*URGENT + 100 POINTS*
A right triangle is shown in the graph.
right triangle on coordinate plane with hypotenuse labeled t and one endpoint of hypotenuse at r comma s and the other endpoint at x comma y, vertical line from point x comma y and horizontal line from r comma s that meet at right angle of triangle, horizontal dotted line from point r comma s to point s on y axis, horizontal dotted line from point x comma y to point y on y axis, vertical dotted line from point r comma s to point r on x axis, and vertical dotted line from right angle to point x on x axis
Part A: Use the Pythagorean Theorem to derive the standard equation of the circle with center at (r, s) and a point on the circle at (x, y). Show all necessary math work. (3 points)
Part B: If (r, s) = (7, –4) and t = 10, determine the domain and range of the circle. (4 points)
Part C: Is the point (9, 1) inside the border of the circle if (r, s) = (7, –4) and t = 10? Explain using mathematical evidence. (3 points)
Answers
Answer:
Part A
[tex](x-r)^2+(y-s)^2=t^2[/tex]
where:
(r, s) is the center of the circle.(x, y) is a point on the circle.t is the radius of the circle.Part B
Domain = [-3, 17]
Range = [-14, 6]
Part C
Point (9, 1) is inside the border of the circle.
Step-by-step explanation:
Part A[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
From inspection of the given triangle:
a = x - rb = y - sc = tSubstitute these values into the formula to derive the standard equation of the circle:
[tex]\boxed{ (x-r)^2+(y-s)^2=t^2}[/tex]
where:
(r, s) is the center of the circle.(x, y) is a point on the circle.t is the radius of the circle.Part BGiven the center of the circle (r, s) is (7, -4) and the radius (t) is 10.
The domain of the circle is the x-value of the center minus and plus the radius:
[tex]\begin{aligned}\implies \textsf{Domain}&=[r-t,r+t]\\&= [7-10, 7+10] \\&= [-3, 17]\end{aligned}[/tex]
The range of the circle is the y-value of the center minus and plus the radius:
[tex]\begin{aligned}\implies \textsf{Range}&=[s-t,s+t]\\&=[-4-10, -4+10]\\& = [-14, 6]\end{aligned}[/tex]
Part CSubstitute (r, s) = (7, –4) and t = 10 into the derived equation from part A:
[tex]\implies (x-r)^2+(y-s)^2=t^2[/tex]
[tex]\implies (x-7)^2+(y-(-4))^2=10^2[/tex]
[tex]\implies (x-7)^2+(y+4)^2=100[/tex]
Substitute the given point (9, 1) into the equation:
[tex]\begin{aligned}\implies (9-7)^2+(1+4)^2&=2^2+5^2\\&=4+25\\&=29\end{aligned}[/tex]
As 29 < 100, the point (9, 1) is inside the border of the circle.
Related Questions
For a pole-vaulter in training, the height the athelete will be able to pole-vault after t months of training is given by
H(t) = 17-11e^-0.02t
After how many months will she be able to vault 11 ft?
Answers
The number of months after which she will be able to vault 11 feet is 30 months.
For a pole-vaulter in training, the height the athlete will be able to pole-vault after t months of training is given by a mathematical formula. The formula is given below.
H(t) = 17 - 11e^(-0.02t)
We need to find the number of months after which she will be able to vault a height equal to 11 feet. We will substitute the value of height in the given equation.
11 = 17 - 11e^(-0.02t)
11e^(-0.02t) = 17 - 11
11e^(-0.02t) = 6
e^(-0.02t) = 6/11
Now, we will take the natural logarithm on both sides of the equation.
-0.02t = -0.606
t = 0.606/0.02
t = 30.3
Thus, she will be able to vault 11 feet in approximately 30 months.
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Please show your work:
-5/2(8a - 2b) =
Answers
[tex]-\frac{5}{2} (8a - 2b) = ...[/tex] the answer for this question is zero (0)
How can the answer is zero?
The first thing that we should do is simplify questions.
[tex]-\frac{5}{2} (8a - 2b) \\= -\frac{5}{2} (8a) + -\frac{5}{2} (- 2b)\\= - 20a + 5b[/tex]
After that, we will search for "a" and "b" to solve the question. For me, i will search for "b" first.
[tex]- 20a + 5b \\5b = 20a\\b = \frac{20}{5} a\\b = 4a[/tex]
Next step is find the "a" by enter the "b" into the same question.
[tex]- 20a + 5b =\\20a = 5b\\20a = 5 (4a)\\20a = 20a\\a = 1[/tex]
After we find the "a" and "b", the next step is to enter it into the main question to find the right answer.
[tex]-\frac{5}{2} (8a - 2b) \\= -\frac{5}{2} (8a - 2[4a]) \\= -\frac{5}{2} (8a) + -\frac{5}{2} (-8a)\\= - 4a + 4a\\= -4 (1) + 4 (1)\\= -4 + 4\\= 0[/tex]
So, the answer is zero (0).
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Answer:
[tex] - 20a + 5b[/tex]
Step-by-step explanation:
[tex] - \frac{5}{2} (8a - 2b)[/tex]
[tex] - \frac{5}{2} (8a) - \frac{5}{2} ( - 2b)[/tex]
[tex] - 20a + 5b[/tex]
The following table summarizes the results of a study on SAT prep courses, comparing SAT scores of students in a private preparation class, a high
school preparation class, and no preparation class. Use the information from the table to answer the remaining questions.
Treatment
Private prep class
High school prep class
No prep class
Number of Observations
Source
Between treatments
Within treatments
60
60
60
Using the data provided, complete the partial ANOVA summary table that follows. (Hint: T, the treatment total, can be calculated as the sample mean
times the number of observations. G, the grand total, can be calculated from the values of T once you have calculated them.)
Sum of Squares (SS)
21,000.00
442,500.00
df
2 ▼
Sample Mean
650
645
625
177
Sum of Squares (SS)
132,750.00
147,500.00
162,250.00
Mean Square (MS)
Answers
The F Test statistic is 1.58.
Given data is
Source SS df MS
between 21,000.00 2 6,000.00
within 442,500.00 117 3,782.00
The SStotal is sometimes easier to calculate than SSbetween Since ......
SSwithin +SSbetween = SStotal you can use SStotal to calculate SSbetween.
= 442,500+21,000 = 463,500
In ANOVA, the F test statistic is the ratio of the between - treatments variance and the within-treatments variance. The value of the F test statistic is 1.58 when the null hypothesis is true , the F test statistics is closer to 1, when the null hypothesis is false, the F test statistic is most likely large in general, you should reject null hypothesis for larger values of the F test statistic.
Hence the answer is the F Test statistic is 1.58.
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Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 33 miles. If the two airplanes are 55 miles apart, how far has the eastbound airplane traveled?
Answers
The eastbound airplane traveled 44 miles far.
Given:
Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 33 miles. If the two airplanes are 55 miles apart.
Let x be the eastbound airplane traveled.
we know that,
north and east direction are perpendicular and with hypotenuse 55 miles it forms a right angle triangle with side = 33 miles.
According to pythagorean theorem:
[tex]33^2+x^2=55^2[/tex]
[tex]x^{2} =55^2-33^2[/tex]
= 3025 - 1089
x = [tex]\sqrt{1936}[/tex]
= 44 miles.
Therefore The eastbound airplane traveled 44 miles far.
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water is poured into a container in the shape of a right circular cone with a radius 4 inches and height 16 inches express the volume v of the water in the cone as a function of the height h of the water
Answers
The volume V of the water in the cone as a function of the height h of the water is,
V = (1/48)πh³
Given, water is poured into a container in the shape of a right circular cone with a radius 4 inches and height 16 inches.
We have to express the volume V of the water in the cone as a function of the height h of the water.
Now, at any instant let the height of the water in the cone be h and radius of the water in the cone be r,
So, using the concept of similarity in triangles
we get, r/4 = h/16
r = h/4
Now, the volume of the water in the cone be,
V = (1/3)πr²h
V = (1/3)π(h/4)²h
V = (1/48)πh³
So, the function expressing the volume of the water in the cone be,
V = (1/48)πh³
Hence, the volume V of the water in the cone as a function of the height h of the water is,
V = (1/48)πh³
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a polytunnel for growing plants has a semi circular opening. the diameter of the opening is 14 feet and the length is 10 feet. the air inside the polytunnel needs to be heated to grow the plants. what is the volume of air inside the polytunnel in cubic feet.
anyone know the answer please it's for a level 2 booklet
Answers
Answer:140
Step-by-step explanation:
14 x 10=
1. A circle is divided into 9 uneven pie shaped pieces. Each section is labeled 1, 2, ... 9. The circle is then spun and the number pointing north recorded. Estimate the probability that the next number pointing north will be between 2 and 7 inclusive.
4 3 2 1 9 8 5 6 7 4 3 2 1 5 8 6 5 4
13/18
5/18
5/8
8/5
2. How many shoppers made a purchase at a big-box store, or online, or at a locally- owned store?
250
359
380
400
3. How many shoppers made a purchase online and at a locally owned store, but not at a big box store?
38
56
109
179
Answers
The probability that the next number pointing north is between 2 and 7 is 13/18 .
1) The circle is divided into 9 parts.
Now the parts are uneven.
So the data about the spins is required to calculate the probability .
The total number of items in the box = 18
favorable outcomes (between 2 and 7) = 13 possible outcome.
Required probability
= favorable outcome / total outcome
= 13 / 18
2) Using the Venn diagram we can calculate the various parts of the problem:
Shoppers at big-box store or online or local store = sum of all the shoppers = 109+57+107+18+17+38+34 = 380 shoppers
3) Shoppers made purchase online or locally owned shop but not at a big box store = 107 + 38 + 34 = 179 shoppers.
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The height of a triangle is 8 centimeters greater than the base. The area of the triangle is 356.5 square centimeters. Find the length of the base and the height of the triangle.
Answers
The lengths of the base of the triangle are 31cm while the height of the triangle is 39cm
A triangle is a polygon that has 3 sides, three angles and three vertices whose sum of angles equal to 180 degrees.
How to find the lengths and height of a triangle?
from the statement,
h=b+8Area = 256.5 square centimetersA=1/2bh
A=1/2 xbx(8+b)
cross multiplying,
713 = 8b+b²
rearrange the function
b²+8b-713=0
Simplyfying the equation,
(b²+23b)-(31b-713)=0
b(b+23)-31(b+23)=0
b-31=0 or b+23 =0
b=31 or b=-23
length cannot be negative. therefore base = 31cm
Recall the h= b+8
this implies that h=8+31
h=39cm
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Stefanie bought a package of pencils for \$ 1.75$1.75 and some erasers that cost \$ 0.25\$0.25 each. She paid a total of \$ 4.25$4.25 for these items, before tax
Exactly how many erasers did Stefanie buy?
Answers
The number of eraser Stefanie bought is 10.
How to find the number of eraser she bought?Stefanie bought a package of pencils for $1.75 and some erasers that cost $0.25 each. She paid a total of $4.25 for these items, before tax.
Therefore, the number of eraser Stefanie bought can be calculated as follows;
let
x = number of eraser bought
Therefore,
4.25 = 1.75 + 0.25x
subtract 1,75 from both sides of the equation
4.25 - 1.75 = 1.75 - 1.75 + 0.25x
2.5 = 0.25x
divide both sides by 0.25
x = 2.5 / 0.25
x = 10
Therefore,
number of eraser = 10
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A survey of 780 was conducted while visiting Sonoma county on vacation. Each person was asked if they drank wine on their trip. 532 people responded "Yes".
What proportion of people did NOT drink wine in the survey? Round your answer to two decimal places.
Answers
Answer:
0.32
Step-by-step explanation:
[tex]\frac{780-532}{780}[/tex] = [tex]\frac{248}{780}[/tex] = 0.32 Rounded.
Use the calculator below to approximate √59 as a decimal to the tenths place.
Note that you must do the approximation without using a square root button.
Your answer must be within a tenth of the actual value.
Answers
The approximate of √59 is 7.7 when rounded off to the nearest tenth place.
Given, √59
we are asked to determine the approximation without using a square root button.
√59 is a decimal number since it is not a perfect square. Therefore, rather than finding a precise value, we will need to obtain a near approximation of 59 in decimal form.
the square root of the number 59 is:
7.6811
You rounded to the nearest tenths place. The 6 in the tenths place rounds up to 7 because the digit to the right in the hundredths place is 8.
= 7.7
When the digit to the right is 5 or greater we round away from 0.
7.6811 was rounded up and away from zero to 7.7
Hence we get the required approximation.
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given f(x)=2x-2, find f (-3)
Answers
Answer:
-8
Step-by-step explanation:
f(-3) = 2(-3)-2
f(-3) = -6 -2
f(-3) = -8
Answer:
[tex] \huge{f( - 3) = - 8}[/tex]
Step-by-step explanation:
[tex]f(x) = 2x - 2[/tex]
In order to find f(-3) we substitute the value of x that's -3 into f(x) that's wherever we find x in f(x) we substitute it with -3 and solve for f(-3).
We have
[tex]f( - 3) = 2( - 3) - 2 \\ = - 6 - 2 \\ = - 8[/tex]
We have the final answer as
[tex] \bold{f( - 3) = - 8}[/tex]
Hope this helps you
A five-gallon tank full of water starts draining at t = 0 at a rate of 1/(1 + t) gallons per hour. How long until the tank is empty?
Answers
1. It will take 5 hours for the five-gallon tank full of water to become empty at the drainage rate of 1/(1 + t) gallons per hour.
2. A correct setup of the equation problem above in terms of function is D. None of these will give the correct solution.
What is an equation?An equation is a mathematical statement that shows that two or more mathematical expressions share equality or are equivalent.
Functions, which are mathematical expressions involving one or more variables known as domain (or the independent variable) and range (or the dependent variable), using the equation symbol to show that two expressions are equal.
Drainage rate per hour = 1/(1 + t) gallons
t = 0
The capacity of the tank = 5 gallons
1/(1 + t) gallons
= 1/(1 + 0) gallons
= 1/1 gallons
= 1 gallon per hour
Since the tank is 5 gallons, it will take = 5 hours (5/1)
Thus, based on the drainage equation rate, the five-gallon tank will empty in 5 hours.
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P(H)=
P(T)=
let me know if you know how to do this
Answers
Answer:
Step-by-step explanation:
they are asking for probability
so they want to know, what's the probability of getting a "heads" on a coin toss. What do you think it is? You probably know this intuitively.
P(H) = ?
P(T) = ?
They are asking for fractions, so just say what you already know to be how often you'll get a "heads" in words, then just make that a fraction. :)
onsider the following equation:
3|x+3|−12=0
Step 1 of 4: Rewrite the equation above in standard form and determine if there is a solution.
Answers
The value of x in the following equation: 3|x+3|−12=0 is -7 and 1
The equation is 3|x+3|−12=0
We need to rewite the equation above in standard form
The modulus function can be positive and negative
Considering it to be negative,
the equation is
3((-1)(x + 3) - 12
3(-x - 3) - 12
-3x - 9 - 12 = 0
3x + 21 = 0
3x = -21
x = -7
3(x + 3) - 12
3x + 9 - 12
3x - 3 = 0
3x = 3
x = 1
Therefore, the value of x in the following equation: 3|x+3|−12=0 is -7 and 1
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Triangles ABD and BCD are right-angled triangles.
A
5 cm
B
x cm
Work out the value of x.
Give your answer correct to 2 decimal places.
10 cm
D
4 cm
C
Answers
In triangle ABC,
Angle ABC = 90 degrees
AD= h = 10 cm
AB= p =5cm
BD = b = ?
Now,
Tan 90= p/b
or, 1/0= 5/BD
or, BD = 0 cm
In triangle BCD,
Let,
BC = h = xcm
BD = b = 0 cm
DC = 4cm
Using the Pythagoras theorem,
h = ( p^2 + b^2)^1/2
or, x = (4^2 + 0^2)^1/2
or, x = (16 + 0)^1/2
or, x = (16)^1/2
or, x = 4 cm
Hence, The value of x is 4 cm.
c. Write two different expressions you could use to represent the
combined area of the Description and Small Diagram sections.
What information about the combined area does each
expression show
Answers
500
Step-by-step explanation:
research
a point lies on the line y=3x-2, its y coordinate is 25. what is the coordinates of this point?
Answers
Answer:
x=9 (9,25)
Step-by-step explanation:
9*7=27
27-2=25
Diego is solving this system of equations:
4x + 3y = 10
-4x + 5y = 6
Here is his work:
4x + 3y = 10
-4x + 5y =6
-4x + 5y = 6 +
0 + 8y = 16
y = 2
4x + 3(2) = 10
4x + 6 = 10
4x = 4
x = 1
Answers
The solutions of the equations are x = 1 and y = 2
The system of equations are
4x + 3y = 10
-4x + 5y = 6
Here we have to use the elimination method. Eliminate the x term and find the value of y term
Add both equation
3y + 5y = 10 +6
Add the like terms
8y = 16
y = 16 / 8
Divide the terms
y = 2
Substitute the value of x in the first equation
4x + 3y = 10
4x + 3×2 = 10
Multiply the terms
4x + 6 = 10
4x = 10 - 6
4x = 4
x = 4 / 4
Divide the terms
x = 1
Hence, the solutions of the equations are x = 1 and y = 2
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Solve the linear programming problem
Answers
After solving the linear programming problem, we get the answer as 250.
What is linear programming?
When a linear function is exposed to multiple restrictions, linear programming maximizes or minimizes the function. This method determines the best way to use the resources that are currently available and aids managers in using the process to make decisions about the most efficient use of scarce resources, such as money, time, materials, and machinery. Linear programming has been useful for guiding quantitative decisions in business planning, industrial engineering, and—to a lesser extent—in the social and physical sciences.
Solution explained:
[tex]\left \{ {{2x + y = 12} \atop {x + y = 7}} \right.[/tex] x=3, y=4 P(3,4) Coordinates of the solution point
[tex]\left \{ {{x + y = 7} \atop {x + 2y = 10}} \right.[/tex] x=4, y=3 Q(4,3)
Coordinates of the simultaneous eqs.
[tex]\left \{ {{2x + y = 12} \atop {x + 2y = 10}} \right.[/tex] x=14/3, y=8/3
Putting the values of x and y in the equation P = 30x + 40y and calculating we get
P(P) = 250, P(Q) = 240, P(R) = 246.67
So, the answer is 250
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Which inequality is represented by the graph?
Answers
Answer:
x<1
Step-by-step explanation:
since the circle is not filled it will not be equal to one but either grater or equal to it
Since the arrow is pointing left than x is less than 1
x<1
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Define the formula for a parabola (a quadratic function) that has horizontal intercepts (roots) at x = −7 and x = 5 and passes through the point (0, −2).
Answers
The formula of the parabola is y = 2/35(x + 7)(x - 5)
How to determine the formula of the parabola?The parameters in the question are given as
Horizontal intercepts (roots) at x = −7 and x = 5 Passes through the point (0, −2).These parameters can be represented as
x₁ = -7 and x₂ = 5
(x, y) = (0, -2)
The quadratic function can be represented as
y = a(x - x₁)(x - x₂)
Substitute the known values in the above equation So, we have the following equation
y = a(x + 7)(x - 5)
At (x, y) = (0, -2), we have
-2 = a(0 + 7)(0 - 5)
Evaluate
a = -2/-35
Divide
a = 2/35
So, we have
y = 2/35(x + 7)(x - 5)
Hence, the equation is y = 2/35(x + 7)(x - 5)
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or Each morning, Sleepwell Hotel offers its guests a free continental breakfast with pastries and orange juice. Last year, the hotel served a total of 490 gallons of orange juice to its guests. This year, the hotel served 931 gallons of orange juice. What is the percent of increase in the amount of orange juice served yearly?
Answers
Answer:
90% increase
Step-by-step explanation:
490-931=-441
931-490=441
931/490=1.9
therefore, 190% of 441=931
Answer:90% increase
A group of 15 athletes participated in a golf competition. Their scores are below:
Score (points) 1 2 3 4 5
Number of Athletes 1 2 3 4 5
Would a line plot or a histogram best represent the data presented here? Why?
Histogram, because a large number of scores are reported as ranges
Histogram, because a small number of scores are reported individually
Line plot, because a large number of scores are reported as ranges
Line plot, because a small number of scores are reported individually
Answers
The solution is Option D.
Line plot , because a small number of scores are reported individually
What is a line plot?
A line plot is a graph that shows data as points or check marks above a number line, indicating the frequency of each value.
If we need to compare two different groups , a line chart does have one advantage for visualizing frequency distributions
Given data ,
Total number of athletes = 15
The score points of athletes = { 1 , 2 , 3 , 4 , 5 }
The number of athletes = { 1 , 2 , 3 , 4 , 5 }
The best option to plot the line plot because of the data set
The line plot , a data visualization chart that more accurately represents the small data set, is chosen.
And , to begin, unlike a line plot , a histogram involves data ranges or intervals. A line plot uses a dot to represent each observation, whereas a histogram divides the data into intervals, masking the identity of each observation
Furthermore, because line plots are one of the simplest statistical plots, they are better suited to small to moderate-sized data sets. They are more effective than histograms at highlighting clusters and gaps
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Answer:D
Step-by-step explanation:
solve the equation for y
5x+7y=35
Answers
Answer:
[tex]y=-\frac{5}{7}+5[/tex]
Step-by-step explanation:
the goal is to isolate y on one side
5x+7y=35
-5x. -5x
7y=35-5x
/7. /7. /7
y=-5/7x+5
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3 years is to 4 , , ,
Answers
Solve for x
-21 = -7x
Simplify your answer as much as possible.
Answers
Answer:
Step-by-step explanation:
-21 = -7x
-21 divided by -7 = -7 divided by -7
3=x
Find the distance between 7 and 18 on a number line.
Answers
Answer:
11
Step-by-step explanation:
Find the difference using subtraction:
18 - 7 = 11
So, the numbers are 11 units apart.
In QRS, RS QR and m/Q = 46°. Find m/S.
Answers
In triangle QRS, ∠S=67⁰ if ∠R =∠S and ∠Q=46⁰
What is meant by an angle?An angle is a figure in Euclidean geometry made up of two rays that share a vertex, or common endpoint, and are referred to as the sides of the angle. Two planes intersecting at an angle also produces an angle.
A measure of an angle or of a rotation is also referred to as an angle. The ratio of a circular arc's length to radius is known as this quantity. When it comes to a geometric angle, the arc's vertex serves as its center and its sides as its boundaries. When there is a rotation, the arc is defined by the rotation's image and any other point other than the rotation's center.
In triangle QRS,
And from the figure,
∠R=∠S
Therefore,
∠R+∠S+∠Q=180⁰
2∠S+46⁰=180⁰
2∠S=134⁰
∠S=67⁰
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Please HELP..
Our class is studying the effects of mindfulness on mental health. To help with data collection, we will have the support of ten grade 9, eleven grade 10, twelve grade 11 and thirteen grade 12 students. Showing your calculations,how many different groups of 12 volunteers can we make if:
a) the group must have the same number of students from each grade?
b) the group must have at least two grade 9 students? [2A]
c) the group must have James and Lucy (both grade 11 students) in it? [2A]
Answers
The number of groups, using the combination formula, for each case, is given as follows:
a) Same number of students from each grade: 1,245,816,000
b) At least two grade nine students: 31,650,886,995.
c) James and Lucy are in it: 2,481,256,778.
How to obtain the number of groups?The number of groups is obtained for the three cases using the combination formula, as the order in which the students are chosen is not important.
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, obtained by the following formula, involving factorials.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
For item a, the same number of students is taken from each class, hence 3 students are taken from each class, as 12 students are taken from 4 classes, hence 12/4 = 3.
Hence the number of groups, considering the number of students in each class, is of:
[tex]C_{10,3}C_{11,3}C_{12,3}C_{13,3} = \frac{10!}{7!3!} \times \frac{11!}{8!3!} \times \frac{12!}{9!3!} \times \frac{13!}{10!3!} = 120 \times 165 \times 220 \times 286 = 1,245,816,000[/tex]
For item b, we use complementary events, first obtaining the total number of students that can be taken, and then removing the combinations that do not have at least two grade nine students.
The total number of groups is obtained as follows:
[tex]C_{46,12} = \frac{46!}{12!34!} = 38,910,617,655[/tex]
(as there are 10 + 11 + 12 + 13 = 46 students, and 12 are taken).
The number of groups with none students from grade 9 is:
[tex]C_{36,12} = \frac{36!}{12!24!} = 1,251,677,700[/tex]
(removing the 10 students from grade 9).
The number of groups with one student from grade 9 is:
[tex]C_{10,1}C_{36,11} = 10 \times \frac{36!}{11!25!} = 6,008,052,960[/tex]
(one from grade 10 and the remaining from the other classes).
This the number of groups with at least two grade 9 students is calculated as follows:
38,910,617,655 - (1,251,677,700 + 6,008,052,960) = 31,650,886,995.
For item c, considering Jamie and Lucy in it, the remaining 10 students are taken from a set containing the 44 remaining students, hence:
[tex]C_{44,10} = \frac{44!}{10!34!} = 2,481,256,778[/tex]
More can be learned about the combination formula at https://brainly.com/question/11732255
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