The quadratic equation with roots 1 and 5 and leading coefficient 2 is 2x^2 - 12x + 10 = 0
To write the quadratic equation with roots 1 and 5 and leading coefficient 2, we can use the fact that the general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants.
Since the roots are 1 and 5, we know that the factors of the quadratic equation are:
(x - 1) and (x - 5)
Multiplying these factors together, we get:
(x - 1)(x - 5) = x^2 - 6x + 5
To satisfy the condition of having a leading coefficient of 2, we can simply multiply the entire equation by 2
2(x^2 - 6x + 5) = 2x^2 - 12x + 10
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isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft
The area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft
Isosceles trapezoid, also known as isosceles trapezium, is defined as a convex quadrilateral, which mainly consists of a line of symmetry that bisects one pair of the opposite side, it is also know that the base angles are of equal measure. It has parallel bases with the legs also being of the equal measure, on the other hand the opposite angles of the isosceles trapezoid are supplementary, which makes it a cyclic quadrilateral.
Area of an Isosceles Trapezoid = [(a+b)h]/2 square units,
when we put these values in the formula, we get:
Area = [(2+5)2.5]/2
Area = (7 × 2.5)/2
Area = 17.5/2
Area = 8.75ft²
therefore, we know that the area of the isosceles trapezoid with base lengths 2ft and 5ft and leg lengths 2.5ft will be 8.75ft
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Given that a function, g, has a domain of -20 ≤ x ≤ 5 and a range of -5 ≤ g(x) ≤ 45 and that g(0) = -2 and g(-9) = 6, select the statement that could be true for g.
g(7) = -1
g(-13)= 20
g(-4)= - 11
g(0) = 2
The statement that could be true for g is g(-4) = -11.
What is a function?
A function is a mathematical relationship between a set of inputs (called the domain) and a set of outputs (called the range), where each input has a unique output. In other words, for every value of x in the domain, there is exactly one corresponding value of g(x) in the range.
For the given function g, the domain is -20 ≤ x ≤ 5 and the range is -5 ≤ g(x) ≤ 45. We also know that g(0) = -2 and g(-9) = 6.
To determine which statement could be true for g, we can check each option against the given domain and range, as well as the known values of g(0) and g(-9):
g(7) = -1: This statement is not necessarily true, as g(7) may fall outside the given range of -5 ≤ g(x) ≤ 45.
g(-13) = 20: This statement is not necessarily true, as g(-13) may fall outside the given domain of -20 ≤ x ≤ 5.
g(-4) = -11: This statement could be true, as -20 ≤ -4 ≤ 5 and -5 ≤ -11 ≤ 45. However, we cannot confirm this without additional information.
g(0) = 2: This statement is not true, as g(0) is known to be -2.
Therefore, the statement that could be true for g is g(-4) = -11.
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What statement describes the decimal equivalent of 5 ? 16 Responses A It is a decimal that terminates after 4 decimal places. It is a decimal that terminates after 4 decimal places. B It is a decimal with a repeating digit of 5. It is a decimal with a repeating digit of 5. C It is a decimal that terminates after 3 decimal places. It is a decimal that terminates after 3 decimal places. D It is a decimal with repeating digits of 6
The statement which describes the decimal equivalent of 5/16 is option(a), It is a decimal that terminates after 4 decimal places.
To convert a fraction to a decimal, we need to divide the numerator by the denominator.
In this case, we want to find the decimal equivalent of the fraction 5/16,
⇒ 5 ÷ 16 = 0.3125
So, the decimal equivalent of 5/16 is 0.3125.
The option(a) states that, It is a decimal that terminates after 4 decimal places.
This statement is correct, because the decimal equivalent of 5/16 terminates after 4 decimal places.
Therefore, the correct statement is option is (a).
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The given question is incomplete, the complete question is
What statement describes the decimal equivalent of 5/16?
(a) It is a decimal that terminates after 4 decimal places.
(b) It is a decimal with a repeating digit of 5.
(c) It is a decimal that terminates after 3 decimal places.
(d) It is a decimal with repeating digits of 6.
336 fewer the the product of u and 258 is 168
Answer:
335.6511628
Step-by-step explanation:
Ux258-336=168
U-336=0.6511628
U=336.6511628
Translate the following to an algebraic expression: r squared minus the product of 6 and d plus 5
The algebraic expression for "r squared minus the product of 6 and d plus 5" is r2 - 6d + 5.
What is algebraic expression?An algebraic expression is an expression that contains at least one variable and uses mathematical operations such as addition, subtraction, multiplication, and division. It can also include exponents and/or roots. Algebraic expressions are used to represent mathematical relationships between different variables or constants. They can be used to represent real-world problems, such as finding the area of a rectangle.
In this expression, r2 represents "r squared", - 6d represents "minus the product of 6 and d", and + 5 represents "plus 5".
So, the final algebraic expression is r2 - 6d + 5.
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Order the numbers from least to greatest
pls edit yours and i will give an answer
Step-by-step explanation:
Answer:
I can't order the numbers without knowing the numbers sorry
Letf(x)=x−5xandg(x)=x3. Find the following functions. Simplify your answers.f(g(x))=g(f(x))=
The f(g(x))=−4x3and g(f(x))=x3−15x2+25x−125x3.
To find f(g(x)), we need to substitute the function g(x) into the function f(x). This means that wherever we see an "x" in f(x), we replace it with the function g(x). So, f(g(x))=g(x)−5g(x)=x3−5x3=−4x3. Similarly, to find g(f(x)), we substitute the function f(x) into the function g(x). So, g(f(x))=f(x)3=(x−5x)3=x3−15x2+25x−125x3.
Therefore, f(g(x))=−4x3and g(f(x))=x3−15x2+25x−125x3.
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Martin buys 2(5)/(6) pounds of peanuts and 4(3)/(4) pounds of almonds. How many pounds of nuts does Martin buy?
Martin buys a total of 2(5)/(6) + 4(3)/(4) = 10/6 + 19/4 = 40/24 + 57/24 = 97/24 pounds of nuts.
To find the total amount of nuts Martin buys, we need to add the amount of peanuts and the amount of almonds he buys.
First, we need to convert the mixed numbers to improper fractions. To do this, we multiply the whole number by the denominator and add the numerator. For 2(5)/(6), we multiply 2 by 6 and add 5 to get 10/6. For 4(3)/(4), we multiply 4 by 4 and add 3 to get 19/4.
Next, we need to find a common denominator in order to add the two fractions. The least common denominator for 6 and 4 is 24, so we multiply the numerator and denominator of 10/6 by 4 to get 40/24 and the numerator and denominator of 19/4 by 6 to get 57/24.
Finally, we add the two fractions by adding the numerators and keeping the same denominator: 40/24 + 57/24 = 97/24.
Therefore, Martin buys a total of 97/24 pounds of nuts.
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Three equivalent fractions for 30/48
1) 5/8
2) 15/24
3) 300/480
Answer: 5/8
10/16
15/24
Step-by-step explanation:
vv^(1) 2 4 Solve by using the quadratic formula. Express the solution set in exact simplest form. x^(2)-5x-5=0 The solution set is
To solve the equation x^(2)-5x-5=0 using the quadratic formula, we can use the formula x = (-b ± √(b^(2)-4ac))/(2a), where a, b, and c are the coefficients of the equation. In this case, a=1, b=-5, and c=-5.
Plugging in the values into the formula, we get:
x = (-(-5) ± √((-5)^(2)-4(1)(-5)))/(2(1))
Simplifying the equation, we get:
x = (5 ± √(25+20))/2
x = (5 ± √45)/2
x = (5 ± 3√5)/2
Therefore, the solution set in exact simplest form is x = (5 ± 3√5)/2.
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A quadratic function models the graph of a parabola. The quadratic functions, y=x and y=x² +3, are modeled in the
graphs of the parabolas shown below.
-10-9-8-76
y M
10
-2-1
0
8
40
6
-7
-8
9
-10-
y=x²+3
y=x²
。
Determine which situations best represent the scenario shown in the graph of the quadratic functions, y=x² and y=x²
+3. Select all that apply.
The situations that best represent the scenario shown in the graph of the quadratic functions, y = x² and y = x² + 3 include the following:
B. "From x = -2 to x = 0, the average rate of change for both functions is negative."
C. "For the quadratic function, y = x² + 3, the coordinate (2, 7) is a solution to the equation of the function."
D. "The quadratic function, y = x², has an x-intercept at the origin."
What is the x-intercept?In Mathematics, the x-intercept of the graph of any function simply refers to the point at which the graph of a function crosses or touches the x-coordinate (x-axis) and the y-value or value of "y" is equal to zero (0).
In this context, we can logically deduce that the x-intercepts of the graph of the given equation y = x² is at the origin:
x = (0, 0)
For the the coordinate (2, 7), we would evaluate the quadratic function, y = x² + 3 as follows;
7 = 2² + 3
7 = 4 + 3
7 = 7 (True).
By critically observing the graph of both functions, we can logically deduce that their average rate of change is negative from x = -2 to x = 0.
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A geneticist models the occurrence of defective genes in the kidney of mouse exposed to a toxin as a Poisson process with an average rate of 3 defective genes per 20 cc of kidney tissue. He wants to examine the variation in the occurrence of defective genes by simulating the number, D, of defective genes in 100 cc of kidney tissues. One such realization of D is generated by using U = 0.71 as a random observation from a Uniform distribution on (0,1). The corresponding value
of D is?.
The corresponding value of D is 15 defective genes in 100 cc of kidney tissue.
The Poisson distribution is used to model the occurrence of rare events in a fixed interval of time or space. In this case, the geneticist is using the Poisson process to model the occurrence of defective genes in kidney tissue exposed to a toxin. The average rate of occurrence is given as 3 defective genes per 20 cc of kidney tissue.
To simulate the number of defective genes in 100 cc of kidney tissue, we need to scale the average rate accordingly. Since 100 cc is five times the size of 20 cc, the average rate for 100 cc would be 15 defective genes.
The geneticist uses a uniform distribution on (0,1) to generate a random observation, U = 0.71. To obtain the corresponding value of D, we can use the inverse transform method. Let X be the number of defective genes in 100 cc of kidney tissue, then we have:
P(X = k) = e^(-λ) (λ^k / k!)
where λ is the average rate of occurrence, which is 15 in this case.
By plugging in λ = 15 and solving for k, we get:
P(X = k) = e^(-15) (15^k / k!) = 0.71
Using a calculator or a statistical software, we can find that the value of k that satisfies this equation is k = 15. Therefore, the corresponding value of D is 15 defective genes in 100 cc of kidney tissue.
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find the slope of the line that contains each pair of points (2,6) and (0,1)
5/2 is the slope of line .
What is slope, exactly?
Typically, a line's slope provides information about the steepness and direction of the line. Finding the difference between the coordinates of the locations will allow you to quickly compute the slope of a straight line connecting two points, (x1,y1) and (x2,y2).
The letter "m" is frequently used to signify slope. A line's steepness can be determined by its slope. In mathematics, slope is calculated as "rise over run" (change in y divided by change in x).
points (2,6) and (0,1)
Slope = 1 - 6/0 - 2
= -5/-2
= 5/2
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In 1 hour 32 cars pass through a particular intersection. At the same rate, how long would it take for 96 cars to pass through the intersection?
It wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.
We can use the cοncept οf prοpοrtiοnaIity tο sοIve this prοbIem. The number οf cars passing thrοugh the intersectiοn is directIy prοpοrtiοnaI tο the amοunt οf time taken. This means that if we dοubIe the number οf cars passing thrοugh, we wiII aIsο dοubIe the amοunt οf time taken.
Let x be the amοunt οf time it takes fοr 96 cars tο pass thrοugh the intersectiοn. Then, we can set up the fοIIοwing prοpοrtiοn:
32 cars / 1 hοur = 96 cars / x hοurs
Tο sοIve fοr x, we can crοss-muItipIy and simpIify:
32x = 96
x = 3
Therefοre, it wοuId take 3 hοurs fοr 96 cars tο pass thrοugh the intersectiοn at the same rate as 32 cars in 1 hοur.
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Write the equation in general form of a parabola with zeros of 3 and 9 and goes through the point \( (6,-18) \). \[ \begin{array}{l} y=-2 x^{2}-24 x+54 \\ y=-2 x^{2}+24 x+54 \\ y=2 x^{2}-24 x+54 \\ y=
The equation of a parabola with zeros at 3 and 9 and goes through the point \( (6,-18) \) is
y = 2x^2 - 24x + 54.
To write the equation in general form of a parabola with zeros of 3 and 9 and goes through the point (6,-18), we can use the fact that the equation of a parabola can be written in the form y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola and a determines the width of the parabola.
First, we can use the zeros to find the vertex of the parabola. The vertex is located halfway between the zeros, so the x-coordinate of the vertex is (3 + 9)/2 = 6. We can plug this value into the equation to find the y-coordinate of the vertex:
y = a(6 - 6)^2 + k = k
Since the parabola goes through the point (6,-18), we know that k = -18.
Now we can plug in the zeros and the vertex into the equation to find the value of a:
0 = a(3 - 6)^2 - 18
0 = a(9) - 18
18 = 9a
a = 2
So the equation of the parabola is y = 2(x - 6)^2 - 18.
To write this equation in general form, we can expand the squared term and simplify:
y = 2(x^2 - 12x + 36) - 18
y = 2x^2 - 24x + 72 - 18
y = 2x^2 - 24x + 54
So the equation in general form is y = 2x^2 - 24x + 54.
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What are the x-intercepts of the quadratic function?
f(x)=x^2+6x−27
enter your answer in the boxes
blank and blank
Step-by-step explanation:
To find the x-intercepts of the quadratic function f(x) = x^2 + 6x - 27, we need to set f(x) equal to zero and solve for x.
So we have:
f(x) = 0
x^2 + 6x - 27 = 0
We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Where a = 1, b = 6, and c = -27.
Substituting these values into the quadratic formula, we get:
x = (-6 ± sqrt(6^2 - 4(1)(-27))) / 2(1)
x = (-6 ± sqrt(180)) / 2
x = (-6 ± 6sqrt(5)) / 2
x = -3 ± 3sqrt(5)
So the x-intercepts of the quadratic function are approximately -10.16 and 4.16.
The diagonals of a rhombus measure 8cm and 6cm .what is the length of a side of the rhombud
All sides of the given rhombus measure 5 cm respectively.
What is a rhombus?A rhombus is a quadrilateral with four equal-length sides in planar Euclidean geometry.
The term "equilateral quadrilateral" refers to a quadrilateral whose sides all have equal lengths.
Quadrilaterals having four congruent sides are called squares.
Squares are rhombuses by definition since they are quadrilaterals with four congruent sides.
So, consider the following values: AC = 8 cm, BD = 6 cm, OD = OB = 3 cm, and OA = OC = 4 cm.
Now,
OB² + OC² = BC²
3²+ 4² = BC²
9 + 16 = BC²
25 = BC²
5 = BC
A rhombus has an equal number of sides.
Therefore, all sides of the given rhombus measure 5 cm respectively.
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PLEASE HELP I NEED HELP PLEASE
Answer:
Step-by-step explanation:
You take 528- divided by 2000- and then you get your answer.
Knowledge Check Divide. (x+6)/(3x+9)-:(7)/(9x)+42 Simplify your answer as much as possible.
(3x²+18x)/(7x+21) is the simplified answer of (x+6)/(3x+9)-:(7)/(9x)+42.
To divide the given expressions, we need to first simplify the expressions as much as possible and then use the rule for dividing fractions. The rule for dividing fractions is to multiply the first fraction by the reciprocal of the second fraction.
Simplifying the expressions:
(x+6)/(3x+9) = (x+6)/(3(x+3)) = (x+6)/3(x+3)
(7)/(9x) = 7/9x
Now, using the rule for dividing fractions:
((x+6)/3(x+3)) ÷ (7/9x) = ((x+6)/3(x+3)) * (9x/7)
= (9x(x+6))/(3(x+3)*7)
= (3x(x+6))/(x+3)*7
= (3x²+18x)/(7x+21)
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If the simple interest on $7000 for 7 years is $3430, then what is the interest rate?
The interest rate will be 7%
What is simple interest?A quick and simple way to figure out interest on money is to use the simple interest technique, which adds interest at the same rate for each time cycle and always to the initial principal amount.
Any bank where we deposit our funds will pay us interest on our investment. One of the different types of interest charged by banks is simple interest. Now, before exploring the idea of basic curiosity in further detail,
The formula for simple interest is:
S = p*t*r/100
the simple interest on $7000 for 7 years is $3430
r = S*100/(p*t)
r = (3430*100)/(7000*7)
r = 7%
Hence the interest rate will be 7%
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area of a triangle that is 34in base. and 14in height.
Answer:
Step-by-step explanation:
A=[tex]\frac{1}{2}[/tex]bh
=[tex]\frac{1}{2}[/tex]×34×14
=238 in²
Expressio
5. The teacher of Class 705 gave out permission
slips for students to go to MoMath, the math
museum in NYC.
On the first day, 12 students returned their
permission slips.
On the second day, 15% of the remaining
students returned their slips.
A total of three permission slips were
returned on the second day.
What is the total number of students in Class 705
The total number of students in Class 705 is 32.
How to calculated the total number of students in Class 705
Let's start by using algebra to solve the problem. Let's assume that the total number of students in Class 705 is "x". We know that 12 students returned their permission slips on the first day, so the number of students who have not returned their slips is (x-12).
On the second day, 15% of the remaining students returned their slips, which means that 0.15(x-12) students returned their slips. We also know that a total of three permission slips were returned on the second day, so we can write an equation:
0.15(x-12) = 3
Simplifying this equation:
0.15x - 1.8 = 3
0.15x = 4.8
x = 32
Therefore, the total number of students in Class 705 is 32.
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Triangle ABC is enlarged with a scale factor of -2 and the origin of the centre to give triangle ABC . WORK OUT THE COORDINATES OF A and B
The coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
What is a dilation?Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.
here, we have,
From the given graph, triangle ABC have A(1, 3), B(7, 1) and (1, 1).
Triangle ABC is enlarged with a scale factor of -2 and the origin of the Centre to give triangle A'B'C'.
Now, A(1, 3) → -2(1, 3)→A'(-2, -6)
B(7, 1) → -2(7, 1) → B'(-14, -2)
C(1, 1) → -2(1, 1) → C'(-2, -2)
Therefore, the coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
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A fair coin is tossed 100 times. Using normal approximation to binomial probabilities, find the probability that the number of heads would take a value within one standard deviation from the mean. a. 0.7698 b. 0.7286 C. 0.6826 d. 0.6318
The answer is C
The probability that the number of heads would take a value within one standard deviation from the mean is 0.6826. This can be found by using the normal approximation to binomial probabilities.
First, we need to find the mean and standard deviation of the binomial distribution. The mean of a binomial distribution is given by np, where n is the number of trials and p is the probability of success. In this case, n = 100 and p = 0.5 (since it is a fair coin), so the mean is 100 * 0.5 = 50.
The standard deviation of a binomial distribution is given by √(np(1-p)). In this case, the standard deviation is √(100 * 0.5 * (1-0.5)) = √(25) = 5.
Now, we can use the normal approximation to find the probability that the number of heads is within one standard deviation from the mean. This is equivalent to finding the probability that the number of heads is between 45 and 55 (since the mean is 50 and the standard deviation is 5).
Using the normal approximation, we can find the z-scores for 45 and 55:
z = (x - μ) / σ
z1 = (45 - 50) / 5 = -1
z2 = (55 - 50) / 5 = 1
Now, we can use a z-table to find the probability that the number of heads is between 45 and 55. The probability that the number of heads is less than 55 is 0.8413, and the probability that the number of heads is less than 45 is 0.1587. So, the probability that the number of heads is between 45 and 55 is 0.8413 - 0.1587 = 0.6826.
Therefore, the answer is C. 0.6826.
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The number of coins in a person's collection changes based on buying, selling, and trading coins. A function defined as f(t) = t³ - 6t² + 9t
is modeled by the table, which represents the number of coins in the coin collection t years since the person began collecting coins.
(Picture has the rest of the problem)
The statements that are true are
The relative minimum function is (3, 0)
When t > 3 the function will increase
Finding the maximum and minimum function:To find the maximum and minimum of a function, find the points where the function's derivative is equal to zero or undefined. These points are called critical points.
We then evaluate the function at these critical points and at the endpoints of the interval to determine the maximum and minimum values. To find the critical points of the function find the derivatives of the given function
Here we have a graph
The graph shows the number of coins in a person's collection
The function is defined as f(t) = t³ - 6t² + 9t
To find the maximum and minimum of the function
Find the critical points where the derivative is equal to zero or undefined.
Differentiate f(t) with respect to t
=> f'(t) = 3t² - 12t + 9
Now, set the derivative equal to zero and solve for t
=> 3t² - 12t + 9 = 0
=> t² - 4t + 3 = 0 divide by 3
On Factoring the quadratic equation, we get:
=> (t - 3)(t - 1) = 0
=> t = 3 and t = 1
Therefore,
The critical points of the graph are t = 1 and t = 3.
Now differentiate f'(t) with respect to t
=> f''(t) = 6t - 12
At t = 1, f''(1) = 6 - 12 = - 6, which is less than zero.
Hence, f(t) has a local maximum at t = 1.
At t = 3, f''(3) = 18 - 12 = 6, which is greater than zero.
Hence, f(t) has a local minimum at t = 3.
At t = 4, f''(4) = 24 - 12 = 12
At t = 5, f''(5) = 30 - 12 = 18
Hence, f(t) will increase when t > 3
Therefore,
The statements that are true are
The relative minimum function is (3, 0)
When t > 3 the function will increase
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Five different stores sell a bag of flour for one of the following prices: $1.45, $1.55, $1.90, $1.95, $1.45. What are the mean, median, and mode of the data?
The mean, median and mode of the prices that the bags of flour are sold at are:
Mean - $ 1.66Median - $ 1. 55Mode - None How to find the mean, mode and median ?To find the mean, we add up all the prices and divide by the total number of prices:
Mean:
= (1.45 + 1.55 + 1.90 + 1.95 + 1.45) / 5
= 1.66
To find the median, we need to arrange the prices in order from lowest to highest:
1.45, 1.45, 1.55, 1.90, 1.95
There are five prices, so the median is the middle value, which is 1.55.
To find the mode, we need to look for the price that appears most frequently in the data set. In this case, there are two prices that appear twice, 1.45 and 1.55. So there is no unique mode for this data set.
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What’s the answer ?
if f(x) = 3*2ˣ + 2 is vertically stretched by a factor of 2, then the new function g(x) is 6*2ˣ + 4.
What is the new function g(x)?
To vertically stretch the function f(x) by a factor of 2, we need to multiply the entire function by 2.
This will stretch the function vertically, making it twice as tall as before.
Therefore, if f(x) = 3*2ˣ + 2 is vertically stretched by a factor of 2, then the new function g(x) is:
g(x) = 2*f(x)
g(x) = 2*(3*2ˣ + 2)
g(x) = 6*2ˣ + 4
So the new function g(x) is 6*2ˣ + 4.
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Convert the Cartesian coordinates (1,−5) to polar coordinates
with r>0 and 0≤θ<2π. give theta answer in radiants
The Cartesian coordinates (1, -5) are equivalent to the polar coordinates (√26, 4.910188838655984) with r > 0 and 0 ≤ θ < 2π.
To convert the Cartesian coordinates (1, -5) to polar coordinates with r>0 and 0≤θ<2π, we need to use the following formulas:
r = √(x^2 + y^2)
θ = tan^-1(y/x)
First, let's find the value of r:
r = √(1^2 + (-5)^2)
r = √(1 + 25)
r = √26
Now, let's find the value of θ:
θ = tan^-1((-5)/1)
θ = tan^-1(-5)
θ = -1.373400766945016
Since we need the angle to be between 0 and 2π, we can add 2π to the negative angle to get the positive equivalent:
θ = -1.373400766945016 + 2π
θ = 4.910188838655984
Therefore, the polar coordinates of the Cartesian coordinates (1, -5) are (r, θ) = (√26, 4.910188838655984).
So the answer is:
r = √26
θ = 4.910188838655984
The Cartesian coordinates (1, -5) are equivalent to the polar coordinates (√26, 4.910188838655984) with r > 0 and 0 ≤ θ < 2π.
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that the denominators are different. Simplify the result, if possible. (2)/(y-5)+(7)/(y+7)
The simplified expression is (3y-7)/(y^2-35)/3.
To simplify the given expression, we need to find the least common denominator (LCD) of the two fractions. The LCD of (y-5) and (y+7) is (y-5)(y+7).
Next, we need to multiply each fraction by the LCD in order to get the same denominator for both fractions.
(2)/(y-5) * (y+7)/(y+7) + (7)/(y+7) * (y-5)/(y-5)
= (2y+14)/(y^2-35) + (7y-35)/(y^2-35)
Now, we can combine the two fractions since they have the same denominator.
(2y+14+7y-35)/(y^2-35)
= (9y-21)/(y^2-35)
Finally, we can simplify the fraction by factoring out a 3 from the numerator.
= (3)(3y-7)/(y^2-35)
= (3y-7)/(y^2-35)/3
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Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers.
f(x) = x²+1, g(x) = √x+5
f(g(x)) =
g(f(x)) =
The solutions of the given functions are:
f(g(x)) = x+10√x+26 and g(f(x)) = √(x²+6)
f(x) = x²+1, g(x) = √x+5 are given.
To find f(g(x)), we need to substitute g(x) into the function f(x):
f(g(x)) = f(√x+5) = (√x+5)²+1 = x+10√x+26
To find g(f(x)), we need to substitute f(x) into the function g(x):
g(f(x)) = g(x²+1) = √(x²+1+5) = √(x²+6)
Therefore, the simplified answers are:
f(g(x)) = x+10√x+26
g(f(x)) = √(x²+6)
Note: It is important to include the parentheses when substituting one function into another to ensure the correct order of operations.
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