25
How to determine the sum of the natural numbers?From the question, we have the following parameters that can be used in our computation:
Number type = natural number
Condition = 5 times bigger than their last digit
This means that
Number = 5 * Last digit
Because the number is a natural number, then the number cannot end in 0
This is so because 0 is not a natural number
Solving further, the last digit of the number is at most, 9
Using the above as a guide, we have the following:
The number is at most 45 i.e. 5 * 9
Also, it means that it can not have more than 2 digits.
The above highlights in a nutshell is that the number is a multiple of 5 that does not end in 0 and the last digit is 5
Since the last digit is 5, then we have the number to be 25 i.e. 5 * 5
Hence, the number is (a) 25
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How do you find the X and y-intercept of a function?
To find the x-intercept and y-intercept of a function, one can set the x or y value equal to zero and solve for the other variable.
The x-intercept of a function is the point where the graph of the function crosses the x-axis. To find the x-intercept, one can set the y-value equal to zero and solve for the x-value. For example, if the function is y = 2x + 3, the x-intercept can be found by setting y equal to zero and solving for x. In this case, 0 = 2x + 3, which means x = -3/2. So the x-intercept of this function is (-3/2,0).
The y-intercept of a function is the point where the graph of the function crosses the y-axis. To find the y-intercept, one can set the x-value equal to zero and solve for the y-value. For example, if the function is y = 2x + 3, the y-intercept can be found by setting x equal to zero and solving for y. In this case, y = 2(0) + 3 = 3, so the y-intercept of this function is (0,3).
Knowing the x and y intercepts of a function gives an idea of where the graph will start and the behavior of the function in the x and y-axis.
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Rewrite the expression (2t+6)+7t in the simplest form show your work.
Answer:
14t²+42t
Step-by-step explanation:
So you're basically going to use distributive property.
(2t+6)+7t is the same as, (2t)7t + 6(7t).
2t(7t)= 14t² and 6(7t)= 42t.
Combine the two answers and you get 14t²+42t.
14t²+42t is the simplest form you can get to.
What is the sum of 3 sides?
The sum of the three sides of the triangle is 24.
The sum of three sides of a triangle is calculated using the formula S = a + b + c, where a, b, and c are the lengths of each side of the triangle.
To calculate the sum of three sides of a triangle, first measure the lengths of each side and assign a value for each side. For example, if the lengths of the sides are 6, 8, and 10 respectively, then the values for each side would be a = 6, b = 8, and c = 10.
Next, plug in the values of the sides into the formula S = a + b + c to calculate the sum:
S = 6 + 8 + 10
S = 24
Therefore, the sum of the three sides of the triangle is 24.
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Determine if the triangles can be proved congruent by SSS, SAS, ASA, AAS, or HL.
SSS
The triangles can be proved congruent by SAS theorem.
Properties of congruent triangles.Two or more triangles are said to be congruent if and only if their corresponding length of sides and measure of internal angles are equal. Therefore, congruency is a property of equality of the triangles.
To prove that two or more triangles are congruent, then their corresponding sides are compared if they are equal. Also the measure of their corresponding internal angles are considered if they are equal.
Therefore. in the given question, to determine if the triangles can be proved congruent by the appropriate theorem;
It can be seen that the two sides of the triangles are equal. Also the measure of their included angles are equal. Therefore, the triangles can be proved to be congruent by SAS.
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A taste test asks people from Texas and California which pasta they prefer brand A or brand B. This table shows the results. Brand A Brand B Total 125 150 105 275 45 Texas California Total 80 90 170 60 A person is randomly selected from those tested. What is the probability that the person is from California, given that the person prefers brand A? Round your answer to two decimal places
Therefore , the solution of the given problem of probability comes out to be thus brand A is more prefer by the person as probability of B IS 0.43.
Define probability.A probability is a number that represents the likelihood that a certain event will occur. Probabilities may be expressed as percentages
range from 0% to 100% or from 0 to 1.
Here,
Given :
Which brand of pasta do Texas and California prefer
Because brand B is preferred by 45 persons in Texas, the answer would be A.
There are 105 persons that favor brand B overall.
=> 45/105
=> 15/ 35
=> 0.42857...
So ,
rounded it would be 0.43,
thus brand A is more prefer by the person
Therefore , the solution of the given problem of probability comes out to be thus brand A is more prefer by the person as probability of B IS 0.43.
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What is parallel lines Class 7?
Parallel lines are those lines that lie in the same plane and never intersect each other. They are equidistant from each other at every point and have the same slope.
Two lines are stated to be parallel when they do not meet at any point in a plane. This means that lines which do not have a common intersection point and never cross paths with each other are parallel to each other. The symbol used for representing parallel lines is ‘||’.
Two parallel lines are represented as:
AB║CD
That is line AB is parallel to CD.
The perpendicular length between the two parallel lines is always constant.
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Put these numbers in order from least to greatest.
-9 11/20,0.25,9/20,9 1/25, 9
Answer:
From Least to Greatest
-9, 1/25, 0.25, 9/20, 11/20, 9
Step-by-step explanation:
Answer:
-9, 0.25, 9/20, 11/20, 9, 9 1/25
Step-by-step explanation:
You can convert everything in decimals or into fractions with a common denominator. Using what you have, you then put them in order.
I converted them all into decimals, getting:
-9, 0.55, 0.25, 0.45, 9, 9.04
Then I put them in order
-9, 0.25, 9/20, 11/20, 9, 9 1/25
Write an equivalent expression by distributing the "-" sign outside the parentheses:
-(-10p - 0. 7q) – 4. 6
After distribution of the negative sing in the parenthesis, the equivalent expression is 10.7p - 4.6
The expression that is given to us is -(-10p - 0. 7q) – 4. 6.
Now, we know that the meaning of equivalent expression is the it has the same value as that of the original expression.
We have to use the "-" and distribute it over the expression to form an equivalent relation,
So, we know,
-(-10p - 0.7q) – 4.6
Here, p is a constant,
Multiplying the negative sign inside of the bracket,
= (-(-10p)-(-0.7p)-4.6
= 10p + 0.7p - 4.6
= 10.7p - 4.6
So, the equivalent expression is 10.7p - 4.6.
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A ship travels 50 km on a bearing of 27°, and then travels on a bearing of 117° for 140 km. Find the distance traveled from the starting point to the ending point
The distance traveled from the starting point to the ending point is 30.35 km.
What is a bearing?
A bearing tracks the rotation of an angle around the north line in a clockwise fashion. The line connecting the compass's centre to a location and measuring its bearing in degrees as it moves clockwise from north is known as the bearing.
The ship travels 50 km on a bearing of 27°, and then travels on a bearing of 117° for 140 km.
The x-coordinate of the final point is -
50 cos (27°) + 140 cos (117°) = -19
The y-coordinate of the final point is -
50 sin (27°) + 140 sin (117°) = 23.67
The distance from (0,0) to the final point is as follows -
d = √(-19)^2+(23.6)^2)
= √(921.27)
= 30.35 km
Therefore, the distance traveled is 30.35 km.
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On a negatively skewed curve, which is true?
The answer is On a negatively skewed curve ,The mean is lower than the median, which is lower than the mode
What is Negatively skewed curve?When a distribution is negatively skewed in statistics, more values are concentrated on the right side (tail) of the distribution graph while the left tail of the distribution graph is longer.
A longer or fatter tail on the left side of the distribution is referred to as negative skew, whereas a longer or fatter tail on the right is referred to as positive skew. The direction or weight of the distribution is indicated by these two skews.
The distribution's mean, median, and mode are referred to as its central tendency. The fact that the usually skewed data's mean, median, and mode are all equal illustrates how equally distributed wealth and income are, as well as the importance of government initiatives and economic growth.Because the negative side is so weighted, the distributions have a large gap. The data, for instance, includes income distributionin negatively skewed distribution, data shows the unequal distribution, the central tendency shows as under:Mode > Median > Mean.
Examples of negatively skewed curveUniversity exams. The tests are the same, yet a small fraction of students receive zero, average, or high percentage scores, indicating a negative skewed distribution of the data.The human Life cycle ,As many people live ordinary lives, some live very short lives, and some live very long lives in terms of age, the human life cycle is another illustration of a negatively skewed distribution.What causes a negatively skewed distribution?When the long tail is on the distribution's left side, the distribution is said to be left skewed. They are also referred to as negatively skewed by statisticians. Probabilities taper off more slowly for decreasing values, which causes this circumstance.
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How many zeros are at the end of the product $25 \times 240$?
Answer: There are three 0s
240 x 25 = 6000 and there are three 0s in 6000
Makayla and Addison are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Makayla is 290 miles away from the stadium and Addison is 345 miles away from the stadium. Makayla is driving along the highway at a speed of 40 miles per hour and Addison is driving at speed of 51 miles per hour. Let MM represent Makayla's distance, in miles, away from the stadium tt hours after noon. Let AA represent Addison's distance, in miles, away from the stadium tt hours after noon. Write an equation for each situation, in terms of t,t, and determine the interval of hours, t,t, for which Makayla is closer to the stadium than Addison.
The linear functions for each situation are given as follows:
Makayla: 290 - 40x.Addision: 345 - 51x.Makayla is closer than Addision when:
x < 5.
How to define the linear functions?The linear functions in slope-intercept format for this problem are given as follows:
y = mx + b.
In which:
m is the slope, representing the velocity, as a negative as they are getting closer to the stadium with time.b is the intercept, representing the initial distance.Hence the functions are defined as follows:
Makayla: 290 - 40x.Addision: 345 - 51x.Makayla is closer when:
290 - 40x < 345 - 51x
11x < 55
x < 5.
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Think back to the pyramids in Egypt. For a Social Studies project, you created a scale figure of one pyramid. The dimensions are shown. How much paint will you need to paint the entire pyramid for your project?
The surface area of the pyramid is 568 square inches.
What is the surface area of a square pyramid?The surface area of the square pyramid is defined as the total area covered by the surfaces required to make the square pyramid. The area is the space occupied by the square pyramid in two-dimensional planes.
The formula to calculate surface area will be calculated by the formula below,
SA=B+12(P × l)
Here, B is the area of the base, P is the perimeter of the base, and l is the slant height.
SA = ( 8 x 8 ) + 12 ( 8 x 4 + 10 )
SA = 64 + 12 ( 32 + 10 )
SA = 64 + 12 x 42
SA = 568 square inches
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How do you describe a logarithmic function?
A logarithmic function is a type of inverse function.
A logarithmic function is a type of inverse function that is commonly used in mathematics and science. It is typically written in the form of y = log base b (x), where b is the base of the logarithm, and x is the input value. The most common bases for logarithms are 10 (common logarithm) and e (natural logarithm).
The logarithmic function is the inverse of the exponential function with base b. This means that if y = log base b (x) then x = b^y
One way to describe a logarithmic function is to say that it represents the exponent to which the base number must be raised in order to produce the input value x. For example, log base 10 of 100 is 2, because 10^2 = 100.
Another way to describe a logarithmic function is to say that it is a way of expressing very large or small numbers in a more manageable form. For example, instead of expressing a number like 10^9 as 1 billion, it can be expressed as 9 using a logarithm to base 10.
It is also important to note that logarithmic function have some key characteristics such as:
The domain of a logarithmic function is x>0, since the logarithm of a non-positive number is undefined.
The range of a logarithmic function is all real numbers.
The graph of a logarithmic function has a vertical asymptote at x=0, since the logarithm of a non-positive number is undefined.
A logarithmic function is increasing over its domain and concave down.
The x-intercept of the graph is (1,0), since the logarithm of 1 is always 0.
Overall, logarithmic functions are mathematical tools that allow us to simplify the representation and manipulation of very large and small numbers, and are used in many areas of science and engineering.
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Can somebody help me solve this equation? Thanks.
Answer:
15w^2
Step-by-step explanation:
The reason being is note how it says portrait, so assume the shape of a square, because it says 3 inches on all sides, and it also mentioned 1.5 times w, so in turn, 15w^2
Refer to the attached image.
Tell whether 30 is divisible by 2
Answer: Yes
Step-by-step explanation
Since the number being divided has a 0, it can be divided by 2
F(1)=120,f(n)=f(n-1)+_n>_2
The value of next two terms are,
⇒ 0 and - 60
What is Function?A relation between a set of inputs having one output each is called a function.
Given that;
The values are,
⇒ f (1) = 120,
⇒ f (2) = 60
And, The function is,
⇒ f (n) = f (n - 1) + ___
For n ≥ 2
Now,
Since, The values are,
⇒ f (1) = 120,
⇒ f (2) = 60
And, The recursive formula is,
⇒ f (n) = f (n - 1) + b
For n ≥ 2
Substitute n = 2;
⇒ f (2) = f (2 - 1) + b
⇒ 60 = f (1) + b
⇒ 60 = 120 + b
⇒ b = 60 - 120
⇒ b = - 60
So, The recursive formula is,
⇒ f (n) = f (n - 1) - 60
Substitute n = 3;
⇒ f (3) = f (3 - 1) - 60
⇒ f (3) = f (2) - 60
⇒ f (3) = 60 - 60
⇒ f (3) = 0
Substitute n = 4;
⇒ f (4) = f (4 - 1) - 60
⇒ f (4) = f (3) - 60
⇒ f (4) = 0 - 60
⇒ f (4) = - 60
Thus, The next two term are 0 and - 60.
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Benji's flight from Chicago to San Jose took 0.5 hour longer than his return flight from San Jose
to Chicago. If Benji's flight from Chicago to San Jose was 4.5 hours long, how long was his return
flight, in hours?
Choose 1 answer:
A
3.5
B 4
C 4.5
D
assroom
5
The time taken for his return flight, in hours could be 4 hours.
What is time ?
Time is defined as the sequence of events that occurs which can be from past through present or to future.
Given,
Benji's flight from Chicago to San Jose took 0.5 hour longer than his return flight from San Jose to Chicago.
If Benji's flight from Chicago to San Jose was 4.5 hours long
let,
chicago to san jose = san jose to chicago+0.5
4.5 - 0.5 = san jose to chicago
san jose to chicago = 4 hours
Hence, How long was his return flight, in hours is 4 hours.
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Please help me respond this
The closest estimate of x when the functions satisfy the condition f(x) = g(x) is 1.65
What is a function?
Function is a mathematical law that establishes the relationship between an independent variable and a dependent variable.
Given,
[tex]f(x) =[/tex] [tex]\log\x_{3} (2x)[/tex]
[tex]g(x) = -4x^{2} + 3x +7[/tex]
Now plotting both the graphs in Desmos, we got the result as in the below figures.
Given f(x) = g(x)
So from the graphs, we can find an intersecting point. This intersecting point is common to both f(x) and g(x)
Therefore from the graph, the closest estimate of x is 1.65.
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How many triangles are there in the following figure 18 17 16 15?
There are 15 triangles in the given figure. Each of the four shapes in the figure can be divided into four triangles.
The figure contains four triangles, each with three sides. To calculate the total number of triangles in the figure, we can multiply the number of sides of each triangle by the number of triangles. Therefore, the total number of triangles in the figure is 3 x 4 = 12.
However, the figure also contains two additional triangles that are formed when three of the triangles intersect. To calculate the total number of triangles, we must add the two additional triangles to the 12 triangles we calculated earlier. Thus, the total number of triangles in the figure is 12 + 2 = 14.
Since the figure also has three more sides, it can be divided into one more triangle. Therefore, the total number of triangles in the figure is 14 + 1 = 15.
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What is Y 1 on a graph?
The graph of y is a straight line parallel to the x−axis through the point (0,1)
Now, According to the question:
Let's know:
What is slope and example?
Whenever the equation of a line is written in the form y = mx + b, it is called the slope-intercept form of the equation. The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b). For example, for the equation y = 3x – 7, the slope is 3, and the y-intercept is (0, −7).
The equation is Y = 1
A straight line in slope (m) and intercept (c) form is:
y = mx + c
y is a straight line of slope 0 and y- intercept of 1
y = 0.x + 1
y = 1
Hence, the graph of y is a straight line parallel to the x−axis through the point (0,1)
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Ali is collecting signatures for a petition. He currently has 520 signatures. He has 6 more weeks to collect the remaining signatures he needs. He needs a total of at least 1,000 signatures before he can submit the petition. Ali wants to collect the same number of signatures each week. Write and solve the inequality for the situation
An inequality that represents the number of signatures Ali needs to collect in each of the remaining weeks so that he will have enough signatures to submit the petition: 520 + 6m ≥ 1000
and the minimum number of signatures he needs to collect = 80
Given that Ali wants to collect the same number of signatures each week.
Assume that Ali collects m number of signatures each week.
He has 6 weeks to collect the remaining signatures he needs.
So, by unitary method in 6 weeks the number of signatures he would collect:
6 × m
He currently has 520 signatures and he needs a total of at least 1,000 signatures before he can submit the petition.
so, we get an inequality.
520 + 6m ≥ 1000
We solve above inequality.
6m ≥ 1000 - 520
m ≥ 480/6
m ≥ 80
Ali needs to collect at least 80 signatures, so that he will have enough signatures to submit the petition.
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Is 8 15 17 are Pythagorean triples?
No, 8, 15, 17 are not Pythagorean triples.
No, 8 15 17 are not Pythagorean triples. A Pythagorean triple is a set of three positive integers a, b, and c such that a2 + b2 = c2. In other words, the sum of the squares of two integers must equal the square of the third integer.
In the case of 8 15 17, if we calculate the sum of the squares of the two smaller integers, 8 and 15, we get
[tex]8^2 + 15^2 = 64 + 225 = 289[/tex].
This does not equal the square of the third integer,
[tex]17^2 = 289,[/tex]
so 8 15 17 are not Pythagorean triples.
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What is an unsolvable problem?
As a result, it may be shown that the answer to the algorithm problem at hand is a halting problem, which is indecidable and thus impossible to solve.
Algorithm : what is it ?No matter the circumstance, algorithms are fundamentally problem solvers of expression ; they exist to identify, resolve, and frequently automate of solution to a specific problem. Algorithms are frequently introduced in area introductory textbooks by characterizing them as "a series of steps to complete a problem."
Here,
one of the most common problems that hasn't been solved is the stopping problem. It asks the following question:
Does M come to a standstill when given the input w for any Turing machine with the parameters alphabet = a, b, and any string w over
It can be shown that the halting problem cannot be resolved since it is indecisive.
As a result, it may be shown that the answer to the algorithmic problem at hand is a halting problem, which is indecidable and thus impossible to solve.
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Task #2: Jack makes $32 when he sells 4 pumpkins. How many pumpkins will he have to sell to make $48? If Jack finds that he has $120 at the end of the day, how many pumpkins did he sell?
Answer:
1. 6
2. 15
Step-by-step explanation:
4 pumpkins: $32
=> 1 pumpkin = 32 : 4 = $8
- He will have to sell 48 : 8 = 6 pumpkins to make $48
- He sold: 120 : 8 = 15 pumpkins that day
Which statement is correct?
Answer:
B. ST and TK intersect at a 90 degree angle.
I know this because you can see that in between STK, there is a square in the corner indicating a 90 degree angle.
Step-by-step explanation:
Hope it helps! =D
A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly.After 11months he weighed 140 Kilograms. He gained weight at rate of 5.5 kilograms per month.
Let y represent the sumo wrestler’s weight (in kilograms)after x month.
Complete the equation for the relationship between the weight and number of months.
The equation that shows the relationship between the weight and number of months. y = 79.5 + 5.5x
What is weight ?In science and technology, the weight of an object is the force acting on an object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on an object. Others define weight as a scalar quantity, the magnitude of gravitational force. Still others define it as the size of the reaction force applied to the body by mechanisms resisting the influence of gravity: weight is a quantity measured, for example, by a spring balance.
Given that ; Total Month = 11 months
Gained weight at the rate = 5.5 kilograms per month.
Assume initial weight W,
The initial weight + total month x rate = 140
w + 11 x 5.5 = 140
w = 79.5
if y is total weight after x months
y = 79.5 + 5.5x
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