The factoring the equation (a + 1)² - 4b² we get (a+1+2b)(a+1-2b).
What is the factoring function of (a + 1)² - 4 b²?Given: (a + 1)² - 4b²
Apply Perfect Square Formula, and we get
(a + b)² = a² + 2 a b + b²
(a + 1)² = a² + 2a1+1²
= a² + 2a1 + 1² - 4 b²
simplifying the equation, we get
a² + 2a1 + 1² = a² + 2a + 1
= a² + 2a + 1 - 4b²
Factor, a² + 2a + 1 = (a + 1)²
= (a + 1)² - 4 b²
Simplify 4 b² = (2 b)²
= (a + 1)² - (2b)²
Apply the Difference of Two Squares Formula, and we get
x² - y² = (x + y)(x - y)
(a + 1)² - (2b)² = ((a + 1) + 2b)((a + 1) - 2 b)
simplifying the equation, we get
= ((a + 1) + 2b)(a + 1) - 2 b)
= (a + 1 + 2b)(a + 1 - 2b)
Therefore, the correct answer is (a+1+2b)(a+1-2b).
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What is the standard form polynomial representing the volume of this shipping container?
The image shows a blue shipping container with the numbers:
4x2 + 3x(along the length of the bottom)
x2 - 8 (Along the bottom of the 'front')
6x + 15 (going up the length of the 'front')
The standard form polynomial representing the volume of this shipping container is determined as: 24x^5 + 78x^4 - 147x^3 - 624x^2 - 360x.
What is a Standard Form Polynomial?A standard form polynomial is a polynomial expression written whereby the term with the highest degree or power on a variable is written first in the expression, followed by the least, then the constant of the polynomial comes last.
What is the Volume of a Rectangular Prism?
The Volume of a rectangular prism = (length)(width)(height).
The shipping container is a rectangular prism with the following dimensions:
Length of container = 4x² + 3x
Width of container = x² - 8
Height = 6x + 15
Plug in the values
Volume of container = (4x² + 3x)(x² - 8 )(6x + 15)
Expand
Volume of container = 24x^5 + 78x^4 - 147x^3 - 624x^2 - 360x
Thus, using the formula for the volume of a rectangular prism, the standard form polynomial representing the volume of this shipping container is determined as: 24x^5 + 78x^4 - 147x^3 - 624x^2 - 360x.
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Carlos went to PetSmart to get his hamster an exercise ball. There was a small one with a 5 inch diameter and a 13 inch ball. How much more space (volume) is in the larger than the smaller? Round to the nearest tenth.
The amount of space (volume) that is in the larger spherical ball than the smaller = 1,084.8 in.³
What is the Volume of a Sphere/Spherical Solid?To find the volume of a sphere, which is the amount of space the solid contains, the formula used is given as: 4/3 πr³, where r is the radius of the sphere, which is half the measure of the diameter. Volume of sphere (V) = 4/3 πr³.
Diameter of the smaller spherical ball = 5 inches
Radius of the smaller spherical ball = 5/2 = 2.5 inches
Volume of the smaller spherical ball = 4/3 πr³ = 4/3 × π × 2.5³
Volume of the smaller spherical ball ≈ 65.5 in.³
Diameter of the larger spherical ball = 13 inches
Radius of the larger spherical ball = 13/2 = 6.5 inches
Volume of the larger spherical ball = 4/3 πr³ = 4/3 × π × 6.5³
Volume of the larger spherical ball ≈ 1,150.3 in.³
The amount of space (volume) that is in the larger spherical ball than the smaller = 1,150.3 - 65.5
The amount of space (volume) that is in the larger spherical ball than the smaller = 1,084.8 in.³
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The following are the distances (in miles) to the nearest airport for 11 families. 9, 11, 11, 16, 16, 24, 28, 30, 34, 42, 45 Notice that the numbers are ordered from least to greatest. Give the five-number summary and the interquartile range for the data set.
Here is the five-number summary, range and interquartile range:
Minimum: 9
Lower quartile: 11
Median: 24
Upper quartile: 34
Maximum: 45
Interquartile range: 23
What is the five-number summary?The minimum number is the smallest number in the data. The minimum number is 9. The maximum number is the biggest number in the dataset. The maximum number is 45.
The first quartile, third quartile and the interquartile range are used to measure dispersion.
The first quartile : 1/4(n + 1)
1/4 x (12) = 3rd term = 11
The third quartile : 3/4 x (n + 1)
3/4 x (12) = 9th term = 34
Interquartile range = third quartile - first quartile
34 - 11 = 23
Median can be described as the number that is at the center of the dataset when the number is arranged in either ascending or descending order.
Median = 25
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Find the critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne. ) f(x) = x4/5(x − 8)2
The critical values of the function are x = 0 and x = 64
The critical values of a function f(x) are the values of x for which f'(x) = 0.
Given,
f(x) = [tex]\frac{x^{4} }{5(x-8)^{2} }[/tex]
The derivative is found as follows, applying the quotient rule:
f'(x) = [tex]\frac{[5(x^{4} )(x-8)^{2}-5x^{4} [(x-8)^{2} ]' }{[5(x-8)^{2} ]^{2} }[/tex]
= [tex]\frac{2x^{3}(x-64) }{5(x-8)^{3} }[/tex]
The zeros of the function are the zeros of the numerator, thus:
[tex]2x^{3} (x-64) =0\\2x^{3} =0-------- > x =0\\[/tex]
x - 64 = 0------------->x = 64
The critical values are x = 0 and x = 64
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What is the surface area of a rectangular prism that has a length of 12 mm, a width of 15 mm, and a height of 16 mm?
Find the surface area of the cylinder and round to the nearest tenth. (Use the π button on your calculator)
The surface area of the cylinder with the given height and radius is 835.2 square inches
How to determine the surface area?The given parameters are
Height, h = 12 inches; this is represented by the distance between the curved surfaces of the cylinderRadius, r = 7 inches; this is represented by the distance between the center of the circle to its circumferenceThe surface area is then calculated using the following formula
A = 2πr² + 2πrh
Substitute the given values in the above equation
So, we have:
A = 2 * 3.14 * 7^2 + 2 * 3.14 * 7 * 12
Evaluate the exponents
A = 2 * 3.14 * 49 + 2 * 3.14 * 7 * 12
Evaluate the products
A = 307.72 + 527.52
Evaluate the sum
A = 835.2
Hence, the surface area of the cylinder with the given height and radius is 835.2 square inches
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Adrian has a bag full of pebbles that all look about the same. he weighs some of the pebbles and finds that their weights are normally distributed, with a mean of 2.6 grams and a standard deviation of 0.4 grams. what percentage of the pebbles weigh more than 2.1 grams? round to the nearest whole percent.
89% of pebbles weigh more than 2.1 grams.
What is a percentage?
Percentage, a relative value indicating hundredth parts of any quantity. One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity. percentage.Given,
Mean = 2.6
SD = 0.4
As we have to find the percentage of pebbles weighing more than 2.1, we have to find the z-score for 2.1 first.
[tex]z = \frac{x- mean}{SD}[/tex]
[tex]z = \frac{2.1 - 2.6}{0.4}[/tex]
[tex]z = -1.25[/tex]
Now we have to use the z-score table to find the percentage of pebbles weighing less than 2.1
So,
[tex]P ( x < 1.25 ) = 0.10565[/tex]
This gives us the probability of P(z<-1.25) or P(x<2.1)
To find the probability of pebbles weighing more than 2.1
[tex]P ( x > 2.1 ) = 1 - P( x < 2.1 ) = 1 - 0.10565 = 0.89435[/tex]
Converting into percentage
0.89435 × 100 = 89.435 %
Rounding off to nearest percent =89%
Therefore, 89% of pebbles weigh more than 2.1 grams.
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It costs $0.50 to mail a postcard to Canada and $0.60 to mail a one ounce letter to Canada. Ahmad wrote to 21 friends and spent $12.00 for postage. How many letters and how many postcards did he write?
It costs $0.50 to mail a postcard to Canada, cost of mailing a one ounce letter to Canada is $0.60 and Ahmad wrote to 21 friends and spent $12.00 for postage. This means that he wrote 15 letters and 6 postcards.
Given Information:
Cost of mailing a postcard = $0.50
Cost of mailing a letter = $0.60
Total number of friends Ahmad wrote to = 21
Total cost of postage = $12.00
Let the number of postcards written by Ahmad be x and that of letters be y.
Then, x + y = 21 ............... (1)
0.50x + 0.60y = 12 ................ (2)
Multiplying equation (1) by 0.50, we get,
0.50x + 0.50y = 10.5 ................. (3)
Subtracting equation (3) from (2), we get,
0.1y = 1.5
⇒ y = 15
From equation (1), x = 21 - y
x = 21 - 15
x = 6
Thus, Ahmad wrote 6 postcards at a mailing cost of $0.50 each and 15 letters at a mailing cost of $0.60 each.
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The functions f(x) = −(x − 1)2 5 and g(x) = (x 2)2 − 3 have been rewritten using the completing-the-square method. apply your knowledge of functions in vertex form to determine if the vertex for each function is a minimum or a maximum and explain your reasoning.
The vertex of the function f(x) exists (1, 5), the vertex of the function g(x) exists (-2, -3), and the vertex of the function f(x) exists maximum and the vertex of the function g(x) exists minimum.
How to determine the vertex for each function is a minimum or a maximum?Given:
[tex]$\mathrm{f}(\mathrm{x})=-(\mathrm{x}-1)^{2}+5$[/tex] and
[tex]$\mathrm{g}(\mathrm{x})=(\mathrm{x}-2)^{2}-3$[/tex]
The generalized equation of a parabola in the vertex form exists
[tex]$y=a(x-h)^{2}+k[/tex]
Vertex of the function f(x) exists (1, 5).
Vertex of the function g(x) exists (-2, -3).
Now, if (a > 0) then the vertex of the function exists minimum, and if (a < 0) then the vertex of the function exists maximum.
The vertex of the function f(x) exists at a maximum and the vertex of the function g(x) exists at a minimum.
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Answer:
- For the function f(x) = -(x - 1)^2/5, the vertex is a maximum point.
- For the function g(x) = x^4 - 3, the vertex is a minimum point.
Step-by-step explanation:
To determine if the vertex for each function is a minimum or a maximum, we need to analyze the leading coefficient of each function.
Let's start with the function f(x) = -(x - 1)^2/5. This function is already in vertex form, where the vertex is (1, 0).
Since the leading coefficient is negative (-1/5), the graph of f(x) opens downward. In this case, the vertex is a maximum point. You can visualize it as a peak on a graph.
Now, let's move on to the function g(x) = (x^2)^2 - 3. This function can be rewritten in vertex form as g(x) = x^4 - 3.
The leading coefficient is positive (1), so the graph of g(x) opens upward. In this case, the vertex is a minimum point. You can visualize it as a valley on a graph.
In summary:
- For the function f(x) = -(x - 1)^2/5, the vertex is a maximum point.
- For the function g(x) = x^4 - 3, the vertex is a minimum point.
Remember, the sign of the leading coefficient determines whether the vertex is a minimum or maximum point.
Assume thst y varies directly with x then solve if y=4 when x=12 find y when x=-24
Answer:
-8
Step-by-step explanation:
y=(1/3)x
4 = (1/3)12
y=(1/3)(-24)
y= -8
Find the value of x if a, b, and c are collinear points and b is between a and c. ab=x 6,bc=3x−5,ac=36−x
The value of x if a, b, and c are collinear points and b is between a and c : 7
What is collinear points?Collinear points are those that are situated on the same line of sight or within a single line. In Euclidean geometry, two or more points are said to be collinear if they are located on a line either close to or far from one another.
According to the given information:
if A, B, and C are col-linear points and B is between A and C
then
AB+BC=AC
AB=x+6
BC=3x-5
AC=36-x
substituting value:(x+6)+(3x-5)=36-x
(x+3x)+(6-5)=36-x
4x+1=36-x
4x+x=36-1
5x=35
x=35/5
x=7
The value of x is 7
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PLS HELP>>>>solving functions
Answer:
The first option:
[tex]x^2-5x+9[/tex]
Step-by-step explanation:
Trust me I am right.
If you need work please ask!
Have a nice day!
How many solutions does the equation −3y 3y 4 = 4 have? none infinitely many one four
A given linear equation has infinitely many solutions.
What is a linear equation?There are only one or two variables in a linear equation. No variable can be multiplied by a number larger than one or used as the denominator of a fraction in a linear equation. All of the points fall on the same line when you identify the values that make a linear equation true and plot those values on a coordinate grid.As much as possible, reduce the equation to y = mx + b. Verify the exponents in your equation. It is nonlinear if it has exponents. Your equation is linear if there are no exponents in it.The definition of a linear equation is an equation with a maximum degree of one.Given: −3y 3y 4 = 4
3y+4=3y+4
The given linear equation satisfies any value of 'y'.
So, A given linear equation has infinitely many solutions.
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In the power function f(x) = -2x, what is the end behavior of f(x) =-2x^3 as x goes to [infinity]?
The end behavior of a polynomial function is as x tends to infinity, f(x) tends to negative infinity.
In this question,
The power function is f(x) =-2x^3
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.
The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
The graph below shows the behavior of the function f(x).
The above equation has the degree of 3, which is odd and the leading coefficient has the negative coefficient.
Then the end behavior is
As x -> ∞,
[tex]\lim_{x \to \infty} f(x)[/tex]
⇒ [tex]\lim_{x \to \infty} -2x^{3}[/tex]
⇒ [tex]-2(\infty)^3[/tex]
⇒ - ∞
Hence we can conclude that the end behavior of a polynomial function is as x tends to infinity, f(x) tends to negative infinity.
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List the sides in order from the smallest to the largest.
L
K
27°
OA. LJ, KL, JK
OB. LJ, JK, KL
OC. KL, LJ, JK
D. JK, LJ, KL
Reset Selection
78°
75°
Answer:
A
Step-by-step explanation:
the side opposite of the angle is related to the size of this angle. Smaller angles have less space for the line to "spread out," so it will be shorter. Vice versa, bigger angles will have opposite sides longer because of more space.
Anyone know the anwser to this
Answer:
4112
Step-by-step explanation:
When you plug in 3 for x you get 6.
Then you just do 4^6 which is 4096.
Then add 16 which gets you 4112.
Find the next two terms in the sequence 3, 6, 12, 24, 48, ...
Answer:
96 and 192
Step-by-step explanation:
This is a geometric sequence with a common ratio of 2 :
3 × 2 = 6
6 × 2 = 12
12 × 2 = 24
24 × 2 = 48
48 × 2 = 96
96 × 2 = 192
Therefore our final answers are 96 and 192
Hope this helped and have a good day
A man bought a car for 8500 and sold it for 9500. find his percentage gain
Answer:
11.7647%
Hope this helps!
The formula for __________________________ is x = t e or raw score (x) equals the true score (t) plus error (e).
The formula for Observed Score is; Observed score (X) = True score (T) + measurement error (e)
What is the observed score formula?The formula for Observed Score is;
Observed score (X) = True score (T) + measurement error (e)
1) True score is the score that would be obtained if an individual took a test an infinite amount of times and those test scores were averaged. The concept of true score is theoretical because you can't give someone something an infinite number of times.
2) Standard error of measurement is the standard deviation of multiple test scores (how far the sample mean of the data is likely to be from the true population mean).
The less random error (e) in the measure, the more the observed score X approximates the true score T.
Thus;
Observed score = True Score + Error
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Work out the circumference of this circle.
Take it to be 3. 142 and give your answer to 1 decimal place.
4 cm
The circumference of the given circle is 25.1 cm.
According to the statement
we have given that the the radius of circle which is 4cm.
And we have to find the circumference of the circle.
So, For this purpose
The circumference is the length of any great circle, the intersection of the sphere with any plane passing through its center.To calculate the circumference of a circle, multiply the diameter of the circle with π (pi).
The formula to calculate it is
C = 2(pi)r
So, substitute the values in it then
Here use pi value is 3.14
C = 2 *3.14*4
C = 25.12.
The circumference of the given circle is according to one decimal place is 25.12 cm.
So, The circumference of the given circle is 25.1 cm.
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Find the surface area of the prism
The surface area of the prism is 256in².
Given that the sides of the prism is 5in, 12in and 4in.
The total area occupied by the surfaces of an object is called the area.
Here, l=12in, b=5in and h=4in
Firstly, we will find the area of the base, we get
B=l×b
B=12×5
B=60in²
Now, we will find the perimeter of the base, we get
P=2(l+b)
P=2(12+5)
P=2×17
P=34in
Further, we will find the surface area of the prism by using the formula SA=2B+Ph.
Here, we will substitute the values from above, we get
SA=2×60in²+34in×4in
SA=120in²+136in²
SA=256in²
Hence, the surface area of the prism where the sides of the prism is 5in, 12in and 4in is 256in².
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Midpoint of US=
Help me please thanks so much
Given the point U and point V, the midpoint between of the line segment UV is ( 0, -5/2 ).
What is the midpoint of line UV?
The midpoint formula is used to find the midpoint of a line segment.
It is expressed as;
[ (x₁ + x₂ )/2 , (y₁ + y₂ )/2 ]
Given the data in the graph,
Point U = ( 3, -1 ) and Point V = ( -3, -4 )
x₁ = 3x₂ = -3y₁ = -1y₂ = -4We substitute into the formular above.
[ (x₁ + x₂ )/2 , (y₁ + y₂ )/2 ]
[ (3 + (-3) )/2 , ( (-1) + (-4) )/2 ]
[ 0/2, (-1-4)/2 ]
[ 0/2, -5 /2 ]
( 0, -5/2 )
Given the point U and point V, the midpoint between of the line segment UV is ( 0, -5/2 ).
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Chestnut Hill Coffee Cafe offers two kinds of espresso: single-shot and double-shot. Yesterday afternoon, the cafe sold 46 espressos in all, 26 of which were single-shot. What percentage of the espressos were double-shot? Round to the nearest hundredth.
Given the number of espresso sold at Chestnut Hill Coffee Cafe as either a single-shot or a double-shot, the percentage of double-shot espressos sold is 43.48%.
What percentage of the espressos were double-shot?Percentage is simply number or ratio expressed as a fraction of 100.
It is expressed as;
Percentage = ( Part / Whole ) × 100%
Given the data in the question;
Total number of espressos sold or Whole = 46Number of single-shots or Part single = 26Number of double-shots or Part double = 46 - 26 = 20Percentage of double-shot = ?Percentage = ( Part / Whole ) × 100%
Percentage = ( Part double / Whole ) × 100%
Percentage = ( 20 / 46 ) × 100%
Percentage = 0.43 × 100%
Percentage = 43.48%
Given the number of espresso sold at Chestnut Hill Coffee Cafe as either a single-shot or a double-shot, the percentage of double-shot espressos sold is 43.48%.
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What is the name for the threshold p-value that determines when the null hypothesis is rejected?.
Alpha level. the minimum P-value at which we reject the null hypothesis
What is Null hypothesis ?That two options are identical is the null hypothesis. According to the null hypothesis, the observed difference is solely the result of chance. The probability that the null hypothesis is correct can be determined via statistical testing.
What is alpha level?The "significance level," often known as the alpha level, must be established before performing any statistical test. The likelihood of rejecting the null hypothesis when it is true is what is meant by the term "alpha level" in statistics. The likelihood of making a poor decision is what this phrase means.
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The diameter of a circle is 8mm. What is the circumference of the circle?
Answer: 8π mm (in terms of π)
Step-by-step explanation:
1) The formulas for solving the circumference of a circle is πd (π times diameter).
2) 8 times π equals 8π mm.
Hopes this helps!
Answer:
12.56 [tex]mm^{2}[/tex]
Step-by-step explanation:
c = 2[tex]\pi[/tex]r
or c =[tex]\pi[/tex]D
The diameter is the measure of a line segment going from one side the circle to the other and going through the middle of the circle. A radius is a line 1/2 a diameter. It is a line segment that goes from the center of a circle to a point on the edge of the circle.
C = (3.14)(4)
C = 12.56
I need to know how to solve this
Complete the square to transform the expression x2 − 2x − 2 into the form a(x − h)2 k.
The given expression is equivalent to the expression (B) [tex](x-1)^{2} -3[/tex].
What is an equivalent function?Two functions are equivalent if they share the same domain and codomain and have the same values for all domain components.To complete the square to transform the expression:
The given expression is [tex]x^{2} -2x-2[/tex].Then the expression can be written as, add and subtract one.Then we have:[tex]x^{2} -2x+1-1-2\\x^{2} -2x+1-3\\(x-1)^{2} -3[/tex]Therefore, the given expression is equivalent to the expression (B) [tex](x-1)^{2} -3[/tex].
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The correct question is shown below:
Complete the square to transform the expression x^2− 2x − 2 into the form a(x − h)^2+ k.
(A) (x − 1)2 + 3
(B) (x − 1)2 − 3
(C) (x − 2)2 − 3
(D) (x − 2)2 + 3
What's the perimeter or how do you find it?
The perimeter of inside of the track is 536.44m
Perimeter of a circle and rectangleThe perimeter of a circle is also known as the circumference of the circle. The formula for calculating the circumference is expressed as:
C = 2πr
where
r is the radius
From the given diagram
r = 46/2 = 23m
C = 2(3.14)(23)
C = 144.44m
Find the perimeter of the rectangle
P = 2(l+w)
p = 2(46+150)
P = 2(196)
P =392m
The perimeter of the inside track = 144.44 + 392
The perimeter of the inside track = 536.44m
Hence the perimeter of inside of the track is 536.44m
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Does x^-2 equal 2/x?
Answer:
No,
Step-by-step explanation:
X^-2 is equal to 1/X^2. so any number or letter to negative power, is the same as 1 divided by that number or letter to the positive power..(power remains with denominator
I'm not sure how to do both of them
The total surface area of the shape is 10053 cm²
Calculating Surface areaFrom the question, we are to determine the surface area of the shape
The surface area of the shape = Surface area of hemispherical part + Surface are of the cylindrical part
∴ Surface area of the shape = 2πr² + 2πrh + πr²
Where r is the radius
and h is the height of the cylindrical part
From the given information.
r = 20
h = 50
Putting the parameters into the equation, we get
Surface area of the shape = 2π(20)² + 2π(20)(50) + π(20)²
Surface area of the shape = 2π×400+ 2π×1000 + π×400
Surface area of the shape = 800π+ 2000π + 400π
Surface area of the shape = 3200π cm²
Surface area of the shape = 10053.096 cm²
Surface area of the shape ≈ 10053 cm²
Hence, the total surface area of the shape is 10053 cm²
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