Answers
This, we can form the following equation to find the measure of angle 1:
38°+∠1=180°
Subtract 38 from both sides:
∠1=180-38
∠1=142°
Now, we know the measure of angle 1. Angle 1 and angle 5 are alternating interior angles. By the Alternate Interior Angles Theorem, alternate interior angles are always congruent.
Therefore, ∠1=∠5; 142°=∠5
Answer: m∠5=142°
Related Questions
Use point-slope form to write the equation of a line that passes through the point (−11,−13) with slope -2/3 .
Answers
Answer:
y + 13 = - [tex]\frac{2}{3}[/tex] (x + 11)
Step-by-step explanation:
the equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b ) a point on the line
here m = - [tex]\frac{2}{3}[/tex] and (a, b ) = (- 11, - 13 ) , then
y - (- 13) = - [tex]\frac{2}{3}[/tex] (x - (- 11) ) , that is
y + 13 = - [tex]\frac{2}{3}[/tex] (x + 11)
Michelle is now 50 miles ahead of John.
Michelle is traveling at a constant rate. John is traveling in the same direction, at a rate 10 miles per hour faster than Michelle. In how many hours will John catch up to Michelle?
A. 6
B. 5
C. 2
D. 0
E. John can't catch up to Michelle
Answers
Let's call Michelle's speed "M" and John's speed "J". We know that John's speed is 10 miles per hour faster than Michelle's speed, so we can express this as:
J = M + 10
We also know that Michelle is 50 miles ahead of John, so we can express this as:
Distance = 50 miles
Now we can use the formula:
Distance = Rate x Time
We want to know how long it will take John to catch up to Michelle, so we can call this time "t". We can use the formula for both Michelle and John, and set their distances equal to each other since they will meet at the same point:
M * t + 50 = J * t
Now we can substitute J with M + 10, and simplify:
M * t + 50 = (M + 10) * t
M * t + 50 = M * t + 10t
50 = 10t
t = 5
Therefore, John will catch up to Michelle in 5 hours (answer choice B).
can you solve this question?
x=?
the value of this limit=?
y=?
Answers
The derivative of f(x) = 3·x² + 7·x + 6, at x = 4, f'(4) is presented as follows;
f'(4) is the limit as x → 4 of the expression 6·x + 7.
The value of this limit is 31
The equation of the tangent line to the parabola y = 3·x² + 7·x + 6 at the point (4, 82) is y = 31·x - 42
What is the derivative of function?The derivative of a function is a measure of how much the output values of the function changes as the input value is changed. The derivative is the limit of the difference quotient as the change in input approaches zero. The limit is the instantaneous rate of change of the function at a specified input variable value.
The value of f'(4) using the definition of derivative, can be obtained using the following definition;
f'(x) = lim(h → 0)[f(x + h) - f(x)]/h
Plugging in x = 4, and f(x) = 3·x² + 7·x + 6, we get;
f'(4) = lim(h → 0)[f(4 + h) - f(4)]/h
f'(4) = lim(h → 0)[3·(4 + h)² + 7·(4 + h) + 6 - (3·(4)² + 7·(4) + 6)]/h
f'(4) = lim(h → 0)[(3·h + 31)·h]/h
f'(4) = lim(h → 0)[(3·h + 31)]
Therefore;
f'(4) = lim(h → 0)[(3·h + 31)] = 31
f'(4) = 31
Therefore; f'(4) is the limit as x → 4 of the expression 6·x + 7, therefore. The value of this limit is 31
The point-slope form of the equation of a line can be used to find the equation of the parabola as follows;
y - y₁ = m·(x - x₁)
The point (x₁, y₁) and the slope of the line is m
The point on the parabola of the tangent is; (4, 82)
The slope of the tangent line at x = 4, f'(4) = 31
The tangent equation is therefore;
y - 82 = 31·(x - 4)
y = 31·(x - 4) + 82 = 31·x - 42
The equation of the tangent line to the parabola, y = 3·x² + 7·x + 6, at the point (4, 82) is; y = 31·x - 42
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Find the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010. Round your answer to three decimal places, if necessary.
Answers
The value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.
With its bell-shaped structure and heavier tails, the t-distribution, commonly referred to as the Student's t-distribution, is a kind of probability distribution that resembles the normal distribution. When there are insufficient samples or unknown variances, it is used to estimate population parameters. T-distributions have broader tails than normal distributions because they are more likely to contain extreme values.
To find the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010, we can use a t-table or a calculator. Using a calculator, we can use the inverse t-distribution function. The inverse t-distribution function gives us the value of t for a given probability and degrees of freedom.
Using this function, we have:
t = invT(0.010, 45) ≈ -2.326
Rounding this to three decimal places gives us the answer:
t ≈ -2.326
Therefore, the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.
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Marie plants flowers in a planter that is 3 1/2 feet long and 2 2/3feet wide. She plans to cover the entire area with fertilizer. How much area will she need to spread with fertilizer?
Answers
Answer:
The answer would be 9 and 1/3
Step-by-step explanation:
3 and 1/2 times 2 and 2/3 gives you 9.3333333 which is rounded to 9 and 1/3.
f(x)=3x^3+5x^2-11x+3
Answers
Polynomials are functions that are constructed from a sum of powers of the independent variable x, multiplied by coefficients. In this case, we have powers of x from 0 to 3, and the coefficients are 3, 5, -11, and 3.
To evaluate the function for a particular value of x, we substitute that value in place of x and perform the necessary arithmetic. For example, to find f(2), we substitute x = 2 in the expression for f(x):
f(2) = 3(2)^3 + 5(2)^2 - 11(2) + 3
= 24 + 20 - 22 + 3
= 25
Therefore, f(2) = 25. We can similarly evaluate the function for other values of x.
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please try to answer all questions below** TYYY
Answers
1.The equation that represents the proportional relationship is y = 4x + 3.
2. The corresponding equation that represents a proportional relationship is y = (1/5)x.
3. y = 7/2x.
4. when y = 21 is x = 6.
5. y = 8 is x = 3.33.
What is slope?The slope of a function is the rate of change in the function's output (y-value) relative to the change in its input (x-value).
The equation that represents a proportional relationship is y = mx + b, where m is the slope of the equation.
In this equation, x and y are in direct proportion.
1.The equation that represents the proportional relationship is y = 4x + 3. This equation is in the form of y = mx + b, with m being the coefficient of x, which is 4, and b being the constant, which is 3.
2. The corresponding equation that represents a proportional relationship is y = (1/5)x.
This equation is in the form of y = mx + b, with m being the coefficient of x, which is 1/5, and b being the constant, which is 0.
3. The equation that represents this relationship is y = 7/2x.
This equation is in the form of y = mx + b, with m being the coefficient of x, which is 7/2, and b being the constant, which is 0.
4. The value of x when y = 21 is x = 6.
This is because the equation representing the proportional relationship is y = 7/2x, and
when y = 21, 21 = 7/2x,
so x = 6.
5. The value of x when y = 8 is x = 3.33.
This is because the equation representing the proportional relationship is y = 12/5x, and when y = 8, 8 = 12/5x, so x = 3.33.
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Now that you have learned about the addition and subtraction of polynomials, it is time to learn about multiplication. What is the process for adding and subtracting polynomials? Do you think that process will be the same for multiplication?
Answers
No, the process for multiplying polynomials is different from adding and subtracting.
What exactly are polynomials?
Polynomials are algebraic expressions made up of variables and coefficients that are joined using the addition, subtraction, and multiplication operations. The variables in a polynomial can be raised to non-negative integer powers.
For example, the expression 3x² - 2x + 1 is a polynomial, where 3, -2, and 1 are the coefficients, and x², x, and 1 are the variables with their respective powers.
Now,
The process for adding and subtracting polynomials involves combining like terms. To add or subtract two polynomials, we simply combine the coefficients of the same degree terms.
For example, to add the polynomials 2x² + 3x + 4 and 4x² - 2x - 1, we group the like terms and add the coefficients:
(2x² + 4x²) + (3x - 2x) + (4 - 1) = 6x² + x + 3
To subtract the polynomial 4x² - 2x - 1 from the polynomial 2x² + 3x + 4, we change the sign of the second polynomial and then combine the like terms:
(2x² + 3x + 4) - (4x² - 2x - 1) = 2x² + 3x + 4 - 4x² + 2x + 1 = -2x² + 5x + 5
The process for multiplying polynomials is different from adding and subtracting. When we multiply two polynomials, we need to distribute each term of polynomials, and then combine the like terms.
For example, to multiply the polynomials (x + 2) and (x - 3), we use the distributive property:
(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x² - 3x + 2x - 6 = x² - x - 6
As we can see, the process for multiplying polynomials is different from adding and subtracting, but all three operations involve combining like terms in some way.
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
Answers
Answer:
x=14
Step-by-step explanation:
80+58=138
180-138=42
42÷3=x
14=x
Answer:
x= 14
Step-by-step explanation:
80 + 58 = 138°
A triangle has 180°
180-138= 42°
Put into equation
42=3x
42/3=14
X=14
How do you solve the equation absolute value of K +7 equals three
Answers
Answer:
k=-4
Step-by-step explanation:
k+7=3
take way 7 from both sides
k=-4
can you pls answer this for me im really struggling with this
Answers
Answer:
The slope of this line is 2.
Step-by-step explanation:
Start at (1, 0). Go up 4 units, then right 2 units. You will end at (3, 4). The slope of this line is 2.
Use the definition of a logarithm to solve the equation. ln ( − 5 z ) = ln ( z^ 2 − 7 z )
Answers
To solve for z, we can subtract 7 from both sides to get -5/z = -6. Finally, we can multiply both sides by -1 to get z = -7. Therefore, the solution to this equation is z = -7.
A logarithm is an equation that expresses the relationship between an exponent and its base. In this equation, we have two logarithms, ln (-5z) and ln [tex](z^2-7z)[/tex], which are both equal to each other. To solve this equation, we can use the properties of logarithms to isolate the variable. First, we can rewrite the equation as ln (-5z) - ln [tex](z^2-7z)[/tex]= 0, which can then be simplified to ln (-5/z+7) = 0. We can then take the inverse of both sides to get -5/z+7 = 1. To solve for z, we can subtract 7 from both sides to get -5/z = -6. Finally, we can multiply both sides by -1 to get z = -7. Therefore, the solution to this equation is z = -7.
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Nathan is driving to a concert and needs to pay for parking. There is an
automatic fee of $8 just to enter the parking lot, and when he leaves
the lot, he will have to pay an additional $2 for every hour he had his
car in the lot. How much total money would Nathan have to pay for
parking if he left his car in the lot for 6 hours? How much would
Nathan have to pay if he left his car in the lot for t hours?
Cost of parking for 6 hours:
Cost of parking for t hours:
Answers
Answer:
Nathan would have to pay 20 dollars if he parked for 6 hours.
2t+8
Step-by-step explanation:
Use a t-distribution to find a confidence interval for the difference in means =1−2
using the relevant sample results from paired data. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using =1−2
.
A 99% confidence interval for
using the paired data in the following table:
Case 1 2 3 4 5
Treatment 1 23 29 32 24 27
Treatment 2 17 32 24 22 21
Give the best estimate for
, the margin of error, and the confidence interval.
Enter the exact answer for the best estimate, and round your answers for the margin of error and the confidence interval to two decimal places.
best estimate = Enter your answer; best estimate
margin of error = Enter your answer; margin of error
The 99% confidence interval is Enter your answer; The 99% confidence interval, value 1
to Enter your answer; The 99% confidence interval, value 2
.
Answers
The range of the difference in means' 99% confidence level is from -1.42 to 10.02. The true mean difference between Treatments 1 and 2 falls between these two numbers, we can claim with 99% certainty for t-distribution.
We can use a t-distribution to determine a confidence interval for the difference in means using paired data. Prior to determining the mean difference and standard deviation of the differences, we first compute the difference between the paired observations.
Because we are only interested in the mean difference between Treatments 1 and 2, we compute the differences for each pair and get the following outcomes:
6 -3 8 2 6
The sample mean and sample standard deviation of these differences are then computed. The average of these variations is the sample mean.
(6 - 3 + 8 + 2 + 6)/5 = 3.8
The square root of the sum of squared differences divided by the degrees of freedom yields the sample standard deviation.
[tex]\sqrt{[(6 - 3.8)^2 + (-3 - 3.8)^2 + (8 - 3.8)^2 + (2 - 3.8)^2 + (6 - 3.8)^2]/(5-1)) } = 3.06[/tex]
The 99% confidence interval for the mean difference can then be determined using the t-distribution. Our sample size is tiny (n=5), so we utilise a t-distribution with four degrees of freedom.
The sample mean, which is 3.8, provides the most accurate approximation of the mean difference.
We must determine the crucial value of t for a 99% confidence interval with 4 degrees of freedom in order to determine the margin of error. The crucial value, which we determine using a t-table, is 4.604.
The margin of error is:
[tex]4.604 * (3.06/\sqrt{5}) = 5.22[/tex]
Lastly, by deducting and adding the margin of error from the sample mean, we can determine the confidence interval:
3.8 - 5.22 = -1.42
3.8 + 5.22 = 10.02
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A laboure digs a pit 6.5 m long, 3 m wide and 1.6 m deep. How much earth is du. out from it ?
Answers
Answer:
Volume =
Step-by-step explanation:
Volume = length x width x depth
Volume = (6.5 x 3 x 1.6)m
Volume = 31.2m
Will give brainly
Trig
Answers
Step-by-step explanation:
angle c = 180 -19 - 139 = 22 degrees ( interior angles of a triangle sum to 180 degrees)
Now you can use law of sines to find the missing side lengths
12 / sin 22 = DC /sin19
DC = sin 19 * 12 / sin 22 = 10.4 units
12/sin 22 = BC / sin 132
BC = sin132 * 12 / sin 22 = 23.8 units
John ran up and $88 Bill last Saturday the service was excellent so we decided to leave a 30% tip for the waitress how much was his tip
Answers
$26.40
ten percent is 88 divided by 10= 8.8
8.8 multiplied by 3 is 26.40
x^2+10x-1
x^2+8x-2
find the perfect square it should be in (x+/-_)(x+/-_) form
Answers
In the diagram below, ABC~ DBE. If AD = 24, DB = 12, and DE = 4, what is the length of
AC?
Answers
Answer:
Step-by-step explanation:
because 110
I’ll give you lots of points for these last two questions
Answers
Answer: 1. (8b + 5)
2. (22p - 9)
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Step-by-step explanation:
A bag contains white marbles and yellow marbles, 49 in total. The number of white marbles is 1 more than 5 times the number of yellow marbles. How many white marbles are there?
Answers
Answer:
there are 41 white marbles in the bag.
Step-by-step explanation:
Let's use the variable w to represent the number of white marbles and y to represent the number of yellow marbles.
From the problem, we know that:
w + y = 49 (since there are 49 marbles in total)
And we also know that:
w = 5y + 1 (since the number of white marbles is 1 more than 5 times the number of yellow marbles)
Now we can use substitution to solve for w:
w + y = 49
(5y + 1) + y = 49 (substitute w = 5y + 1)
6y + 1 = 49 (combine like terms)
6y = 48 (subtract 1 from both sides)
y = 8 (divide both sides by 6)
Now we know there are 8 yellow marbles. We can use this information to find the number of white marbles:
w = 5y + 1
w = 5(8) + 1
w = 41
A shuffleboard disk is accelerated to a speed of 5.6 m/s and released. If the coefficient of kinetic friction between the disk and the concrete court is 0.34, how far does the disk go
before it comes to a stop? The courts are 14.3 m long.
Answers
Answer:
Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).
Step-by-step explanation:
We can start by using the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy. In this case, we can assume that the initial kinetic energy of the disk is entirely converted to work done by friction, which causes the disk to come to a stop. The equation can be written as:
Work done by friction = Change in kinetic energy
The work done by friction can be calculated using the formula:
Work = force x distance
The force of friction can be found using the formula:
Force of friction = coefficient of friction x normal force
The normal force is equal to the weight of the disk, which can be found using the formula:
Weight = mass x gravity
Substituting the values given in the problem, we get:
Weight = mass x gravity = 0.75 kg x 9.81 m/s^2 = 7.3575 N
Force of friction = coefficient of friction x normal force = 0.34 x 7.3575 N = 2.4985 N
Work done by friction = Force of friction x distance
We can solve for the distance by rearranging the equation as:
Distance = Work done by friction / Force of friction
The initial kinetic energy of the disk can be found using the formula:
Kinetic energy = 0.5 x mass x velocity^2
Substituting the values given in the problem, we get:
Kinetic energy = 0.5 x 0.75 kg x (5.6 m/s)^2 = 11.76 J
Using the work-energy principle, we know that the work done by friction is equal to the change in kinetic energy, which is:
Work done by friction = Kinetic energy = 11.76 J
Substituting this value and the force of friction into the distance formula, we get:
Distance = Work done by friction / Force of friction = 11.76 J / 2.4985 N = 4.71 m
Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).
What is the nth term for the sequence 1, 8, 15, 22, 29
Answers
Answer:
[tex]a_{n}[/tex] = 7n - 6
Step-by-step explanation:
there is a common difference between consecutive terms , that is
8 - 1 = 15 - 8 = 22 - 15 = 29 - 22 = 7
this indicates the sequence is arithmetic with nth term
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 1 and d = 7 , then
[tex]a_{n}[/tex] = 1 + 7(n - 1) = 1 + 7n - 7 = 7n - 6
Answer:
7n-6
Step-by-step explanation:
Work out the difference of the sequence:
8-1=7
Now you have the first part of the equation: 7n
n is the number that the integer is on the sequence
In this case:
1 = 1 as 1 is the first number of the sequence
And 2 = 8 as 8 is the 2nd number of the sequence
To find the full equation:
Do 7x1 to get you 7
Now see how far the 1st number is from 7
In this case you would do:
7-1 which gives you 6
Since you subtracted it to find the difference, it would be:
- 6
Therefore your answer would be 7n-6
To check it:
Times 7 by let's say 3 to get you 21
Then subtract 6 to get 15.
This is proven right as the 3rd number of the given sequence is 15.
Hope this helped
there is 30 students in tthe gym if there are at least 16 girls write an inequalitly
Answers
The number of girls in the gym must be: g ≥ 16
How to write the in equality?Let's define the variable "g" to be a representation of the number of girls in the gym.
We know that there are 30 students in total. Therefore, the number of boys in the gym will be:
b = 30 - g
We also know that there are at least 16 girls in the gym. So, we can write the inequality:
g ≥ 16
This inequality means that the number of girls in the gym must be greater than or equal to 16.
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emily invests $6,398 in a retirement account with a fixed annual interest rate compounded continuously .After 16 years the balance Reaches $9,483.80. What is the interest rate of the account?
Answers
Consequently, the retirement account's income rate is roughly 3.8%. (rounded to one decimal place).
What is an interest example?Consider borrowing $1,000 at a 10% interest rate for seven years. Your interest for the first year would be $100. Your interest payment for the following year would be made up of the original sum plus interest, or $1,100. As a result, your income for the following year would be $110 ($1,100 multiplied by 0.10).
The interest rate can be calculated using the continuous compounding formula:
[tex]A=P e^{r t}[/tex]
where:
A = final balance = $9,483.80
P = initial investment = $6,398
r = rate
t = time in years = 16
Substituting the given values, we have:
$9,483.80 = $6,398[tex]e^{r16}[/tex]
Dividing both sides by $6,398, we get:
1.4829 = [tex]e^{r16}[/tex]
Using the simple logarithm of both parts, the following is obtained:
ln(1.4829) = r × 16
Solving for r, we get:
r = ln(1.4829)/16
r ≈ 0.038
Therefore, the interest rate of the retirement account is approximately 3.8% (rounded to one decimal place).
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Enter an expression equivalent to
d^8
——
d^3
in the form, d^n
Answers
From the expression, the form of the d⁵ is provided by the stated assertion.
What does an arithmetic the expression mean?A group of words joined with the actions +, -, x, or form an expression, such as 4 x 3 or 5 x 2 3 x y + 17. A statement containing the equals symbol, such as 4 b 2 = 6, says that two formulas are equivalent in value and is known as an equation.
Describe expression using an illustration.As an illustration, the expression x + y is one where both x and y have words with an addition function in between. There are two kinds of expressions in mathematics: numerical expressions, which only comprise integers, and algebraic expressions, which also include variables.
[tex]d^{(8-3)} = d^5[/tex]
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Enter an expression equivalent to (d^(8))/(d^(3)) in the form, d^(n).
Simplify (Write each expression without using the absolute value symbol)
|x+3| if x>5
Answers
we can simplify |x+3| to x+3 when x is greater than 5. This is the final answer.The absolute value of a number is the distance of the number from zero on a number line, regardless of whether the number is positive or negative.
For example, the absolute value of -5 is 5, because 5 is the distance of -5 from zero on the number line.
In this problem, we are asked to simplify the expression |x+3| without using the absolute value symbol. We are also given the condition that x is greater than 5.
When x is greater than 5, we know that x+3 is also greater than 5+3=8. This is because x is already greater than 5, and adding 3 to it makes it even larger. So, we can say that x+3 is positive when x is greater than 5.
Now, let's consider what the absolute value of x+3 means in this context. Since x+3 is positive when x is greater than 5, the absolute value of x+3 is just x+3 itself. This is because the absolute value of a positive number is just the number itself.
Therefore, we can simplify |x+3| to x+3 when x is greater than 5. This is the final answer.
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Porter is buying t ride tickets at the country fair. He spends d dollars and receives 3 tickets for every dollar he spends. Which is the independent variable and which is the dependent variable?
Answers
The independent variable and the dependent variable are the number of dollars spent and the number of tickets bought
How to determine the independent variable and the dependent variable?Given that we have the following statement:
Porter is buying t ride tickets at the country fair. He spends d dollars and receives 3 tickets for every dollar he spends.The independent variable is the input value
i.e. the number of dollars spent
Similarly, the dependent variable is the output value
i.e. the number of tickets bought
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Use the Pythagorean Theorem to find the lengths of the
sides of the triangle.
26
2x-14
Answers
[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{26}\\ a=\stackrel{adjacent}{2x}\\ o=\stackrel{opposite}{2x-14} \end{cases} \\\\\\ (26)^2= (2x)^2 + (2x-14)^2\implies 676 = (4x^2)+(4x^2-56x+14^2) \\\\\\ 676=4x^2+4x^2-56x+14^2\implies 676=8x^2-56x+196 \\\\\\ 0=8x^2-56x-480\implies 0=8(x^2-7x-60) \\\\\\ 0=x^2-7x-60\implies 0=(x-12)(x+5)\implies x= \begin{cases} ~~ 12 ~~ \checkmark\\ -5 ~~ \bigotimes \end{cases}[/tex]
now, -5 is a valid value for "x", however in this case we can't use it, because that makes one of our sides negative and all sides must be a positive value.
[tex]\stackrel{ 2(12) }{\text{\LARGE 24}}\hspace{5em}\stackrel{ 2(12)-14 }{\text{\LARGE 10}}\hspace{5em}\text{\LARGE 26}[/tex]
A sine function has the following key features:
Period = 4
Amplitude = 3
Midline: y=−1
y-intercept: (0, -1)
The function is not a reflection of its parent function over the x-axis.
Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.
Answers
The resulting graph should have a period of 4, an amplitude of 3, a midline of y=-1, and no reflection over the x-axis.
What are the period and amplitude of a graph?
The period of a function is the smallest distance over which the function repeats itself. In other words, it is the length of one complete cycle of the function. For a sine or cosine function of the form f(x) = a sin(bx) or f(x) = a cos(bx), the period is given by 2π/b.
The most simple sine function considered the parent function, is:
y = sin(x)
That function has:
Midline, also known as rest or equilibrium position: y = 0
Minimum: - 1
Maximum: 1
Amplitude: the distance between a minimum or a maximum and the midline = 1
period: the interval of repetition of the function = 2π
The more general sine function is:
y = Asin(Bx + C) + D
That function has:
Midline: y = D (it is a vertical shift from the parent function)
Minimum: - A + D
Maximum: A + D
Amplitude: A
period: 2π/B
phase shift: C (it is a horizontal shift of from the parent function)
Now, you have to draw the sine function with the given key features:
Period = 4 ⇒ 2π/B = 4 ⇒ B = π/2
Amplitude, A = 3
midline y = - 1 ⇒ D = - 1
y-intercept = (0, -1)
Substitute the know values and use the y-intercept to find C:
y = 3sin(2x/π + C) -1
Substitute (0, -1)
-1 = 3sin(2(0)/π + C) -1
3sin(C) = 0
sin(C) = 0
C = 0
Hence, the function to graph is:
y = 3sin(2x/π ) -1
To draw that function use this:
Maxima: 3(1) - 1 = 3 - 1 = 2, at x = 1 ± 4n (n = 0, 1, 2, 3, ...)
Minima: 3(-1) - 1 = - 3 - 1 = -4
y-intercept: (0, - 1)
x-intercepts: the solutions to 0 = 3sin(πx/2) = - 1
first point of the midline: (0, -1) it is the same y-intercept
Hence, the resulting graph should have a period of 4, an amplitude of 3, a midline of y=-1, and no reflection over the x-axis.
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