As per the area of rectangle, the required length of rope is 36.66 square meter.
The formula to calculate the area of rectangle is written as,
=> A = l x b
Where l refers the length and b refers the breadth.
Here we have given that a florist want to put a rope around his rectangular garden with the dimensions 7. 8m by 4. 7m.
In order to put the rope we have to find the area of the garden,
The area of the garden is calculated by using the area of rectangle formula, as 7.8 as length and 4.7 as breadth
=> A = 7.8 x 4.7
=> A = 36.66
Therefore, 36.66 square meter is required.
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The solution of the cubic function f(x) are −4,53, and 1. Identify a factor of this function.
Select one:
a. 3x+5
b. 5x+3
c. x+1
d. x+4
A factor of the polynomial function is (d) x + 4
How to determine the factor of the function?The solution of the cubic function f(x) are given as:
−4, 5/3, and 1.
This means that:
x = -4, x = 5/3 and x = 1
Set the above equations to 0
x + 4 = 0, x - 5/3 = 0 and x - 1 = 0
From the list of options, we have
(d) x + 4
Hence, a factor of the polynomial function is (d) x + 4
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What's the difference between sampling with replacement and sampling without replacement? How should you account for this
using TI-84's or the "hat method"?
When sampling with replacement is used, an element that has already been chosen is put back into the sample frame and given the chance to be chosen again.
In the case of sampling without replacement, an element is eliminated from the sample frame after it has been chosen.
In this question, we are supposed to state the differences between sampling with replacement and sampling without replacement.
Generally speaking, sampling without replacement is more effective than producing representative samples using replacement sampling. It prevents the inclusion of the same population element in the sample several times.
By "drawing from a hat" or by using a computer program that simulates the same process, you select the sample at random using the lottery method.
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Use the function f(x) = 5x² + 2x-3 to answer the questions.
Part A: Completely factor f(x). (2 points)
Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part C: Describe the end behavior of the graph of f(x). Explain. (2 points)
Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to draw the graph. (4 points)
A)f(x)=3/5 and -1.
B)f(x)=(3/5, 0) and (-1, 0).
C)f(x)=(-0.2, -3.2)
D)f(x)= = -0.2.
What is factorization and example?Factorization in mathematics is dividing a large number into smaller ones that, when multiplied together, give you the original number. Factorization is the division of a number into its factors or divisors.For instance, the factorization of the integer 12 might appear as 3 times 4.Function: f(x) = 5x² + 2x-3.
a) Completely factor f(x)
f(x) = 5x² + 2x-3.
= 5x² + 5x-3x − 3
= 5x(x+1)-3(x+1)
=(5x-3)(x+1)
f(x)=3/5 and -1.
b)The x-intercepts of the graph of f(x):
f(x) = 5x² + 2x-3.
At f(x) = 0, the x-intercepts are present.
Therefore, x-intercepts are (3/5, 0) and (-1, 0).
c) The end behavior of the graph of f(x):
f(x) = 5x² + 2x-3.
The middle of the zeros is the vertex's x-coordinate.
=-0.2
Find the y-coordinate of the vertex, and substitute the found value of x into the given equation:
f(-0.2)= 5(-0.2)² + 2(-0.2) − 3
=-3.2
d)The steps you would use to graph f(x):
The axis of symmetry is the x value of the vertex.
Therefore, the graph is symmetrical at about x = -0.2.
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Which of the following correlation coefficients expresses the weakest relationship between two variables
The correct option is a) -0.15.
The weakest correlation relationship between two variables is given by -0.15.
What is correlation?With the aid of correlation, the strength of the relationship between two variables can be determined. We can tell from this study whether there is no correlation, weak correlation, or a strong correlation between the two variables.
Two different types of correlation exist:
positive correlation: Two variables that move together, or in the same direction, are said to have a positive correlation. When one variable rises while the other rises or when one variable falls while the other falls, there is a positive correlation.negative correlation: When two variables are correlated negatively, the other variable's value falls as the value of the first variable rises. The correlation coefficient, which expresses correlation on a scale from +1 to -1, is used. Values less than 0 indicate a negative connection.The correlation value (r) always lies between -1 and 1.
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The complete question is -
Which of the following is the weakest correlation relationship between two variables?
a) -0.15
b) 0.27
c) -0.70
d) 0.55
Charlotte is driving at 52.7 mi/h and receives a text message. She looks down at her phone and takes her eyes off the road for 4.99 s. How far has Charlotte traveled in feet during this time
Answer:
385.7 ft
Step-by-step explanation:
The relation between speed, distance, and time is ...
distance = speed × time
Compatible unitsThe speed is given in miles per hour, but the time is given in seconds and the distance is wanted in feet. This suggests a conversion of speed to feet per second is appropriate.
[tex]\dfrac{52.7\text{ mi}}{\text{h}}\times\dfrac{1\text{ h}}{3600\text{ s}}\times\dfrac{5280\text{ ft}}{1\text{ mi}}=\dfrac{52.7\cdot22\text{ ft}}{15\text{ s}}=77.29\overline{3}\text{ ft/s}[/tex]
DistanceThe distance traveled is the product of this speed and the time period.
[tex]\dfrac{77.29\overline{3}\text{ ft}}{1\text{ s}}\times4.99\text{ s}=385.6937\overline{3}\text{ ft}\approx\boxed{385.7\text{ ft}}[/tex]
If AH=13mm, A F=15mm, and FC=37mm, what would be the distance between A and D? Round your answer to two decimal places if necessary.
The measure of segment AD in the similar triangle is 45.07 mm
What are similar triangles?Two triangles are said to be similar if they have the same shape and the ratio of their corresponding sides are in the same proportion. Also, their corresponding angles are congruent.
Triangle FEA and triangle ABC are similar triangles by the angle angle similarity postulate, therefore:
13/AD = 15 / (15 + 37)
AD = 45.07 mm
The measure of segment AD in the similar triangle is 45.07 mm
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Total slack is calculated by ____
Total slack is calculated by the late finish minus early finish and the late start minus the early start
What is the total slack?Total slack is simply the amount of time taken for a task to be delayed before the project finish date is delayed. It can be positive or negative.
It is calculated as the calculated as the value of the Late Finish minus the Early Finish field, and the Late Start minus the Early Start field.
Total slack = Late finish - early finish + late start - early start
Thus, the total slack is calculated by the late finish minus early finish and the late start minus the early start.
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A group of scientists is conducting an experiment on the effects of media on children. they randomly select 100 children and randomly assign each child to one of four treatment groups. each treatment group has a specific amount of screen time during a one-week time frame. the first group has no screen time, the second group has two hours of screen time, the third group has four hours of screen time, and the fourth group has six hours of screen time. after the first week, the scientists conduct the same experiment, with the same subject groups, for three more weeks so that each group experiences each of the four treatments. which statements about this study are true?
The statements that are true about this study are
This study uses random samplingThis study uses a control groupThis study uses a repeated measures designThis is further explained below.
What is an experiment?Generally, a scientific technique is used to create a discovery, test a theory or show an established truth.
In conclusion, The following are accurate claims concerning this study:
Random sampling is used in this investigation. In this investigation, a control group is used. Repeated measurements are employed in this investigation.
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The figure shows lines r, n, and p intersecting to form angles numbered 1, 2, 3, 4, 5, and 6. All three lines lie in the same plane.
not to scale
Based on the figure, which of the individual statements would provide enough information to conclude that line r is perpendicular to line
p?
Select all that apply.
A m2 3 = m/6
OB. m2 3+m/4= 90°
Answer:
B.
Step-by-step explanation:
If the measures of angles 3 and 4 add to 90, then lines r and p meet at right angles making them perpendicular.
Answer: B.
Answer:
Step-by-step explanation:
The graph of F(x) shown below resembles the graph of G(x) = x4, but it has
been changed somewhat. Which of the following could be the equation of
F(x)?
Using translation concepts, the equation of F(x) is:
A. [tex]F(x) = \frac{1}{3}x^4 - 2[/tex]
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
The parent function is:
[tex]G(x) = x^4[/tex]
For function F(x), we have that:
It was shifted down 2 units, hence F(x) = G(x) - 2.It was vertically compressed by a factor of 3, hence [tex]F(x) = \frac{1}{3}G(x) - 2[/tex]Then the equation is:
A. [tex]F(x) = \frac{1}{3}x^4 - 2[/tex]
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Alissa is analyzing an exponential growth function that has been reflected across the y-axis. She states that the domain of the reflected function will change because the input values will be the opposite sign from the reflected function. Simon disagrees with Alissa. He states that if an exponential function is reflected across the y-axis, the domain will still be all real numbers.
Which student is correct and why?
Alissa is correct because the domain will change from negative to positive x-values.
Alissa is correct because a reflection across the y-axis will change the possible input values of the reflected function.
Simon is correct because even though the input values are opposite in the reflected function, any real number can be an input.
Neither student is correct.
Simon is correct because even though the input values are opposite in the reflected function, any real number can be an input for an exponential growth function.
Verification of Choice
Let us consider the following exponential growth function.
y = Aeˣ ...........................(1)
This equation changes when it is reflected across the y axis and becomes,
y = Ae⁻ˣ ...........................(2)
Here, the set of all x values for which y is defined is known as the domain of this exponential growth function.
Therefore, the domain of exponential growth function 1 is the set of all real numbers, given by, x ∈ [-∞,∞]
The domain of function 2, which is a reflection of function 1, will also consist entirely of real numbers and will be given by x ∈ [-∞,∞ ]
Therefore, between Alissa and himself, Simon is right when he asserts that even if an exponential growth function is reflected across the y-axis, the domain will still consist only of real values.
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Last year, nicholas started babysitting to earn extra spending money. over the summer, he tracked how much money he earned at each babysitting job. this box plot shows the results.
The percent of the time Nicholas earned is the percent of the time 25%
The 5 values of the given box plot are;
The minimum fee = 20
In the first quartile, the 25th percentile point, Q₁ = 25
In the median or second quartile, the fiftieth percentile factor, Q₂ = 30
The 1/3 quartile, the 75th percentile point, Q₃ = 40
The maximum value = 55
The range = The most value - The minimal fee
The range = 55 - 20 = 35
We are aware that Nicholas earned $25 or less from the minimum value up to the first quartile, Q₁, which is the 25th percentile point, in which we've got 25 percent of the statistics points.
Consequently, the percentage of the time Nicholas earned $25 or much less is 25 percent of the time.
In mathematics, a percentage is a variety of ratios that can be expressed as a fraction of 100. If we have to calculate the percent of various, divide the wide variety by way of the whole and multiply by way of a hundred. For this reason, the shared approach is a component in line with a hundred. The phrase according to cent means in keeping with one hundred.
A relative cost indicates the hundredth component of any amount. One percent (symbolized 1%) is the hundredth part; for this reason, a hundred percent represents the whole lot and two hundred percent specifies two times the given amount.
Multiply the entire number amount through the decimal equal to the proportion. The end result is the share portion quantity. (consequently, this answers the question “How a whole lot is 24 percentage of 500,” you examine that 24 percent of 500 is a hundred and twenty).
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If $2000 is invested at 4% per annum compounded semiannually, how much is in the account after 8 years?
Brian is 8 and his brother is twice his age. How old will his brother be when brian is 11?.
Hence, the age of his brother when brain is [tex]11[/tex] is [tex]19.[/tex]
What is the simplification?
Simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.
Here given that,
Brian is [tex]8[/tex] and his brother is twice his age.
As age of brain is [tex]8[/tex].
His brother is twices than his age is [tex]8(2)=16[/tex]
When the age of brain is [tex]11[/tex] then [tex]3[/tex] added to his present age.
Then the age of his brother is [tex]16+3=19[/tex]
Hence, the age of his brother when brain is [tex]11[/tex] is [tex]19.[/tex]
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A yard with a tree is to be concreted. The owner wants to concrete the whole yard, but keep the tree in place and so concretes around the tree to its circumference. The tree has a circular cross section, with a diameter of 1 m. The yard is a rectangle, with sides of length 3 m and 4 m. What is the area of concrete required (to 1 d.p.)?
Answer:
11.2 square meters
Step-by-step explanation:
This is a very simple question. Basically, it's the area of a rectangle with the area of a circle subtracted. The area of a circle is calculated with the following formula:
[tex]\pi \times r^{2} [/tex]
Where r is the radius. We know the diameter is 1m, so we can simply halve this value to find the radius of 0.5m. Substituting in this value, we find the area of the circle to be:
[tex]\pi \times {0.5}^{2} \approx0.78539816339744830961566[/tex]
Now let's calculate the area of the rectangle, which is simply 3x4=12. Subtracting our earlier value from 12, we are left with an answer of 11.2 (to 1 d.p)
help im in a rush!!!!!!!
Answer:
Your answer should be 52!!
Step-by-step explanation:
Work included on the picture below.
What is the image point of (-8, 3) after the transformation R2700 0 T-2,-1?
The transformation of the coordinate (-8, -3) R 270° and T-2,-1 is (-5, -9)
What is transformation?Transformation is the change of the coordinates of a shape either by rotation, reflection or translation.
Analysis:
When coordinates (x, y) rotate 270° the resulting coordinates become (-y, x)
so coordinates (-8, -3) after 270° rotation becomes ( 3, -8)
When it is translated 2 units to the left and 1 unit down, that is -3 + -2 = -5 and -8 + -1 = -9
Therefore the transformed coordinate becomes (-5, -9)
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The projectile motion of an object can be modeled using s(t) = gt2 + v0t + s0, where g is the acceleration due to gravity, t is the time in seconds since launch, s(t) is the height after t seconds, v0 is the initial velocity, and s0 is the initial height. The acceleration due to gravity is –4.9 m/s2.
An object is launched at an initial velocity of 19.6 meters per second from an initial height of 24.5 meters. Which equation can be used to find the number of seconds it takes the object to hit the ground?
A. 0 = –4.9t2 + 19.6t + 24.5
B 0 = –4.9t2 + 24.5t + 19.6
C. 19.6 = –4.9t2 + 24.5t
D. 24.5 = –4.9t2 + 19.6t
Answer: 0=-4.9t²+19.6t+24.5
Step-by-step explanation:
[tex]s(t)=gt^2+v_0t+s_0\\v_0=19,6\ \frac{m}{s} \ \ \ \ s_0=24,5\ m\ \ \ \ g=-4,9\ \frac{m}{s^2} \ \ \ \ S(t)=0\ (the\ object \ hitts\ the \ ground)\ \Rightarrow\\0=-4,9t^2+19,6t+24,5.[/tex]
Firm valuation schultz industries is considering the purchase of arras manufacturing. arras is currently a supplier for schultz, and the acquisition would allow schultz to better control its material supply. the current cash flow from assets for arras is $8 million. the cash flows are expected to grow at 9 percent for the next five years before leveling off to 6 percent for the indefinite future. the cost of capital for schultz and arras is 13 percent and 11 percent, respectively. arras currently has 3 million shares of stock outstanding and $25 million in debt outstanding. what is the maximum price per share schultz should pay for arras? (do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) price per share $
Based on the calculations, the maximum price per share that Schultz should pay for Arras is approximately equal to $53.00.
How to calculate the maximum price per share?First of all, we would determine the cash flows for the company until the growth rate changes at a constant perpetual rate:
Year 1: $8,000,000(1 + 0.09) = $8,720,000.
Year 2: $8,720,000(1 + 0.09) = $9,504,800.
Year 3: $9,504,800(1 + 0.09) = $10,360,232.
Year 4: $10,360,232(1 + 0.09) = $11,292,652.88.
Year 5: $11,292,652.88(1 + 0.09) = $12,308,991.6392.
Year 6: $12,308,991.6392(1 + 0.06) = $13,047,5311.137552.
Next, we would calculate the terminal value in Year 5:
TV₅ = CF₆/(RWACC - g)
TV₅ = CF₆/(0.11 - 0.06)
TV₅ = $12,308,991.6392/0.05
TV₅ = $246,179,832.784.
By using Arras' cost of capital, we would determine its present value:
V₀ = $8,720,000/(1 + 0.11) + $9,504,800/(1 + 0.11)² + $10,360,232/(1 + 0.11)³+ $11,292,652.88/(1 + 0.11)⁴+ ($12,308,991.6392 + $246,179,832.784)/(1 + 0.11)⁵.
V₀ = $8,720,000/(1.11) + $9,504,800/(1.11)² + $10,360,232/(1.11)³+ $11,292,652.88/(1.11)⁴+ ($12,308,991.6392 + $246,179,832.784)/(1.11)⁵.
V₀ = 7,855,855.8559 + 7,714,308.9035 + 7,575,312.3467 + 7,438,820.2323 + 153,400,536.1422.
V₀ = $183,984,855.4806.
For the market value of the equity, we have:
S = 183,984,855.4806 - 25,000,000
S = $158,984,855.4806.
Now, we can find the maximum price per share:
Maximum price per share = 158,984,855.4806/3,000,000
Maximum price per share = $52.9949 ≈ $53.00.
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Consider this quadratic equation.
x2 + 2x + 7 = 21
The number of positive solutions to this equation is
.
The approximate value of the greatest solution to the equation, rounded to the nearest hundredth, is
Answer:
1 positive solution (and 1 negative)
the greatest solution is about 2.87
Step-by-step explanation:
so, it is
x² + 2x + 7 = 21
x² + 2x - 14 = 0
the general solution to such a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = 2
c = -14
x = (-2 ± sqrt(2² - 4×1×-14))/(2×1) =
= (-2 ± sqrt(4 + 56))/2 = (-2 ± sqrt(60))/2 =
= (-2 ± sqrt(4×15))/2 = (-2 ± 2×sqrt(15))/2 =
= -1 ± sqrt(15)
x1 = -1 + sqrt(15) = 2.872983346... ≈ 2.87
x2 = -1 - sqrt(15) = -4.872983346...
The cost of a flight is related to the distance traveled:
Miles 225 1,100 1,375 1,675 1,950 2,250
Cost ($) 50.3 111 143 187 214 196
Find the linear regression equation that models this data.
The linear regression equation that models the cost of a flight in function of the distance traveled is given as follows:
C(m) = 0.08316m + 31.36331
How to find the equation of linear regression using a calculator?To find the equation, we need to insert the points (x,y) in the calculator.
For this problem, the points (x,y) are given as follows:
(225, 50.3), (1100, 111), (1375, 143), (1675, 187), (1950, 214), (2250, 196).
Hence, inserting these points in the calculator, the linear regression equation that models the cost of a flight in function of the distance traveled is given as follows:
C(m) = 0.08316m + 31.36331
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Which side lengths form a right triangle?
Choose all answers that apply:
A:
11,13,√290
B:
7, 24, 25
C: 2, 5, 7
Answer:
B
Step-by-step explanation:
Use the Pythagorean theorem, 7^2+24^2=25^2
49+576=625
625=625
All the other answers do not satisfy the Pythagorean theorem (a^2+b^2=c^2)
Answer: A, B
Step-by-step explanation:
A right triangle's sides have a special property that the sum of the squares of the two legs is equal to the square of the hypotenuse. This is also known as the Pythagorean Theorem.
Let's check if this theorem holds true for each triangle.
A[tex]11^2+13^2=(\sqrt{290})^2\\ 121 + 169=290\\ 290 = 290[/tex]
The Pythagorean Theorem holds true, so this is a right triangle.
B[tex]7^2+24^2=25^2\\ 49 + 576 = 625\\ 625=625[/tex]
The Pythagorean Theorem holds true, so this is a right triangle.
C[tex]2^2+5^2=7^2\\ 4+25=49\\ 29 = 49[/tex]
This is a false statement, which doesn't make this triangle a right triangle. We could have use the triangle inequality theorem to prove that this isn't a triangle at all, and therefore not a right triangle, but I encourage you to learn about it and try it yourself.
The table shows data from a survey about the number of times families eat at restaurants during a week. The families are either from Rome, Italy or New York, New York:
Maximum Minimum Q1 Q3 IQR Median Mean σ
Rome 16 0 3 13 10 8.5 8 5.4
New York 20 1 4.5 6 1.5 5.5 7.25 5.4
Which of the choices below best describes how to measure the center of this data? (5 points)
Group of answer choices
Both centers are best described with the mean.
Both centers are best described with the median.
The Rome data center is best described by the mean. The New York data center is best described by the median.
The Rome data center is best described by the median. The New York data center is best described by the mean.
It's NOT C "The Rome data center is best described by the mean. The New York data center is best described by the median." already tried, its incorrect.
The best description on measurement of the center of this data is; C. The Rome data center is best described by the mean. The New York data center is best described by the median.
How to find the center of the data?If we make the box and whisker plot of the Rome and New York data, we will see that there will clearly be an outlier in the New York data.
Thus;
Due to the fact that the data set for New York has an outlier and the mean is skewed, it means that we would use the median.
Finally, we see that the Rome data set is symmetrical because it has no outlier, and as such we can use the mean.
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Which point is a solution to the inequality shown in this graph? (urgent)
Answer: D
Step-by-step explanation:
sorry if its wrong
Two friends are planning for a gathering. The food budget is modeled by 1/2(x-120)=10,where x is the amount spent on food. What are the least and greatest amounts that the two friends could spend on food
The least and greatest amounts that the two friends could spend on food are 100 and 140 respectively
Functions and values
Functions are expressions written in terms of letters known as variables. Given the equation that represents the food budget of two friends as shown;
1/2(x-120)=10
where x is the amount spent on food.
In order to determine the least and greatest amounts that the two friends could spend on food,
For the least amount
-(x-120) = 20
-x+120 = 20
x = 100
For the greatest amount
(x-120) = 20
x = 140
Hence the least and greatest amounts that the two friends could spend on food are 100 and 140 respectively
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If you apply the changes below to the quadratic parent function, f(x) = x²
what is the equation of the new function?
• Shift 3 units right.
• Vertically stretch by a factor of 4.
• Reflect over the x-axis.
A. g(x) = -4(x+3)²
B. g(x) = -4(x-3)²
C. g(x) = (-4x - 3)²
D. g(x) = 4x²+3
The equation of the unique function exists g(x) = -4(x-3)².
What is a quadratic equation?The polynomial of degree two exists named a quadratic polynomial and the equation corresponding to a quadratic polynomial P(x) exists named a quadratic equation.
Given: F(x) = x²
The vertical stretch by a factor of 4 exists can be satisfied by (b)
if we use a vertical stretch, it transforms the y-values which causes it to seem skinnier when graphed.
Multiply 4 by f(x) which provides 4x².
As the reflection over the x-axis, so multiply it by -1 to f(x), which results in -4x².
Shift the graph right 3 which exists by moving it right, so by adjusting the x values indicating use f(x-3), to obtain this subtract the value from x when you move right, and add the value to x when you move left.
Hence, the unique graph would be g(x) = -4(x-3)²
Therefore, the correct answer is option B. g(x) = -4(x-3)².
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f(x) 2 square root -x^2+10 x what’s the domain
The domain of the function are all real numbers greater than 10
Domain of a functionDomain are values of the independent variables for which the function exist.
Given the function below;
[tex]f(x)=2\sqrt{-x^2+10x}[/tex]
For the function to exist, then;
-x^2+10x = 0
-x^2 = -10x
x = 10
Hence the domain of the function are all real numbers greater than 10
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A pile of cards contains 3 diamonds,
4 hearts, and 2 spades. Marianna randomly
selects two cards without replacing the
first. What is P(hearts, then spades)?
F. 8/81
G. 1/9
H. 2/3
J. 1/18
Answer:G
Step-by-step explanation:
4/9 times 2/8= 1/9
Answer:
G 1/9
Step-by-step explanation:
add all of the cards together that will get you 9 then you take the hearts overing it by total then times it by 2 overing by 8 or
4/9x2/8
a. Find the linear approximating polynomial for the following function centered at the given point a. b. Find the quadratic approximating polynomial for the following function centered at the given point a. c. Use the polynomials obtained in parts a. and b. to approximate the given quantity. -1/1.06
The required answers are:
a. The linear approximating polynomial at a = 1: P₁(x) = x - 2
b. The quadratic approximating polynomial at a = 1: P₂(x) = -3 + 3x - x²
c. Appromating the given quantity -1/1.06 for P₁(x) and P₂(x): -3.06 and -7.30 respectively.
Using Taylor polynomial series, the required approximating polynomials are calculated.
What is the Taylor polynomial series?The Taylor polynomial series is
Pₙ(x) = ∑ [tex]\frac{f^n(a)}{n!}[/tex](x - a)ⁿ; limits from 0 to n
Where fⁿ(a) is the nth derivation of f(x) at a
The first order(linear) Taylor polynomial series is
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
The second order(quadratic) Taylor polynomial series is
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
Calculation:The given function is f(x) = -1/x at a = 1
f(a) = f(1) = -1
f'(x) = -(-1/x²) = 1/x²; f'(a) = f'(1) = 1
f''(x) = -2/x³; f''(a) = f''(1) = -2
a. Linear approximating polynomial:
From the Taylor series, we have
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
On substituting,
P₁(x) = -1 + (1/1!)(x - 1)¹
= -1 + (x - 1)
= x - 2
b. Quadratic approximating polynomial:
From the Taylor series, we have
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
On substituting,
P₂(x) = f(1) + (1/1!)(x - 1) + (-2/2!)(x - 1)²
= -1 + (x - 1) - (x - 1)²
= -1 + x - 1 - (x² - 2x + 1)
= x - 2 - x² + 2x - 1
= -3 + 3x - x²
c. Approximating the given quantity -1/1.06:
Substituting x = -1.06 in P₁(x) and P₂(x),
P₁(x) = x - 2
⇒ P₁(-1.06) = (-1.06) - 2 = -3.06
P₂(x) = -3 + 3x - x²
⇒ P₂(-1.06) = -3 + 3(-1.06) - (-1.06)² = -7.30
Therefore, the approximating polynomials are calculated by using Taylor polynomial series.
Disclaimer: The given question is incomplete. Here is the complete question.
Question:
a. Find the linear approximating polynomial for the following function centered at the given point a.
b. Find the quadratic approximating polynomial for the following function centered at the given point a.
c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.
f(x) = -1/x , a = 1; approximate -1/1.06
Learn more about approximating polynomials here:
https://brainly.com/question/15540322
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A bookshelf is 4-3/4 feet wide. how many 3-inch-wide books will fit on the shelf?
19 books can fit into a bookshelf that has a width of 19/4 feet
What is an equation?An equation is an expression that shows the relationship between two or more number and variables.
The bookshelf is 4 3/4 feet wide = 19/4 feet
1 ft = 12 in, hence:
19/4 feet = 19/4 feet * 12 in per feet = 57 inches
Number of books = 57 inches / 3 in = 19
19 books can fit into a bookshelf that has a width of 19/4 feet
Find out more on equation at: https://brainly.com/question/2972832
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