Answer:
I think that the answer might be A
Answer:
A
Step-by-step explanation:
^^
Is asymptote the same as discontinuity?
An asymptote is not the same as discontinuity.
What is asymptote and discontinuity?An asymptote is a line or curve to which a function's graph draws closer without touching it. A vertical asymptote is a line that a function approaches but never quite touches. A vertical asymptote isn't always a discontinuity. It only matters if there's something "on the other side" since then how are you supposed to get across that gap without picking up your pencil? For instance ln(x) has a vertical asymptote at x=0 but it's no big thIng. But ln(|x|) (which is just ln(x) together with its "mirror image" on the negative side) is discontinuous.
An asymptote is not the same as discontinuity.
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The height of a triangle is 8 centimeters greater than three times its base. The area of the triangle is 128 square centimeters. What is the base of the triangle?
(Hint: area of triangle=12⋅base⋅height)
40 cm
8 cm
5 1/3 cm
16 cm
Answer:
To solve this problem, we can use the formula for the area of a triangle, which is A=1/2bh, where b is the base and h is the height. In this problem, we are given that the area of the triangle is 128 square centimeters, so we can set up the following equation:
A = 1/2bh
128 = 1/2bh
We are also given that the height of the triangle is 8 centimeters greater than three times the base, so we can set up the following equation:
h = 3b + 8
We can substitute the second equation into the first equation to get the following:
128 = 1/2bh
128 = 1/2(3b + 8)b
128 = 3/2b^2 + 4b
We can then use the quadratic formula to solve for b, the base of the triangle:
b = (-4 +/- sqrt(4^2 - 4(3/2)(128)))/(2(3/2))
b = (-4 +/- sqrt(64 - 192))/3
b = (-4 +/- sqrt(-128))/3
Since the square root of a negative number is not a real number, we know that there is no solution to this equation, which means that the given information is not sufficient to determine the base of the triangle.
Therefore, the correct answer is that the base of the triangle cannot be determined from the information given.
To solve this problem, we can use the formula for the area of a triangle, which is A=1/2bh, where b is the base and h is the height. In this problem, we are given that the area of the triangle is 128 square centimeters, so we can set up the following equation:
A = 1/2bh
128 = 1/2bh
We are also given that the height of the triangle is 8 centimeters greater than three times the base, so we can set up the following equation:
h = 3b + 8
We can substitute the second equation into the first equation to get the following:
128 = 1/2bh
128 = 1/2(3b + 8)b
128 = 3/2b^2 + 4b
We can then use the quadratic formula to solve for b, the base of the triangle:
b = (-4 +/- sqrt(4^2 - 4(3/2)(128)))/(2(3/2))
b = (-4 +/- sqrt(64 - 192))/3
b = (-4 +/- sqrt(-128))/3
Since the square root of a negative number is not a real number, we know that there is no solution to this equation, which means that the given information is not sufficient to determine the base of the triangle.
Therefore, the correct answer is that the base of the triangle cannot be determined from the information given.
Lily finds some dimes and quarters in her piggy bank. How much money (in dollars) does she have if she has 5 dimes and 12 quarters? How much money (in dollars) does she have if she has x dimes and y quarters?
Answer: 7 dollars
Step-by-step explanation:
i think
Answer: A dollar is 100 cents, so you would add the 5 dimes and 12 quarters together. A dime is 10 cents so 10x5 is 50, so you would have 50 cents in dimes. A quarter is 25 cents, so you would multiply 12x25 and you would get 300 and you add those both together to get 350 cents. A dollar is 100 cents so you would have 3 dollars and the remaining 50 cents would stay 50 cents! <33
Step-by-step explanation:
Ridgid motions to prove
Step-by-step explanation:
Kamla tablet in other developed its to the previous yield
Given that D(x) = 2x, select all of the following that are true statements. D( x) is a direct variation. D( x) is a function. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). x is the dependent variable. D(6) = 3
Answer:
Step-by-step explanation:
I reformatted the question to clarify the statements. Please check for accuracy.
Given that D(x) = 2x, select all of the following that are true statements.
1. D( x) is a direct variation.
2. D( x) is a function.
3. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4).
4. x is the dependent variable.
5. D(6) = 3
----------
1. D( x) is a direct variation. YES. D(x) will always be 2 times x.
2. D( x) is a function. YES. Each calculated D(x) will have a unique point.
3. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). YES. Each (x,y) shows a value of y that is 2 times x, as per the equation's instructions.
4. x is the dependent variable. YES. y depends on what value is chosen for x.
5. D(6) = 3 NO. d(6) = 2*(6) or 12
i need help with this
Answer:
v = 6
w=13
Step-by-step explanation:
v = 6 because we know that both of the triangles are the same, they just rotate, but the measure is still the same.
To calculate w, we have the formula
2w + v + 11 = w - 3v +48
We already know v = 6, so put 6 in the v spot
2w + 6 + 11 = w -3(6) + 48
2w + 17 = w - 18 + 48
2w + 17 = w +30
Subtracted 17 from both sides
2w = w + 13
Subtract w from both sides
w= 13
Step-by-step explanation:
to add to the other correct answer :
the symbol means that both triangles are congruent.
and that means that when you turn then so that both can be laid on top of each other in the same direction, they would perfectly cover each other without any overlap whatsoever.
and that means that they must have the same angles and the same side lengths (and also other lengths like heights, bisectors, ...).
therefore,
v + 9 = 15
v = 6
and then
2w + v + 11 = w - 3v + 48
giving us w = 13
as the other answer already shows how.
and ED = QR, of course.
What is the total net worth if the banker pays off the student loans? $219,869 $247,517 $233,693 $252,304.
Given the credit union banker's assets and liabilities, their total net worth is C. $233,693.
The difference between a person's long-term and short-term assets and obligations is that person's net worth (long-term and short-term).
The equity of a company, or the net assets possessed by the stockholders after all obligations have been settled, is comparable to the net value of the company.
Assets:
Car Value = $29,850
Savings Accounts = $12,409
Treasury Bonds = $10,000
Checking Account Balance = $19,419
Home Value = $194,450
Total assets = $266,128
Liabilities:
Car loan = $10,560
Student Loans = $13,824
Credit Card Balance = $8,051
Total liabilities = $32,435
Net worth = $233,693 ($266,128 - $32,435)
Thus, the banker's net worth is $233,693.
Complete question:
The assets and liabilities of a credit union banker are listed below.
What is the total net worth if the banker pays off the student loans?
A. $219,869
B. $247,517
C. $233,693
D. $252,304
(Refer to the image for the table)
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Car Value = $29,850
Savings Accounts = $12,409
Treasury Bonds = $10,000
Checking Account Balance = $19,419
Home Value = $194,450
Total assets = $266,128
Liabilities:
Car loan = $10,560
Student Loans = $13,824
Credit Card Balance = $8,051
Total liabilities = $32,435
Net worth = $233,693 ($266,128 - $32,435)
Thus, the banker's net worth is $233,693.
Congruent triangles. What does x equal?
The value of x for the congruent triangles will be equal to 10.
What is congruency?In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other. The meaning of congruency is that the two shapes are similar as well as equal in size.
Given that the angles are ∠E = 95, ∠F = 32 and ∠R=4x +13. The value of x will be calculated as,
Calculate the value of angle ∠G,
∠G = 180 - 32 - 95
∠G = 53
The value of x will be,
4x + 13 = 53
4x = 40
x = 10
Therefore, the value of x for the congruent triangles will be equal to 10.
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Two piece of metal meaure1 1/6 yard and3/12 yard each. Three time the amount of metal i needed for a project. How many total yard of metal i needed?
The total number of yards needed is (4 + 1/4) yards of metal.
In this case, there are two pieces measuring (1 + 1/6) yard and (3/12) yard; adding the two, the total length is:
L = (1 + 1/6) yard + (3/12) yard
= (1 + 2/12) yard + 3/12yardd
= (1 + 5/12) yard
Since three times that amount of metal is needed, then the total amount of metal that is needed is found by multiplying the total length obtained above by three:
3×L = 3×(1 + 5/12) yard = (3 + 15/12) yard
Writing this as a mixed number:
(3 + 12/12yard + 3/12yard) = (4 + 3/12) yard = (4 + 1/4) yard
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If x = 6, y = 4 and z = 1, evaluate x(y to the power of 2 - z)
The solution to the evaluation of the expression x(y² - z) x = 6, y = 4 and z = 1 is 90
How to solve equationEvaluate:
x(y² - z)
When
x = 6, y = 4 and z = 1
x(y² - z)
substitute the value of x, y and z
= 6(4² - 1)
Applying PEMDAS:
P = parenthesis
E = Exponential
M = Multiplication
D = Division
A = Addition
S = Subtraction
= 6(16 - 1)
= 96 - 6
= 90
In conclusion, the expression x(y² - z) when evaluated gives 90.
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Predict the next three numbers in the pattern 25,5,1,1/5,1/25
The next three number is 1/125, 1/625, 1/3125.
What is pattern?
A recurring arrangement of numbers, forms, colours, and other elements constitutes a pattern in mathematics. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series.
Given pattern is 25, 5, 1, 1/5, 1/25.
Assume first term is a₁ = 25
The second term is a₂ = 5
The third term is a₃ = 1
The fourth term is a₄ = 1/5
The fifth term is a₅ = 1/25
The ratio of each consecutive term is
a₂/a₁= a₃/a₂ = a₄/a₃ = a₅/a₄ = 1/5.
The 6th term is a₆ = (1/25)× (1/5) = 1/125
The 7th term is a₇ = (1/125) × (1/5) = 1/625
The 8th term is a₈ = (1/625) × (1/5) = 1/3125
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What is the median if there are 11 numbers?
The Median for n = 11 numbers will be 6th term of the Observation set.
One measure of central tendency is the median. The median is the number that lies in the middle of the specified range of integers. It separates the higher half from the lower half of a particular data sample. At least half of the observations are above or equal to the median, and at least half of the observations are below the median.
The median can be calculated as follows when there are odd numbers of observations in a data collection:
Step 1: Arrange the sample of data in ascending or descending order.
Step 2: If the number of observations (let's say n) is odd, the middle observation is the median of the given data.
Median = [tex](\frac{n+1}{2} )^{th}[/tex] term.
Here n = 11, So median will be 6th term of Observation set.
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How can you describe the behavior of the graph of a polynomial function at its intercept with even number of multiplicities?
At its intercept with even number of multiplicities, the graph of a polynomial function will have a smooth curve that passes through the intercept point. The curve will loop around the intercept point, increasing and then decreasing in a cyclic manner, before repeating the pattern.
At its intercept with even number of multiplicities, the graph of a polynomial function will have a smooth curve that passes through the intercept point. This is because the multiplicities of the intercept point are even, meaning that the graph will not have a hard turn or corner at the intercept point. Instead, the graph will loop around the intercept point, gradually increasing until it reaches a peak before gradually decreasing and eventually reaching the same point it started at. This cyclic pattern will repeat itself as the graph moves away from the intercept point. The even number of multiplicities means the graph will not have any sharp turns or corners, only a smooth curve that passes through the intercept point.
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a physicist examines 28 sedimentary samples for potassium chloride concentration. the mean potassium chloride concentration for the sample data is 0.158 cc/cubic meter with a standard deviation of 0.0171. determine the 98% confidence interval for the population mean potassium chloride concentration. assume the population is approximately normal. step 1 of 2 : find the critical value that should be used in constructing the confidence interval. round your answer to three decimal places.
The critical value that should be used in constructing the confidence interval is (0.150 , 0.166).
The given parameter is as follows,
t\alpha /2,df = 2.473
Margin of error = E = t\alpha/2,df * (s sqrt n)
= 2.473 * (0.0171 / \sqrt28)
Margin of error = E = 0.008
If the populace preferred deviation (σ) is known, a speculation take a look at carried out for one populace imply is known as one-imply z-take a look at or definitely z-take a look at. A z-take a look at is a speculation take a look at for trying out a populace imply, μ, in opposition to a intended populace imply, μ0.
The 98% confidence interval estimate of the population mean is,
\bar x - E < \mu < \bar x + E
0.158 - 0.008 < \mu < 0.158 + 0.008
0.150 < \mu < 0.166
(0.150 , 0.166)
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The clearing house has resistors that sell for $3.50 each and circuit boards that sell for $2.25 each. Write an inequality that represents how many of each type of electronic equipment can be bought with at most $7.
Answer:
3.50x + 2.25y smaller than or equal 7
Step-by-step explanation:
Let x represent the resistor and y represent the circuit board
Resistor: $3.50
Circuit board: $2.25
We know it can be bought at most for $7, so it will be smaller than or equal to $ 7. So, we have equation
3.50x + 2.25y smaller than or equal 7
, if "y varies directly as x", find the constant of variation and write an equation of direct variation that
relates the two variables.
y = 12 when x = 1/4
Answer:
k = 48 , y = 48x
Step-by-step explanation:
given y varies directly as x then the equation relating them is
y = kx ← k is the constant of variation
to find k use the condition y = 12 when x = [tex]\frac{1}{4}[/tex] , then
12 = [tex]\frac{1}{4}[/tex] k ( multiply both sides by 4 to clear the fraction )
48 = k
constant of variation is then k = 48 , so
y = 48x ← equation of variation
How do you solve algebraic expressions Grade 7?
Algebraic expressions can be solved by using the order of operations. The order of operations is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
To solve an algebraic expression, start by simplifying any parts inside parentheses, then move on to the exponents. Next, perform any multiplications and divisions from left to right. Finally, complete any additions and subtractions from left to right.How solve algebraic expressions Grade 7?The best way to solve algebraic expressions in Grade 7 is to use the Order of Operations (also known as PEMDAS) or BEDMAS. The Order of Operations states that you must first complete any operations inside parentheses, followed by Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Once you have completed the operations in the correct order, you will have the solution to the algebraic expression.
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The figure(Figure 1) is a graph of Ex. The potential at the origin is -140 V . What is the potential at x=3.0m? Expert Answer. Who are the experts?
As per the given origin, the potential at x = 3.0m is 143.0V
The term origin in math is referred as the initial point or the starting point from where we begin our calculations or measurements.
Here we have given that the potential at the origin is -140 V.
And we need to find the potential at x = 3.0m.
While we looking into the given question, we have identified that the region of space has a non-uniform electric field that points in the +x-direction and has magnitude and as the reference point, it take the potential at the origin to be -140 V .
Then the electric potential at x=3.0m is calculated as,
=> V(3) - 140 = 3.0
=> V(3) = 143.0
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y is the number of tickets purchased for a randomly chosen olympic event. is the random variable y discrete or continuous?
The number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
y is the number of tickets purchased for a randomly chosen Olympics event. is the random variable y discrete or continuous?
A random variable is a numerical quantity that is generated by a random experiment.
A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.
A random variable is a number generated by a random experiment.
A random variable is called discrete if its possible values form a finite or countable set.
A random variable is called continuous if its possible values contain a whole interval of numbers.
In Olympics , number of event are there . that are not countable.
therefor, the number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
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A random variable is a numerical quantity that is generated by a random experiment.
A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.
A random variable is called continuous if its possible values contain a whole interval of numbers.
The number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
P(x)= 1/2x^2+3x-4x^3x+6x^4-1 write in standard form, is it polynomial, what degree, type, and leading coefficient is it?
Answer is 1. Polynomial Function is a quadratic, cubic, and of any degree function. It involves only a non - negative powers of x.
Find the solution ?Things to Remember: Leading coefficient is the numerical coefficient of the term with the highest degree.To write a polynomial function in standard form, simply arrange the terms according to the degree of the variable. The number of turning points/solutions of the polynomial function is based on the highest degree of the polynomial function.Polynomial Function is a quadratic, cubic, and of any degree function. It involves only a non - negative powers of x. The roots of the function are the values that when substituted equates the polynomial to 0.These are also the values that intercepts the x-axis with the graph. involves only non-negative integer powers or only positive integer exponents of a variable in a equation like the quadratic equation, cubic equation, and others.To learn more about Polynomial Function refer
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32 x 79 =
PLEASE HELP QUICK
Answer:
2528
Step-by-step explanation:
[tex]32\times79[/tex]
Multiply:
[tex]32\times79[/tex]
[tex]=\fbox{2528}[/tex]
What is parallel to the line y 3x 1?
Equation y = 3x is parallel to the line y = 3x - 1.
Describe parallel lines.Lines in a plane that are consistently spaced apart are known as parallel lines. Parallel lines don't cross each other. Perpendicular lines are those that cross at a straight angle of 90 degrees. If two lines have different y-intercepts and equal slopes, they are said to be parallel. In other words, the reciprocals of perpendicular slopes are negative. Pairs of parallel lines cross one another. If they went on, they'd never run into each other. The separation between parallel lines is always the same.
Given
A line that passes through (0,0) has the equation Y=mX, where m is the slope of the line.
Now that this line is perpendicular to the line Y=3X-1, m is equal to 3.
As a result, the line's equation is Y=3X.
Equation y = 3x is parallel to the line y = 3x - 1.
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Sharonda uses a blend of dark chocolate and milk chocolate to make the ice cream topping at her restaurant. She needs to buy 120\,\text{kg}120kg120, start text, k, g, end text of chocolate in total for her next order, and her recipe calls for twice the amount of dark chocolate as milk chocolate. Let ddd be the number of kilograms of dark chocolate she buys and mmm be the number of kilograms of milk chocolate she buys. Which system of equations represents this situation?.
The equations m = 2d and m+d = 120 represent this situation.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign.
The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals sign. Typically, we consider an equation's right side to be zero.
d = kilograms of dark chocolate
m =kilograms of milk chocolate
So, her recipe calls for twice the amount of dark chocolate as milk chocolate.
Mathematically speaking:
m = 2d
She needs to buy 120 kg, of chocolate in total. So;
m + d = 120
The following system of equations represents the situation:
m = 2d
m+d = 120
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The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals sign. Typically, we consider an equation's right side to be zero.
So, her recipe calls for twice the amount of dark chocolate as milk chocolate.
m = 2d
She needs to buy 120 kg, of chocolate in total. So;
m + d = 120
The following system of equations represents the situation:
m = 2d
m+d = 120
May I ask what’s the answer to this MC question because what I've calculated is 36 to the power 222 which giant matched the above choices
Answer:
B
Step-by-step explanation:
I just answered this question. you posted it multiple times ?
again
16 = 2⁴
9 = 3²
so, we have
(2⁴)¹¹¹ × (3²)²²² = 2⁴⁴⁴ × 3⁴⁴⁴ = (2×3)⁴⁴⁴ = 6⁴⁴⁴
remember, you can only combine either the exponents of the same base, or the bases of the same exponents (like in our case here).
Find the circumference of a semicircle with diameter d cm
Answer:
Circumference of the semi-circle is d(π/2 + 1)
Step-by-step explanation:
Circumference of a semicircle is πr + 2r
So, since r = d/2,
therefore, π*d/2 + 2*d/2 = πd/2 + d = d(π/2 + 1).
A student was attempting to write an inequality that she could use to find how many more hours she had to work at her job to buy a bicycle. The student had already
saved $55 and earns $9 per hour at their job. The student will need at least $199 to buy the bicycle.
Look at the student's inequality below. Find and explain the student's error in writing her inequality.
h = the amount of hours worked
55+9h ≤ 199
The correct Inequality that shows the above situation is 55 + 9h ≥ 199, the error in the given expression is taking less than or equal to.
What is Inequality:Inequality simply means that two things aren't equal. One of the items might be lower too, greater than, or equivalent to the others.
Mathematical expressions with inequalities are those in which the two sides are not equal.
the signs we use in Inequalities are
≠ means not equal to
< means less than
> means greater than
≤ means less than or equal to
≥ means greater than or equal to
Here we have
The student already saved $ 55
She earns $ 9 per hour
The money that she needs to buy the bicycle is at least $ 199
Let's assume that she worked for h hours to earn at least $ 199
The amount that she earned after h hours = 9h
As we know she already saved $ 55
Finally the amount that she had = 55 + 9h
She wants at least $ 199 but not less than that, so the amount 55 + 9h must be greater than $ 199
=> 55 + 9h ≥ 199
The expression that the student used is 55 + 9h ≤ 199, here error is taking less than or equal to.
Therefore,
The correct Inequality that shows the above situation is 55 + 9h ≥ 199, the error in the given expression is taking less than or equal to.
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The acceleration y, in meters per second squared, of an object after x seconds is given by y =. During the first 10 seconds, over which intervals is the acceleration increasing?.
During the first 10 seconds, in the intervals (0, 2) and (6, 10) the acceleration is increasing.
The acceleration is: y = 7*sin( x*pi/4).
Sin(θ) increases between -pi/2 and pi/2
Sin(θ) decreases between pi/2 and (3/2)*pi.
And so on.
Here we have:
θ = x*(pi/2)
When x = 0, θ = 0.
And sin(θ) is increasing in θ = 0.
Then we must choose one of the options that start with x = 0.
now it will stop increasing when:
θ = pi/2 = x*(pi/4)
x = 2.
So the acceleration is increasing in the segment (0,2)
And it will start increasing again when:
θ = (3/2)*pi = x*(pi/4)
(3/2)*pi*(4/pi) = 6 = x.
So at x= 6 the acceleration starts increasing again, and the acceleration is only defined until x = 10, then
the correct option is: (0, 2) and (6, 10)
Thus, during the first 10 seconds, in the intervals (0, 2) and (6, 10) the acceleration is increasing.
Complete question:
The acceleration y, in meters per second squared, of an object after x seconds is given by y = 7 sin (pi/4 x). During the first 10 seconds, over which intervals is the acceleration increasing?
(2, 6)
(4, 8)
(0, 2) and (6, 10)
(0, 4) and (8, 10)
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The acceleration is: y = 7*sin( x*pi/4).
Sin(θ) increases between -pi/2 and pi/2
Sin(θ) decreases between pi/2 and (3/2)*pi.
And so on.
Here we have:
θ = x*(pi/2)
When x = 0, θ = 0.
And sin(θ) is increasing in θ = 0.
θ = pi/2 = x*(pi/4)
x = 2.
θ = (3/2)*pi = x*(pi/4)
(3/2)*pi*(4/pi) = 6 = x.
So at x= 6 the acceleration starts increasing again, and the acceleration is only defined until x = 10, then
Thus, during the first 10 seconds, in the intervals (0, 2) and (6, 10) the acceleration is increasing.
(2, 6)
(4, 8)
(0, 2) and (6, 10)
(0, 4) and (8, 10)
Tan 32 deg + tan 88 deg / 1 - tan32 tan88 = -√3
Answer:
Step-by-step explanation:
[tex]\displaystyle\\\frac{tan32^0+tan88^0}{1-tan32^0tan88^0} \ \ \ \ (1)\\\\Let\ \alpha =32^0\ and\ \beta =88^0\\\\Hence,\ expression\ (1)\ will \ look\ like\ this:\\\\\frac{tan\alpha +tan\beta }{1-tan\alpha tan\beta }\\[/tex]
The trigonometric formula for adding angles for a tangent is as follows:
[tex]\displaystyle\\\\\boxed {tan(\alpha +\beta )=\frac{tan\alpha +tan\beta }{1-tan\alpha tan\beta }}[/tex]
Thus,
[tex]\displaystyle\\\frac{tan32^0 +tan88^0 }{1-tan32^0 tan88^0 }=tan(32^0+88^0)\\\\\\\frac{tan32^0 +tan88^0 }{1-tan32^0 tan88^0 }=tan120^0[/tex]
Calculate the value of tan120°:
[tex]tan120^0=tan(90^0+30^0)\\\\tan120^0=-cot30&^0\\\\tan120^0=-\sqrt{3}[/tex]
So,
[tex]\displaystyle\\\frac{tan32^0 +tan88^0 }{1-tan32^0 tan88^0 }\equiv-\sqrt{3}[/tex]
How do you describe the relationship of a square root?
Therefore , the relationship of square root can be illustrated as [tex]y^{2}[/tex] = x.
What is square root ?A number's square root is that factor of the number that, when multiplied by itself, yields the original number. Specifically, squares and square roots are exponents. Think of the number nine. This can be expressed as [tex]3^{2}[/tex] or as 3 x 3.
Here,
In math, the square root of a number x is a number y such that y2 = x, or a number y whose square
(the result of multiplication by itself, or y), is x.
As an illustration, 4 and 4 are the square roots of 16, since 42 = (4)2 = 16.
Therefore , the relationship of square root can be illustrated as [tex]y^{2}[/tex] = x
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Indefinite Integrals
f"(x)=6x, f'(-3)=20. f(-3) = -5
f(x)= ?
The function f(x) = x³ - 7x + 1 for the given f''(x) = 6x, f'(-3) = 20 and f(-3) = -5. By using the antiderivative formula, the required function is evaluated.
What is the antiderivative formula?The antiderivative formula for indefinite integrals is
[tex]\int f'(x)dx = f(x) + c[/tex]
or we can also write this as
[tex]\int f^n(x)dx = f^{n-1} + c[/tex]
Calculation:It is given that,
f''(x) = 6x, f'(-3) = 20, f(-3) = -5
Then, by the antiderivative formula, we can write
[tex]\int f''(x)dx = f'(x) + c[/tex] ...(1)
On substituting, we get
[tex]\int f''(x)dx = \int 6x dx = 6\int xdx[/tex]
Since we have the formula for the indefinite integral, [tex]\int x^n dx = \frac{x^{n+1}}{n+1}+c[/tex], we get
[tex]\int f''(x)dx = 6(\frac{x^{1+1}}{1+1})+c[/tex]
⇒ [tex]\int f''(x)dx = 6(\frac{x^2}{2})+c=3x^2+c[/tex]
So, from (1), we get f'(x) = 3x² + c
But we have f'(-3) = 20. So, on substituting x = -3 in the above function f'(x), we get
f'(x) = 3x² + c
⇒ f'(-3) = 3(-3)² + c
⇒ 20 = 3(9) + c
⇒ 20 - 27 = c
∴ c = -7
So, the second derivative function is f'(x) = 3x² - 7.
From the antiderivative expression, we can write
[tex]\int f'(x)dx = f(x) + c[/tex] ...(2)
On substituting f'(x) in the above integral we get
[tex]\int f'(x) dx = \int (3x^2 +-7)dx[/tex]
⇒ [tex]\int f'(x)dx = 3\int x^2dx -7\int dx + c[/tex]
⇒ [tex]\int f'(x) dx = 3(\frac{x^3}{3})-7x+c[/tex]
⇒ [tex]\int f'(x) dx = x^3 - 7x + c[/tex]
From (2), we can write
f(x) = x³ - 7x + c
But we have f(-3) = -5;
Then,
f(-3) = (-3)^3 - 7(-3) + c
⇒ -5 = -27 + 21 + c
⇒ -5 = -6 + c
∴ c = -5 + 6 = 1
Thus, the function f(x) = x³ - 7x + 1.
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