BMED is a trapezoid and OD is equals to BD/2. Proof of these is given below.
How to explain the proofIt should be noted that to prove that BMED is a trapezoid, we need to show that either BM || ED or BM || DE. We will show that BM || DE.
Since O is the midpoint of AM, we have OA = OM, and thus triangle ODM is isosceles. Therefore, OD = DM.
Also, since E is the midpoint of DC, we have DE || AB (because AB is also a mid-segment of triangle ADC). Therefore, angle ADE = angle ABC.
Using the fact that angles in a triangle add up to 180 degrees, we have:
angle EDC + angle ADE + angle CED = 180 degrees
Substituting angle ADE = angle ABC, we get:
angle EDC + angle ABC + angle CED = 180 degrees
But angle CED = angle BCD (because they are alternate interior angles formed by transversal BD cutting parallel lines DC and AB). Therefore:
angle EDC + angle ABC + angle BCD = 180 degrees
This means that angles BCD, ABC, and EDC are all on the same line, and thus angle BCD + angle ABC = 180 degrees.
Now, using the fact that angles in a triangle add up to 180 degrees, we have:
angle BAC + angle ABC + angle ACB = 180 degrees
Substituting angle BCD + angle ABC = 180 degrees, we get:
angle BAC + angle BCD + angle ACB = 180 degrees
Since triangle BCD is isosceles (because BD is an angle bisector), we have angle BCD = angle CBD. Therefore:
angle BAC + angle CBD + angle ACB = 180 degrees
But angle ACD = angle ACB + angle CBD, so:
angle BAC + angle ACD = 180 degrees
This means that triangle ABC is similar to triangle ACD.
Now, since BM is a median of triangle ABC, we have:
BM = 1/2 AC
But triangle ADC is similar to triangle ABC, so:
AC = 2 AD
Therefore:
BM = AC/2 = AD
So we have BM || DE, and BM = DE, which means that BMED is a trapezoid.
2. To prove that D is the midpoint of AE, we will use the fact that BMED is a trapezoid. Since BM || DE, we have:
BD/DM = BE/EM
But DM = OD (since triangle ODM is isosceles) and EM = DC/2 = AC/4 (since E is the midpoint of DC and AC is a mid-segment of triangle ADC).
Therefore:
BD/OD = BE/(AC/4)
But we also have:
BE = BD + DE = BD + BM = 2BD
Substituting this into the previous equation, we get:
BD/OD = 2BD/(AC/4)
Simplifying:
BD/OD = 8BD/AC
This means that:
OD = AC/8
But we also know that AC = 2AD, so:
OD = AD/4
Since O is the midpoint of AM, we also have:
OD = OM = AM/2
Therefore:
AD/4 = AM/2
This means that:
AD = 2AM/4 = AM/2 = OD
So D is the midpoint of AE, and CD = 2AD.
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In a triangle ABC. [AM] is the median relative to [BC] and O is the midpoint of [AM]. (BO) cuts [AC] at D. Let E be the midpoint of [DC]. 1) Prove that BMED is a trapezoid. 2) Prove that D is the midpoint of [AE); deduce that CD = 2AD. 3) Prove that OD == BD.
Write an equation using the parent function y = √x that has a domain of 3≤x<∞ and a range of -2≤y<∞.
The equation using the parent function y = √x that has a domain of 3≤x<∞ and a range of -2≤y<∞ is y = √(x - 3) - 2.
What is equation?An equation is a mathematical statement that indicates the equality of two expressions. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Equations are used to solve problems in various fields of mathematics, science, engineering, and economics. They can be written in various forms, such as standard form, slope-intercept form, point-slope form, and more.
Here,
To shift the parent function y = √x to the right by 3 units and down by 2 units, we use the transformation:
y = √(x - 3) - 2
This equation has a domain of 3 ≤ x < ∞, since we subtract 3 from x inside the square root function. The range is -2 ≤ y < ∞, since we subtract 2 from the square root function.
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A stained glass window is shaped like a semicircle. The bottom edge of the window is 4 feet long.
What is the area of the stained glass window?
Answer: 12.57
Step-by-step explanation:
the radius is 2 so you 2 times pi squared
1/2 + 1/6 least multiple that is the same add using renamed fraction
Answer: 2/3
Step-by-step explanation: 1/2 converted to 6ths is 3/6 add them together and it’s 4/6 simplify that and it’s 2/3
determine if each of the following statements is true or false. if a statement is false, explain which part of the statement is incorrect. a) skipped b) as the degree of freedom increases, the t distribution curve becomes more similar to the standard normal curve c) the two mean non-pooled test is used for 2 independent populations under the assumption that the two populations share the same standard deviation. d) the standard error is always equal to the sample standard deviation.
a) Given statement: skipped : I'm not sure what statement you're referring to, as there is no statement labeled "a" in question.
Hence answer provided.
b) As the degrees of freedom increase, the t distribution curve does become more similar to the standard normal curve.
True. Because the t distribution resembles the normal distribution when the degrees of freedom are high.
c) The two mean non-pooled test is used for 2 independent populations under the assumption that the two populations have different standard deviations.
False - Because the non-pooled t-test can be used to compare the means of the two populations when the assumption of equal variances is true.
d) The standard error is the standard deviation of the sampling distribution and is calculated by dividing the sample standard deviation by the square root of the sample size.
False - Because the standard error, which is determined as the sample standard deviation divided by the square root of the sample size, is a measure of the variability of the sample mean.
a) Given statement: skipped : I'm not sure what statement you're referring to, as there is no statement labeled "a" in question.
b) Given statement : As the degrees of freedom increase, the t distribution curve does become more similar to the standard normal curve.
The given statement is true.
Because, When the degrees of freedom are large, the t distribution approaches the standard normal distribution.
c) Given statement: The two mean non-pooled test is used for 2 independent populations under the assumption that the two populations share the same standard deviation.
The given statement is true.
Because, When the assumption of equal variances is met, the non-pooled t-test can be used to compare the means of the two populations.
d) Given statement: The standard error is not always equal to the sample standard deviation.
The given statement false.
Because, The standard error is a measure of the variability of the sample mean, and it is calculated as the sample standard deviation divided by the square root of the sample size.
The sample standard deviation is a measure of the variability within the sample itself.
The standard error tends to be smaller than the sample standard deviation because it accounts for the effect of the sample size on the variability of the sample mean.
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Solve the quadratic equation 9x2 − 16 = 0
Answer:
x = ± [tex]\frac{4}{3}[/tex]
Step-by-step explanation:
9x² − 16 = 0
9x² = 16
x² = [tex]\frac{16}{9}[/tex]
x = ± √[tex]\frac{16}{9}[/tex]
x = ± [tex]\frac{4}{3}[/tex]
So, the answer is: x = [tex]\frac{4}{3}[/tex] , - [tex]\frac{4}{3}[/tex]
19. You are laying on the ground looking up at a building. The distance between your head and the top of the
building is 60 feet. You need to look up at an angle of elevation of 55° to see the top of the building.
(Hint: draw a picture.)
a. How tall is the building?
b. How far is your head from
the base of the building?
The building is approximately 74.8 feet tall, and your head is approximately 117.7 feet away from the base of the building.
We can use trigonometry to solve this problem. Let's call the height of the building "h" and the distance from your head to the base of the building "d".
Using the tangent function;
tan(55°) = h/d
We know that h + 60 = d, so we can substitute h with d - 60:
tan(55°) = (d - 60)/d
Simplifying this equation, we get;
d = (h + 60)/tan(55°)
Substituting this value of d in first equation, we have;
tan(55°) = h/[(h + 60)/tan(55°)]
Simplifying this equation, we get;
h = 60 tan(55°)
Using a calculator, we find that h ≈ 74.8 feet.
Therefore, the building is approximately 74.8 feet tall.
To find the distance from your head to the base of the building, we can use the Pythagorean theorem;
d² = h² + 60²
Substituting h with 60 tan(55°), we get;
d² = (60 tan(55°)² + 60²
Using a calculator, we find that d ≈ 117.7 feet.
Therefore, your head is approximately 117.7 feet away from the base of the building.
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Choose the correct phrase in the piece.
Find the marginal revenue curve under monopoly market. 1. When the demand curve is P = -5Q + 25, the Marginal Revenue curve is MR = Q+ 2. When the demand curve is P = -0.5Q + 12, the Marginal Revenue curve is MR = Q +
The marginal revenue curves for the given demand curves are: MR = -0.5 Q + 12 In a monopoly market, the marginal revenue curve is always below the demand curve. This is because a monopolist can only increase their revenue by decreasing the price of their product, and therefore they must lower the price for all units sold, not just the last one.
For the first demand curve, P = -5Q + 25, the marginal revenue curve can be found by taking the derivative of the demand curve with respect to Q. This gives us:
MR = dP/d Q = -5
So the marginal revenue curve is a horizontal line at MR = -5. Adding the intercept of the demand curve at P = 25, we get:
MR = -5Q + 25
For the second demand curve, P = -0.5Q + 12, the marginal revenue curve can be found in the same way:
MR = dP/dQ = -0.5
So the marginal revenue curve is a horizontal line at MR = -0.5. Adding the intercept of the demand curve at P = 12, we get:
MR = -0.5Q + 12
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A bag contains pennies, nickels, dimes, and quarters. There are 50 coins in all. Of the coins,16% are pennies and 42% are dimes There are 6 more nickels than pennies. How much money does the bag contain?.
The bag contains 4.63 dollars.
To find :
Total amount of the coins bag contains.
Solution:
According to the given statement, a bag contains 50 coins.
Number of pennies = 16% of 50
= 0.16 × 50
= 8 pennies
Number of dimes = 42% of 50
= 0.42 × 50
= 21 dimes
Number of nickels = 6 more than pennies
= 8 + 6
= 14 nickels
Number of quarters = Rest of the coins
= 50 - (8 + 21 + 14)
= 50 - 43
= 7 quarters
Since,
1 penny = 0.01 dollar
8 pennies = 0.01 × 8 = 0.08 dollar
1 dime = 0.10 dollar
21 dimes = 0.10 × 21 = 2.1 dollar
1 nickel = 0.05 dollar
14 nickel = 0.05 × 14 = 0.70 dollar
1 quarter = 0.25 dollar
7 quarter = 0.25 × 7 = 1.75 dollars
Total value of the coins in the bag = 0.08 + 2.10 + 0.70 + 1.75
= 4.63 dollars
Hence, The bag contains 4.63 dollars.
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The graph of line m is shown. What is the
equation of the line that is perpendicular to line
m and passes through the point (3,2)?
WRITE IN POINT SLOPE FORM
Answer:6
Step-by-step explanation:
The value of houses in Livingston goes up by 7% each year. Ben owns a house in Livingston that is currently worth $281,320. How much will it be worth in 9 years?
Ben's house will be worth $490,523.96 in 9 years if the value continues to increase at a rate of 7% per year.
Formula for Compound InterestWe can use the formula for compound interest, which is:
A = P × (1 + r)ⁿ
Where:
A = final amount
P = principal (starting value)
r = annual interest rate (expressed as a decimal)
n = number of compounding periods
In this case, the principal is $281,320, the annual interest rate is 7% (or 0.07 as a decimal), and the number of compounding periods is 9 years. Plugging in these values, we get:
A = $281,320 × (1 + 0.07)⁹
A = $281,320 × 1.743
A = $490,523.96
Therefore, Ben's house will be worth approximately $490,523.96 in 9 years if the value continues to increase at a rate of 7% per year.
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Consider a cone with a height of 9 in and a base diameter of
6 in.
Using a calculator, approximate the volume of the cone.
To do the approximation, do not round any intermediate
steps, and use the button on the calculator. Round your
answer to the nearest hundredth.
Make sure that you use the correct units in your answer.
A cone with a height of 9 in and a base diameter of 6 in has the volume 84.78 cubic inches.
The formula for the volume of a cone is:
V = (1/3)π[tex]r^{2}[/tex]h, where r is the radius of the base and h is the height.
In the question it is given height = 9 in and a base diameter = 6 in.
Radius = diameter/2
r = d/2 = 6/2 = 3 inches
Substituting the values in the formula:
V = (1/3)π[tex](3\ in)^2[/tex](9 in)
Simplifying the expression:
V ≈ 84.78 [tex]in^3[/tex]
Rounding to the nearest hundredth, we get:
V ≈ 84.78 [tex]in^3[/tex]
Therefore, the approximate volume of the cone is 84.78 cubic inches.
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A teacher poses this problem: I am thinking of four numbers, a, b, c, and d, where a 0, b 0, c 0, and d 0 What else do you need to know to plot the four numbers in the correct order on a number line? What two questions should you ask? Explain how the answers would help you plot the numbers on a number line
The two questions should you ask are:
What is the smallest number among the values (a, b, c, and d)What is the largest number among the values (a, b, c, and d)Why ask the question above?The given answers to the above questions, one can be able to know the ranges of values that the four numbers have, as well as their relative positions.
In the event that we know the smallest and biggest numbers, we can be ready to mark those points on the number line and after that put the other two numbers in between them agreeing to their relative positions.
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Rewrite 84 + 36 in the form
a(b+c)
where a is the greatest common factor of 84 and 36
The GCF of 84 and 36 is 12.
The GCF of two non-zero integers, x(36) and y(84), is the greatest positive integer m(12) that divides both x(36) and y(84) without any remainder.
GCF of 36 and 84 by Listing Common Factors
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
There are 6 common factors of 36 and 84, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 36 and 84 is 12.
Rewrite in the form of a(b + c), a is the greatest common factor of 84 and 36
84/ 12 = 7
36/12 = 3
So, 12(84 + 36)
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solve the equation -75=y^3+50
Answer: y = -5
Step-by-step explanation:
We move the constants to one side to get y^3 = -125.
Then, we cube root both sides to get y = -5.
Triangulation
Here are two-five pointed stars. Both figures A and B are polygons.They are both composed of line segments and are
two dimensional. Neither have curves. Do you agree with the statement?
A plane is an infinite two-dimensional figure.
We have,
When two planes intersect, they form a line.
The vector equation for the line of intersection is given by
r = r₀ + tₓ
where r₀ is a point on the line
tₓ is the cross product of the normal vectors of the two planes.
The parametric equations for the line of intersection are given by
x = a , y = b and z = c
where a, b and c are the coefficients from the vector equation
r = a (i) + b (j) + c (k)
The line of intersection will be perpendicular to the normal of both the planes
The line of intersection lies on both the planes
Therefore , a line is observed at the intersection of two planes
Given data ,
A line is one-dimensional, a segment is a part of a line and is also one-dimensional, and a point is zero-dimensional.
A plane, on the other hand, is a flat, two-dimensional surface that extends infinitely in all directions.
It has length and width, but no thickness, and can be thought of as an infinitely large sheet of paper.
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complete question:
Which of the following is an infinite two-dimensional figure?
A. A line
B. A plane
C. A segment
D. A point
Question 9 (5 points)
Use 0= 30° to write x^2/4 - y^2/9=1
in the xy plane. Then identify the conic
1. 7 x^2 - 9 y^2 = 343 is a hyperbola
2. x^2 + y^2 - 4 x + 6 y - 5 = 1 is a circle
3. 4 x^2 + 9 y^2 = 1 is a ellipse
4. x^2 - 6 y = 0 is a parabola
What is a conic section?A conic section conic or a quadratic curve is described as a curve obtained from a cone's surface intersecting a plane.
The three types of conic section are :
the hyperbola the parabola, and the ellipseAll conics are written in terms of the following equation:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
In conclusion, a conic is the intersection of a plane and a right circular cone.
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#Complete question:
Match the following equations with the conic sections formed by them. 1. x 2 + y 2 - 4x + 6y - 5 = 0 hyperbola 2. x 2 - 6y = 0 circle 3. 4x 2 + 9y 2 = 1 ellipse 4. 7x 2 - 9y 2 = 343 parabola
How can you write the explicit rule for a geometric sequence if your know the recursive rule for the sequence?
The explicit formula for nth term geometric sequence is aₙ = a x rⁿ⁻¹.
let a geometric series is of the form
a, a², a³, a⁴ a⁵, ....
where a₁ = a = first term and other terms are obtained by multiplying by
r.
Now, each term is r times of previous term.
So, to get the nth term we have to multiply (n-1) by r.
Thus, the explicit formula for nth term geometric sequence is
aₙ = a x rⁿ⁻¹
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find the midpoint between -3,5 and 1,6
Step-by-step explanation:
Let A(x1,y1)=(-3,1), B(x2,y2)=(5,6) be the points of a line segment.
Mid point = [tex]\sf{ \dfrac{x1+x2}{2} , \dfrac{y1+y2}{2} }[/tex]
Mid point = [tex]\sf{ \dfrac{-3+1}{2} , \dfrac{1+6}{2} }[/tex]
Mid point = [tex]\sf{ \dfrac{-2}{2} , \dfrac{7}{2} }[/tex]
[tex]\large\sf\purple{ 1, \dfrac{7}{2} }[/tex]find the slope of (6,1) and (6,-3)
Answer:
Step-by-step explanation:
10. Johnny is looking at a map and measures the distance between Chattanooga and Nashville to be 3.5 inches. He wants to know the actual distance between the two cities. According to the scale on the map, 1 inch = 38 miles. How many miles is there between Chattanooga and Nashville?
according to the question the actual distance between Chattanooga and Nashville is 133 miles.
What is distance ?Distance is an object's total, aimless movement. Without regard to whether anything has a beginning or an end, distance can be defined as the amount of space that something has traversed. Distance is referred to be the breadth or magnitude of the gap between two locations. It should be highlighted that the distance between two points and the length travelled between the two are not the same thing. The total length of a path connecting two locations counts as the distance travelled. Distance is the distance in reality that a body travels. It also goes by the name "path length." For instance, the length of a path for the thing passing via point O and arriving at position P would be OP = 360 m.
given,
If 1 inch on the map represents 38 miles in reality, then 3.5 inches on the map would represent:
3.5 inches * 38 miles/inch = 133 miles
Therefore, according to the scale on the map, the actual distance between Chattanooga and Nashville is 133 miles.
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The zoom feature on a camera lens allows you dilate what appears on the display. When you change from 100% to 400%, the new image on your screen is an enlargement of the previous image with a scale factor of 4. If the new image is 38 millimeters wide, what was the width of the previous image?
the width of the previous image is 9.5mm.
What is the width?The widthh can be defined as the horizontal measurement or distance measured from side to side.
In geometry, the shorter side of an object is frequently referred to as its "width". For instance, both of the given rectangles are identical rectangles, but one of them has been rotated to demonstrate that the shorter side of the rectangle is referred to as its width.
In the given problem,
if the width of the previous image is x,
4x=38
⇒x=9.5mm
So, the width of previous image is 9.5mm.
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Which is not a legal python statement
a) print("x")
b) 3 + 4 = x
c) x = 3 + 4
d) x = eval(input("Enter x: "))
Chester has 49 CDs. Tony has 1/7 as many as Chester. How many CDs does Tony have?
Answer:7
Step-by-step explanation: you divide 49 by 7 which gives you 7
Answer:
7
Step-by-step explanation:
1/7 of 49 would be equal to divided 49 by 7 to find 1 seventh of 49
49/7 = 7
so 1/7 of Chesters 49 CDS is equal to 7 CDS that Tony has.
hope this helps :)
suppose the confidence interval (0.6358, 0.7532) represents the proportion of skyline college students who say math is their favorite subject. a. based on the confidence interval, is it likely 60% of skyline college students say math is their favorite subject?
Answer: No, it is not likely that 60% of Skyline College students say math is their favorite subject based on the given confidence interval of (0.6358, 0.7532).
The confidence interval tells us that we are 95% confident that the true proportion of Skyline College students who say math is their favorite subject lies between 0.6358 and 0.7532. This means that if we were to take many random samples from the population of Skyline College students and construct 95% confidence intervals for the proportion of students who say math is their favorite subject, 95% of those intervals would contain the true population proportion.
Since 0.60 is not within the interval (0.6358, 0.7532), it is unlikely that the true proportion of Skyline College students who say math is their favorite subject is 60%. However, we cannot make any definitive statements about the population proportion based on this one confidence interval alone.
Step-by-step explanation:
rewrite the following standard form quadratic into vertex form: f(x) = -2(x + 1 )^2 + 8
Step-by-step explanation:
f(x) = - 2 (x+1)^2 + 8 is already in vertex form
quadratic form can be found by expanding it :
f(x) = -2 ( x^2 + 2x +1) + 8
f(x0 = -2x^2 -4x -2 + 8
f(x) = - 2x^2 - 4x + 6 <=====quadrtaic form
please explain every step clearly
As a result, for the two schools, the measures of variability are:
Mountain View School:
Range = 4, Variance = 9.1733, Standard deviation = 3.029
Bay Side School:
Range = 10, Variance = 7.84, Standard deviation = 2.8.
How do you define standard deviation?The degree of variance or dispersion in a set of data values is measured by standard deviation. The degree to which the data values deviate or vary from the mean or average value of the data set is indicated by this statistical indicator.
We must ascertain the range, variance, and standard deviation for each dataset in order to determine the measures of variability for the two schools.
Mountain View School:
Range: The smallest class size is 5 and the largest is 9, so the range is 9-5 = 4.
Variance: To calculate the variance, we need to find the mean first.
Mean = (5+6+8+9+8+2+0+10+2+4+5+6+8)/15
= 5.6
Then, we can find the variance using the formula:
Variance = Σ(xi - μ)²/n
= [(5-5.6)² + (6-5.6)² + (8-5.6)² + (9-5.6)² + (8-5.6)² + (2-5.6)² + (0-5.6)² + (10-5.6)² + (2-5.6)² + (4-5.6)² + (5-5.6)² + (6-5.6)² + (8-5.6)²]/15
= 9.1733
Standard deviation = √(9.1733) = 3.029
Bay Side School:
Range: The smallest class size is 0 and the largest is 10, so the range is 10-0 = 10.
Variance: To calculate the variance, we need to find the mean first.
So,
Mean will be = (8+7+6+5+5+4+4+3+1+0+2+0+0+2+3+5)/15
= 3.6
Then, we can find the variance using the following formula:
Variance = Σ(xi - μ)²/n
= [(8-3.6)² + (7-3.6)² + (6-3.6)² + (5-3.6)² + (5-3.6)² + (4-3.6)² + (4-3.6)² + (3-3.6)² + (1-3.6)² + (0-3.6)² + (2-3.6)² + (0-3.6)² + (0-3.6)² + (2-3.6)² + (3-3.6)² + (5-3.6)²]/15
= 7.84
Standard deviation will be = √(7.84)
= 2.8
As a result, for the two schools, the measures of variability are:
Mountain View School:
Range = 4, Variance = 9.1733, Standard deviation = 3.029
Bay Side School:
Range = 10, Variance = 7.84, Standard deviation = 2.8
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Katie's claim can be represented by the equation: a² ¢= 4(1/2ab) + b²
How to explain the equationKatie's claim can be represented by the equation:
Outer square area = Sum of areas of four triangles + Inner square area or mathematically, a² = 4(1/2ab) + b²
where "a" and "b" are the lengths of the legs of the right triangle and "a²" represents the area of the outer square, while "b²" represents the area of the inner square.
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Maria claims that any fraction located between
1
5
5
1
and
1
7
7
1
on a number line must have a denominator of
6
6.
Enter a fraction that shows Maria's claim is incorrect.
However, 2/7 is not equivalent to any fraction with denominator 6. Therefore, Maria's claim is incorrect.
Fraction calculation.
A fraction is a way of representing a part of a whole or a ratio between two quantities. It consists of two numbers, the numerator (located above the fraction line) and the denominator (located below the fraction line), separated by a horizontal fraction bar. The numerator represents the number of parts of the whole or the ratio, while the denominator represents the total number of equal parts or the denominator of the ratio.
For example, the fraction 3/4 represents three out of four equal parts or three parts out of a total of four parts. It can also be interpreted as a ratio of three to four. Fractions can be simplified by dividing both the numerator and denominator by their greatest common factor. For instance, the fraction 6/8 can be simplified to 3/4 by dividing both the numerator and denominator by 2.
A counterexample to Maria's claim is the fraction 2/7.
To see this, note that 1/5 is equivalent to 14/70 and 1/7 is equivalent to 10/70. Therefore, any fraction between 1/5 and 1/7 can be expressed in the form n/70, where 10 < n < 14.
However, 2/7 is not equivalent to any fraction with denominator 6. Therefore, Maria's claim is incorrect.
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In 1995, 13,000 Internet users were surveyed and asked about their willingness to pay fees for access to websites. Of these, 2,938 were definitely not willing to pay such fees. How large a sample is necessary to estimate the proportion of interest to within 2 percent in a 95 percent confidence interval?
A sample size of at least 1,521 Internet users would be necessary to estimate the proportion of interest (the proportion of Internet users who are willing to pay fees for website access) to within 2 percent in a 95 percent confidence interval.
To calculate the sample size required to estimate a proportion with a certain level of confidence and margin of error, we can use the following formula:
[tex]n = (Z^2 \times p \times (1 - p)) / E^2[/tex]
where:
n is the required sample size
Z is the Z-score for the desired level of confidence (95% confidence corresponds to a Z-score of 1.96)
p is the estimated proportion of interest (the proportion of Internet users who are willing to pay fees for website access)
E is the desired margin of error (2% in this case, expressed as a decimal)
We are given that out of a sample of 13,000 Internet users surveyed in 1995, 2,938 were definitely not willing to pay fees for website access. This means that the proportion of interest (p) is:
p = (13,000 - 2,938) / 13,000 = 0.774
Now we can plug in the values and solve for n:
[tex]n = (1.96^2 \times 0.774 \times (1 - 0.774)) / 0.02^2[/tex]
n ≈ 1,521
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