Answer: Let us call the centers of the four circles C1, C3, C5, and C7, respectively, where the subscript refers to the radius of the circle. Without loss of generality, we can assume that the tangent point A lies to the right of all the centers, as shown in the diagram below:
C7
o-----------o
C5 / \ C3
/ \
o-----------------o
C1
|
|
| l
|
A
Let us first find the coordinates of the centers C1, C3, C5, and C7. Since all the circles are tangent to line l at point A, the centers must lie on the perpendicular bisector of the line segment joining A to the centers. Let us denote the distance from A to the center Cn by dn. Then, the coordinates of Cn are given by (an, dn), where an is the x-coordinate of point A.
Using the Pythagorean theorem, we can write the following equations relating the distances dn:
d1 = sqrt((d3 - 2)^2 - 1)
d3 = sqrt((d5 - 4)^2 - 9)
d5 = sqrt((d7 - 6)^2 - 25)
We can solve these equations to obtain:
d1 = sqrt(16 - (d7 - 6)^2)
d3 = sqrt(4 - (d7 - 6)^2)
d5 = sqrt(1 - (d7 - 6)^2)
Now, let us consider the region S that lies inside exactly one of the four circles. This region is bounded by the circle of radius 1 centered at C1, the circle of radius 3 centered at C3, the circle of radius 5 centered at C5, and the circle of radius 7 centered at C7. Since the circles are all tangent to line l at point A, the boundary of region S must pass through point A.
The maximum possible area of region S occurs when the boundary passes through the centers of the two largest circles, C5 and C7. To see why, imagine sliding the circle of radius 1 along line l until it is tangent to the circle of radius 3 at point B. This increases the area of region S, since it adds more points to the interior of the circle of radius 1 without removing any points from the interior of the other circles. Similarly, sliding the circle of radius 5 along line l until it is tangent to the circle of radius 7 at point C also increases the area of region S. Therefore, the boundary of region S must pass through points B and C.
Using the coordinates we obtained earlier, we can find the x-coordinates of points B and C as follows:
x_B = a - 2 - sqrt(9 - (d7 - 6)^2)
x_C = a + 6 + sqrt(9 - (d7 - 6)^2)
To maximize the area of region S, we want to maximize the distance BC. Using the distance formula, we have:
BC^2 = (x_C - x_B)^2 + (d5 - d3)^2
Substituting the expressions we derived earlier for d3 and d5, we get:
BC^2 = 32 - 2(d7 - 6)sqrt(9 - (d7 - 6)^2)
To maximize BC^2, we need to maximize the expression inside the square root. Let y = d7 - 6. Then, we want to maximize:
f(y) = 9y^2 - y^4
Taking the derivative of f(y) with respect to y and setting it equal to zero, we get:
f'(y) = 18y - 4y^3 = 0
This equation has three solutions: y = 0, y = sqrt(6)/2, and y = -sqrt(6)/2. The only solution that gives a maximum value of BC^2 is y = sqrt(6)/2, which corresponds to d7 = 6 + sqrt(6)/2.
Substituting this value of d7 into our expressions for d1, d3, and d5, we obtain:
d1 = sqrt(16 - (sqrt(6)/2)^2) = sqrt(55/2)
d3 = sqrt(4 - (sqrt(6)/2)^2) = sqrt(19/2)
d5 = sqrt(1 - (sqrt(6)/2)^2) = sqrt(5/2)
Using these values, we can compute the coordinates of points B and C as follows:
x_B = a - 2 - sqrt(9 - (sqrt(6)/2)^2) = a - 2 - sqrt(55)/2
x_C = a + 6 + sqrt(9 - (sqrt(6)/2)^2) = a + 6 + sqrt(55)/2
The distance between points B and C is then:
BC = |x_C - x_B| = 8 + sqrt(55)
Finally, the area of region S is given by:
Area(S) = Area(circle of radius 5 centered at C5) - Area(circle of radius 7 centered at C7)
= pi(5^2) - pi(7^2)
= 25pi - 49pi
= -24pi
Since the area of region S cannot be negative, the maximum possible area is zero. This means that there is no point that lies inside exactly one of the four circles. In other words, any point that lies inside one of the circles must also lie inside at least one of the other circles.
Step-by-step explanation:
Change the subject of each formula to the letter given in brackets.
Here he formula for v is:
v = √(2gh)
What is meant by the term formula?
A formula is a mathematical relationship or rule that is expressed using symbols and mathematical operations. It is used to represent a relationship between quantities or to calculate a value based on given variables or inputs. Formulas are often used in various branches of mathematics, science, and engineering.
According to the given information
To change the subject of the formula mgh = (1/2)mv² to v, we need to isolate v on one side of the equation.
First, we can multiply both sides of the equation by 2 to eliminate the fraction:
2mgh = mv²
Next, we can divide both sides of the equation by m:
(2mgh) / m = v²
Simplifying the left side, we get:
2gh = v²
Finally, we can take the square root of both sides of the equation to solve for v:
v = √(2gh)
Therefore, the formula for v is:
v = √(2gh)
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A rectangle is seven times as long as it’s width. One way to write an expression to find the perimeter would be m+7m+m+7m. Write the expression in two other ways.
Step-by-step explanation:
Perimeter = 2 m + 2 * 7m
Perimeter = 2 ( m + 7m)
Perimeter = 2m + 14m
Perimeter = 16 m Pick any of them
A 35 foot ladder is set against the side of a house so that it reaches up 21 feet. If Elijah grabs the ladder at its base and pulls it 4 feet farther from the house, how far up the side of the house will the ladder reach now? (The answer is not 17 ft.) Round to the nearest tenth of a foot.
The side of the house which the ladder will reach now is equal to 14.2 feet.
What is Pythagorean theorem?In Mathematics and Geometry, Pythagorean's theorem is represented or modeled by the following mathematical equation:
x² + y² = z²
Where:
x, y, and z represents the length of sides or side lengths of any right-angled triangle.
In order to determine how far up the side of the house will the ladder reach, we would have to apply Pythagorean's theorem as follows;
21² + y² = 35²
y² = 1,225 - 441
y² = 784
y = √784
y = 28 feet.
Since Elijah grabs the ladder at its base and pulls it 4 feet farther from the house, we have:
New Length = 28 + 4 = 32 feet.
Therefore, the required length is given by;
Distance = √(35² - 32²)
Distance = 14.2 feet.
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If you needed only 1cup of milk, what is your best choice at the grocery store-a quart container, a pint container or a 1/2 gallon container?
Since 1 gallon = 4 quarts → (1/2) gallons = 2 quarts
1 gallon = 8 pints → (1/2) gallons = 4 pints
1 gallon = 16 cups → (1/2) gallons = 8 cups
Therefore from the given relation with gallons we can describe the relation between the cups, quarts and pints.
(1/2) Gallon > quarts > pints > cups
If we need 1 cup of milk then a pint container will be the best choice at a grocery store.
Answer: 1 pint would because a quart has 4 cups and 1/2 a gallon would be way to much. and a pint would be 2.
Step-by-step explanation:
1 pint would be because a quart has 4 cups and 1/2 a gallon would be way to much. and a pint would 1 cup more than you need so 1 pint is right
50 POINTS
A study was conducted to investigate whether local car mechanics charge women more than men for a transmission repair. The researcher selected randomly one man and one woman from everyone who had used the same mechanic for the same transmission repair. The process was repeated for a total of seven selected randomly cars. The repair prices and the differences are shown in the table.
Car 1 Car 2 Car 3 Car 4 Car 5 Car 6 Car 7
Women $3,550 $3,200 $1,850 $2,000 $3,000 $1,950 $2,250
Men n $3,285 $3,100 $1,975 $2,150 $2,850 $1,750 $2,175
Difference$265 $100 −$125 −$150 $150 $200 $75
Mean Standard Deviation
Women $2,542.86 $691.27
Men $2,469.29 $599.77
Difference $73.57 $157.37
Dotplots of the data and the differences are shown.
(image)
Do the data provide convincing statistical evidence that women pay more than men for the same transmission repair?
we do not have convincing statistical evidence that women pay more than men for the same transmission repair
How to determine if data provide convincing statistical evidence that women pay more than men for the same transmission repairTo determine whether the data provide convincing statistical evidence that women pay more than men we do not have convincing statistical evidence that women pay more than men for the same transmission repair for the same transmission repair, we can perform a hypothesis test.
Null hypothesis: The mean repair cost for women is the same as the mean repair cost for men.
Alternative hypothesis: The mean repair cost for women is greater than the mean repair cost for men.
We can use a two-sample t-test for the difference in means, assuming equal variances.
The test statistic is given by:
t = (xbar1 - xbar2) / (s_p * sqrt(1/n1 + 1/n2))
where xbar1 and xbar2 are the sample means, s_p is the pooled standard deviation, and n1 and n2 are the sample sizes.
The degrees of freedom is given by:
df = n1 + n2 - 2
Using the data given, we have:
xbar = 2542.86, xbar2 = 2469.29
s1 = 691.27, s2 = 599.77
n1 = n2 = 7
First, we can calculate the pooled standard deviation:
s_p = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 + n2 - 2))
s_p = sqrt(((6)*691.27^2 + (6)*599.77^2) / (7 + 7 - 2))
s_p = 644.47
Then, we can calculate the t-statistic:
t = (xbar 1 1 - xbar2) / (s_p * sqrt(1/n1 + 1/n2))
t = (2542.86 - 2469.29) / (644.47 * sqrt(1/7 + 1/7))
t = 0.97
Using a t-distribution table with df = 12 and a significance level of 0.05, the critical value for a one-tailed test is 1.7823. Since our t-statistic is less than the critical value, we fail to reject the null hypothesis.
Therefore, we do not have convincing statistical evidence that women pay more than men for the same transmission repair
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What is the equation of the trend line in
the scatter plot?
Use the two yellow points to write the
equation in slope-intercept form. Write
any coefficients as integers, proper
fractions, or improper fractions in
simplest form.
Answer:
y= 7/3x -7
Step-by-step explanation:
Slope is 7/3 (rise/run or up seven over 3) and when you continue the graph the line will eventually hit your y-intercept at -7
PLEASE HELP ME ITS URGENT MY GRADE NEEDS HELP! THIS IS MULTIPLE CHOICE QUESTION!
Which ordered pairs are solutions to the inequality 2x+3y≥−1?
Select each correct answer.
Responses
(0, −1)
begin ordered pair 5 comma negative 1 end ordered pair
(−2, 1)
begin ordered pair negative 2 comma 1 end ordered pair
(0, 1)
begin ordered pair 0 comma 1 end ordered pair
(−6, 0)
begin ordered pair negative 6 comma 0 end ordered pair
(2, −1)
Answer:
b, c, d, f
Step-by-step explanation:
Which ordered pairs are solutions to the inequality?
2x+3y≥−1?
a≥b means that A must be equal to or greater than B.
Let's do this the long, but easy way, by plugging it in!
(x,y)
a. (0,-1) -> -3≥-1 -> false
b. (5,-1)-> 7≥-1 -> true
c. (-2,1) -> -1≥-1 -> true
d. (0,1) -> 3≥-1 -> true
e. (-6,0) -> -12≥-1 - > false
f. (2,-1) -> 1≥-1 -> true
Let me know if it is incorrect!
- a friendly 8th grader :)
Given: Circle with center D.
Construct: Equilateral triangle ABC so that points A, B, and C are on circle D.
The circle with equilateral triangle is constructed.
What is triangle?
A triangle is a form of polygon with three sides; the intersection of the two longest sides is known as the triangle's vertex. There is an angle created between two sides. One of the crucial elements of geometry is this.
Here we need to construct the circle with center D.
Now we know that in equilateral triangle , side length of the all sides are equal.
Then, AB=BC=CA.
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can someone js help me w these questions
Answer:
a=60 (12*5=60)
b=7 (45/9=5 5+2=7)
c=62
d=25 (11*5=55 55-30=25)
the average score of 100 students taking a statistics final was 70, with a standard deviation of 7. assuming a normal distribution, what test score value separates the top 2.5% of the students from the rest of the students? (show your work)
The average score of 100 students taking a statistics final was 70, with a standard deviation of 7. The test score esteem that isolates the beat 2.5% of normal distribution from the rest of the understudies is roughly 83.72.
To discover the test score esteem that isolates the best 2.5% of the understudies, we got to discover the z-score comparing to that rate utilizing the standard ordinary conveyance table.
z = (x - μ) / σ
To discover the z-score compared to the best 2.5%, we see up the region of the right-hand tail of the standard normal distribution table, which is 0.025. This compares to a z-score of roughly 1.96.
Presently ready to utilize the z-score equation to unravel for x:
1.96 = (x - 70) / 7 Increasing both sides by 7, we get:
x - 70 = 13.72
Including 70 to both sides, we get:
x = 83.72
Hence, the test score esteem that isolates the beat 2.5% of the understudies from the rest of the understudies is roughly 83.72.
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If Joanne has three hours and 45 minutes before swim practice, she wants to go to the store which will take 220 minutes. She also wants to take her brother to the park for 230 minutes which activity does Joanne have time to do before practice
Q3. (Calculator)
Abox contains only red, blue and green pens.
The ratio of red pens to blue pens is 5:9.
The ratio of blue pens to green pens is 1:4.
Calculate the percentage of pens that are blue.
The percentage of pens that are blue =18%
Let's assume that r represents the red pens, b represents the blue pens and g represents the green pens.
From the statements we can observe that the blue appears in both ratio.
The ratios are,
r:b=5:9 and b:g=1:4
Consider ratio b:g,
b:g
= 1:4
= (1×9) : (4×9)
=9:36
i.e., b:g = 9:36 and r:b = 5:9
So, we can conclude that, b=9, g=36, r=5
so, the total number of pens would be,
n = b + g + r
n = 9 + 36 + 5
n = 50
Now we need to find the percentage of pens that are blue.
Using percentage formula,
p = (b/n) × 100
p = (9/50) × 100
p = 18%
Hence, the percentage of blue pens are 18%.
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put on pair of brackets into each calculation to make it correct
a. 6×7-5 +4= 16
b. -2[tex]x^{2}[/tex]+24÷12-4=2
(a) The simplified and correct expression is (6×7)-(5 +4) = 16
(b) The simplified and correct expression is (-2x²) + (24÷12)-4 = 2
What is the simplified expression?
a. 6×7-5 +4= 16
In this calculation, we need to use brackets to ensure the correct order of operations, which is typically remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division [from left to right], Addition and Subtraction [from left to right]).
Without brackets, the calculation would be evaluated as follows:
6×7-5 +4 = 42-5 +4 (applying multiplication first)
= 37 +4 (applying subtraction)
= 41 (applying addition)
However, the desired result is 16. To achieve this, we can use brackets to group the addition and subtraction operations together, like this:
(6×7)-(5 +4) = 42 - 9 (applying addition within the brackets first)
= 33 (applying subtraction)
b. -2x²+24÷12-4=2
Similarly, in this calculation, we need to use brackets to ensure the correct order of operations.
Without brackets, the calculation would be evaluated as follows:
-2x²+24÷12-4 = -2x²+2-4 (applying division first)
= -2x²-2 (applying addition and subtraction)
However, the desired result is 2. To achieve this, we can use brackets to group the division and subtraction operations together, like this:
-2x²+(24÷12)-4 = -2x²+2-4 (applying division within the brackets first)
= -2x²-2 (applying addition and subtraction)
By adding brackets to each calculation to ensure the desired order of operations, we arrive at the correct results of 16 and 2, respectively.
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the cunninghams are moving across the country. mr. cunningham leaves 4 hours before mrs. cunningham. if he averages 46mph and she averages 62mph , how long will it take mrs. cunningham to overtake mr. cunningham?
Answer: Let's call the time it takes for Mrs. Cunningham to overtake Mr. Cunningham "t".
In that time, Mr. Cunningham will have traveled for 4 more hours than Mrs. Cunningham, so he will have traveled 4 + t hours.
We can set up an equation to represent the distance each person traveled:
distance = rate x time
For Mr. Cunningham:
distance = 46 mph x (4 + t) hours
For Mrs. Cunningham:
distance = 62 mph x t hours
Since they end up at the same place, their distances must be equal:
46 mph x (4 + t) = 62 mph x t
Simplifying this equation:
184 + 46t = 62t
16t = 184
t = 11.5
Therefore, it will take Mrs. Cunningham 11.5 hours to overtake Mr. Cunningham.
Step-by-step explanation:
1/4 exponent 4 equal to in fraction form
Answer:
(1/256)
Step-by-step explanation:
[tex](\frac{1}{4})^{4} = \frac{1^{4} }{4^{4} } = \frac{1}{256}[/tex]
Using laws of exponents I distributed the exponent to the fraction and solved.
WHAT IS THE RANGE OF THIS PIECEWISE FUNCTION?
The given piecewise function has a range of:
Range = {46, 48, 50, 52, 54, 56}.
What is range of a function?The range of a function is the set of all possible output values, or the set of all y-coordinates that correspond to the x-coordinates in the domain of the function.
It is, in other words, the entire set of values that the function is capable of returning as its result.
From the given piecewise function, we can see that the y-coordinates of the function are limited to the values 46, 48, 50, 52, 54, and 56.
These values correspond to the y-intercepts of each of the line segments in the graph.
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Find the quotient. Assume that no denominator has a value of 0.
(6x-24)÷x^2-16/6x
The quotient of the expression (6x-24)÷x^2-16/6x is 36x/(x + 4)
Finding the quotient of the expressionFrom the question, we have the following parameters that can be used in our computation:
(6x-24)÷x^2-16/6x
Assume that no denominator has a value of 0, we have
(6x-24)÷x^2-16/6x = 6(x - 4) ÷ (x - 4)(x + 4)/6x
Express as products
So, we have the following representation
(6x-24)÷x^2-16/6x = 6(x - 4) * 6x/(x - 4)(x + 4)
When the factors are evaluated, we have
(6x-24)÷x^2-16/6x = 36x/(x + 4)
Hence, the solution is 36x/(x + 4)
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The graph of EFG is shown. Graph the image of EFG after translation of 3 units left 1 unit down write the coordinates of the image
The coordinates for the image after the translation of EFG would be E'(-2, 3), F'(-4, 0), and G'(-1, -2).
How to translate an image ?To translate a point, you simply add or subtract the specified units from its x and y coordinates. In this case, you are asked to translate the points 3 units left and 1 unit down.
To translate 3 units left, subtract 3 from the x-coordinate, and to translate 1 unit down, subtract 1 from the y-coordinate.
For point E (1, 4):
E' = (1 - 3, 4 - 1) = (-2, 3)
For point F (-1, 1):
F' = (-1 - 3, 1 - 1) = (-4, 0)
For point G (2, -1):
G' = (2 - 3, -1 - 1) = (-1, -2)
The translated points are E'(-2, 3), F'(-4, 0), and G'(-1, -2).
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refer to exercise 3.91. in this exercise, we determined that the mean and variance of the costs necessary to find three employees with positive indications of asbestos poisoning were and , respectively. do you think it is highly unlikely that the cost of completing the tests will exceed ?
The probability that the cost will exceed $600 is approximately 0.057, or 5.7%. which is relatively small, so it is highly unlikely that the cost of conducting the test will exceed $600.
we need to use the information provided to calculate the probability that the test execution cost will exceed a certain value.
Let X be the cost of conducting a test to find three of her employees with positive signs of asbestos poisoning. From the information provided, we can see that:
E(X) = $550
Var(X) = $500
Using these values, we can standardize X to a standard normal distribution.
Z = (X - E(X)) / sqrt(Var(X))
= (X - $550) / square meters ($500)
Accepting that the costs take after an ordinary conveyance, able to utilize the standard ordinary conveyance to compute the likelihood that the fetched surpasses a certain esteem. For case, to find the likelihood that the taken toll surpasses $600, we are able to compute:
P(X > $600) = P(Z > ($600 - $550) / square meters ($500))
= P(Z > 1.58)
Using a standard normal table or calculator, we find that P(Z > 1.58) is approximately 0.057. hence, the probability that the cost will exceed $600 is approximately 0.057, or 5.7%.
This likelihood is generally little, so it is profoundly impossible that the fetch of conducting the test will surpass $600.
Be that as it may, the precise limit for what is considered exceedingly improbable may change depending on the setting and the particular criteria utilized.
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please help
The terms of a geometric sequence are −4, [tex]\frac{4}{3}[/tex], [tex]-\frac{4}{9}[/tex], [tex]\frac{4}{27}[/tex]. Please write the formula for the nth term an.
The common ratio is -1/3 and the initial term is -4, then the formula is:
A(n) = -4*(-1/3)^(n - 1)
How to find the formula for the nth term?Remember that in a geometric sequence, to get the next term we need to multiply the previous term by the common ratio.
To get the common ratio, take the quotient between two consecutive terms.
We will get:
(4/3)/(-4) = -1/3
Then the n-th term of the geometric sequence is:
A(n) = -4*(-1/3)^(n - 1)
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Answer:
[tex](-4)\left(\frac{1}{3}\right)^{n-1}[/tex]
Step-by-step explanation:
The common ratio between any two terms in a geometric sequence is constant. Let this ratio be denoted by r. Then, we have:
[tex]r = \frac{\frac{4}{3}}{-4} = \frac{-\frac{4}{9}}{\frac{4}{3}} = \frac{\frac{4}{27}}{-\frac{4}{9}} = \frac{1}{3}[/tex]
Using this value of r, we can write the formula for the nth term an as:
[tex]an=[/tex][tex](-4)\left(\frac{1}{3}\right)^{n-1}[/tex]
Time plots are special scatterplots where the explanatory variable, x, is a measure of time
The statement " Time plots are special scatterplots where the explanatory variable, x, is a measure of time" is true because time plots are scatterplots where the x-axis represents time, and they are commonly used to visualize trends and patterns in time series data.
A time plot is a type of scatterplot where the x-axis represents time and the y-axis represents a response variable. Time plots are useful for displaying data that change over time and can reveal trends, patterns, and seasonality in the data. They are commonly used in fields such as economics, finance, and social sciences to analyze time series data.
By visualizing data over time, time plots can help to identify relationships, outliers, and potential forecasting models. Overall, time plots are a powerful tool for analyzing and understanding trends and patterns in time series data.
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The given question is incomplete, the complete question is:
Time plots are special scatterplots where the explanatory variable, x, is a measure of time. True or false
can you find the probability of one thing happening before another based on their expected time until occurance
Yes, we can determine the probability of one thing happening before another based on their expected time to occur.
A coin has two sides: one side ("heads") is side A and the other ("tails") is side B. A coin is tossed into the air and the question is what are the chances, the "probability" if you want it to land on side A. The probability of an event occurring describes the number of chances that the event occurred. The total probability of hitting A before B is the sum: P(A before B) = p + rp + r² + r³p + r⁴p + … = p[1 + r + r² + r³ + r⁴ + … ] and this determines the result. Thus, it is possible to determine the probability of one thing happening before another, based on their expected time to occur.
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find the product of (3-2x)(2x+1)(x-5)
Answer:
the product of (3-2x)(2x+1)(x-5) is -4x^3 + 26x^2 - 19x - 75.
Step-by-step explanation:
To find the product, we need to multiply these three expressions:
(3 - 2x)(2x + 1)(x - 5)
Let's start by multiplying the first two expressions using the distributive property:
(3 - 2x)(2x + 1) = 6x - 4x^2 + 3 - 2x
Now we can multiply this result by the third expression using the distributive property again:
(6x - 4x^2 + 3 - 2x)(x - 5) = 6x^2 - 34x + 15x - 75 - 4x^3 + 20x^2
Simplifying this expression by combining like terms, we get:
-4x^3 + 26x^2 - 19x - 75
Therefore, the product of (3-2x)(2x+1)(x-5) is -4x^3 + 26x^2 - 19x - 75.
Find the solution of the system of equations. -2x-y=6 over -2x+8y=-39
The solution of the system of equations. -2x-y=6 over -2x+8y=-39 is the ordered pair (-0.5, -5).
How to graphically solve this system of equations?In order to to graph the solution to the given system of equations on a coordinate plane, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection;
-2x-y=6 ......equation 1.
-2x+8y=-39 ......equation 2.
In this exercise, we would use an online graphing calculator to plot the given system of equations as shown in the graph attached below.
Based on the graph shown in the image attached below, we can logically deduce that the solution to this system of equations is the point of intersection of the lines on the graph representing each of them, which is given by the ordered pair (-0.5, -5).
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URGENT - Will also give brainliest to simple answer
Step-by-step explanation:
Radius = 70 cm then diameter = 140 cm
Circumference = rope length = pi * diameter
= 140 * pi = 439.8 cm long (round as needed)
A. Find the average rate of change over the interval (-1, 2] for each function. Show all steps and work for credit.
B. Which function has the greatest average rate of change over the interval [1, 2]
the average rate of change over the interval (-1, 2] in the given function is 1.17.all three functions have the same average rate of change over the interval [1, 2], which is 2
what is function and average rate?A function is a mathematical rule that takes an input and produces an output. The average rate of change of a function over an interval is the average amount the output changes per unit change in the input over that interval.
According to given informationFor the function f(x) = 2x + 3:
Let x1 = -1 and x2 = 2
f(x1) = 2(-1) + 3 = 1
f(x2) = 2(2) + 3 = 7
The average rate of change is:
[f(x2) - f(x1)] / [x2 - x1] = [7 - 1] / [2 - (-1)] = 6 / 3 = 2
Therefore, the average rate of change of f(x) over the interval (-1, 2] is 2.
For the function g(x) = x^2 - 1:
Let x1 = -1 and x2 = 2
g(x1) = (-1)^2 - 1 = 0
g(x2) = 2^2 - 1 = 3
The average rate of change is:
[g(x2) - g(x1)] / [x2 - x1] = [3 - 0] / [2 - (-1)] = 3 / 3 = 1
Therefore, the average rate of change of g(x) over the interval (-1, 2] is 1.
For the function h(x) = 2^x + 1:
Let x1 = -1 and x2 = 2
h(x1) = 2^(-1) + 1 = 1.5
h(x2) = 2^2 + 1 = 5
The average rate of change is:
[h(x2) - h(x1)] / [x2 - x1] = [5 - 1.5] / [2 - (-1)] = 3.5 / 3 = 1.17 (rounded to 2 decimal places)
Therefore, the average rate of change of h(x) over the interval (-1, 2] is approximately 1.17.
b)To determine which function has the greatest average rate of change over the interval [1, 2], we need to calculate the average rate of change for each function over that interval and compare the results.
Let's assume the functions are f(x), g(x), and h(x).
The average rate of change for f(x) over the interval [1, 2] is:
[f(2) - f(1)] / [2 - 1] = (2(2) + 3 - 2(1) - 3) / 1 = 2
The average rate of change for g(x) over the interval [1, 2] is:
[g(2) - g(1)] / [2 - 1] = (2^2 - 1 - 1^2 + 1) / 1 = 2
The average rate of change for h(x) over the interval [1, 2] is:
[h(2) - h(1)] / [2 - 1] = (2^2 + 1 - 2^1 - 1) / 1 = 2
Therefore, all three functions have the same average rate of change over the interval [1, 2], which is 2.
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17. Corina makes $27 for every 2 hours that she works. If she worked 36
hours this week, how much money did she make?
If she worked 36 hours this week, then Corina made $486 this week.
From the question, we have the following parameters that can be used in our computation:
Corina makes $27 for every 2 hours that she works.
So she makes:
27/2 = $13.50 per hour
If she worked 36 hours this week, she would make:
36 hours x $13.50 per hour = $486
Therefore, Corina made $486 this week.
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Suppose On a Sunny Day the Temperature decreases 5.4 F° for each 1,000- foot rise in elevation. If the temperature at the base of a 3,000 foot mountain is 27 F°, what is the temp at the mountain summit?
GUYS PLS EXPLAINNN
Therefore, the temperature at the summit of the 3,000-foot mountain on a sunny day is estimated to be 10.8 F°.
What is equation?In mathematics, an equation is a statement that indicates the equality of two expressions. An equation typically contains one or more variables, which are placeholders for unknown values or quantities. The variables can take on different values, and the goal is often to find the values that satisfy the equation.
Here,
Let's start by calculating the rate of change of temperature with respect to elevation:
=-5.4 F° / 1,000 ft
This means that for every 1,000-foot increase in elevation, the temperature will decrease by 5.4 F°.
Next, we can calculate how much the temperature will decrease from the base of the mountain to the summit:
3,000 ft / 1,000 ft = 3
This means that the elevation difference between the base of the mountain and the summit is 3,000 - 0 = 3,000 ft.
So, the temperature decrease from the base to the summit is:
-5.4 F° / 1,000 ft * 3,000 ft = -16.2 F°
This means that the temperature at the summit will be:
27 F° - 16.2 F° = 10.8 F°
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Help me calculate this integral!
Answer:
Step-by-step explan 456
Which equation matches the table?
An equation that matches the table include the following: y = x + 5.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
m represent the slope.x and y represent the points.First of all, we would determine the slope of this line;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (6 - 5)/(1 - 0)
Slope (m) = 1/1
Slope (m) = 1.
At data point (0, 5) and a slope of 1/, a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 5 = 1(x - 0)
y - 5 = x
y = x + 5
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