The horizontal asymptote is y = -2
The horizontal asymptote is an equation of the form y = k, for some constant k, is a straight line parallel to x-axis.
Horizontal asymptotes are found for rational functions of polynomials when x becomes extremely large, or, as x → ∞.
The function given is, f(x) = (-2x²+3x+5)/(x²+1)
Horizontal asymptote is given by,
y = [tex]\lim_{x \to \infty} \frac{-2x^2+3x+5}{x^2+1}[/tex]
= [tex]\lim_{x \to \infty} \frac{x^2(-2+\frac{3}{x} +\frac{5}{x^2}) }{x^2(1+\frac{1}{x^2}) }[/tex]
= [tex]\lim_{x \to \infty} \frac{(-2+\frac{3}{x} +\frac{5}{x^2}) }{(1+\frac{1}{x^2}) }[/tex]
= -2
So the horizontal asymptote of f(x) is, y = -2
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Answer:
A) y = -2
Step-by-step explanation:
The rules to find the horizontal asymptote (HA) is shown below
Y = A if the numerator and denominator have same highest exponent for example
A^3
B^3
Y = 0 if the numerator have lower exponent and denominator have higher exponent for example
A^1
B^2
None HA if the numerator have higher exponent and denominator have lower exponent for example
A^2
B^1
And the equation shown in the question have same exponent so the answer is -2
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if three buckets of capacities 15 litre,18 litre and 24 litre can fill a drum in exact number of times, find the least capacity of the drum.
There are two questions to be solved, therefore, I'll solve the buckets one first and then I'll solve the other one. To solve both we need to apply LCM (least common multiple).
There are three buckets with differente capacities, one with 15 litres, one with 18 litres and one with 24 litres. Since they can fill a drum in an exact number of times, this means that the capacity of the drum is a multiple of the capacity of each bucket, therefore we should break each number into their multiples and find the common ones as shown below.
[tex]\begin{gathered} 15\text{ = 3}\cdot5 \\ 18\text{ = 2}\cdot3\cdot3 \\ 24\text{ = 2}\cdot2\cdot2\cdot3 \end{gathered}[/tex]We now need to multiply all the common multiples:
[tex]\text{least capacity = 2}\cdot2\cdot2\cdot3\cdot3\cdot5\text{ = }360\text{ litres}[/tex]For the second problem we have three items a book, a pen and a box. They cost respectively 36, 45 and 54. We need to then find the smallest sum of money that could buy a exact number o each item. We need to apply LCM once again as shown below:
[tex]\begin{gathered} 36\text{ = 2}\cdot2\cdot3\cdot3 \\ 45\text{ = 3}\cdot3\cdot5 \\ 54\text{ = 2}\cdot3\cdot3\cdot3 \end{gathered}[/tex]We now need to multiply all the common multiples:
[tex]\text{least sum of money = 2}\cdot2\cdot3\cdot3\cdot3\cdot5\text{ = }540[/tex]8. Find the area of the regular polygon.
14 in
5.2 in
The area of a regular polygon is 84.87 in²
What is the regular polynomial?A polygon with congruent sides and equal angles is referred to as a regular polygon. Equilateral triangles, squares, regular pentagons, and more are examples of regular polygons.
In this case Formula of the area of the equilateral triangle is ( [tex]\sqrt{3\\}[/tex]×a²)/4
where a is the length of the side of triangle.
Here the length of the side of the polygon(triangle) is 14in.
So the area of the equilateral triangle is (√3xa²)/4.
And a=14 in the above equation.
So the area of the regular polygon is: (√3x14x14)/4 =84.87 in²
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PLEASE HELP ASAP! THANK YOU!
The scalar factor is of 6.
What is the scalar factor?
A scale factor is the ratio between corresponding measurements of an object and a representation of that object. in simple words how big or small is one object compared to another.
We are given two triangle and the measurements of the corresponding sides of the triangle.
side of one triangle is 4 and side of another triangle is 24.
To find the scalar factor we divide the two
We get the answers as 6
Hence the scalar factor is of 6
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4. Simplify the radical expression.
Answer: Hello Love I believe the answer is the LAST one
Step-by-step explanation:
Sean is buying 9/16 pound of tea at a tea shop. Write the amount as a decimal.
Step-by-step explanation:
A decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point.
Answer: 9/16 as a decimal is 0.5625
9-16-as-a-decimal-is-0.5625
Let's look into the two methods to write 9/16 as a decimal.
Explanation:
Method 1: Writing 9/16 as a decimal using the division method
To convert any fraction to decimal form, we just need to divide its numerator by denominator.
Here, the fraction is 9/16 which means we need to perform 9 ÷ 16
This gives the answer as 0.5625. So, 9/16 as a decimal is 0.5625
Method 2: Writing 9/16 as a decimal by converting the denominator to powers of 10
Step 1: Find a number such that we can multiply by the denominator of the fraction to make it 10 or 100 or 1000 and so on.
Step 2: Multiply both numerator and denominator by that number to convert it into it's an equivalent fraction.
Step 3: Then write down just the numerator, putting the decimal point in the correct place, that is, one space from the right-hand side for every zero in the denominator.
9/16 = (9 × 625) / (16 × 625) = 5625/10000 = 0.5625
Irrespective of the methods used, the answer to 9/16 as a decimal will always remain the same.
You can also verify your answer using Cuemath's Fraction to Decimal Calculator.
Thus, 9/16 as a decimal is 0.5625
State the power function that the graph of f resembles for large values of x. Find the end-behavior for the function. Write your results using limit notation
Notice that:
[tex]\begin{gathered} \lim _{x\rightarrow\infty}\frac{2x^2(x-5)^2}{x^4}=\lim _{x\rightarrow\infty}\frac{2(x-5)^2}{x^2} \\ =\lim _{x\rightarrow\infty}\frac{2(x^2-10x+25)}{x^2} \\ =\lim _{x\rightarrow\infty}(\frac{2x^2}{x^2}-\frac{10x}{x^2}+\frac{25}{x^2}) \\ =\lim _{x\rightarrow\infty}(2-\frac{10}{x}+\frac{25}{x^2}) \\ =\lim _{x\rightarrow\infty}2-\lim _{x\rightarrow\infty}\frac{10}{x}+\lim _{x\rightarrow\infty}\frac{25}{x^2} \\ =2-0+0 \\ =2 \end{gathered}[/tex]Which means that, for large values of x:
[tex]2x^2(x-5)^2\approx2x^4[/tex]Since the function:
[tex]f(x)=2x^4[/tex]is a 4th degree monomial, with positive coefficient, then it keeps growing as x grows. Then, we know that:
[tex]\lim _{x\rightarrow\infty}2x^2(x-5)^2=\infty[/tex]Which means that the end behavior is such that f(x) approaches infinity as x approaches infinity.
On the other hand, for large negative values of x, the function is also positive. Then:
[tex]\lim _{x\rightarrow-\infty}2x^2(x-5)^2=\infty[/tex]What is the slope of the line on the graph
What would you multiply by to decrease a number by 13%?
Answer:
To decrease it by 13% we will multiple (100-13 = 87) .
Step-by-step explanation:
Multiply the decimal form of the number (in this case 13) to the number.
For example: 13% of 50
13% can also be said as 0.13 in decimal form.
50*0.13=6.5
13% of 50 is 6.5.
. The price of a new TV is £540, which includes 20% VAT.
Find the cost of the TV excluding VAT.
Answer:
450
Step-by-step explanation:
Based on the given conditions, formulate:
Calculate the sum or difference:
Convert decimal to fraction:
Divide a fraction by multiplying its reciprocal:
Cross out the common factor:
Calculate the product or quotient:
Answer:
you are constructing a 12 cubic feet box that is open at the top. the material used to construct the bottom costs $3 per square foot and the material used to construct the sides is $1 per square foot. what dimensions should you use to minimize the cost of the materials? set up and solve using the lagrange method.
The length and breadth should be 2 feet each and height should be 3 feet to minimize the cost of the materials.
Let consider the length of the box =x
considering a square bottom ,
length = breadth = x
and let the height =h
The surface area of the box is
SA = x² + 4xh ( box is open from top )
cost of 1sq feet material for bottom =$3
cost of 1sq feet material for sides =$1
The cost functions can be written as
C = 3x² + 1(4xh)
C = 3x² + 4xh
Also the volume equation is
x²h = 12
h = 12/x²
substituting the value of h into the cost function.
C = 3x² + 4x (12/x²)
C = 3x² + 48/x
C = (3x³ + 48)/ x
Take the derivative of C and set it equal to zero. So we get
6x³ - 48 = 0
x³ = 48/6 = 8
x = 2
∴ h = 12/x²
h = 12/4
h = 3
This is the value of x and h that minimizes the cost.
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Solve the simultaneous equation
y= 9 - x
y= 2x² + 4x + 6
Using substitution method, For the given equations, the value of x is 1/2 or x = -3
Describe substitution.A fundamental operation in computer algebra is substitution. In computer algebra systems, it is frequently abbreviated as "subs" or "subst."
When a variable (or set of letters) in an algebraic statement is substituted, its numerical value is used instead. The expression's total value can then be calculated.
Given,
y= 9 - x
y= 2x² + 4x + 6
Substituting the value of y from 1 equation to other.
⇒ 9 - x = 2x² + 4x + 6
Now factorizing the equation,
⇒ 2x² + 5x - 3 = 0
⇒ 2x² + 6x - 1x - 3
⇒ 2x(x + 3) -1 (x + 3)
⇒ (2x - 1) (x + 3)
x = 1/2 or x = -3
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if -(x-1)=5 find the value of -4x
if 9-6x=45 find the value of x-4
Answer:
value of -4x=16.
value of -4x=10.
Can someone help me and explain this please. I don’t understand what to do.
Answer:
The answer is 3 or 3/1
Step-by-step explanation:
The equation you use is m=y2-y1/x2-x1
Answer:
Slope = 3
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{4.4cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\\\end{minipage}}[/tex]
Define two points on the line from the given table:
[tex]\textsf{Let}\:(x_1,y_1)=(2,7)[/tex][tex]\textsf{Let}\:(x_2,y_2)=(4,13)[/tex]Substitute the defined points into the formula:
[tex]\implies \textsf{Slope}=\dfrac{13-7}{4-2}=\dfrac{6}{2}=3[/tex]
Note:
The change between each y-value is 6.The change between each x-value is 2.Therefore, the question may require you to enter 6/2 into the answer boxes. However, the slope of the line is 3.
Find the value of r that gives the line passing through (3,2) and (r,-4) a slope that is undefined.
The value of r is calculated as (m/-6) + 3 for the line with undefined slope passing through (3,2) and (r,-4).
Describe slope intercept form using an example.The slope-intercept form of an equation is when the equation of a line is written in the form y = mx + b. The slope of the line is given by m. And b is the value of b in the y-intercept point (0, b). For example, the slope of the equation y = 3x - 7 is 3, and the y-intercept is (0, 7).We must determine the value of r that gives the line passing through (3,2) and (r,-4) an undefined slope.
Let,
x₁ = 3
y₁ = 2
x₂ =r
y₂ = -4
slope = m
Using the slope equation we get,
m = (y₂ - y₁)/(x₂ - x₁ )
m = (-4-2)/(r-3)
m/-6 = r - 3
r = (m/-6) + 3
Therefore, the value of r is calculated as (m/-6) + 3 for the line passing through (3,2) and (r,-4) with undefined slope.
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I need answer to this equation with working ASAP
5.104 ÷ 0.016
Answer:
319
Step-by-step explanation:
hope this helps
If cx-d+f, then x is equal to
(f + d)/c.
First get rid of -d on the left side by adding d to both sides
cx - d = f --> cx = f + d
Then remove c from the left side by dividing both sides by c.
cx = f + d --> x = (f + d)/c
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a scientist observes a congenital birth defect in a breed of cats and predicts that in a cross between heterozygotes, 25% of the kittens will be born with the birth defect. she surveys several litters, and finds that 44 out of 128 kittens have the defect. to see if her hypothesis is correct, she decides to use chi-square analysis and determines that the expected number of unaffected kittens should be:
The expected number of unaffected kittens should be 84
Find the below calculation of chi square:
Category Affected Unaffected Total
Observed values 44 84 128
Expected Ratio 1 3 4
Expected Values, E 32 96
Deviation, D 12 -12
Deviation Squared 14 144
D^2/E 4.5 1.5 6
X^2 6
Degrees of freedom - 1
Unaffected number of kittens:
Observed values for Affected=44
Total=128
25% of the 128 are affected.
The unaffected proportion is,
Total-Observed values for Affected kittens
=128-44
=84
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A wood board 100 cm long is cut into two pieces. One piece is 26 cm longer than the other. What are the lengths of the two pieces?
Answer:
the little peice (x) is 37 cm
the second piece(x +26) = 37 + 26 = 63 cm
Step-by-step explanation:
think the little or second piece is x
and another piece is (x + 26)
according to question,
x + x+26 = 100
2x + 26 = 100
2x = 100 - 26
2x = 74
x = 74 ÷ 2
x= 37
Would be nice if answered soon.
Simplify the expression.
√8r³y
Enter your answer in the box. Enter variables in alphabetical order when within roots.
√8x³y=
Answer:
2√2 x ³y
Step-by-step explanation:
1) Write the number in exponential form with the base of 2
2) Rewrite the exponent as a sum where one of the addends is a multiple of the index
3) Use a m+n m n = a xa to expand the expression
4) Any expression raised to the power of 1 equals itself
5) The root of a product is equal to the product of the roots of each factor
6)Reduce the index of the radical and exponent with 2
Then you get your answer!! Hope this helps
graph the proportion y = -4x
Answer in the attachment
In a math class with 22 students, a test was given the same day that an assignmentwas due. There were 14 students who passed the test and 13 students who completedthe assignment. There were 6 students who failed the test and also did not completethe assignment. What is the probability that a student chosen randomly from theclass passed the test and completed the homework?
Let:
A = students who passed the test = 14
B = students who completed the assignment = 13
N = Total students = 22
(A⁺ ∩ B⁺) = 6
P(A ∪ B) = 1 - 6/22 = 8/11
P(A ∩ B) = P(A) + P(B) - P(A ∪ B) = 14/22 + 13/22 - 8/11 = 1/2
P(A|B) = P(A ∩ B)/P(B) = (1/2)/(13/22) = 11/13 ≈ 0.85 ≈ 85%
Find the area and perimeter of a rectangle with vertices at (-1, 2), (3, 2),
(3.-5), and (-1,-5).
The area of the rectangle is A = L w, A = 4 * 3√5 = 12√5 Unit²
What is rectangle short answer?
A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. Because of this, it is also known as an equiangular quadrilateral. The term "parallelogram" can also be used to describe a rectangle because the opposing sides are equal and parallel.the points (-1 , 2) and (3 , 2 ) form one the of the rectangle w = [tex]\sqrt{( 3 + 1 )^{2} + ( 2 - 2 )^{2} }[/tex]
= [tex]\sqrt{4^{2} + 0 }[/tex]
= √ 16
= 4
The length can be obtained by finding distance between the points (-1,2) and ( -1, - 5) = [tex]\sqrt{( -1 -2 )^{2} + ( -5 + 1)^{2} }[/tex]
= [tex]\sqrt{(-3)^{2} + ( -6)^{2} }[/tex]
= √9 + 36
= √45 = 3√5
The perimeter of the rectangle is: P = 2( w + L) = 2( 4 + 3√5) ⇒ 8 + 6√5 unit
The area of the rectangle is A = L w, A = 4 * 3√5 = 12√5 Unit²
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If a+ 1/a = 5 then find the value of; i) a² + 1/a² ii) a³ + 1/a³
Answer:
23
110
Step-by-step explanation:
We have a + 1/a = 5
(a + 1/a)² = a² + (1/a²) + 2.a.1/a = a² + (1/a²) + 2
But (a + 1/a)² = 5² = 25
So a² + (1/a²) = 25 - 2 = 23
(a + 1/a)³ = a³ + 3a²(1/a) + 3a(1/a)² + (1/a)2
= a³ + 1/a³ + 3a + 3/a
= a³ + 1/a³ + 3(a + 1/a)
= a³ + 1/a³ + 3(5)
= a³ + 1/a³ + 15
But (a + 1/a)³ = 5³ = 125
So
a³ + 1/a³ + 15 = 125
a³ + 1/a³ = 110
Answer:
23 and 110
Step-by-step explanation:
using the expansion
(a + b)² = a² + 2ab + b² , then
a + [tex]\frac{1}{a}[/tex] = 5 ( square both sides )
(a + [tex]\frac{1}{a}[/tex] )² = 5²
a² + 2(a × [tex]\frac{1}{a}[/tex] ) + [tex]\frac{1}{a^2}[/tex] = 25
a² + 2(1) + [tex]\frac{1}{a^2}[/tex] = 25
a² + 2 + [tex]\frac{1}{a^2}[/tex] = 25 ( subtract 2 from both sides )
a² + [tex]\frac{1}{a^2}[/tex] = 23
---------------------------------------------------------------------
using the expansion
(a + b)³ = a³ + b³ + 3ab(a + b) , then
a + [tex]\frac{1}{a}[/tex] = 5 ( cube both sides )
(a + [tex]\frac{1}{a}[/tex] )³ = 5³
a³ + [tex]\frac{1}{a^3}[/tex] + 3(a × [tex]\frac{1}{a}[/tex] )(a + [tex]\frac{1}{a}[/tex] ) = 125
a³ + [tex]\frac{1}{a^3}[/tex] + 3(1)(5) = 125
a³ + [tex]\frac{1}{a^3}[/tex] + (3 × 5) = 125
a³ + [tex]\frac{1}{a^3}[/tex] + 15 = 125 ( subtract 15 from both sides )
a³ + [tex]\frac{1}{a^3}[/tex] = 110
how dose a value that is greater than 100 of the orangianl value less than1 of the orangilal value copare the oraginal value
If a value that is is greater than 100% of the original value will always be larger than original value.
If a value is less than 1% of the original value will always be less than the original value.
Given,
A value that is greater than 100% of the original value.
And it is 1% less than the original value.
We have to compare these values with the original value;
That is,
A value that is greater than 100% of the original value,
eg;
130% of 20 = 20 × 130/100 = 26
26 is greater than 20.
That is,
If a value that is is greater than 100% of the original value will always be larger than original value.
Next,
It is 1% less than the original value.
eg;
1% of 20 = 1/100 × 20 = 0.2
0.2 is less than 20
That is,
If a value is less than 1% of the original value will always be less than the original value.
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Which point represents -(-7)−(−7)minus, left parenthesis, minus, 7, right parenthesis on the number line?
The final point where -(-7)−(−7) number after simplification lie is 14
In the above question it is given,
-(-7)−(−7)
We need to find the position of this number on the number line.
A number line is a visual depiction of numbers on a straight line in mathematics. A number line's numerals are arranged in a sequential manner at equal intervals along its length. It is often displayed horizontally and can be infinitely expanded in any direction.
To do that, we will first simplify the given numbers by using the sign rules of the real numbers
According to the sign rule of real numbers,
(i) (-) x (-) = +
(ii) (-) x (+) = -
(iii) (-) +(-) = -
(iv) (+) + (+) = +
Now using the above sign rules, we will simplify the given number
-(-7)−(−7)
-(-7)+ [-(−7)]
7 + 7
Now, on the number line, we'll move 7 steps in the positive direction of 0, we'll be on 7. Again from 7 we'll take next 7 steps in the same direction and we'll reach the point 14.
Hence, 14 is the final point where -(-7)−(−7) number after simplification lie.
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Scientists are drilling a hole in the ocean floor to learn more about the Earth's history. Currently, the hole is in the shape of a cylinder whose volume is approximately 4200 cubic feet and whose height is 2.1 miles. Find the radius of the hole to the nearest hundredth of a foot. (Hint: Make sure the same units of measurement are used.)
The radius of the hole to the nearest hundredth of a foot is 2.88 feet.
What is radius?
The separation between a circle's centre and circumference is called the radius.Its diameter is halved that of the circle.The path that leads from a circle's centre to one of its points.The separation between a circle's centre and a certain point upon that circle.The volume of the cylinder, V = 4200 cubic feet
Height, H = 2.1 miles = 11088 feet
The volume of the cylinder is given by the formula, V =πr²H
Putting values in the formula to find out radius,
4200=πr²(11088)
r²=[(11088)π]/4200
r²=8.2938
r=2.879~2.88 feet
The radius of the hole to the nearest hundredth of a foot is 2.88 feet.
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A 16-foot board is to be cut into 4 pleces. Three of the pleces will be the same length, and one piece will be 4 feet
longer than the other three.
Which equation could be used to solve for the lengths?
x+x+x+x+4=16
x+x+x+4x=16
x+x+x+X-4=16
x+x+x+x=16+4
Answer:
I think it's B
Step-by-step explanation:
It's the only one with 3 x's to represent the part where it's cut into four pieces the x represents the 3 numbers and then the 4x means how the x = the number that the 3 represent plus the 4 feet longer
Determine whether the equation below has a one solutions, no solutions, or an infinite number of solutions. Afterwards, determine two values of xx that support your conclusion.
x/-2=-9
The equation x/-2 = -9 has only one solution and is x = 18.
We are given an equation x/-2 = -9 for which we need to check whether it has a solution, no solution or infinite solutions.
For this we first solve this equation.
x/-2 = -9
⇒ x = -9 x -2
⇒ x = 18
So this equation has only one solution and it is unique.
Hence we cannot find two values of to support this conclusion, since x = 18 is the only value which satisfy this equation.
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16. Given that Angle 1 and Angle 2 form a linear pair, find the measure of both angles if m<1 = (5x + 9)° and
m<2= (3x + 11).
x =
m<1 =
m<2=
Answer: [tex]x=20, m\angle 1=109^{\circ}, m\angle 2=71^{\circ}[/tex]
Step-by-step explanation:
Angles that form a linear pair add to 180 degrees, so:
[tex]5x+9+3x+11=180\\\\8x+20=180\\\\8x=160\\\\x=20\\\\\implies m\angle 1=5(20)+9=109^{\circ}\\\\\implies m\angle 2=180^{\circ}-109^{\circ}=71^{\circ}[/tex]
through: (4, -3) and (-5, -4)
Find point-slope form of the equation.
Answer:
y + 3 = (1/9) (x - 4)
Step-by-step explanation:
To find slope:y2-y1 / x2-x1
-4 -(-3) / -5 - 4
Double negative switches the # to a positive
-4+3 / 5 - 4
= -1 / -9
Two negatives equal a positive
m=1/9
point-slope form:y-y1=m(x-x1)
y-(-3)=(1/9)(x-4)
Solving:Double negative=positive #
y+3=(1/9)(x-4)
Distribute (1/9) to the x and -4
y+3= (1/9)x - (4/9)
Subtract 3 from both sides
y = -3.4