They may bake 20 additional bags of cookies. The problem is represented by the inequality x < 80.
What is inequality?In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.
Given, A bakery wants to bag at most 100 cookies into cookie packs. Each cookie pack has 4 cookies. They have already bagged 20 cookies.
Since each cookie pack can hold 4 cookies
Thus, 100 cookies will require = 100/4 packs
100 cookies will required = 25 packs
Since They have already bagged 20 cookies.
Thus,
The total pack they used = 20/4
The total pack they used = 5
Inequality:
let "x" be the number of cookies they can make
80 > x
Therefore, 20 more bags of cookies they could make. the inequality that represents the situation is x< 80.
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2c-10<3c-9 solve the following inequality
Answer:
[tex]c > -1[/tex]
Step-by-step explanation:
[tex]2c-10 < 3c-9\\-10 < c-9\\-1 < c\\c > -1[/tex]
If You write all numbers from 30 to 65 including 30 and 65 how many times would you write the digit 7?
Answer:
3 times.
Step-by-step explanation:
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,40,41,42,43,44,45,46,47,48,49,50,51,
52,53,54,55,56,57,58,59,60,61,62,63,64,65.
37,47,57.
3 times
Jack spins a fair six-sided spinner 240 times. What is the probability that he will score more than 45 sixes?
Give your answer to 2 d.p.
The probability that Jack will score more than 45 sixes in 240 spins of the spinner is about 0.13%.
What is the probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that an event is certain.
The probability of getting a six on one spin of a fair six-sided spinner is 1/6. If Jack spins the spinner 240 times, the expected number of sixes he will get is 240 * 1/6 = 40.
We can use a normal approximation to the binomial distribution to find the probability that Jack will get more than 45 sixes. The mean of the distribution is 40 and the standard deviation is the square root of the variance, which is the product of the number of spins and the probability of getting a six on each spin, divided by the number of trials:
Standard deviation = sqrt(240 * 1/6 * 5/6 / 240) = sqrt(40/240) = sqrt(1/6)
Using the normal approximation, the probability that Jack will get more than 45 sixes is equal to the probability that a standard normal random variable is greater than (45 - 40) / sqrt(1/6) = 5 / sqrt(1/6) = 5 * sqrt(6).
Using a standard normal table or a calculator, this probability can be found to be approximately 0.0013, or 0.13%.
Hence, the probability that Jack will score more than 45 sixes in 240 spins of the spinner is about 0.13%.
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Help pls
A consumer purchased a computer after a 12% price reduction. If x represents the computer's original price, the reduced price can be represented by
If x represents the computer's original price, the reduced price can be represented by
We want to reduce x by 12%. Thus, the reduced price is:
x - 0.12x = 0.88x
More informally, if you take away 12%, you're left with 88%.
Answer:
We want to reduce x by 29%. Thus, the reduced price is:
x - 0.29x = 0.71x
More informally, if you take away 29%, you're left with 71%.
The constellation whose stars are used as pointers to the north celestial pole in the northern hemisphere at this time in history is:
A) Ursa Minor, the Little Bear, containing the bright star Polaris.
B) Leo, the Lion, containing the bright star Regulus.
C) Boötes, the Herdsman, containing the bright star Arcturus.
D) Ursa Major, the Big Dipper.
The correct option is (A). The correct answer is Ursa Minor, the Little Bear, containing the bright star Polaris.
Whereas stars spin or revolve from the northern hemisphere is the center of the sky, also known as the North Celestial Pole. Additionally, it marks the precise location of the Polaris or Northern Star constellation in full view. In addition, this served as a point of reference for the construction of sundials in earlier times.
When people go outside, especially at night, and try to find the Polaris constellation, it is evident that the stars close to this star formation circle in an east-to-west direction, in contrast to the northern star, which makes no movement whatsoever as the evening wears on.
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i need the answer and explain how you got it
The amount of money that the worker will earn in a week if he works 4 hours overtime is; $3180
How to solve Algebra Word Problems?We are given;
Number of work hours per week = 40 hours
Regular rate of work = $15.90 per hour
Overtime rate of work = 1.5(15.90) per hour
Since the regular rate is $15.90 per hour, then for a week, the money earned is; (15.90 * 40)
Now, he also works 4 hours overtime in a week and as such amount earned for overtime in the week = 4 * 1.5(15.90)
Thus;
Total amount earned in the entire week = (15.90 * 40) + (4 * 1.5(15.90))
= $3180
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Part A
In a major league baseball park, the pitching rubber, where the pitcher stands, is always 60.5 feet from home plate, where the batter stands. In Widget Stadium, the distance from home plate to the wall in center field is 450 feet.
A major league baseball pitcher throws a baseball with a horizontal velocity of 132 feet/second. The batter then hits the ball with a horizontal velocity of 141 feet/second, which results in a home run when the baseball passes over the center field fence. The ball passes over the pitching rubber on its way out of the stadium
Ignoring gravity and the air resistance that would slow the baseball down, create an equation to model the distance that the baseball is from the pitching rubber once it is thrown. (Hint: distance = velocity × time.)
Answer:
The distance that the baseball is from the pitching rubber, D, can be modeled using the equation:
D = V*t + 60.5
Where V is the horizontal velocity of the baseball in feet per second and t is the time in seconds that the baseball is in the air.
In this scenario, the baseball is thrown by the pitcher with a horizontal velocity of 132 feet/second, and then hit by the batter with a horizontal velocity of 141 feet/second. The baseball passes over the pitching rubber on its way out of the stadium and eventually passes over the center field fence, 450 feet away. So, the equation will be
D = (132+141) * t + 60.5
9 less than 10 and 6 times the letter n
The statement '9 less than 10 and 6 times the letter n' in the mathematical form will be 9 < 10 + 6n.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
The act of converting a specified statement into an expression or equation is known as a sentence-to-equation transformation.
The statement is '9 less than 10 and 6 times the letter n'.
Convert the statement into a mathematical form. Then we have
9 < 10 + 6n
Simplify the inequality, then we have
9 < 10 + 6n
6n > - 1
n > - 1/6
The statement '9 less than 10 and 6 times the letter n' in the mathematical form will be 9 < 10 + 6n.
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Fluffy laundry detergent reduced its regular size from 4.2 kg to 3.6 kg. The retail price dropped from $10.29 to $9.18. What was the percent change in the unit price?
Original Unit Price:
$10.29 / 4.2 kg = $2.45/kg
New Unit Price:
$9.18 / 3.6 kg = $2.55/kg
The price went up by $0.10/kg. If we compare that (divide that) by the starting unit price of $2.45, we get:
$0.10 / $2.45 ≈ 0.040816326530612
≈ 4.0816326530612 %
(You can round as accurately as your teacher needs.)
The gate code at your apartment complex is three numerical digits followed by the "#" sign. If each code
contains non-repeating digits, how many possible gate codes does your apartment complex have
If each code contains non-repeating digits there would be 720 possible gate codes.
What is a combination in mathematics?
Combination in mathematics is a way of selecting items from a set, without regard to the order in which they were chosen.
In your example, there are 10 possible digits (0-9) that can be used for each of the three numerical digits, but since the digits cannot repeat, the total number of possible gate codes is 1098 = 720. Since the "#" sign is not included in this calculation, the final answer is 720 possible gate codes.
Hence, If each code contains non-repeating digits there would be 720 possible gate codes.
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Virus C has 4 strains (a, b, c, d). It is known that in a sample of 10 people, each of them was sick with the C virus exactly 1 time, and someone was sick with each of the strains. In how many ways can the statistics of the disease be arranged under these conditions (it is important who was ill with which strain).
The number of ways that the people can be arranged is given as follows:
120,960.
What is the permutation formula?The number of possible permutations of x elements from a set of n elements is given by:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
To choose four people from a set of 10, as the order matters, the permutation formula is used, hence:
10!/6! = 5040.
Then we have to consider the four strains, which can be arranged in 4! = 24 ways, hence the number of ways for which the statistics can be arranged is of:
24 x 5040 = 120,960.
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URGENT!!! PLS HELP....
Answer:
Ex1
[tex] ln( \frac{ {y}^{4} { {e}^{2} } }{ \sqrt[3]{ {x}^{2} } } ) \\ = ln( {y}^{4} {e}^{2} ) - ln( \sqrt[3]{ {x}^{2} } ) \: \: \: \: (by \: quotient \: rue) \\ = ln( {y}^{4} ) + ln( {e}^{2} ) - ln( \sqrt[3]{ {x}^{2} } ) (by \: product \: rule) \\ = 2 + ln( {y}^{4} ) - ln( \sqrt[3]{ {x}^{2} } ) \\ = 2 + 4 ln(y) - \frac{3}{2} ln(x) \: \: \: \: (power \: rule)[/tex]
Ex2
[tex] log_{4}(x + 2) + log_{4}(x - 1) = 1 \\ log_{4}((x + 2)(x - 1)) = 1 \: (by \: product \: rue) \\ log_{4}( {x}^{2} + x - 2 ) = 1 \\ {x}^{2} + x - 2 = {4}^{1} \\ {x }^{2} + x - 6 = 0 \\ x = 3 \: \: or \: x = - 2[/tex]
I need to understand quadratic graphs. eg. y=x^2 +2x -3
A quadratic graph is a type of parabolic graph that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The graph of this equation is a parabola that opens either up or down, depending on the value of a.
The equation y = x^2 + 2x - 3 is an example of a quadratic graph. It can be written in the standard form of a quadratic equation as y = x^2 + 2x - 3, where a = 1, b = 2 and c = -3.
The x^2 term in the equation is known as the quadratic term, and it is responsible for the parabolic shape of the graph. The x term and the constant term, 2x and -3, respectively, determine the location of the vertex and the direction of the parabola.
The vertex of the parabola is the point where the parabola changes direction. It can be found by using the formula x = -b/2a, which in this case is x = -2/2 = -1 and y = f(-1) = -1.
The graph of y = x^2 + 2x - 3 is a parabola that opens upwards and the vertex of the parabola is (-1,-1)
In summary, a quadratic graph is a parabolic graph represented by the equation y = ax^2 + bx + c, where a, b, and c are constants. The x^2 term gives the parabolic shape to the graph, the x term and the constant term determine the vertex and the direction of the parabola respectively.
The perimeters of a triangle is 19 feet. One side of the triangle is 3 times the second side. The third side is 4 feet longer than the second side. Find the kength of each side?
Answer:
9 ft, 3 ft 7, ft
Step-by-step explanation:
Perimeter = 19 ft
Let side 2 = x
Side 1 = 3x
Side 3 = x + 4
perimeter = 3x + x + x + 4 = 19
5x + 4 = 19
5x = 15
x = 3
Side 1: 3x = 3(9) = 9
Side 2: x = 3
Side 3: x + 4 = 3 + 4 = 7
Answer:
9 ft, 3 ft 7, ft
Which of the following equations are equivalent? Select three options. 2 + x = 5 x + 1 = 4 9 + x = 6 x + (negative 4) = 7 Negative 5 + x = negative 2
Answer:
2 + x = 5, x + (negative 4) = 7, Negative 5 + x = negative 2 are equivalent equations.
In the first equation 2+ x = 5, x = 3
In the second equation x + (negative 4) = 7, x = 11
In the third equation Negative 5 + x = negative 2, x = negative 3
What would be the difference in interest received from an investment of $750 from an account paying 4% simple interest per annum and an account paying 0.3% monthly compound interest, over a one-year period? and which account paid the most interest
Answer:
The difference in interest received from an investment of $750 from an account paying 4% simple interest per annum and an account paying 0.3% monthly compound interest, over a one-year period is $19.75. The account paying 0.3% monthly compound interest will pay the most interest, yielding $102.50 in total.
Find the general solution, in implicit form, to the differential equation dy/dx = 9 + 5x ^1/2 / 8+ (13y)^1/2 x>0, y>0Then, rearrange the solution that you get into the form f (x,y) = C, for some constant C.
A general solution is one which involves the integer 'n' and gives all solutions of a trigonometric equation.
What is general and particular solution of differential equation?A specific differential equation solution is one with the formula y = f(x), which lacks any arbitrary constants. The arbitrary constants a and b are used in the differential equation's general solution, which has the form y = f(x) or y = axe + b.The differential equation's solution can be quickly and easily found by integrating each variable once it has been separated. The formulas are: f(x)dx+g(y)dy=0, where f(x) and g(y) are respectively constants or functions of x and y.Any equation involving the unknown function y=f(x) and one or more of its derivatives is known as a differential equation. A function y=f(x) that satisfies the differential equation when f and its derivatives are replaced into the equation is a solution to a differential equation.Given data :
dy / dx = 9 + [tex]\frac{9 + (5x)^{1/2} }{8 + (13y)^{1/2} }[/tex] ; x > 0
The general solution of a differential equation in implicit form is obtained as follows
dy / dx = h(x) / g(y)
g(y)dy = {8 + (13y)^{1/2}
h(x) dx = {9 + (5x)^{1/2}
F(x,y) = C
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The baggage handling services of On-Time Airlines is interested in how many baggage handlers they need on duty at various time of the day to ensure that passengers do not wait an unreasonable amount of time for their baggage. An airport executive performed a study and found that there is a correlation between the number of passengers arriving at given times and the number of baggage handlers needed. She sampled various times during the day and different days of the week including weekend. She recorded the number of passengers arriving within any 1-hour time block. The computer output from the regression equation analysis is shown below
The baggage handling services of On-Time Airlines are conducting a study to determine the correlation between the number of passengers arriving at a given time and the number of baggage handlers needed.
The study is performed by an airport executive who samples different times during the day and different days of the week including weekends. The main objective is to ensure that passengers do not wait an unreasonable amount of time for their baggage.
The study records the number of passengers arriving within any 1-hour time block. The data collected is then analyzed using a statistical technique called regression equation analysis, which is a method to estimate the strength and direction of the relationship between two or more variables.
The outcome or the result of the analysis is not provided in the given information.
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The gate code at your apartment complex is three numerical digits followed by the "#" sign. If each code
contains non-repeating digits, how many possible gate codes does your apartment complex have?
There are a total of 720 possible gate codes for your apartment complex.
What is gate code?Gate codes are security codes used to control access to a restricted area. Gate codes are used in both residential and commercial settings to ensure the safety of the occupants and to protect the property. Gate codes can be used to unlock and lock a gate, door, or other access point, and typically consist of a combination of letters, numbers, and/or symbols.
This is because each code consists of three non-repeating digits (0-9).
For the first digit, there are 10 possible choices (0-9).
For the second digit, there are 9 possible choices (excluding the number chosen for the first digit).
For the third digit, there are 8 possible choices (excluding the numbers chosen for the first and second digits).
This means that the total number of possible gate codes is 10 x 9 x 8 = 720.
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In a circle with centre O, the chord BC is intersected by a radius OA. If arc AC = arc AB and BC is an angle bisector of angle ABO, prove that triangle ABO is an equilateral triangle.
When two lines are perpendicular to one another, they are said to intersect at a right angle or a 90-degree angle.
What is meant by angle bisector?An angle is split in half equally by an angle bisector. Thus, divide the angle's degree count by 2 to determine the location of the angle bisector. At 80 degrees of the angle, the angle bisector is located.
An equal split of a line is created by a bisector. As a result, when we refer to a line segment's perpendicular bisector as AB, we mean that it bisects or divides AB into two equal halves.
According to the angle bisector theorem, if a point is on an angle bisector, it is equidistant from the angles' sides. The opposite is also true; if a point is equally far from each of the triangle's sides, it is on the angle's bisector.
Two lines are said to intersect at a right angle or a 90-degree angle when they are perpendicular to one another. A bisector, on the other hand, is a line that splits a line into two equally sized half.
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Question 3 (5 points)
(05.02)
Solve log x = 2.
O a
Ob
Oc
Od
2
20
100
1,000
For logx = 2, the value of x would be 100.
Option (C) is correct.
What is a logarithm?
A logarithm is a mathematical function that calculates the power to which a given number (called the base) must be raised to produce a given value. The most common base used in logarithms is base 10, but logarithms can be defined with any base. If the base of a logarithm is not specified, it is assumed to be base 10. The logarithm of a number is represented by the symbol log.
The value of x in the equation logx = 2 can be found by solving for x. The logarithm of a number is the exponent that the base must be raised to in order to produce that number. So, in the equation logx = 2, x is equal to the base raised to the power of 2.
So,
logx = 2
x = 10^2
x = 100
if the base of the logarithm is not specified it is assumed to be base 10.
So, logx = 2 means x = 10^2 = 100
Hence, for logx = 2, the value of x would be 100.
Option (C) is correct.
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2. In the third paragraph, line number 10, it says "since each location on Earth has a
different magnetic signature...." what does it mean that each location on Earth has
its own magnetic signature?
Iron, which is a magnetic substance, makes up most of the core of the Earth, where the magnetic field of the planet first appears. The fluid outer core's movement around the solid inner core causes electric currents, which in turn creates and maintains the Earth's magnetic field.
How do you calculate the Earth's magnetic field?V(r) = m • r / (4 r3) = m cos / (4 r2) = scalar magnetic potential of dipole field. A spherical harmonic field expansion is used. Dipole is the first phrase (above). An approximate range of 25,000 to 65,000 nT is the Earth's magnetic field intensity (. 25 - . 65 gauss). The distance between magnetic north and true north is known as magnetic declination.The geomagnetic field, also referred to as the earth's magnetic field, is the magnetic field that emanates from the interior of the Earth into space and collides with the solar wind, a stream of charged particles that is emitted from the Sun.Fluxgate magnetometers, which provide information about the field's vector components, or proton precession magnetometers, which measure scalar field intensity, are typically used to measure the earth's magnetic field.To learn more about Earth's magnetic, refer to:
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The quarter circle of radius R in the first quadrant (x' + y' = R2 for x ) 0 and
y > 0) is revolved about the x-axis to produce a hemisphere. Find the volume of the hemisphere in the following three ways:
a. Apply the disk method and integrate with respect to x.
b. Apply the shell method and integrate with respect to y.
c. Apply the general slicing method and integrate with respect to y.
The volumes of the hemisphere are (π/12)R³, 2π/3(R³) and π/3(R³)
How to determine the volumes of the hemisphereFrom the question, we have the following parameters that can be used in our computation:
x' + y' = R²
Using the disk method
Start by calculating the area of a single disk is given by the quarter circle of radius R in the first quadrant.
The area of the single disk is:
Area = π(R/2)²
The volume of the hemisphere using the disk method is given by:
V = ∫(π(R/2)² dx from 0 to R
This gives
V = (π/4) * ∫R^2 dx from 0 to R
Differentiate
V = (π/12)R³
Using the shell methodHere, we first need to find the area of a single shell.
The area of the single shell is:
Area = 2πx * dx
The volume of the hemisphere using the single shell method is given by:
V = ∫(2π x * dx)dy from 0 to R
So, we have
V = 2π * ∫xdy*dx from 0 to R
Differentiate
V = 2π/3(R³)
Using the general slicing methodHere, we can use any slicing method, such as slicing the hemisphere parallel to the y-axis.
The volume of the hemisphere using the general slicing method is given by:
V = ∫(π * (x^2) * dy) from 0 to R
So, we have
V = π * ∫(y^2)dy from 0 to R
Differentiate
V = π/3(R³)
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Robert earns $512 less than noel, john earns times as much as robert. if their total salaries sum to 47,645.25, how much is roberts salary?
n + (n - 512) + 3(n - 512) = 47645.25
Answer:
Robert's salary = $ 9426.65
Step-by-step explanation:
First we need to solve for n ( noel's salary)
And from there we can just sub n value into Robert's salary equation which is (n - 512) [ he earns $512 less than noel]
there is an angle $\theta$ in the range $0^\circ < \theta < 45^\circ$ which satisfies \[\tan \theta \tan 2 \theta \tan 3 \theta
The solution of the below trignometric equation are θ= mπ , θ= nπ/3
Given tan θ + tan 2θ = tan 3θ
=> tan θ + tan 2θ - tan 3θ = 0
(tanθ + tan2θ)−(tanθ + tan2θ)/(1 - tanθ tan2θ) =0
or (tanθ + tan2θ)(1 − tanθ tan2θ−1)=0
or tanθ tan2θ(tanθ + tan2θ) = 0
tanθ = 0
∴θ=nπ, tan2θ=0,
∴2θ=nπ or θ=nπ/2
tanθ + tan2θ=0
or sin(θ+2θ)=0
∴θ=nπ/3
However, the values of given by =n/2 for odd values of n do not fulfill the preceding equation as it will result in ∞ = ∞
Hence the required solution is:
θ= 2mπ/2=mπ
θ= nπ/3
where m and n is an integer
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Complete question:
Find the general solution of the equation tan θ + tan 2θ = tan 3θ
The following data show the amount (in ounces) of beverage in randomly selected 16-ounce beverage cans. Create a Stem-Leaf plot for these data.
15.8
16.1
15.2
14.8
15.8
15.9
16.0
15.5
16.1
15.7
Fill in the blanks of table below to complete the Stem-Leaf plot for these data. The key is 15|2 = 15.2. Sort the values for Leaf in increasing order and do not keep spaces or commas between leaves.
Stem
Leaf
14
15
16
(a)
(b)
(c)
The 5th value is 15.5, and the 6th value is 15.7. The median is then (15.5 + 15.7)/2 = 15.6.
A Stem-Leaf plot is a graphical way to organize and summarize numerical data. It is composed of two columns, the stem and the leaf. The stem column contains the leading digits of the data values, while the leaf column contains the remaining digits.
For the given data set, the stem-leaf plot looks like this:
Stem Leaf
14 8
15 2 8 5 7 9
16 0 1
To create this stem-leaf plot, the data values were first sorted in ascending order, and then the leading digits were placed in the stem column. The remaining digits were then placed in the leaf column, in increasing order. The key, 15|2=15.2, is an example of a data value from the set.
To calculate the median of the data set, we first need to find the number of values in the set. Since there are 10 values, the median will be the average of the 5th and 6th values.
The 5th value is 15.5, and the 6th value is 15.7. The median is then (15.5 + 15.7)/2 = 15.6.
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Line q passes through the point 4,6 and 0,6 what is the equation for the line which is perpendicular to q and passes through the point -2,3
A x= -2
B x= -3
C y= -3
D y= -2
The equation for line q is y = mx + b where m is the slope and b is the y-intercept. Since the line passes through the point (4,6) and (0,6), we can find the slope by using the formula m = (y2 - y1) / (x2 - x1) = (6 - 6) / (4 - 0) = 0.
Since the line is perpendicular to line q, the slope of the new line is the negative reciprocal of 0, which is undefined. Therefore, the new line is a vertical line, and the equation is in the form x = a constant. Since the line passes through the point (-2,3), the equation is x = -2. Therefore, the answer is A: x = -2.
I need help please asap, it's missing
Answer:
f(g(8)) = 26
Step-by-step explanation:
to calculate f(g(8)) , evaluate g(8) , then substitute the value obtained into f(x)
g(8) = 8 - 6 = 2 , then
f(2) = 3(2)² + 2 + 12 = 3(4) + 14 = 12 + 14 = 26
The monthly charge for a waste collection service is 470 dollars for 100 kg of waste and 770 dollars for 175 kg of waste. (a) Find a linear model for the cost, C, of waste collection as a function of the number of kilograms, w.
The linear model for the cost, [tex]C[/tex], of waste collection as a function of the number of kilograms, [tex]w[/tex] is [tex]C = 4w +70[/tex] .
Given,
The monthly charge for a waste collection service for 100 kg of waste = $470
so, [tex](x_{1} , y_{1} ) = (100, 470)[/tex]
The monthly charge for a waste collection service for 175 kg of waste = $770
so, [tex](x_{2} , y_{2} ) = (175, 770)[/tex]
In this question, We are supposed to find a linear model for the cost, C, of waste collection as a function of the number of kilograms, w.
So, we will use two point slope form :
[tex]y - y_{1} = \frac{y_{2} - y_{1} }{x_{2}- x_{1} } (x-x_{1} )[/tex]
Substitute the values:
[tex]y - 470 = \frac{770- 470 }{175- 100 } (x-100)[/tex]
[tex]y - 470 = \frac{300 }{75} (x-100)[/tex]
[tex]y - 470 = 4 (x-100)[/tex]
[tex]y - 470 = 4x-400[/tex]
[tex]y = 4x +70[/tex]
Now, replace [tex]y[/tex] with C and [tex]x[/tex] with w
[tex]C[/tex] [tex]= 4w +70[/tex]
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Verify that every member of the family of functions y=(lnx+C)/x is a solution of the differential equation x2y' + xy = 1
a) Find a solution of the differential equation that satisfies the initial condition y(7)=6. y=
b) Find a solution of the differential equation that satisfies the initial condition y(6)=7. y=
The solutions are:
a) y = (lnx + 42 - ln7)/x
b) y = (lnx + 42 - ln6)/x
What are differential equations?
A differential equation is an equation that relates a function to its derivatives.
To verify that every member of the family of functions y=(lnx+C)/x is a solution of the differential equation x^2y' + xy = 1, we can substitute the function into the equation and see if it holds true for all values of x.
y=(lnx+C)/x
y'= (1/x - lnx - C)/x^2
x^2(1/x - lnx - C)/x^2 + x(lnx+C)/x = 1
(1 - xlnx - xC) + (lnx+C) = 1
lnx + C = 1/x
The equation holds true for all x, so it is a solution of the differential equation.
a) To find a solution of the differential equation that satisfies the initial condition y(7)=6, we can use the fact that y=(lnx+C)/x is a solution and plug in the initial condition.
y(7) = (ln7+C)/7 = 6
ln7 + C = 42
C = 42 - ln7
y = (lnx + 42 - ln7)/x
b) To find a solution of the differential equation that satisfies the initial condition y(6)=7, we can use the fact that y=(lnx+C)/x is a solution and plug in the initial condition.
y(6) = (ln6+C)/6 = 7
ln6 + C = 42
C = 42 - ln6
y = (lnx + 42 - ln6)/x
Hence, the solutions are:
a) y = (lnx + 42 - ln7)/x
b) y = (lnx + 42 - ln6)/x
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