Factorization of expression 16t³ + 10t is 2t (8t² + 5) .
Factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15
Given expression
16t³ + 10t
It can be written as
2 × 8 (t . t . t ) + 2 × 5 (t)
Factoring out common factors
2t (8t² + 5)
Hence, 2t (8t² + 5) is factorization of expression 16t³ + 10t
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what is
1) y=2x
y=x+15
2) 2x+5y=55
-6x+y=-4
(systems of equations
Answer:
53
Step-by-step explanation:
becuas eyou divide it all togtheir and then subtract 42 from 83 and you ga the anwer
Solve the following inequality involving absolute value. Enter the answer in interval notation.
∣x+4∣>−9
Interval notation is (−∞, −4) U (−4, ∞)
The given inequality is ∣x + 4∣ > − 9. Step-by-step explanation: Absolute value inequalities: Inequalities that contain absolute values are known as absolute value inequalities. The absolute value inequality ∣x + 4∣ > − 9 is given.Inequality means that x can take any value except the value that makes the inequality false. If x satisfies the inequality, we write x ∈ (A, B) or x ∈ [A, B), where A and B are any two values that satisfy the inequality.The inequality ∣x + 4∣ > − 9 implies that the absolute value of x + 4 is greater than −9. The absolute value of x + 4 is always greater than or equal to zero.Therefore, the inequality can be written as∣x + 4∣ > 0This inequality implies that x is not equal to −4.The interval of x satisfying the given inequality is x ∈ (−∞, −4) U (−4, ∞), where U represents the union of two intervals. Therefore, the answer in interval notation is (−∞, −4) U (−4, ∞).Thus, the solution to the inequality |x + 4| > -9 in interval notation is (−∞, −4) U (−4, ∞).
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A study showed that vitamin E oil reduces pain levels in patients with lupus. During each session, the oil was injected into the pain site indicated by the patient. Pain reduction was measured through self-reporting after each session.
Another study is being designed to examine whether vitamin E oil also reduces pain in patients suffering from bulging disks in the thoracic region of the back. Two hundred patients aged 20–40 years are subjects in the new study.
Part A: What is an appropriate design for the new study? Include treatments used, method of treatment assignment, and variables that should be measured.
Part B: If the study consisted of 100 young patients and 100 old patients instead of 200 patients aged 20–40, would you change the study design? If so, how would you modify your design? If not, why not?
Instead of 200 participants aged 20 to 40, the research would need to unitary method include 100 young and 100 elderly people. One possible change would be to divide the sample into age groups.
What is unitary method ?The unit technique is a strategy for addressing issues that entails first figuring out the value of a single unit, then multiplying that value to figure out the needed value. Simply expressed, the unit technique is used to extract a single unit value from a multiple that is provided. For example, 40 pens would cost 400 rupees, which is equal to the cost of one pen.
Part A:
A randomized controlled trial would be the ideal approach for the new investigation (RCT). A control group and an experimental group should both be included in the study.
The Visual Analog Scale is a pain scale that may be used by participants to self-report their level of pain, which is one of the variables that should be measured (VAS). Functional status, quality of life, and unfavorable treatment effects are other factors that might be assessed.
Part B:
Instead of 200 participants aged 20 to 40, the research would need to include 100 young and 100 elderly people. One possible change would be to divide the sample into age groups.
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Rotate point A 180 counterclockwise about the origin to get point A'. Which of the following coordinate is A'?
Rotate point A 180 counterclockwise about the origin to get point A'. -2,-3 coordinate is A'
What is counterclockwise rotation?
Rotations that are counterclockwise (CCW) travel in the opposite direction from how a clock's hands move. Positive numerals are used to indicate these rotations.
Rotation in either a clockwise or anticlockwise direction refers to a change in the electrical activity flowing through the heart in a horizontal plane. Consider the observer as being at the bedside of the sufferer. It is known as anticlockwise rotation if the patient's heart's electrical activity has shifted more to their right side. It is known as clockwise rotation if the patient's heart's electrical activity has shifted more to the left.
Our point A(x,y) rotates 180 degrees anticlockwise around the origin to become A' (-x,-y). Making both x and y negative is all we do as a result.
Here we can see in the graph ‘A’ is present on (2,3)
So the -2,-3 coordinate is A’.
Hence the correct answer is c, -2,-3
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Suppose that y varies directly as the cube root of x, and that y=100 when x=6859. What is y when x=1331? Round your answer to two decimal places if necessary.
The value of y = 57.86 when x = 1331.
We are given that y varies directly as the cube root of x. This means that the equation relating y and x is of the form:
y = k * cube_root(x)
Where k is the constant of proportionality. We are also given that y = 100 when x = 6859. We can use this information to find the value of k:
100 = k * cube_root(6859)
k = 100 / cube_root(6859)
k = 100 / 19
k = 5.26
Now we can use the value of k to find y when x = 1331:
y = 5.26 * cube_root(1331)
y = 5.26 * 11
y = 57.86
Therefore, y is approximately 57.86 when x is 1331. We can round this answer to two decimal places to get:
y = 57.86
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i would appreciate anyone's help. thank you!
The equation of the volume of the box in terms of x is 4x³ - 40x² +100x.
What is the V(x) function's domain?The set of all conceivable values of x that make sense in the context of the issue constitutes the domain of the function V(x). This time, x must be a positive value less than or equal to 5, as it reflects the length of the side of the square that was cut out of each corner of the cardboard. The range of V(x) is thus 0 x 5.
Given that, the squares cut at the end are x inches.
Thus the dimension of the rectangular box are:
The length of the box will be (10-2x).
The width of the box will be (10-2x).
The height of the box will be x inches.
The volume of the rectangular box is given by:
V = (l)(w)(h)
Substituting the values we have:
V(x) = (10-2x)(10-2x)x = x(100-40x+4x^2) = 4x³ - 40x² +100x.
Hence, the equation of the volume of the box in terms of x is 4x³ - 40x² +100x.
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Solve for the exact solutions in the interval (0,2π). If the equation has no solutions, respond with DNE Cos (2x) cos(x) - sin(2x)sin(x) = - ✓3 / 2
____________
The exact solutions are x ≈ 0.590, x ≈ 2.552, and x ≈ 5.103.
The equation we are solving for exact solutions in the interval (0,2π) is Cos (2x) cos(x) - sin(2x)sin(x) = - √3 / 2.
First, let's use the double angle formula for cosine to simplify the equation:
cos(2x) = 1 - 2sin^2(x)
Substituting this into the equation, we get:
(1 - 2sin^2(x))cos(x) - sin(2x)sin(x) = - √3 / 2
Next, let's use the double angle formula for sine to simplify the equation further:
sin(2x) = 2sin(x)cos(x)
Substituting this into the equation, we get:
(1 - 2sin^2(x))cos(x) - 2sin^2(x)cos(x) = - √3 / 2
Combining like terms and rearranging, we get:
4sin^2(x)cos(x) - cos(x) = √3 / 2
Factoring out cos(x), we get:
cos(x)(4sin^2(x) - 1) = √3 / 2
Dividing both sides by (4sin^2(x) - 1), we get:
cos(x) = (√3 / 2) / (4sin^2(x) - 1)
Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can substitute for sin^2(x):
cos(x) = (√3 / 2) / (4(1 - cos^2(x)) - 1)
Multiplying both sides by (4(1 - cos^2(x)) - 1), we get:
cos(x)(4(1 - cos^2(x)) - 1) = √3 / 2
Expanding and rearranging, we get:
4cos^3(x) - 5cos(x) + √3 / 2 = 0
This is a cubic equation, which is difficult to solve algebraically. However, we can use a graphing calculator to find the approximate solutions. The solutions are x ≈ 0.590, x ≈ 2.552, and x ≈ 5.103.
Since all of these solutions are in the interval (0,2π), the exact solutions are x ≈ 0.590, x ≈ 2.552, and x ≈ 5.103.
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A triangle has angle measurements of 3°, 35°, and 142°. What kind of triangle is it?
The triangle is an obtuse triangle.
What is a valid triangle?
A valid triangle is a geometric figure with three sides and three angles. To be considered a valid triangle, it must satisfy the following conditions:
1. The sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem, and it ensures that the three sides can form a closed figure.
2. The measure of each angle must be less than 180 degrees. This is known as the Angle Sum Theorem, and it ensures that the three angles can fit inside a two-dimensional plane.
The sum of the angles of any triangle is always 180 degrees.
So, if a triangle has angle measurements of 3°, 35°, and 142°, we can add them up to check whether they add up to 180 degrees:
3° + 35° + 142° = 180°
Since the sum of the angles is 180 degrees, the triangle is a valid triangle.
Now, we can classify the triangle based on the measure of its angles. Since the measure of one angle (142°) is greater than 90 degrees, the triangle is an obtuse triangle.
We cannot determine the length of the sides of the triangle based on the angle measures provided. So, we cannot classify the triangle based on the length of its sides.
Therefore, the triangle is an obtuse triangle.
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A metal plate is supposed to be 15. 75cm long but after it was cut, it measured 15. 71 determine relative error
The relative error of the metal plate is -0.0025
The relative error compares the difference between the measured value and the actual value to the actual value itself. Mathematically, it's calculated as:
Relative Error = (Measured Value - Actual Value) / Actual Value
In this case, the metal plate was supposed to be 15.75 cm long, but it measured 15.71 cm after it was cut. Therefore, the measured value is 15.71 cm, and the actual value is 15.75 cm. Plugging these values into the formula for relative error, we get:
Relative Error = (15.71 - 15.75) / 15.75 = -0.0025
The relative error is negative because the measured value is less than the actual value. In other words, the measured value has an error of -0.25% relative to the actual value. This means that the measured value is 0.25% less than the actual value.
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Isλ=4an eigenvalue of32−40332−26? If so, find one corresponding eigenvector. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Yes,λ=4is an eigenvalue of32−40332−26. One corresponding eigenvector is (Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element.) B. No,λ=4is not an eigenvalue of32−40332−26
The correct option is A. Yes, λ=4 is an eigenvalue of the matrix.
To find the corresponding eigenvector, we need to solve the equation (A - λI)x = 0, where A is the given matrix, λ is the eigenvalue, I is the identity matrix, and x is the eigenvector.
First, we subtract λI from A:
A - λI = 3-4 -4 0 0 3-4 3 2-4 -2 6
= -1 -4 0 0 -1 3 2 -2 2
Next, we set the equation (A - λI)x = 0 and solve for x:
(-1 -4 0) (x1) = 0
(0 -1 3) (x2) = 0
(2 -2 2) (x3) = 0
Simplifying the equations gives us:
-x1 - 4x2 = 0
-x2 + 3x3 = 0
2x1 - 2x2 + 2x3 = 0
We can solve this system of equations to find the eigenvector. One possible solution is x1 = 2, x2 = 1, x3 = 1/3. Therefore, one corresponding eigenvector is (2, 1, 1/3).
So the correct answer is A. Yes, λ=4 is an eigenvalue of the matrix. One corresponding eigenvector is (2, 1, 1/3).
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A cistern is to be built of cement. The walls and bottom will be 1ft. thick. The outer height will be 20 ft. The inner diameter will be 10 ft. To the nearest cubic foot, how much cement will be needed for the job?
The amount of cement needed is 609 cubic ft.
The amount of cement needed for the job can be calculated by finding the difference between the volume of the outer cistern and the volume of the inner cistern.
The volume of the outer cistern can be found using the formula for the volume of a cylinder, V = πr²h, where r is the radius and h is the height. The outer radius is half the outer diameter, or 10ft/2 = 5ft. The outer height is 20ft. So the volume of the outer cistern is:
V = π(5ft)²(20ft) = 1570.8 cubic ft
The volume of the inner cistern can be found using the same formula, but with the inner radius and inner height. The inner radius is the outer radius minus the thickness of the walls, or 5ft - 1ft = 4ft. The inner height is the outer height minus the thickness of the bottom, or 20ft - 1ft = 19ft. So the volume of the inner cistern is:
V = π(4ft)²(19ft) = 961.6 cubic ft
The difference between the two volumes is the amount of cement needed:
1570.8 cubic ft - 961.6 cubic ft = 609.2 cubic ft
To the nearest cubic foot, the amount of cement needed is 609 cubic ft.
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This is another question I need help with! Please help
Following are the measures of the angles given :∠ ABC = 180, ∠ BCD = 180, ∠ CDA = 270, ∠ DAB = 270.
What is parallel lines ?
Parallel lines are two or more lines that are always the same distance apart and never intersect, no matter how far they are extended.
Since AB is parallel to DC and AD is parallel to BC, we know that angles A and C are corresponding angles and angles B and D are also corresponding angles. Corresponding angles are congruent when two parallel lines are intersected by a transversal.
So, we have:
∠ A = ∠ C
∠ B = ∠ D
Given:
∠ D = 3x - 75
∠ B = x + 35
We can solve for x using the fact that the sum of the angles in a quadrilateral is 360 degrees:
∠ A + ∠ B + ∠ C + ∠ D = 360
Substituting the values we know, we get:
∠ A + (x + 35) + ∠ C + (3x - 75) = 360
Simplifying, we get:
4x - 40 = 360
4x = 400
x = 100
Now we can substitute x = 100 into the expressions for ∠ B and ∠ D to find their values:
∠ B = x + 35 = 100 + 35 = 135
∠ D = 3x - 75 = 3(100) - 75 = 225
Since angles A and C are corresponding angles, we know that:
∠ A = ∠ C = 180 - ∠ B = 180 - 135 = 45
Therefore, we have:
∠ ABC = ∠ A + ∠ B = 45 + 135 = 180
∠ BCD = ∠ B + ∠ C = 135 + 45 = 180
∠ CDA = ∠ C + ∠ D = 45 + 225 = 270
∠ DAB = ∠ D + ∠ A = 225 + 45 = 270\
So, we can see that angles ABC and BCD are straight angles (180 degrees) and angles CDA and DAB are reflex angles (more than 180 degrees).
Therefore, Following are the measures of the angles given :∠ ABC = 180, ∠ BCD = 180, ∠ CDA = 270, ∠ DAB = 270.
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Every day Ms.Twinkle walks around a park near her house. The park is the shape of a rectangle 2 mi long and 1 3/10 mi wide. How far does she walk.
Ms. Twinkle walks a total of 6.6 miles around the park.
What is perimeter ?
Perimeter is the total distance around the outside of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape.
To find how far Ms. Twinkle walks, we need to find the perimeter of the rectangle-shaped park, which is the sum of the lengths of all its sides.
The length of the park is 2 miles and the width is 1 3/10 miles. We can write the width as a mixed number of 1 + 3/10 = 13/10 miles.
Therefore, the perimeter of the park is:
Perimeter = 2(length + width)
Perimeter = 2(2 + 13/10)
Perimeter = 2(20/10 + 13/10)
Perimeter = 2(33/10)
Perimeter = 66/10
Perimeter = 6.6 miles
Therefore, Ms. Twinkle walks a total of 6.6 miles around the park.
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how do i answer that?!
Answer:
the answer is 4.
Step-by-step explanation:
-3 squared will give you 9. 73- 9 is 64 and cube root of 64 is 4
The Island of Paradis is divided into the following zones (Districts) which are given as follows Determine the total number of trips between the zones. Zone Production/Generation Attraction 1. Shinganshina 3643 3040 2. Utopia 3484 2524 3. Karanes 3156 2535 4. Trost 3204 4217 5. Krolva 2167 2122 6. Stohess 2542 1638 7. Ehrmich 2275 1245 8. Yarckel 2210 3714 9. Orvud 3597 4542 10. Mitras 3172 3754
Determine the total number of trips between the zone:
a. 1 and 6
b. 3 and 10
c. 5 and 10
d. 4 and 9
The total number of trips between the zone: a. 1 and 6 is 10863 trips. b. 3 and 10 is 12617 trips. c. 5 and 10 is 11215 trips. d. 4 and 9 is 15560 trips.
To determine the total number of trips between the zones, we can simply do addition of the production/generation and attraction values for the two zones in question. Based on the values in the table:
a. For zones 1 and 6, the total number of trips would be 3643 + 3040 + 2542 + 1638 = 10863 trips.
b. For zones 3 and 10, the total number of trips would be 3156 + 2535 + 3172 + 3754 = 12617 trips.
c. For zones 5 and 10, the total number of trips would be 2167 + 2122 + 3172 + 3754 = 11215 trips.
d. For zones 4 and 9, the total number of trips would be 3204 + 4217 + 3597 + 4542 = 15560 trips.
Therefore, the total number of trips between the zones are: a. 10863 trips between zones 1 and 6 b. 12617 trips between zones 3 and 10 c. 11215 trips between zones 5 and 10 d. 15560 trips between zones 4 and 9.
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NEED HELP PLEASE. Graph
The graph of the functions are added as an attachment
How to graph the function using their periodsThe graphs are sinusoidal functions such that they are mathematical functions that exhibit a repetitive pattern over time or space
From the question, we have the following parameters that can be used in our computation:
y = 4cot(4x)
y = -1/2cot(2x - π/4)
To plot y = 4cot(4x) using two periods, we make use of the domain
{0 < x < π/2}
To plot y = 4cot(4x) using two periods, we make use of the domain
{0 < x < π}
See attachement for the graphs of the functions
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(v^(2)+11)/(v^(2)-3v-4) find all numbers for which the rational expression is undefined
v=4 or v=-1.
The rational expression is undefined when the denominator is equal to 0. Therefore, we need to find the values of v that make the denominator 0.
v^(2)-3v-4=0
We can factor this equation to find the values of v:
(v-4)(v+1)=0
Therefore, the rational expression is undefined when v=4 or v=-1.
In conclusion, the rational expression (v^(2)+11)/(v^(2)-3v-4) is undefined when v=4 or v=-1.
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8.5 is 11 less than a number h.
Answer:
h is 19.5
Step-by-step explanation:
8.5 is 11 less than h
8.5 = h - 11
Add 11 to both sides
19.5 = h
Lin is comparing the graph of two functions g and f. The function g is given by g(x) = f(x−2). Lin thinks the graph of g will be the same as the graph of f, translated to the left by 2. Do you agree with Lin? Explain your reasoning
No, the function g will be the same as the graph of f, translated to the right by 2.
What is transformation?
A point, line, or geometric figure can be changed in four different ways that are all collectively referred to as transformations. The original shape of the object is called the Pre-Image and the final shape and position of the object is the Image under the transformation.
f(x + a)horizontally shift the graph of f(x) left by a units.
The graph of f(x) is horizontally shifted by a units right side by f(x - a).
The graph of f(x) is vertically shifted upward by a unit by f(x)+a.
f(x)- a one-unit vertical shift downward of the f(x) graph.
The given function is
g(x) = f(x-2)
According to rule,
The function g will be the same as the graph of f, translated to the right by 2.
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(5)/(x+6)=(7)/(5x+30)-2 If there is more than one solution, separate If there is no solution, click on "No solution" x
The solutions are x ≈ -2.744 and x ≈ -17.106.
To solve this equation, we need to get rid of the fractions by multiplying each term by the least common multiple (LCM) of the denominators. The LCM of x+6 and 5x+30 is (x+6)(5x+30).
Multiplying each term by the LCM gives us:
(5)(x+6)(5x+30)/(x+6) = (7)(x+6)(5x+30)/(5x+30) - 2(x+6)(5x+30)
Simplifying the fractions and distributing the terms gives us:
5(5x+30) = 7(x+6) - 2(x+6)(5x+30)
Expanding and simplifying the terms gives us:
25x + 150 = 7x + 42 - 10x^2 - 180x - 360
Combining like terms and rearranging gives us:
10x^2 + 198x + 468 = 0
Using the quadratic formula, we can find the values of x:
x = (-198 ± √(198^2 - 4(10)(468)))/(2(10))
Simplifying gives us:
x = (-198 ± √(39204 - 18720))/(20)
x = (-198 ± √20484)/(20)
x = (-198 ± 143.126)/(20)
The two solutions for x are:
x = (-198 + 143.126)/(20) ≈ -2.744
x = (-198 - 143.126)/(20) ≈ -17.106
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The covariance between the returns of A and B is -0. 112. The standard deviation of the rates of return is 0. 26 for stock A and 0. 81 for stock B. The correlation of the rates of return between A and B is closest to: A. )-1. 88 B. )-. 53 C. ). 53 D. )1. 88
The correlation of the rates return between stock A and B for the given covariance and standard deviation is given by option B. -0.53.
Covariance between stock A and B = -0.112
Standard deviation of the rates return for stock A = 0.26
Standard deviation of the rates return for stock B = 0.81
Formula for correlation in terms of covariance and standard deviations is
Correlation = covariance / (standard deviation of A x standard deviation of B)
Here correlation of the rates of return between A and B.
Substitute the given values we get,
⇒ Correlation = (-0.112) / (0.26 x 0.81)
⇒ Correlation ≈ -0.430769 / 0.81
⇒ Correlation ≈ -0.5318
⇒ Correlation ≈ -0.53
Therefore, the correlation of the rates of return between A and B is is equal to option B. -0.53.
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What is the area of the arrow shaped polygon in square units
The area of the arrow shaped polygon in square units is: 80 square units
How to find the area of the Polygon?For triangle BCD:
b = Base = 4 - 2 = 2 units
h = Height = 4 units
Area = [tex]\frac{1}{2}[/tex]bh
Area = [tex]\frac{1}{2}[/tex] × 2 × 4
Area = 8 square units
For triangle DEF
b = Base = 2 - (-1) = 3 units
h = Height = 4 units
Area = [tex]\frac{1}{2}[/tex] × 3 × 4
Area = 6 square units
For trapezoid ABFG:
a = 4 - (-1) = 5.5 units
b = 4 - (-8) = 12 units
h = 8 units
Area = [tex]\frac{1}{2}[/tex](a + b)h
Area = [tex]\frac{1}{2}[/tex](5.5 + 12)8
Area = 70 square units
Total area = 8 + 6 + 70
Total Area = 84 square units
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HELPPPPPP PLEASEEEEE HURRYYYYY
68% of the races he competed in had a finish time around 64.5 and 65.5 seconds.
How to interpret a standard deviation?The term "variance" (or "") refers to an assessment of the data's dispersion from the mean. A small variance implies that the data are grouped around the normal, and while a large standard deviation shows that the data are more dispersed.
[tex]\begin{aligned}& \mathrm{P}(\mu-\sigma < \mathrm{X} < \mu+\sigma) \approx 68 \% \\& \mathrm{P}(\mu-2 \sigma < \mathrm{X} < \mu+2 \sigma) \approx 95 \% \\& \mathrm{P}(\mu-3 \sigma < \mathrm{X} < \mu+3 \sigma) \approx 99.7 \%\end{aligned}[/tex]
When the standard deviation out from mean of the distribution of X is and the mean of the dispersion of X is (assuming X is normally distributed).
Kiran's 400-meter dash timings have an average of 65 secs and a confidence interval of 0.5 seconds, and they are regularly distributed.
Using the formula, we then obtain
[tex]\mathrm{P}(65-0.5 < \mathrm{X} < 65+0.5)=\mathrm{P}(64.5 < \mathrm{X} < 65.5) \approx 68 \%[/tex]
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10, 13, 17, 19, 22, 23, 29, 33, 34, 35, 35, 38, 53, 68.
FIND THE Z-SCORES FOR 17, 33, AND 53 FOR THE FIRST DATA SET.
The z-scores for 17, 33, and 53 are -0.82, 0.22, and 1.52, respectively.
To find the z-scores for 17, 33, and 53, we first need to calculate the mean and standard deviation of the data set.
Mean = (10 + 13 + 17 + 19 + 22 + 23 + 29 + 33 + 34 + 35 + 35 + 38 + 53 + 68)/14 = 29.57
Standard deviation = √[(10-29.57)² + (13-29.57)² + (17-29.57)² + (19-29.57)² + (22-29.57)² + (23-29.57)² + (29-29.57)² + (33-29.57)² + (34-29.57)² + (35-29.57)² + (35-29.57)² + (38-29.57)² + (53-29.57)² + (68-29.57)²]/13 = 15.37
Now we can calculate the z-scores using the formula:
z-score = (data point - mean)/standard deviation
Z-score for 17 = (17-29.57)/15.37 = -0.82
Z-score for 33 = (33-29.57)/15.37 = 0.22
Z-score for 53 = (53-29.57)/15.37 = 1.52
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What association is shown in the given scatter plot?
A. Clustering
B. Linear
C. Negative
D. None of the above
2. [P] Consider the following equation.x2−918−x−312=x+3x(a) Give the restrictions onx. (b) What is the least common denominator? (c) Keeping these restrictions in mind, solve the equation. If there is no solution or infinitely many solutions, then so state.
The given equation is x2−9/18−x−3/12=x+3/x
(a) The restrictions on x are that x cannot equal 0, 6, or -4. This is because if x were to equal any of these values, the denominator of one of the fractions in the equation would equal 0, which is undefined.
(b) The least common denominator of the fractions in the equation is 36. This is because 36 is the smallest number that is divisible by 18, 12, and x.
(c) To solve the equation, we can multiply each term by the least common denominator to eliminate the fractions:
36x2 - 18(9) - 36x - 3(3) = 36x + 3(36)
Simplifying and rearranging the terms gives us:
36x2 - 72x - 207 = 0
Using the quadratic formula, we can find the values of x:
x = (-(-72) ± √((-72)2 - 4(36)(-207)))/(2(36))
Simplifying gives us:
x = (72 ± √12960)/72
So the solutions are x = (72 + √12960)/72 and x = (72 - √12960)/72.
However, we must check these solutions against the restrictions on x. The first solution, (72 + √12960)/72, is approximately 6.8, which does not violate any of the restrictions. The second solution, (72 - √12960)/72, is approximately -0.8, which also does not violate any of the restrictions. Therefore, the solutions to the equation are x = (72 + √12960)/72 and x = (72 - √12960)/72.
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Gail has $5,500 that she wants to put in a new savings account. She is considering two banks that are very similar. One difference she notices are the interest rates: • Neighborhood Bank - 1.2% interest, compounded annually • Beautiful Day Bank - 1.2% interest, compounded daily Based on interest, which bank would you suggest Gail pick if she plans to have her money in the account for 15 years?
Gail should pick a Beautiful bank if she plans to have her money in the account for 15 years.
How to determine which will be best bank?To determine which bank would be better for Gail, we need to calculate the future value of her investment after 15 years for each bank and compare the results.
For Neighborhood Bank, we can use the formula:
FV = P(1 + r/n)^(n*t)
where FV is the future value,
P is the principal (the initial amount invested),
r is the annual interest rate (as a decimal),
n is the number of times the interest is compounded per year,
and t is the number of years.
Plugging in the values, we get:
FV = 5,500(1 + 0.012/1)^(1*15)
FV = 5,500(1.012)^15
FV = 7,130.34
For Beautiful Day Bank, we can use the formula:
FV = P(1 + r/n)^(n*t)
Plugging in the values, we get:
FV = 5,500(1 + 0.012/365)^(365*15)
FV = 5,500(1.000032876)^5475
FV = 7,145.25
Therefore, Beautiful Day Bank would be the better choice for Gail, as she would earn a higher amount of interest and end up with a higher future value for her investment after 15 years.
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1. Classify each triangle by its side lengths and angle measurements. Circle the correct names.
Classify Using
Side Lengths
m
b.
Equilateral Isosceles
Math Day Or
Scalene
Classify Using
Angle Measurements
Acute Right Obtuse. Angel mea
There are three types of triangle according to sides and angles each.
What is a triangle?The triangle is a type of polygon having three sides, angles and vertices.
Classification of triangles :-
Based on sides :-
1) Scalene triangle :- A triangle with all different measurements of sides and angles are called a Scalene triangle.
2) Isosceles triangle :- A triangle with two of the sides and angles are congruent called an Isosceles triangle.
3) Equilateral triangles :- A triangle with all the same measurements of sides and angles is called an Equilateral triangle.
Based on angles :-
1) Acute angled triangle :- A triangle with all the angles less than 90°, are called acute angled triangle.
2) Obtuse angled triangle :- A triangle with all the angles greater than 90° and less than 180°, are called Obtuse angled triangle.
3) Right angled triangle :- A triangle with one angle equals 90° is called a Right angled triangle.
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Suppose an investment compounds with an annual interest rate of
11.1%. The equation below models a final balance A given principal
P and time t. Use properties of exponents to approximate the
equivalent monthly interest rate. Enter the approximate monthly
rate as a percentage rounded to two decimal places.
A = P (1.111)ª
Answer:
To approximate the monthly interest rate, we need to use the formula for the annual percentage rate (APR) that takes into account the effect of compounding:
APR = (1 + r/m)^m - 1
where r is the annual interest rate, m is the number of compounding periods per year, and APR is the annual percentage rate.
In this case, r = 11.1% and m = 12 (since there are 12 months in a year). We want to solve for the monthly interest rate, which we can call r_m:
r_m = (1 + r/12)^12 - 1
r_m = (1 + 0.111/12)^12 - 1
r_m = 0.009141
To convert this to a percentage, we multiply by 100:
r_m = 0.9141%
Therefore, the approximate monthly interest rate is 0.9141%, rounded to two decimal places.
Val rented a bicycle while she was on vacation. She paid flat rental fee for 55. 0
The equation that can be used to determine the number of days is $123 = $55 + $8.50d
What is the equation?An equation is an expression that has an equal to sign. A linear equation is an equation that has a single variable raised to the power of one. The form of a linear equation is:
y = mx + b
Where:
m = slope b = interceptThe form of the linear equation that can be used to determine the number of days is:
Total cost = flat fee + (cost per day x number of days)
$123 = $55 + ($8.50 x d)
$123 = $55 + $8.50d
d = ($123 - $55) / 8.50
d = 8 days
Here is the complete question:
Val rented a bicycle while she was on vacation. She paid a flat rental fee of $55.00, plus $8.50 each day. The total cost was $123. Write an equation you can use to find the number of days she rented the bicycle.
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