Answer:
The orthocenter of triangle XYZ is (-6,4).
Step-by-step explanation:
To find the orthocenter of triangle XYZ, we need to first find the altitudes of the triangle.
The altitude of a triangle is a line that passes through a vertex of the triangle and is perpendicular to the opposite side.
Let's find the equation of the line passing through X(-9,5) and perpendicular to YZ. The slope of YZ is (8-3)/(-6+6) = 5/0, which is undefined. Therefore, YZ is a vertical line passing through the point (-6,3). The equation of this line is x=-6.
The line passing through X(-9,5) and perpendicular to YZ must be a horizontal line passing through X. Therefore, the equation of this line is y=5.
Similarly, the equation of the line passing through Y(-6,8) and perpendicular to XZ is x=-6.
The equation of the line passing through Z(-6,3) and perpendicular to XY is y=3.
Now, we need to find the intersection of these lines to find the vertices of the triangle's altitude.
The altitude from X passes through the point X(-9,5) and is perpendicular to the line YZ with equation x=-6. Therefore, the intersection of the altitude from X and YZ must have x-coordinate -6. Since the equation of the altitude from X is y=5, we get the point of intersection as (-6,5).
Similarly, the altitude from Y passes through the point Y(-6,8) and is perpendicular to the line XZ with equation x=-6. Therefore, the intersection of the altitude from Y and XZ must have x-coordinate -6. Since the equation of the altitude from Y is y=8, we get the point of intersection as (-6,8).
The altitude from Z passes through the point Z(-6,3) and is perpendicular to the line XY with equation y=3. Therefore, the intersection of the altitude from Z and XY must have y-coordinate 3. Since the equation of the altitude from Z is x=-6, we get the point of intersection as (-6,3).
Now that we have the vertices of the altitudes, we can find the equation of the lines passing through them.
The line passing through (-6,5) and (-6,8) is the line x=-6.
The line passing through (-6,5) and (-6,3) is the line y=4.
The line passing through (-6,8) and (-6,3) is the line x=-6.
Now, we find the intersection of the lines x=-6 and y=4. The intersection point is (-6,4).
Therefore, the orthocenter of triangle XYZ is (-6,4).
Suppose a standard six sided die is rolled 15 times. What is the probability of rolling 1 exactly 4 times? Round to four decimal places.
Therefore , the solution of the given problem of probability comes out to be rounded to four decimal places, is approximately 0.1041.
What precisely is probability?The assessment of the probability that a claim is true or that a particular event will happen is the main objective of the constructions inside style known as criteria. Chance can be symbolised by any number between zero and 1, for which 1 typically denotes certainty range and 0 typically denotes possibility. A probability diagram illustrates the likelihood that a particular occurrence will take place. Decimal digits 0, 1, rationals with just 0% and roughly 100%.
Here,
A typical six-sided die has a 1/6 chance of rolling a 1 on any particular roll.
We can describe this scenario using a binomial distribution with
=> n = 15 and p = 1/6 since we are rolling the die 15 times.
The binomial probability formula can be used to calculate the likelihood of rolling precisely four ones:
=> P(X = 4) = (15 choose 4) (15 choose 4) * (1/6)^4 * (5/6)^11
Calculating the answer, we obtain:
=> P(X = 4) ≈ 0.1041
The likelihood of rolling precisely four one-sided dice, rounded to four decimal places, is approximately 0.1041.
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I will mark you brainiest!
Which quadrilaterals have diagonals that are perpendicular?
A) Kite,Rhombus, and square
B) Trapezoid, rhombus, and square
C) kites trapezoid, and parallelogram
D) rectangle, square, and parallelogram
The quadrilaterals which have diagonals that are perpendicular are Kite, Rhombus, Square.(option A)
What is Diagonal?Diagonals are the part of a shape, in geometry. In Mathematics , a diagonal is a line that connects two vertices of a polygon or a solid, whose vertices are not on the same edge. in general, a diagonal is defined as a sloping line or the slant line , that connects to the vertices of a shape. diagonals are defined as lateral shapes that have sides/ edges and corners.
Kite
Rhombus
Square
these are the quadrilaterals that have diagonals that are perpendicular.
hence the answer is option A.
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Find One third and one fifth of the Following (A) 1/3 of 33 and 1/3 of 1/5 of 45
one third of 33 is 11
one fifth of 45 is 9.
What is the solution to the inequality -6+12p+31 > 7?
O p<-8 or p>5
O-8
The solution to the inequality is p > -2 and p < -1, expressed in interval notation as -2>p>-1.
What is interval notation?Interval notation is a mathematical notation for expressing ranges of values between two numbers. It is used to describe the set of real numbers that are bounded between two specified numbers, such as [x, y]. The interval notation is composed of two numbers (x and y) separated by a comma and enclosed in square brackets. The notation is used to represent the entire set of values that lie between the two boundaries.
This is expressed in interval notation as -2>p>-1. This can be solved by isolating the variable p, first by subtracting 31 from both sides of the equation. This leaves -6 + 12p > -24. Then, divide both sides by 12, which gives p > -2. To check the solution, the original equation can be substituted with the value of p. If the equation is true, the solution is correct.
To solve the inequality -6+12p+31 > 7, first subtract 31 from both sides of the equation to isolate the variable. This leaves -6 + 12p > -24. Then, divide both sides by 12 to get p > -2. Finally, check the solution by substituting the value of p in the original equation. If the equation is true, the solution is correct. The solution to the inequality is p > -2 and p < -1, expressed in interval notation as -2>p>-1.
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At the Chesterfield Pet Adoption Day, there are 5 tents set up where people can play with puppies and senior dogs. In each tent, there are x puppy playpens and y senior dog playpens. There are 4 dogs in each playpen. Pick all the expressions that represent how many dogs are at the event.
Answer: The total number of dogs at the event will depend on the number of playpens in each tent and the number of tents.
In each tent, there are x puppy playpens and y senior dog playpens, and there are 4 dogs in each playpen.
To find the total number of dogs at the event, we need to multiply the number of playpens in each tent by the number of dogs per playpen, and then multiply that by the number of tents.
So the expressions that represent the total number of dogs at the event are:
5xy(4) (multiplying the number of playpens in each tent by the number of dogs per playpen, and then by the number of tents).
20xy (combining the multiplication).
4(5xy) (multiplying the number of dogs per playpen by the number of playpens in each tent, and then by the number of tents).
20xy (combining the multiplication).
Therefore, the expressions that represent the total number of dogs at the event are 20xy and 4(5xy).
Step-by-step explanation:
The number of puppies at the event is represented by the expression 20x, and the number of senior dogs at the event is represented by the expression 20y.
Explanation:The total number of dogs at the event can be calculated by adding the number of puppies and senior dogs. Since there are x puppy playpens and y senior dog playpens in each tent, and there are 5 tents, we would have 5 * x playpens for puppies and 5 * y playpens for senior dogs. Each playpen holds 4 dogs, so the number of puppies would be 4 * 5 * x, and the number of senior dogs would be 4 * 5 * y. Therefore, the expressions that represent the total number of dogs at the event are 20x (for the puppies) and 20y (for the senior dogs).
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If the December 1 balance in the Direct Materials Inventory account was $31,000, the December 31 balance was $38,500, and $185,000 of direct materials were issued to production during December, what was the amount of direct materials purchased during the month?
Answer:
We can use the following formula to calculate the amount of direct materials purchased during the month:
Direct Materials Purchased = Ending Direct Materials Inventory Balance - Beginning Direct Materials Inventory Balance + Direct Materials Issued to Production
Substituting the given values, we get:
Direct Materials Purchased = $38,500 - $31,000 + $185,000
Direct Materials Purchased = $192,500
Therefore, the amount of direct materials purchased during the month was $192,500.
Solve the IVP: y′′+ 2y′= [tex]\frac{1}{e^{2t}-1}[/tex], y(−ln 2) = 0, y′(−ln 2) = ln(9/8).
According to the question the solution to the given IVP is [tex](ln(9/8))(e^{2t} - te^{2t} + 1).[/tex]
What is IVP?IVP stands for Initial Value Problem. It is a mathematical problem where an initial value is given at a certain point in time, and the goal is to find the solution for the function at all other points in time. An IVP is important in the study of differential equations, which are used to model a wide range of phenomena in science, engineering, and economics.
The given IVP is a second order linear differential equation with constant coefficients, so the general solution can be written as
[tex]y(t) = c_1e^{2t} + c_2te^{2t} + d[/tex]
where c_1, c_2, and d are constants to be determined.
Substituting the initial conditions into the general solution yields
[tex]y(-ln2) = c_1(-ln2)^2 + c_2(-ln2) + d = 0[/tex]
and
[tex]y'(-ln2) = 2c_1(-ln2) + c_2e^{2(-ln2)} = ln(9/8).[/tex]
Solving these two equations for c_1 and c_2 yields
[tex]c_1 = -(ln(9/8) + d)/2(-ln2)^2[/tex]
and
[tex]c_2 = (ln(9/8) + 2d)/2(-ln2)[/tex]
Substituting these values into the general solution and solving for d yields
d = ln(9/8).
Therefore, the solution to the given IVP is
[tex]y(t) = -(ln(9/8) + ln(9/8))e^{2t}/2(-ln2)^2 + (ln(9/8) + 2ln(9/8))te^{2t}/2(-ln2) + ln(9/8)[/tex]
= [tex](ln(9/8))(e^{2t} - te^{2t} + 1).[/tex]
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If someone help I’ll be thankful!!!
Answer:
m<v = m<w
Step-by-step explanation:
I do not know how to explain it. CD are parallel with those to lines. They ahve the same degree as the other two angles.
<y = <z are opposite angle.
<w + <x = 180 are suplementary angle which mean they add up to 180.
Ed wants to borrow $20,000 from a bank to open a small gym. Three banks charge different interest rates. To help decide the best loan option, Ed wants to know the percent profit he will make each month.
Part A
The cost to run the gym each month is $5,000. Find Ed’s total monthly expenses for each loan option.
Answer:
Step-by-step explanation:
it will take 4 days to pay of the loan
Can anyone help me with number 3?
Answer:
the least is 40 the greatest is 49
Step-by-step explanation:
Answer:
40 the greatest is 49
Step-by-step explanation:
Need help, thanks!!!!!
When interpreted, the figures of the delivery times of the pizza shops shows that D. The means are nearly equal, but the variation is significantly greater for the second shop than for the first.
What do the delivery times mean ?This means that, on average, the second pizza shop delivers pizzas slightly faster than the first shop. However, the delivery times for the first shop are more consistent (less variable) than the second shop.
The standard deviation for the first shop (4 minutes) is much lower than the second shop (21 minutes), indicating that the delivery times for the first shop are clustered closely around the mean delivery time, whereas the delivery times for the second shop are more spread out. If delivery time is the only consideration, one may prefer the first shop as it has more consistent delivery times.
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Find the equation of the line that passes through the given point and has the given slope. (Use x as your variable.) (- 9, - 9), m = 0
Answer:kwbebrhdjejhehe
Step-by-step explanation:
Classifying a Parallelagram
The coordinates of the fourth vertex are (-3, -1).
How to determine the midpoint of a line segment?In order to determine the midpoint of a line segment with two (2) endpoints, we would add each point together and divide by two (2).
Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]
Substituting the given parameters into the midpoint of a line segment formula, the midpoint of the diagonal is given by;
Midpoint = [(5 - 5)/2, (-3 - 5)/2]
Midpoint = [0/2, -8/2]
Midpoint = (0, -4).
For the fourth vertex, we have:
Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]
(0, -4) = [(3 + x₂)/2, (-7 + y₂)/2]
3 + x₂ = 2(0)
x₂ = -3
-7 + y₂ = 2(-4)
-7 + y₂ = -8
y₂ = -8 + 7
y₂ = -1.
Therefore, the fourth vertex = (-3, -1).
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With the points A(-4, -2) B(1, -3) C(2, -5) D(-1, 7).
What are the new points if the scale factor of dilation is 3?
Answer:
A'(-12, - 6); B'(3, - 9); C'(6, - 15); D'(-3, 21)-----------------------------------
The scale factor k affects the points with the rule:
(x, y) → (kx, ky)Apply the rule to the given points, given k = 3:
A(- 4, - 2) → A'(-12, - 6);B(1, -3) → B'(3, - 9);C(2, - 5) → C'(6, - 15);D(-1, 7) → D'(-3, 21)What is it? 1/2(4x + 8)
Answer: (2x+4)
Step-by-step explanation:
The term (4x+8) is in brackets. That means whatever you do to it will affect every term in the bracket.
This time, you are dividing by 2
Thus, (4x+8)/2 = 4x/2+8/2
=2x+4
Hope this helps! :)
Find the area of the figure below, formed from a triangle and a parallelogram. one triangle has a height of 3mm. Another triangle has a side of 4mm and 5mm. the trapezoid has a side of 5mm, 5mm and height of 4mm.
After answering the presented question, we can conclude that As a triangle result, the figure has a surface area of 24.5 square millimeters.
What precisely is a triangle?Because it has four or more pieces, a triangle is a polygon. It has a straightforward rectangular shape. A triangle ABC is a rectangle with A, B, and C edges. Euclidean geometry provides a single plane and cube when the sides are not collinear. A polygon is a triangle with three components and three angles. The corners of a triangle are the spots where the three edges meet. A triangle's sides add up to 180 degrees.
To calculate the area of the figure, we must first calculate the areas of each shape and then add them together.
c² = a² + b²5² = 4
_g² +b²25 = 16+ b²_9_ = 3
As a result, the base of the second triangle is 3mm. We can now pinpoint its location:
A = 1/2 * 3mm * 3mm * 3mm = 4.5mm2
A is equal to (5mm + 5mm) * 4mm / 2 = 20mm2.
We can now add the regions of all three shapes:
Total surface area = 4.5mm2 + 20mm2 = 24.5mm2
As a result, the figure has a surface area of 24.5 square millimeters.
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A seed company planted a floral mosaic of a national flag. The perimeter of the flag is 2,460 ft Determine the flag's width and length if the length is 250 ft greater than the width.
The flag's width is ___ ft. Its length is ___ ft.
Given:-
Perimeter of the flag is 2460 ft .length is 250 greater than the width .To find:-
length and breadth of the flag .Answer:-
Since a flag is of rectangular shape, its perimeter would be;
P = 2( l + b )
where ,
l is lengthb is breadth.Now let us assume the breadth to be " x " , then its length will be " x + 250 " . Now substitute the respective values in the given formula as ;
2460 = 2 ( x + x + 250)
2460/2 = 2x + 250
1230 = 2x + 250
2x = 1230 - 250
2x = 980
x = 980/2
x = 490
Hence,
length = x + 250 = 490 + 250 = 740ftbreadth = x = 490ftHence the flag's width is 490ft. Its length is 740ft.
and we are done!
_
6thMarch , 3:57 pm
Step-by-step explanation:
the perimeter of a rectangle is
2×length + 2×width
length = width + 250
2×length + 2× width = 2460
now we use the first equation in the second equation abd get
2×(width + 250) + 2×width = 2460
2×width + 500 + 2×width = 2460
4×width = 1960
width = 1960/4 = 490 ft
length = width + 250 = 490 + 250 = 740 ft
Suppose we want to chose 2 objects without replacement from 3 objects how many ways can this be done with choice matter and with choice does not Matter
There are 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters and 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.
What is the factοrial?The prοduct οf all pοsitive integers less than οr equal tο a given pοsitive integer and denοted by that integer and an exclamatiοn pοint. Thus, factοrial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7. Factοrial zerο is defined as equal tο 1.
If the chοice matters (i.e., the οrder οf the chοsen οbjects matters), then there are 3 pοssible chοices fοr the first οbject and 2 pοssible chοices fοr the secοnd οbject.
Thus, there are 3 * 2 = 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters.
If the chοice dοes nοt matter (i.e., the οrder οf the chοsen οbjects dοes nοt matter), then there are 3 chοοse 2 = 3!/((3-2)!*2!) = 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.
Therefοre, there are 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters and 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.
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The function f(x) = 2x3 - 6x2 _ 210x + 7 is increasing on the inverval
(-00, A]U |B, ∞), and decreasing on the interval [4, B].
A =
and B =
The critical points are A = -5 and B = 7, and the function is increasing on the interval (-∞, -5] ∪ [7, ∞), and decreasing on the interval [-5, 7].
What is the value of A and B along the intervalTo find A and B, we need to find the critical points of the function f(x) and determine the intervals of increase and decrease.
First, we find the derivative of f(x) as:
f'(x) = 6x^2 - 12x - 210
Next, we find the critical points by setting f'(x) = 0:
6x^2 - 12x - 210 = 0
Simplifying, we get:
x^2 - 2x - 35 = 0
Factoring, we get:
(x - 7)(x + 5) = 0
So the critical points are x = 7 and x = -5.
Now we can determine the intervals of increase and decrease. We create a sign chart based on the sign of f'(x) in each interval:
Interval | f'(x) | f(x) increasing/decreasing
(-∞, -5) | - | decreasing
(-5, 7) | + | increasing
(7, ∞) | - | decreasing
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translate this phrase into an algebraic expression.
The sum of 15 and twice Jose’s score
use the variable j to represent Jose’s score
Answer:
15 + 2j
Step-by-step explanation:
Sum, which means addition, is used to say that 15 will be added to something.
That "something" is twice Jose's score, which is 2 * j, since j is the score.
15 + 2 * j, or just 15 + 2j
3x + 2y =4 complete the table
[tex] \mathbb{ANSWER:}[/tex]
[tex] \sf 3x + 2y = 4[/tex]
[tex] \sf x = 0 \\ \sf \cancel{3(0) }+ 2y = 4 \\ \sf \frac{2y}{2} = \frac{4}{2} \\ \sf y = \boxed{ \sf 2}[/tex]
[tex] \sf x = - 2 \\ \sf 3( - 2)+ 2y = 4 \\ \sf - 6 + 2y = 4 \\ \sf \cancel{- 6 + 6} + 2y = 4 + 6 \\ \sf \frac{2y}{2} = \frac{10}{2} \\ \sf y = \boxed{ \sf 5}[/tex]
[tex] \sf x = 1 \\ \sf 3(1)+ 2y = 4 \\ \sf 3 + 2y = 4 \\ \sf \cancel{3 - 3} + 2y = 4 - 3 \\ \sf \frac{2y}{2} = \frac{1}{2} \\ \sf y = \boxed{ \sf \frac{1}{2} }[/tex]
[tex] \sf x = 4 \\ \sf 3(4)+ 2y = 4 \\ \sf 12 + 2y = 4 \\ \sf \cancel{- 12 - 12} + 2y = 4 - 12 \\ \sf \frac{2y}{2} = \frac{ - 8}{2} \\ \sf y = \boxed{ \sf - 4}[/tex]
A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h(t)=−4.9t2+19t+19. How many seconds does it take to reach maximum height? Enter the answer with at least 3 decimal places.
t=
This answer can be derived from the equation h(t)=-4.9t²+19t+19.
What is Equation?An equation is a mathematical statement that two expressions are equal. Equations are used to describe relationships between numbers, variables, and constants. They are typically written in the form of an expression on the left of an equal sign and the result of the expression on the right. For example, 2 + 3 = 5 is an equation that states that the sum of two and three is five. Equations can be used to solve problems and express relationships in science, engineering, economics, and many other fields.
The equation represents a parabola, so the maximum height is found at the vertex of the parabola. The x-coordinate of the vertex can be found by setting the derivative of the equation equal to zero, which gives us the equation -4.9×2t + 19 = 0. Solving this equation for t gives us the answer of t = 1.914 seconds.
This is happening because of the kinematic equation which describes the motion of objects in terms of their displacement, velocity, and acceleration. In this equation, h(t) is the displacement of the ball as a function of time. The derivative of the displacement equation with respect to time gives us the equation for the velocity of the ball, which identifies the point at which the ball reaches its maximum height. Since velocity is equal to zero at the highest point of the ball's trajectory, this equation can be used to find the time it takes for the ball to reach its maximum height.
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help please. Geometry N.10 IXL. will mark brainliest
The value of y is given as follows:
y = 35.
How to obtain the value of y?A diagonal trapezoid, also known as a trapezium, is a quadrilateral with two parallel sides that are of different lengths. In a diagonal trapezoid, the non-parallel sides are not perpendicular to the parallel sides. This means that the two non-parallel sides are slanted or tilted, forming a diagonal line that connects them.
In a diagonal trapezoid, the diagonals are congruent, meaning that they have the same length.
The diagonals are HJ and IK, hence the value of y is obtained as follows:
5y - 1 = 4y + 34
y = 35.
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help!! A containers when 3/8 full holds 15 litres of gasoline. how many litres of gasoline would be needed to fill 5 similar containers?
The number of litres of gasoline that would be needed to fill 5 similar containers is 200 litres.
What is a litre?A litre is a measure of volume
Since a containers when 3/8 full holds 15 litres of gasoline. We need to find how many litres of gasoline would be needed to fill 5 similar containers.
Let x be the volume of the container.
Since a containers when 3/8 full holds 15 litres of gasoline, we have that
3x/8 = 15
Solving for x, we have that
3x/8 = 15
Multiplying through by 8, we have that
3x = 15 × 8
Dividing through by 3, we have that
x = 15 × 8/3
x = 5 × 8
x = 40 litres.
Since one container contains 40 litres, then 5 containers would contain V = 5 × 40 litres
= 200 litres.
So, 5 containers would contain 200 litres
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Find the area of the figure below.
Answer:
348 in²
Step-by-step explanation:
[tex] \frac{ad \times (cd + ab)}{2} [/tex]
[tex] \frac{19.2(16 + 24)}{2} [/tex]
384 in²
The graph of the function g(x) contains the point (5, 1/3). What point must be on the graph of y = 3g(x + 1)?
Answer: we move (5, 1/3) to (5-3, 1/3*3) = (2, 1)
Step-by-step explanation:
I WILL GIVE 70 POINTS AND THE BRAINLIEST PLEASE HURRY!!!
Jill solves the equation: 4(x - 7) - 2x = 0
Fill in the blanks with the correct values.
4x + - 2x = 0
2x =
x =
Answer:
1) 4x -28 - 2x = 0
2) 2x = 28
3) x = 14
Step-by-step explanation:
1) 4 * (-7) = -28
2) 4x - 2x = 28
2x = 28
3) x = 28/2
x = 14
HELP HELP NOW.BRO PLEASE. Benjamin is planning what to wear tomorrow evening.
• He has 3 shirts to choose from: red, white, and green.
• He has 2 pairs of pants to choose from: blue jeans and khaki pants.
• He has 3 pairs of shoes to choose from: sneakers, flip flops, and boots.
Benjamin randomly chooses one item from each group. What is the probability that he chooses
sneakers and the green shirt?
Answer:1/9 i gotchu
Step-by-step explanation:
Answer:
4/8
Step-by-step explanation:
matthew is on the basketball team at school. she calculated it hat in 65 minutes he can burn 468 calories. if m is the number of minutes and c is the number of calories burned
Answer:
7.2
Step-by-step explanation:
468/65 = 7.2
Amy is interested in estimating the number of hours of sleep a typical student at her middle school gets on Saturday nights. She asks each of the other students in her science class to write down how many hours of sleep they got last Saturday on a piece of paper passed around the class.
Which of the following statements are true?
Choose all that are correct.
A.
Amy should have included students from one other science class.
B.
To get a good estimate, Amy must have all the students at the school answer the question.
C.
To estimate the value, Amy must first mix up the data from the science class.
D.
Amy's results cannot be used to estimate the value she wishes to estimate.
E.
The sample Amy used is not a true random sample.
The correct statements are A and E.
What is sample?A sample is a subset of a larger population that is selected for research or analysis. The sample should be representative of the population to ensure accurate and unbiased conclusions.
A. Amy should have included students from one other science class because the sample should be representative of the entire middle school population, and including students from only one class may not provide a representative sample.
B. To get a good estimate, Amy does not necessarily need all the students at the school to answer the question. However, a larger sample size would increase the accuracy and precision of the estimate.
C. Mixing up the data from the science class would not be necessary if the sample is random and representative of the middle school population.
D. Amy's results can be used to estimate the average number of hours of sleep a typical student at her middle school gets on Saturday nights, provided the sample is random and representative of the middle school population.
E. The sample Amy used may not be a true random sample if it was only taken from one science class. The sample should be randomly selected from the entire middle school population to be representative.
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