The equation of the tangent line to the graph of y = xlnx at the point (2,2ln2) is y = (ln(2) + 1)x - 2ln2 - 2.
What is tangent?A tangent is a straight line or plane that touches a curve or curved surface at a single point, but does not intersect it at that point. The tangent line can be thought of as the instantaneous rate of change of the curve at that point. In calculus, the slope of the tangent line at a point on a curve is defined as the derivative of the curve at that point. The tangent line is often used to approximate the behavior of the curve near the point of tangency.
To find the equation of the tangent line to the graph of y = xlnx at the point (2,2ln2), we need to use the slope-point form of the equation of a line.
First, we find the slope of the tangent line by taking the derivative of the function y = xlnx:
y = xlnx
y' = ln(x) + 1
At x = 2, we have:
y' = ln(2) + 1
So the slope of the tangent line at (2,2ln2) is ln(2) + 1.
Next, we use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
where (x1, y1) is the given point, and m is the slope we just found.
Substituting x1 = 2, y1 = 2ln2, and m = ln(2) + 1, we get:
y - 2ln2 = (ln(2) + 1)(x - 2)
Expanding and simplifying, we get the equation of the tangent line:
y = (ln(2) + 1)x - 2ln2 - 2
So the equation of the tangent line to the graph of y = xlnx at the point (2,2ln2) is y = (ln(2) + 1)x - 2ln2 - 2.
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James Kamau stocks and sells cabbages, oranges and mangoes in his grocery at Kitengela market. On Monday last week, he sold 55 cabbages, 100 oranges and 95 mangoes making a total sale of sh. 1,625. On Tuesday, he sold 60 cabbages, 120 oranges and 80 mangoes making a total sale of sh. 1,580. On Wednesday, he sold 75 cabbages, 150 oranges and 120 mangoes making a total sale of sh. 2,175. He buys these items from a distributor at sh.3, sh.2 and sh.6 for a cabbage, an orange and a mango respectively. Required: a) Three simultaneous equations connecting the number of units sold and total sales. (3 Marks) b) The selling price for each. (9 Marks) c) The profit that James Kamau made on each of the three days and his total profits.
a) Let x, y, and z be the selling price of one cabbage, one orange, and one mango, respectively.
From the given data, we can write three simultaneous equations:
Monday: 55x + 100y + 95z = 1625
Tuesday: 60x + 120y + 80z = 1580
Wednesday: 75x + 150y + 120z = 2175
b) To find the selling price of each item, we need to solve the system of equations. We can use any method of solving systems of equations, such as substitution or elimination. Here, we will use the elimination method.
Multiplying the first equation by 6, the second equation by -5, and the third equation by 3, we get:
Monday: 330x + 600y + 570z = 9750
Tuesday: -300x - 600y - 400z = -7900
Wednesday: 225x + 450y + 360z = 6525
Adding all three equations, we get:
255x + 450y + 530z = 8385
Dividing both sides by 5, we get:
51x + 90y + 106z = 1677
Now we can use this equation and any of the original equations to solve for one of the variables. Let's use the first equation:
55x + 100y + 95z = 1625
Multiplying both sides by 106 and subtracting 530 times the first equation from it, we get:
76x + 45z = 43
Solving for x, we get:
x = (43 - 45z)/76
Now we can substitute this value of x into any of the previous equations to solve for y and z. Let's use the third equation:
75x + 150y + 120z = 2175
Substituting x, we get:
75[(43-45z)/76] + 150y + 120z = 2175
Simplifying, we get:
43z/2 - 375/2 + 150y = 825
Solving for y, we get:
y = (825 - 43z/2 + 375/2)/150
Now we can substitute the values of x and y into any of the previous equations to solve for z. Let's use the second equation:
60x + 120y + 80z = 1580
Substituting x and y, we get:
60[(43-45z)/76] + 120[(825-43z/2+375/2)/150] + 80z = 1580
Simplifying, we get:
z = 4.6
Substituting z into the equation for y, we get:
y = 3.45
Substituting z and y into the equation for x, we get:
x = 1.5
Therefore, the selling price for one cabbage is sh. 1.5, for one orange is sh. 3.45, and for one mango is sh. 4.6.
c) The profit that James Kamau made on each of the three days and his total profits:
To calculate the profit, we need to subtract the cost of the items from the revenue generated by selling them.
On Monday:
Cost of cabbages = 55 x 3 = 165 shillings
Cost of oranges = 100 x 2 = 200 shillings
Cost of mangoes = 95 x 6 = 570 shillings
Total cost = 935 shillings
Revenue = 1625 shillings
Profit = Revenue - Cost = 1625 - 935 = 690 shillings
On Tuesday:
Cost of cabbages = 60 x 3 = 180 shillings
Cost of oranges = 120 x 2 = 240 shillings
Cost of mangoes = 80 x 6 = 480 shillings
Total cost = 900 shillings
Revenue = 1580 shillings
Profit = Revenue - Cost = 1580 - 900 = 680 shillings
On Wednesday:
Cost of cabbages = 75 x 3 = 225 shillings
Cost of oranges = 150 x 2 = 300 shillings
Cost of mangoes = 120 x 6 = 720 shillings
Total cost = 1245 shillings
Revenue = 2175 shillings
Profit = Revenue - Cost = 2175 - 1245 = 930 shillings
Total profit over three days:
Profit on Monday + Profit on Tuesday + Profit on Wednesday = 690 + 680 + 930 = 2300 shillings
Therefore, James Kamau made a profit of 690 shillings on Monday, 680 shillings on Tuesday and 930 shillings on Wednesday, with a total profit of 2300 shillings over the three days.
a. Find the present value of 350 000 which is due after 10 years if the interest rate is 8% per year
i. compounded annually, or (4 marks) ii. compounded continuously.
The present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, is approximately 157,456.41, while the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, is approximately 147,252.13.
i. To find the present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, we can use the formula:
[tex]PV = FV / (1 + r)^n[/tex]
where PV is the present value, FV is the future value, r is the interest rate, and n is the number of periods.
Substituting the given values, we get:
PV = 350,000 / (1 + 0.08)^10
PV ≈ 157,456.41
Therefore, the present value of 350,000 due after 10 years, compounded annually at an interest rate of 8% per year, is approximately 157,456.41.
ii. To find the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, we can use the formula:
[tex]PV = FV / e^(r * n)[/tex]
where e is the mathematical constant approximately equal to 2.71828.
Substituting the given values, we get:
[tex]PV = 350,000 / e^(0.08 * 10)[/tex]
PV ≈ 147,252.13
Therefore, the present value of 350,000 due after 10 years, compounded continuously at an interest rate of 8% per year, is approximately 147,252.13.
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Choose the three true statements about the angles in the figure
Answer-
answer marked on the picture
Step-by-step explanation:
angle 1 and 55 is not vertical angles thus is the wrong answer
angle 2 can be determined with the given info by subtracting the other two angles to 180
Angle 2 not equal to 125 as the angles are not supplementary from each other
if a particle moves along the curve y=x^(2/3), such that dx/dt=3 for all x, find: a.) dy/dt when x=-1 and c.) lim dy/dt as x goes to infinity
Answer:
Step-by-step explanation:
y=x23 so the dt is moved to perticlulate the dy
Which statement describes the relationships between x and y in these two equations? y = 10x y = x + 10
When graphed, an exponential decay function shows that as x increases, f(x) approaches O; and. as x decreases. f(x) increases. True or false?
the statement that "as x decreases, f(x) increases" is not true for an exponential decay function.
Define exponentAn exponent, also known as a power or index, is a mathematical operation that indicates how many times a number (the base) is multiplied by itself.
When graphed, an exponential decay function shows that
as x increases
f(x) approaches 0
but as x decreases
f(x) also approaches 0.
The function decreases from left to right, meaning that as x increases, f(x) decreases towards 0. However, it does not increase as x decreases.
In general, an exponential decay function can be represented as f(x) = a ×e⁻ᵇˣ, where a and b are constants. As x gets larger and larger, the exponential term e⁻ᵇˣ approaches 0, which causes the overall function value f(x) to approach 0 as well. As x gets smaller and smaller, the exponential term e⁻ᵇˣ becomes larger, but the function value f(x) still approaches 0.
Therefore, the statement that "as x decreases, f(x) increases" is not true for an exponential decay function.
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15 points will mark as brainlist !! please help
Anna borrowed N$20 000 at 5% for three and half years. She wants to pay N$8000 on maturity. To achieve
this, she is planning to pay 2000 in 10 months, 5000 in 16 months from now. How much should she pay in two
and half years from now to meet her obligation?
Anna should pay N$289,711.85 in two and a half years from now to meet her obligation.
What is interest rate?An interest rate is the rate at which a borrower pays to a lender for the use of borrowed money. It is expressed as a percentage of the principal amount borrowed and is typically calculated on an annual basis.
According to question:To solve this problem, we can use the formula for the future value of an annuity:
FV = [tex]P * [(1 + r)^n - 1]/r[/tex]
Where FV is the future value of the annuity, P is the periodic payment, r is the interest rate per period, and n is the total number of periods.
We know that Anna borrowed N$20,000 at 5% for three and a half years, which is equivalent to 42 months. The periodic payment, P, is the sum of N$2,000 and N$5,000, which is N$7,000.
First, we can calculate the future value of the annuity after 42 months:
FV = 7,000 * [(1 + 0.05/12)[tex]^42[/tex] - 1]/(0.05/12)
FV = 7,000 * 56.3743
FV = N$394,120.10
Since Anna wants to pay N$8,000 on maturity, she will need to pay N$386,120.10 at the end of the 42 months to meet her obligation.
Now we can calculate how much Anna should pay in two and a half years from now, which is equivalent to 30 months:
[tex]PV = FV / (1 + r)^n[/tex]
PV = 386,120.10 / (1 + 0.05/12)[tex]^30[/tex]
PV = N$289,711.85
Therefore, Anna should pay N$289,711.85 in two and a half years from now to meet her obligation.
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What is the radian value for 2/3 pi?
Answer:120 degrees
Step-by-step explanation:
2pi = 360 degrees
pi = 180 degrees
2pi/3 = 120 degrees
Which of the following relations is a function? A. (7, 1), (-2, 4), (4, 1), (7, 2) B. (4, 0), (-2, 3), (7, 1), (-2, 5) C. (4, 4), (-2, 2), (7, 1), (-7, 2) D. (4, 4), (-2, 6), (4, 3), (-7, 2)
C. (4, 4), (-2, 2), (7, 1), (-7, 2)
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.[1] The set X is called the domain of the function[2] and the set Y is called the codomain of the function
the breaking strength of a rivet has a mean value of 9,950 psi and a standard deviation of 502 psi. (a) what is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,850 and 10,150? (round your answer to four decimal places.) (b) if the sample size had been 15 rather than 40, could the probability requested in part (a) be calculated from the given information? explain your reasoning. - yes, the probability in part (a) can still be calculated from the given information.
- no, n should be greater than 30 in order to apply the central limit theorem. - no, n should be greater than 20 in order to apply the central limit theorem. - no, n should be greater than 50 in order to apply the central limit theorem.
a) The probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,850 and 10,150 is equal to 0.8903.
b) No, because n should be greater than 30 in order to apply the central limit theorem. So, right choice for answer is option (b).
The normal distribution forms a bell shaped curve. The parameter of normal distribution is [tex]\mu[/tex] and [tex]\sigma ^2 [/tex]. It is a continuous distribution. Now, Mean value of breaking strength of a rivet = 9,950
Standard deviations = 502
Let's X be variable denotes breaking strength of a rivet.
a) Now, a random sample is choosen from X such that, Sample size, n = 40. By using central limit theorem, sample mean follows approximately normal distribution with mean 9950 and standard deviations = 502/√40 = 79.3732 that is [tex]\bar X \: \tilde \: \: N( 9950, ( 79.3732)²)[/tex]
[tex]P( 9,850 < \bar X < 10,150 ) = P( \bar X < 10,150) -
P( \bar X < 9,850) \\ [/tex]
= [tex] P(\frac{\bar X - \mu }{\sigma} < \frac{10,150 - 9950}{79.3732} )- P(\frac{\bar X - \mu }{\sigma} < \frac{10,150 - 9850}{79.3732}) \\ [/tex]
= P( Z < 2.52 ) - P( Z < -1.26)
= 0.9941 - 0.1038
= 0.8903
b) If sample size,n is 15 instead of 40, then the probability requested in part (a) will not be calculated from the provide information. Because, for applying central limit theorem we have sample size must be grater than 30. So, option(b) is correct.
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Every day Jin reads for 0.75
hours in the morning and 1.25
hours in the evening. He uses the expression 0.75d+1.25d
to keep track of the number of hours he has read for any number of days, d
. If Jin reads for 20
days, how many hours has he read?
Answer:
Step-by-step explanation:The number of hours Jin reads in 20 days is 39.8 hours.
The expression for the given situation is 0.75d + 1.24d.
What is an expression?
An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.
In the given expression 0.75d + 1.24d, d is number of days.
Substitute d=20 in 0.75d + 1.24d, we get
0.75(20) + 1.24(20)
15+24.8
=39.8 hours
Hence, the number of hours Jin reads in 20 days is 39.8 hours
Margo borrows $1300, agreeing to pay it back with 3% annual interest after 14 months. How much interest will she pay?
Round your answer to the nearest cent, if necessary.
Margo will pay approximately $45.50 in interest on the loan.
What is interest?
Borrowers must pay lenders simple interest as a fee in exchange for a loan.
First, we need to calculate the interest that Margo will pay on the loan.
Interest = Principal x Rate x Time
where:
Principal = $1300
Rate = 3% per year (or 0.03)
Time = 14/12 years (since there are 12 months in a year)
Plugging in the values, we get:
Interest = $1300 x 0.03 x (14/12)
Interest ≈ $45.50
Therefore, Margo will pay approximately $45.50 in interest on the loan.
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help I don't understand please show work
11 is the value of x in this polygon.
What exactly is a polygon in mathematics?
A closed, two-dimensional, flat, closed polygon is a shape that is constrained by geometry and has straight sides. Its sides don't curve inward at all.
Another term for a polygon's sides is its polygonal edges. The points where two sides converge are known as a polygon's vertices (or corners).
2x - 4/6 = 9/3
(2x - 4)3 = 6 * 9
2x - 4 = 54/3
2x - 4 = 18
2x = 18 + 4
2x = 22
x = 11
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Find the inverse and original steps of the inverse equation
Step-by-step explanation:
To find the inverse of an equation, we need to switch the roles of the dependent variable and the independent variable. In other words, if we have an equation of the form y = f(x), we need to rewrite it as x = f^{-1}(y), where f^{-1}(y) is the inverse function of f.
Once we have found the inverse equation, we can perform the original steps of the inverse equation by plugging in the output of the inverse function into the original equation.
For example, let's say we have the equation y = 2x + 3. To find the inverse equation, we first switch the roles of x and y to get:
x = 2y + 3
Next, we solve for y in terms of x:
x - 3 = 2y
(y = x - 3)/2
So the inverse equation is y = (x - 3)/2.
To perform the original steps of the inverse equation, we can plug the output of the inverse function, (x - 3)/2, into the original equation, y = 2x + 3:
y = 2((x - 3)/2) + 3
y = x - 3 + 3
y = x
We have arrived back at the independent variable, x, so the inverse and original steps have canceled each other out, as expected.
a) The temperature the day after Hurricane Katrina in August 2005 was 91°F. It then fell three degrees, rose 5 degrees and fell another 2°F. What was the final temperature?
23 cats and 27 dogs were entered into a cat and dog show. What percentage of the entries were dogs?
What is the meaning of "we number the vertices counterclockwise 0, 1, ..., n − 1, and number the axes of symmetry counterclockwise, 0, 1, ..., n − 1 "?
The statement "we number the vertices counterclockwise 0, 1, ..., n − 1, and number the axes of symmetry counterclockwise, 0, 1, ..., n − 1" is likely referring to a specific mathematical context, such as a polygon or a regular solid.
What is the meaning of "we number the vertices counterclockwise 0, 1, ..., n − 1?In this context, "numbering the vertices counterclockwise" means that the vertices (or corners) of the shape are given labels or numbers in a particular order, moving counterclockwise around the shape. For example, in a triangle, the vertices might be labeled as 0, 1, and 2, in counterclockwise order around the triangle.
Similarly, "numbering the axes of symmetry counterclockwise" means that the axes of symmetry (or lines of reflection) in the shape are also given labels or numbers in a particular order, moving counterclockwise around the shape. For example, in a regular pentagon, there are 5 axes of symmetry that can be labeled as 0, 1, 2, 3, and 4 in counterclockwise order around the pentagon.
By establishing this numbering convention, mathematicians and scientists can more easily communicate about the properties and characteristics of the shape or object, and can refer to specific vertices or axes of symmetry in a standardized way.
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A university is trying to determine what price to charge for tickets to footbal games. At a price of $20 per ticket, attendance averages 40,000 people per game. Every decrease of $5 adds 10,000 people to the average number.
Every person at the game spends an average of $5.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?
The price that maximizes revenue is $10 per ticket, which will bring in $1,500,000 in revenue. 60,000 people will attend at that price.
What is price?Price is the amount of money that is required to purchase or obtain a good or service. It is a numerical value that is assigned to a particular product or service and is used to indicate its value in the market. The price of a product or service is determined by a variety of factors, including the cost of production, supply and demand, competition, and market conditions.
In the given question,
To maximize revenue, we need to find the price per ticket that will bring in the greatest total revenue. Let's start by figuring out how many people will attend at various price points:
At $20 per ticket, 40,000 people attend.
At $15 per ticket, 50,000 people attend.
At $10 per ticket, 60,000 people attend.
At $5 per ticket, 70,000 people attend.
Now we can calculate the total revenue at each price point:
At $20 per ticket, revenue is 40,000 × $20 + 40,000 × $5 = $1,000,000.
At $15 per ticket, revenue is 50,000 × $15 + 50,000 × $5 = $1,250,000.
At $10 per ticket, revenue is 60,000 × $10 + 60,000 × $5 = $1,500,000.
At $5 per ticket, revenue is 70,000 × $5 + 70,000 × $5 = $700,000.
From these calculations, we can see that the price that maximizes revenue is $10 per ticket, which will bring in $1,500,000 in revenue. 60,000 people will attend at that price.
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I need help with this
Answer:
g(0)= -1/3
g(-1)=0
g(1)=-1
g(2)=-3
g(3)=∞
Step-by-step explanation:
g(0)= -1/3
g(-1)=0
g(1)=-1
g(2)=-3
g(3)=∞
In the xy-plane, a line crosses the y-axis at the point (0, 3) and passes through the point (4, 5). Which of the following is an equation of the line?
Answer:
y = 1/2 x + 3
Step-by-step explanation:
y = mx + b
You need to find the slope (m) and the y-intercept (b) to write the equation
The slope is the change in y over the change in x.
(0,3) (4,5) The y values are 5 and 3. The x values are 4 and 0. You find the change by subtracting.
[tex]\frac{5-3}{4-0}[/tex] = [tex]\frac{2}{4}[/tex] = [tex]\frac{1}{2}[/tex]
The slope (m) is 1/2.
The y-intercept is the point (0,b). They give us this with the point (0,3) The y-intercept (b) is 3
y = 1/2 x + 3
Helping in the name of Jesus.
HELP ASAP PLEASE URGENT
By answering the presented question, we may conclude that Option D is false because function addition is not commutative in general.
what is function?In mathematics, a function seems to be a link between two sets of numbers in which each member of the first set (known as the domain) corresponds to a specific member of the second set (called the range). In other words, a function takes input from one collection and creates output from another. The variable x has frequently been used to represent inputs, whereas the variable y has been used to represent outputs. A formula or a graph can be used to represent a function. For example, the formula y = 2x + 1 depicts a functional form in which each value of x generates a unique value of y.
The correct statement about functions is:
C. If f is a function, then f(2 + h) = f(2) + f(2 + h) (h)
This is known as the additivity property or the additivity rule. It indicates that evaluating a function at a total of two values is equivalent to evaluating the function at each value individually and then adding the results. This is true for a wide range of functions, including linear and polynomial functions.
Option A is false since the order of function composition matters in general.
Option B is false because matrix multiplication is not always defined unless the matrices' dimensions are compatible.
Option D is false because function addition is not commutative in general.
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PLEASE HURRY WILL GIVE BRAINLIEST AND 5 STARS!!!!!!
To add (or subtract) in Scientific Notation, you must have the same Response area. Then you can Response area the co-efficients and Response area the power of 10.
To multiply in Scientific Notation, you must multiply the Response area and Response area the powers of 10.
To divide in Scientific Notation, you must Response area the co-efficients and Response area the powers of 10.
According to the given information, divide in Scientific Notation, you must divide the co-efficients and subtract the powers of 10.
What is Scientific Notation?
Scientific notation, also known as standard form or exponential notation, is a way of expressing numbers that are either very large or very small, using powers of 10. In scientific notation, a number is expressed in the form:
a x 10ⁿ
where "a" is a number between 1 and 10 (usually a decimal), and "n" is an integer (positive or negative) that represents the power of 10 by which the number is multiplied.
To add (or subtract) in Scientific Notation, you must have the same power of 10. Then you can add (or subtract) the co-efficients and keep the power of 10.
To multiply in Scientific Notation, you must multiply the co-efficients and add the powers of 10.
To divide in Scientific Notation, you must divide the co-efficients and subtract the powers of 10.
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Solve the equation for the specified variable.
Scott has a flat screen TV with an area of A
square inches. The width of the TV is 36 inches. Which equation represents x the length of the TV in inches?
A. X = 36/A
B. X= A + 36
C. X = A - 2(36)
D. X= A/36
Explain here:
Answer:
D
Step-by-step explanation:
A=LW
A=L(36)
Divide by 36
L=A/36
(L=X)
:)
If a coordinate system is set up such that the positive x axis points in a direction 60° above the horizontal, what should be the angle between the x axis and the y axis?
What should be the direction of the positive y axis?
(measured from the horizontal)
The two axes should be perpendicular, so the angle betwen them is 90°, and the y-axis is 150° above the horizontal.
What should be the angle between the x axis and the y axis?If we have a coordinate axis, the two axis should be perpendicular between them, this means that the angle between the positive x-axis and the positive y-axis should always be 90°.
Now, what should be the direction of the positive y axis?
We know that the positive x-axis is 60° above the horizontal, and the angle between the two axes is 90°, then the positive y axis is at:
60° + 90° = 150°
The positive y-axis is at 150° above the horizontal line.
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A number, c, rounded to 1 d.p. is 47.3
Another number, d, rounded to 1 d.p. is 4.6
What are the lower and upper bounds of c - d?
The lower bound of c - d is 42.66 and the upper bound of c - d is 42.7.
Calculating the lower and upper bounds of c - d?Given that
c, rounded to 1 d.p. is 47.3d, rounded to 1 d.p. is 4.6To find the upper and lower bounds of c - d, we first need to find the maximum and minimum possible values of c and d.
For c rounded to 1 decimal place, the actual value of c could be between 47.25 and 47.34.
This is because when we round to 1 decimal place, we take the first decimal place and round up or down depending on the second decimal place.
Similarly, we have
d = 4.59 to 4.64
So, we have
c - d = 47.25 - 4.59 to 47.34 - 4.64
Evaluate
c - d = 42.66 to 42.7
Therefore, the lower bound of c - d is 42.66 and the upper bound of c - d is 42.7.
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Can someone explain 1+8×6091+689+8×879+68×9+7
Answer:
Step-by-step explanation:
1+8×6091+689+8×879+68×9+7
We need to do the multiplications first
1+(8×6091)+689+(8×879)+(68×9)+7=
1+(48728)+689+(7032)+(612)+7
Now we add all of them up
48729 + 7721 + 619 =
57069
Answer is 57069
Find the resolved part of the vector p = (6i-3j+9k) in the direction of a = (2i+2j-k)
The calculated resolved part of vector p (6i-3j+9k) in the direction of a is 2i + 2j - k.
Calculating the resolved part of the vectorGiven that
p = (6i-3j+9k)
a = (2i+2j-k)
To find the resolved part of the vector p in the direction of a, we can use the following formula:
r = (p · a) / |a|
where · denotes the dot product and |a| denotes the magnitude of vector a.
First, we need to calculate the magnitude of vector a:
|a| = √(2² + 2² + (-1)²)
|a| = √(4 + 4 + 1)
|a| = √9
|a| = 3
Next, we need to calculate the dot product p · a:
p · a = (6i - 3j + 9k) · (2i + 2j - k)
= 12i + 12j - 6k - 6i - 6j + 3k
= 6i + 6j - 3k
Finally, we can calculate the resolved part of p in the direction of a:
r = (p · a) / |a|
= (6i + 6j - 3k) / 3
= 2i + 2j - k
Therefore, the resolved part of vector p in the direction of a is 2i + 2j - k.
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1) Find fraction numerator d y over denominator d x end fraction for y equals x squared ln open parentheses fraction numerator x over denominator x plus 3 end fraction close parentheses
Halle dy/dx para y = x^2In(x/x+3)
The result is, [tex]\dfrac{dy}{dx} = x \times ln\dfrac{x}{(x+3)} + 2x\times ln\dfrac{x}{(x+3)} - \dfrac{(3x^2)}{(x+3)^2}[/tex].
The derivative of a function is a measure of how much the function changes as its input (often time or space) changes. More specifically, the derivative is the instantaneous rate of change of the function at a particular point.
To find the derivative of y = x^2ln(x/(x+3)), we use the product rule and the chain rule.
y = x^2 * ln(x/(x+3))
[tex]y' = 2x \times ln\dfrac{x}{(x+3)} + x^2 \times \dfrac{d}{dx}ln\dfrac{x}{(x+3)}[/tex]
To find the derivative of ln(x/(x+3)), we use the quotient rule:
[tex]\dfrac{d}{dx} ln\dfrac{x}{(x+3)} = \dfrac{1}{\dfrac{x}{(x+3)}} \times \dfrac{d}{dx}\dfrac{x}{(x+3)} - \dfrac{1}{\dfrac{x+3}{x}} \times \dfrac{d}{dx}\dfrac{(x+3)}{x}[/tex]
= [(x+3)/(x^2)] - [x/(x+3)^2]
= (x^2 + 3x - x^2)/(x^2(x+3)^2)
= 3(x+1)/(x^2(x+3)^2)
Substituting this back into the original equation, we get:
y' = 2x * ln(x/(x+3)) + x^2 * [3(x+1)/(x^2(x+3)^2)]
= 2x * ln(x/(x+3)) + 3(x+1)/(x+3)^2
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