It will take 51 days to fill the Olympic sized pool.
What is relation between time and work?Time and work have a relationship that is
Work Completed is equal to Time x Rate of Work.
Alternately, Rate of Work = Time Taken / 1.
Alternatively, Time Taken = 1 / Rate of Work
Considering that,
Pool volume is 6.6043 x 105 gallons, or 6,600,430 gallons.
9 litres of water are delivered through a hose every minute.
Therefore, the amount of water delivered by hose every hour is equal to (9 * 60) gallons.
Water provided by hosen each day is equal to (9 * 60 * 24) gallons, or 12,960 gallons.
Thus, the amount of time needed to fill the pool is equal to 6,600,430/12960.
There are 50.9 days needed to fill the pool.
51 days were needed to fill the swimming pool.
The Olympic-sized pool will therefore fill in 51 days.
The Complete Question is Olympic sized pool.
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h(n)=2/n -2 solve inverse function
Answer:
2/n+2
Step-by-step explanation:
Determine the intercepts of the line.
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yy-intercept:
(
(left parenthesis
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xx-intercept:
(
(left parenthesis
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A coordinate plane. The x- and y-axes each scale by one-tenth. A graph of a line intersects the points zero, four-tenths and three-tenths, zero.
One canned juice drink is 30% is orange juice, another is 5% orange juice. How many liters of each should be mixed together in order to get 25 Liters that is 27% orange juice?
The amount of juice for 30% orange juice is 22 liters,
and the amount of juice for 5% orange juice is 3 liters mixed together in order to get 25 Liters.
What is a linear equation?When an algebraic equation is graphed, it always results in a straight line since each term in a linear equation has an exponent of 1. For this reason, it is referred to as a "linear equation."
There are both one-variable and two-variable linear equations.
Given One canned juice drink is 30% orange juice, another is 5% orange juice,
Let x and y be the amount of 30% and 5% orange juice respectively,
according to condition
0.30x + 0.05y = 0.27(25) .........(1)
x + y = 25.........(2)
Multiply the second equation by -0.05,
0.30x + 0.05y = 0.27(25)
-0.05x - 0.05y = -0.05(25)
Add equation to eliminate y,
0.30x - 0.05x + 0.05y - 0.05y = 0.27(25) - 0.05(25)
Solve for y,
0.25x = 5.5
x = 5.5/.25
x = 22
substitute value of x in equation,
x + y = 25
(22) + y = 25
y = 25 - 22
y = 3
Hence the amount of 30% orange juice is 22 liters and the amount of 5% orange juice is 3 liters.
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determine its average speed when it goes around the closed path. express your answer to three significant figures and include the appropriate units.
The average speed is 0 m/s since the object does not move.
Since the object does not change its position, it does not move. Therefore, its average speed is 0 m/s.
The object in question does not move in a closed path, meaning that its position remains the same throughout the entire path. As a result, its average speed is 0 m/s. This is because average speed is the total distance traveled divided by the total time taken, and since the object does not move its distance traveled is 0 and its time taken is also 0. Therefore, when the object goes around the closed path, its average speed is 0 m/s.
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Our school purchased 100 new computers. Of those are 1/5 broken. How many computers are broken?
Answer:
20
Step-by-step explanation:
1/5 = 20/100
There are three quarters, five dimes, and twelve pennies in a bag.
Once a coin is drawn from the bag, it is not replaced. If two coins
are drawn at random, determine each probability.
1. P(a quarter and then a penny)
2. P(a nickel and then a dime)
3. P(two pennies)
4. P(a dime and then a quarter)
The probability for these are
Probability(a dime and then a quarter)=3/76
Probability(two pennies)=33/95
Probability(a quarter and then a penny)=9/95
What is probability?
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.
Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes
Now,
Quarters=3, Dimes=5 and Pennies=12
Total=20
Probability(a dime and then a quarter)=5/20*3/19=15/380=3/76
Probability(two pennies)=12/20*11/19=132/380=33/95
Probability(a quarter and then a penny)=3/20*12/19=36/380=9/95
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Find the value for the following determinant
Answer:
42
Step-by-step explanation:
determinant of matrix
= 5(2) - 8(-4)
= 10 - (-32)
= 10 + 32
= 42
Answer:
32
Step-by-step explanation:
the determinant of matrix
[tex]\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right][/tex] = ad - bc
then
[tex]\left[\begin{array}{ccc}5&-4\\8&2\\\end{array}\right][/tex]
= (5 × 2) - (- 4 × 8) = 10 - (- 32) = 10 + 32 = 42
Factor the following examples of difference perfect squares for X to power of 2-9
Answer: X^2 - 9 can be factored into (X - 3)(X + 3) which is the difference of squares.
The difference of squares factorization of X^2 - 9 is (X - 3)(X + 3)
Step-by-step explanation:
Given right triangle ABC, where side "c" is the hypotenuse, angle B measures 42 degrees, and side c measures 18 m, find the length of side b.
The length of side b of the given right angle triangle using law of sines is; b = 12.044 m
How to use the law of sines?The law of sines states that when we divide side "a" by the sine of angle A, it is equal to side "b" divided by the sine of angle B, and also equal to side "c" divided by the sine of angle C
Thus;
a/sin A = b/sin B = c/sinC
The parameters are;
B = 42°
c = 18m
Since c is the hypotenuse, it is the side that will be opposite the right angle and so;
C = 90°
Thus, using sine rule;
c/sinC = b/sin B
18/sin 90 = b/sin 42
b = (18 * sin 42)/1
b = 12.044 m
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Pythagoras is very tall for ancient Greck standards: 1.78 m. He notices that when he stands with his back to the sun, his shadow is 3.08 m in length, while the shadow cast by the lighthouse is 26 m long How tall is the lighthouse?
The lighthouse is 8.88 m tall.
What is the ratio?
A ratio is a mathematical comparison of two or more quantities. It expresses the relationship between the values of the quantities in terms of a numerical fraction.
This problem can be solved using similar triangles. Since the ratio of the heights of Pythagoras and the lighthouse is the same as the ratio of their corresponding shadow lengths, we can set up the following proportion:
(Height of lighthouse) / (Height of Pythagoras) = (Shadow length of lighthouse) / (Shadow length of Pythagoras)
Plugging in the given values:
(Height of lighthouse) / 1.78 = 26 / 3.08
To solve for the height of the lighthouse, we can cross-multiply and divide:
(Height of lighthouse) = (26 / 3.08) * 1.78
The lighthouse is approximately:
(26/3.08)*1.78 = 8.88 m
Hence, The lighthouse is 8.88 m tall.
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Part A Create the smallest cylinder possible with the tool, and record the values o cylinder by the given scale factors, and then record the resulting volumes ( cylinder. BI UX² X₂ 12pt Original Cylinder Radius (units) 2 Scale Factor (k) AV Radius Height (units) funite Height (units) A 6 Volume (V) 13 13 E Volume (cu. unit: 8 Vxk³. scaled cylinders are 2, 3, and 4
The original cylinder has a radius of 2 units and a height of 13 units, and its volume is given by the formula: V = πr²h = π * 2² * 13 = 104π cubic units.
How you can create the smallest cylinder?Original cylinder: The original cylinder has a radius of 2 units and a height of 13 units, and its volume is given by the formula: V = πr²h = π * 2² * 13 = 104π cubic units.
Scaled cylinder 1: The first scaled cylinder has a scale factor of 2, so its radius is 2 * 2 = 4 units and its height is 13 * 2 = 26 units. Its volume is given by: V = πr²h = π * 4² * 26 = 672π cubic units.
Scaled cylinder 2: The second scaled cylinder has a scale factor of 3, so its radius is 2 * 3 = 6 units and its height is 13 * 3 = 39 units. Its volume is given by: V = πr²h = π * 6² * 39 = 1764π cubic units.
Scaled cylinder 3: The third scaled cylinder has a scale factor of 4, so its radius is 2 * 4 = 8 units and its height is 13 * 4 = 52 units. Its volume is given by: V = πr²h = π * 8² * 52 = 3072π cubic units.
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Can someone help is for geometry
A truth table deconstructs a logic function by outlining all possible outcomes the function might achieve.
Explain about the truth table?A truth table offers a way to map out the potential truth values in an expression and predict their results. The table has a row for each conceivable combination of truth values and a column for each variable in the expression. There is also a column that displays the results of each set of values.
For propositional logic formulations, this tool builds truth tables. There are numerous formats available for entering logical operators. As an illustration, the propositional formula p q r could be represented as p / q -> r, p and q => not r, or p && q ->!r. You can enter the connectives and as T and F.
P Q (Q ∧ P)
F F F
F T F
T F F
T T T
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please help me with this math problem i’ll give you brainlist
Answer:
42=3.5x+2y
Step-by-step explanation:
3.5 is the cost of the cakes and 2 is the cost of the oreos. 42 is how much Francis can spend
Write two different expressions you could use to represent the
combined area of the Activities and Quotations sections. What
information about the sections does each expression show?
The area expressions are (2x + 1) * (x^2 + 4x - 5) and 2x^3 + 9x^2 - 6x - 5
How to determine the area expressionsThe missing figure is added as an attachment
From the question, we have the following parameters that can be used in our computation:
Length = 2x + 1
Width = x^2 + 4x - 5
The area is calculated as
Area = (2x + 1) * (x^2 + 4x - 5)
When expanded, we have
Area = 2x^3 + 8x^2 -10x + x^2 + 4x - 5
Evaluate the like terms
Area = 2x^3 + 9x^2 - 6x - 5
Hence, the area expression is 2x^3 + 9x^2 - 6x - 5
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Please answer this question for me
Using properties of triangles, Correct options are A,B and D. AE = EB, EG || AC and AC = 2 EG.
A triangle has three sides, three angles, and three vertices, which are its characteristics.
A triangle's total internal angles are always equal to 180 degrees. This is referred to as the triangle's angle sum property.
Any two triangle sides can add up to a length that is longer than the third side.
A triangle's total angles are always 180 degrees.
A triangle's outside angles are always a sum of 360 degrees.
The total of the parallel interior and exterior angles is additional.
From the given figure,
As E is the mid point of BA,
So, BE = EA
As it is clearly seen that EG is parallel to AC = EG||AC
Further,
As, EG is intersecting AB and BC at mid points,
⇒ AC = 2EG
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Santiago has planned a basketball game for the weekend.
The chance of snow on Saturday is 35%, with a 70% chance on
Sunday. If these probabilities are independent, what is the
chance that it will snow on both days?
The probability that it will snow on both days, given the chances of snow on Saturday and Sunday, is 24.5%.
How to find the probability ?The chance of it snowing on both days is calculated by multiplying the probability of it snowing on Saturday by the probability of it snowing on Sunday. If the probabilities are independent, we can assume that one event occurring doesn't affect the other event, therefore the probability of both events happening is just the product of the individual event probabilities.
So, the chance of it snowing on both days is:
P(Saturday and Sunday) = P(Saturday) x P(Sunday)
P(Saturday and Sunday) = 0.35 x 0.70
P(Saturday and Sunday) = 24.5 %
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Which object is about 15 centimeters long? Responses pencil, desk, paper, clip or television
The object that is about 15 cm is pencil.
What is Measurement?The fundamental idea in the study of science and mathematics is measurement. The qualities of an object or event can be quantified so that we can compare them to those of other objects or occurrences. When discussing the division of a quantity, measurement is the word that is used the most frequently.
Given:
Measurement = 15 cm
As, the average length for the pencil varies from 7 cm - 17 cm.
and, average length for the desk is 120 cm.
and, the average length for the paper is 30.3 x 40.6cm.
and, the average length for the clip is 4 cm.
and, the average length for the television is 40 cm.
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Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. ||u||= 4, θu =6 , ||v|| = 1, θv = 4
The component of u+v along the length of u and v is (6.3, 0.66).
A vector is quantity which has both magnitude and direction. A magnitude is the length of the vector.
Here in the question u and v makes angle with the positive x axis.
||u||= 4 and θu =6 and ||v|| = 1, θv = 4
To determine the component of u = |u||(cosu, sinu)
Component of u = 6(0.9, 0.1) = (5.4, 0.6)
Component of v = ||v||(cosv sinv)
Component of v along the x axis = 1(0.9, 0.06 ) = (0.9, 0.06)
In the question value of u+v needs to be find.
Value of u + v = (5.4 + 0.9 , 0.6+ 0.06)
Value of the component form of u + v = (6.3, 0.66).
Hence, the component of u+v along the length of u and v is (6.3, 0.66).
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What is the solution to the system of equations?
6x-2y=-14
4x-3y=-31
A:(2,1)
B:(-6,11)
C:(2,13)
D:(-2,1)
The solution to the system of equations is 6x – 2y = -14 with x = -4 and y = -5
How to get perfect solution?4x – 3y = -31
To find x or method, you must first make one of the words the same so that they may cancel out.
To determine the x terms, we will make the y terms the same in this example.
To do so, multiply the first equation by three, yielding:
18x – 6y = - 42
And multiply the second equation by two to get:
8x – 6y = - 62
Now that we have the y terms, you may cancel them out.
You now have the following two equations:
18x = -42
8x = -62
The next step is to combine them, since the preceding – and – form a +.
This implies that you are left with one equation
26x = -104
Now divide by 26 on each side to get:
x = -4
As you have x, you substitute it into one of the ORIGINAL equations.
6x – 2y = -14
Substitute x in:
6(-4) – 2y = -14
-24 – 2y = -14
Add 24 on each side to get:
-2y = 10
y = -5
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need the answer please
The number of feet that canyon wall away from the person will be 3,300 feet.
What is speed?Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.
We know that the speed formula
Speed = Distance/Time
Sound travels about 750 miles per hour. If you stand in a canyon and sound a horn, you will hear an echo.
Convert the speed in feet per second. Then we have
Speed = 750 x 5280 / 3600
Speed = 1,100 feet per second
Let 'n' be the number of seconds. Then the distance is given as,
1,100 = (d / 2) / n
d = 2,200 n
If n = 1.5, then we have
d = 2,200 x 1.5
d = 3,300 feet
The number of feet that canyon wall away from the person will be 3,300 feet.
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Si el costo de 1 Kg de azucar es $8?cual es el precio de 3 3/4 kg?
On evaluating the fraction 3 3/4 the cost of 3 3/4 kg of sugar is $30.
What is fraction?
In mathematics, a fraction is used to denote a portion or component of the whole. It stands for the proportionate pieces of the whole. Numerator and denominator are the two components that make up a fraction. The numerator is the number at the top, and the denominator is the number at the bottom.
The cost of 1 kg of sugar is = $ 80
The amount of sugar is given in fractional form - 3 3/4 kg
The equation for cost of 3 3/4 kg sugar is -
Cost = 3 3/4 × 8
Solve the fractional part first -
Cost = 15/4 × 8
Simplify the equation -
Cost = 15/4 × 8
Cost = 120/4
Cost = 30
Therefore, the cost value is obtained as $ 30.
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If the cost of 1 kg of sugar is $8, what is the price of 3 3/4 kg?
List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8. (b) the set S = (x1x2 + 4x - 5 -0. (c) the set of outcomes when a coin is tossed until a tail or three heads appear. (d) the set S = {x|2x - 4 20 and x < 1)
a)The set of integers between 1 and 50, divisible by 8 are given as.
S={ 8,16,24,32,40,48}
b ) Hence, the set {-5,1}
c) The list of set of outcomes when a coin is tossed until a tail or three heads appear will be.
S={T,HT,HHT,HHH}
d) S={Europe, Asia, Australia, North America, South America, Africa, Antarctica.}
a) We have to find a list of integers between 1 and 50, divisible by 8.
Keep in mind that the S is the event that represents the sample space. The set of integers between 1 and 50, divisible by 8 are given as.
S={ 8,16,24,32,40,48}
b) We need to list the set S= {[tex]x| x^2+4x-5=0}[/tex]}
[tex]x^2+4x-5=0\\\\x^2+5x-x-5=0\\x(x+5)-(x+5)==\\(x+5)(x-1)=0\\\\x=-5 , x=1[/tex]
Hence, the set {-5,1}
c) c) We have to find a list of the set of outcomes when a coin is tossed until a tail or three heads appear.
Let T denote the event that tail appeared.
Let H denote the event that head appeared.
The list of set of outcomes when a coin is tossed until a tail or three heads appear will be.
S={T,HT,HHT,HHH}
d) We need to list the set
S={x ∣ x is a continent}.
There are 7 continents:
S={Europe, Asia, Australia, North America, South America, Africa, Antarctica.}
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Sales of portable MP3 players grew approximately exponentially from $0.08 billion in 1999 to $0.88 billion in 2004. Predict the sales of MP3 players in 2008.
The sales of MP3 players in 2008 is $ billion.
(Type an integer or a decimal. Round the final answer to one decimal place as needed. Round all intermediate values to three decimal places as needed.)
Based on the exponential function, the sales of MP3 players in 2008 is $1.82 billion.
What is an exponential function?An exponential function is a function that has the general form y = abˣ
where
a ≠ 0,
b is the base b is a constant (b is a positive real number and b ≠ 1)
x is the independent variable whose domain is the set of real numbers.
Using the equation for exponential growth
y = ab^x
y = 0.88
a = 0.08
0.88 = 0.08bˣ
bˣ = 0.88/0.08
bˣ = 11
x = 4
Hence;
b⁴ = 11
b = 1.82
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Your kite is stuck in a tree that is 45 feet tall. The angle your string makes with the ground is 68°. Rather than worrying about the kite, you decide to calculate how much string you have let out. Assuming the string is held tight and makes a straight line to the ground, how much string have you let out?
Answer:
48.53 = 49 feet
Step-by-step explanation:
sin 0 = opposite/hypotenuse
sin 68 = AB/AC = 45/x
x = 45/sin 68 = 45/0.9272 = 48.53
48.53 rounded = 49 feet
Answer:
You have let out 48.5 feet of string
Step-by-step explanation:
Attached is a sketch of the problem.
We can use SOH CAH TOA to find our answer.
In this acronym, O is the opposite side, A is the adjacent side, and H is the hypotenuse. S is for the SIN function. C is for the COS function. T is for the TAN function.
We can calculate the length of the string by using the SIN function.
So we can say the sine of angle x is the opposite side divided by the hypotenuse.
[tex]sin(x)=\frac{O}{H}[/tex]
Lets solve for [tex]H[/tex].
Multiply each term by [tex]H[/tex].
[tex]H*sin(x)=\frac{O}{H} *H[/tex]
Simplify the right side by cancelling the common factor of [tex]H[/tex].
[tex]H*sin(x)=O[/tex]
Divide both sides of the equation by [tex]sin(x)[/tex].
[tex]\frac{H*sin(x)}{sin(x)} =\frac{O}{sin(x)}[/tex]
Simplify the right side by cancelling the common factor of [tex]sin(x)[/tex].
[tex]H=\frac{O}{sin(x)}[/tex]
Now lets evaluate the length of the string.
In this example we are given
[tex]x=68\\O=45[/tex]
[tex]H=\frac{45}{sin(68)}[/tex]
[tex]H=48.5341[/tex]
How do I do this problem I do not get it do I add when I found the volume of both
Answer:
206
Step-by-step explanation:
cube volume = 5*5*5 +3*3*3 = 125+81 = 206
A grocery chain determines the cost and revenue models using the following functions: C(x)=1.2x−0.012x2, 0≤x≤150 R(x)=3.6x−0.06x20≤x≤150, where x is the number of unit items sold. Determine the interval on which the profit function P(x) = R(x) − C(x) is increasing.
Answer:
Yesterday I have received amount of be with you i need to pay
The interval on which the profit function is increasing is [0, 25).
Given that:
The cost function is, C(x) = 1.2x - 0.012x²
The revenue function is, R(x) = 3.6x - 0.06x²
Here, x i is the number of units sold, and 0 ≤ x ≤ 150.
The profit function is:
P(x) = R(x) - C(x)
= (3.6x - 0.06x²) - (1.2x - 0.012x²)
= 2.4x - 0.048x²
Find P'(x).
P'(x) = 2.4 - 0.096x
The profit function is increasing when P'(x) > 0.
2.4 - 0.096x > 0
2.4 > 0.096x
25 > x
That is x < 25.
Hence the function is increasing in the interval [0, 25).
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Answer and Explain Please
a) The final coordinates are given as follows:
X(-5,-4) -> X'(-4,5).Y(2,-2) -> Y'(-2,-2).Z(-3,-6) -> Z'(-6,3).b) The rotation rule is given as follows: (x,y) -> (y,-x).
What are the rotation rules?The five more known rotation rules are given as follows:
90° clockwise rotation: (x,y) -> (y,-x)90° counterclockwise rotation: (x,y) -> (-y,x)180° clockwise and counterclockwise rotation: (x, y) -> (-x,-y)270° clockwise rotation: (x,y) -> (-y,x)270° counterclockwise rotation: (x,y) -> (y,-x)The rotation in this problem is the last rotation, 270º counterclockwise about the origin, hence the rule is given as follows:
(x,y) -> (y,-x).
The rule is applied to each of the vertices X, Y and Z to obtain the vertices of the reflected triangle X', Y' and Z'.
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what is the hypothesis of the following conditional statement?
If we walk home from school, it takes 30 minutes.
a. we walk home from school
b. it takes 30 minutes
c. we do not walk home from school
d. we walk
The hypothesis of the conditional statement is;
Option A: we walk home from school
How to identify the hypothesis?A hypothesis is defined as an educated guess while using reasonable thought patterns, about the answer to a scientific question. Now, the hypothesis can either be supported or not supported at all, but then it depends on the data gathered.
A conditional statement is defined as a set of rules performed provided a certain condition is met. Thus, it is sometimes referred to as If-Then statement because if a condition is said to be met, then an action is said to have been performed " .
Thus, the conditional statement here is: " we walk home from school ".
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1. You are trying to decide where the best place to get a pizza delivered from as quickly as possible. Here are the several delivery times of some local pizza places (in minutes).
Ronny’s: 24, 38, 31, 27, 40, 21, 34, 19
John and Mary’s: 18, 69, 22, 27, 32, 25, 16, 25
Penora’s: 19, 42, 23, 42, 24, 39, 20, 25
Use the information above to answer all five questions below. Be detailed in your answer and provide work for all but the five number summary.
A. Find the five number summary, mean, standard deviation, range and interquartile range for all three local pizza places.
B. Determine if there are any outliers for the three pizza places. List the outliers if any for each pizza place.
C. What place do you think Allison and Emily should order from? How did you determine this?
D. For each pizza place, which measure of central tendency (mean or the median) represents the data the best and why?
E. If you were to plot the data for each pizza company, how would the data be described? (Symmetric, Skewed left or Skewed right)
2. Ms. McGregor is a teacher at Wendover High. She surveyed her students to find out how far from school her students lived. The following data is the distance between the school and her students' homes, in miles:
9
10
7
7
8
10
11
10
5
9
8
7
5
2
10
10
14
8
7
4
Use the data provided above to answer the questions below:
Create a boxplot for Ms. McGregor’s class and display it on a number line. Make sure to label your number line clearly so it easy to determine the accuracy of your calculations.
Create a frequency table and histogram of the number of miles students are from the school in the space provided below for Mrs. McGregor’s class
Intervals
Frequency
0 - 3
3 - 6
6 - 9
9 - 12
12 - 15
3. Jonathan compared the ounces in and prices of several different cereals at the grocery store. Here are his results:
ounces
Price (in dollars)
32
$3.89
28
$2.79
42
$4.19
8
$1.00
27
$3.15
12
$2.25
Use the information listed above to answer all questions:
A. Find the line of best fit. Round any decimals to the hundredths place:
y = _____________________
Find the correlation coefficient and explain what it means in the context of the problem:
Correlation coefficient: _____ Explanation:
C. Estimate the price when you have a weight of 48 ounces. Show all work for credit.
D. Estimate the ounces when you have a price of $3.50. Show all work for credit.
E. Find the residuals for your data. You can provide a screenshot from your calculator or
show all work to identify your residuals.
F. Provide a screenshot of your residual plot from your calculator below.
G. Explain if your equation is a good fit for your data based on your screenshot from your
residual plot
Answer:
Step-by-step explanation:
In a right-skewed distribution, which of the following is true? The mean tends to be greater than the median.
The number N of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)Year 2004 2005 2006 2007 2008N 8,574 10,238 12,440 15,006 16,676a. Find the average rate of growth between each pair of years.2004 to 2006 _____ locations/ year2006 to 2007 _____ locations/ year2005 to 2006 _____ locations/ yearb. Estimate the instantaneous rate of growth in 2006 by taking the average of the last two rates of change in part (a).c. Estimate the instantaneous rate of growth in 2006 by measuring the slope of the tangent line through (2005, 10,238) and (2007, 15,006).d. Estimate the instantaneous rate of growth in 2007 by measuring the slope of the tangent line through (2006, 12,440) and (2008, 16,676).e. Compare the growth rates you obtained in part (c) and (d). What can you conclude?1. The rate of growth is increasing.2. The rate of growth is decreasing.3. The rate of growth is constant.4. There is not enough information.
a. To find the average rate of growth between each pair of years, we can use the formula: (end value - start value) / (end year - start year).
2004 to 2006: (12,440 - 8,574) / (2006 - 2004) = 3,866 / 2 = 1,933 locations/year
2006 to 2007: (15,006 - 12,440) / (2007 - 2006) = 2,566 / 1 = 2,566 locations/year
2005 to 2006: (12,440 - 10,238) / (2006 - 2005) = 2,202 / 1 = 2,202 locations/year
b. To estimate the instantaneous rate of growth in 2006, we can take the average of the last two rates of change from part (a).
(2,566 + 2,202) / 2 = 2,384 locations/year
c. To estimate the instantaneous rate of growth in 2006 by measuring the slope of the tangent line through (2005, 10,238) and (2007, 15,006), we can use the formula:
(15,006 - 10,238) / (2007 - 2005) = 4,768 / 2 = 2,384 locations/year
d. To estimate the instantaneous rate of growth in 2007 by measuring the slope of the tangent line through (2006, 12,440) and (2008, 16,676), we can use the formula:
(16,676 - 12,440) / (2008 - 2006) = 4,236 / 2 = 2,118 locations/year
e. Comparing the growth rates from part (c) and (d), we can see that the growth rate in 2007 is less than the growth rate in 2006. So we can conclude that 2. The rate of growth is decreasing.
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