A committee of 6 is to be chosen from the 28 students in a class. If there are 10 males and 18 females in the class, in how many ways can this be done if there must be at least three females on the committee? A: 339864B: 816720C: 3060D: 18564
Hello! First, let's write some important information contained in the exercise:
committee = 6 students
class: 28 students:
- 10 males
- 18 females
Let's consider the rule: At least three females must be on the committee, so we have some cases, look:
_F_ * _F_ * _F_ * __ * __ * __
1st option:
3 females and 3 males
_F_ * _F_ * _F_ * _M_ * _M_ * _M_
2nd option:4 females and 2 males
_F_ * _F_ * _F_ * _F_ * _M_ * _M_
3rd option:5 females and 1 male
_F_ * _F_ * _F_ * _F_ * _F_ * _M_
4th option:6 females and 0 male
_F_ * _F_ * _F_ * _F_ * _F_ * _F_
Now, we have to use the formula below and find the number of possible combinations:
[tex]C_{n,p}=\frac{n!}{p!\cdot(n-p)!}[/tex]Let's calculate each option below:
1st:3 females:
[tex]C_{18,3}=\frac{18!}{3!\cdot(18-3)!}=\frac{18\cdot17\cdot16\cdot15!}{3\cdot2\cdot1\cdot15!}=\frac{4896}{6}=816[/tex]3 males:
[tex]C_{10,3}=\frac{10!}{3!\cdot(10-3)!}=\frac{10\cdot9\cdot8\cdot7!}{3\cdot2\cdot1\cdot7!}=\frac{720}{6}=120[/tex]3 females and 3 males: 816 * 120 = 97920
2nd option:4 females:
[tex]C_{18,4}=\frac{18!}{4!\cdot(18-4)!}=\frac{18\cdot17\cdot16\cdot15\cdot14!}{4\cdot3\cdot2\cdot1\cdot14!}=\frac{73440}{24}=3060[/tex]2 males:
[tex]C2=\frac{10!}{2!\cdot(10-2)!}=\frac{10\cdot9\cdot8!}{2\cdot1\cdot8!}=\frac{90}{2}=45[/tex]4 females and 2 males: 3060* 45 = 137700
3rd option:5 females:
[tex]C_{18,5}=\frac{18!}{5!\cdot(18-5)!}=\frac{18\cdot17\cdot16\cdot15\cdot14\cdot13!}{5\cdot4\cdot3\cdot2\cdot1\cdot13!}=\frac{1028160}{120}=8568[/tex]1 male:
[tex]C_{10,1}=\frac{10!}{1!\cdot(10-1)!}=\frac{10!}{1\cdot9!}=\frac{3628800}{362880}=10[/tex]5 females and 1 male = 8568 * 10 = 85680
4th option:6 females and 0 male:
[tex]C_{18,6}=\frac{18!}{6!\cdot(18-6)!}=\frac{18\cdot17\cdot16\cdot15\cdot14\cdot13\cdot12!}{6\cdot5\cdot4\cdot3\cdot2\cdot1\cdot12!}=\frac{13366080}{720}=18564[/tex][tex]C_{10,0}=\frac{10!}{0!\cdot(10-0)!}=\frac{10!}{10!}=1[/tex]6 females and 0 male: 18564 * 1 = 18564
To finish the exercise, we have to sum the four options:
97920 + 137700 + 85680 + 18564 = 339864
So, right answer A: 339864.
dentashboard/home
Town policy requires that a certain number of trees be planted for every tree that is cut down.
For example, if 8 trees are cut down, 48 trees will be planted. A homeowner is going to cut down
5 trees on his property.
Solve Problems with Ratios and Unit Rates-Instruction-Level F
How many trees will be planted when 5 are cut down?
Trees Planted
Trees Cut Down
48
8
5
This can be solved using ratio and proportions.
What is ratio?
A ratio in mathematics illustrates how many times one number contains another. For example, if a dish of fruit contains eight oranges and six lemons, the orange-to-lemon ratio is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the proportion of lemons to oranges is 6:8 (or 3:4), while the proportion of oranges to overall fruit is 8:14. (or 4:7). A ratio's numbers can be any quantity, such as a count of persons or things, or measures of lengths, weights, time, and so forth. In most situations, both numbers must be positive. A ratio in mathematics illustrates how many times one number contains another.
First, we find the ratio of trees planted to trees cut
= 48/8 = 6
So, no. of trees to be planted when 5 trees are cut = 5x6 = 30
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Find the vertex of the function given below.
y = 3x² + 6x +1
A. (1,7)
B. (-1,-2)
C. (-4,9)
D. (-1,-1)
[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{3}x^2\stackrel{\stackrel{b}{\downarrow }}{+6}x\stackrel{\stackrel{c}{\downarrow }}{+1} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{ 6}{2(3)}~~~~ ,~~~~ 1-\cfrac{ (6)^2}{4(3)}\right) \implies \left( - \cfrac{ 6 }{ 6 }~~,~~1 - \cfrac{ 36 }{ 12 } \right) \\\\\\ (-1~~,~~1-3)\implies {\Large \begin{array}{llll} (-1~~,~~-2) \end{array}}[/tex]
Find all X such that -13 < 5x - 3 ≤ 17.
Answer:
-5 < x < 17/5
Step-by-step explanation:
add three onto all sides -10 < 5x < 17
then divide all sides by -5 < x < 17/5
What is the measure of a?
In the figure below, C D bisects ∠A C B
AB=B C
∠ B E C=90° and
∠D C E=42°
Find the measure of ∠A
The measure of "a" in the above-given right-angled triangle is 32°
What is a triangle?Note that a triangle is a three-sided polygon that is sometimes (but not always) referred to as the trigon. Every triangle has three sides and three angles, which may or may not be the same.
Key properties of a right-angled triangle to note are:
One angle is always 90° or a right angle.The side opposite angle of 90° is the hypotenuse.The hypotenuse is always the longest side.The sum of the other two interior angles is equal to 90°.The other two sides adjacent to the right angle are called base and perpendicular.The Sum of all three interior angles is equal to 180°To solve for "a" recall the following. It is given that:
AB = BC
∠BEC=90°
∠DCE=42°
Because CD bisects ∠ACB, Hence,
∠ACD = ∠DCB = x.............................1
∠DCE= ∠BCE + ∠DCB .....................2
= ∠BCE + x = 42° ................................3
To find ∠A, we use deductive reasoning to state:
∠A +∠E +∠ACD + ∠DCB + ∠BCE = 180° (Sum of Interior Angles)
Recall equation 1 hence, replacing ∠ACD with ∠DCB we have:
∠A +∠E +∠DCB + ∠DCB + ∠BCE = 180°
Recall equation 2, where ∠DCB = x, hence
∠A +∠E +x + x + ∠BCE = 180°
Recall equation three where 3 where ∠BCE + x = 42°, hence
∠A +∠E +x + (x + ∠BCE) = 180°
⇒ ∠A +∠E + x + 42° = 180°
Recall that ∠E = 90° so
⇒ ∠A +90° + x + 42° = 180° (Collect like terms)
⇒ ∠A + x = 180° - 90° - 42° = 48°; Hence
∠A + x = 48°...........................4
Recall that in Δ ABC
AB = BC (given)
Hence, ABC is an Isosceles triangle. Since this is true, then
⇒ ∠A = ∠ACB = 2x ...........................5 (Two angles opposite the equal sides are equal)
Substituting 2x in equation 5 into equation 4, we have
2x + x = 48°
3x = 48
x = 48/3
x = 16°
Recall that
∠A = 2x...........equation 5, hence,
∠A = 2 * 16
∠A = 32°
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I need help finding the area I have a huge test tomorrow and I’m nervous.
The given figure can be divided into a rectangle and a triangle
The rectangle has the dimensions 10 m by 12 m
so, the area of the rectangle = 10 x 12 = 120 square meters
Now, we will find the area of the triangle:
Base = 12 - 10 1/4 = 1 3/4 m
Height = 15 - 10 = 5 m
Area = 1/2 x base x height =
[tex]\frac{1}{2}\cdot1\frac{3}{4}\cdot5=\frac{1}{2}\cdot\frac{7}{4}\cdot5=\frac{35}{8}=4\frac{3}{8}m^2[/tex]So, the total area =
[tex]120+4\frac{3}{8}=124\frac{3}{8}m^2[/tex]This diagram shows a pre-image A ABC, and its image, A A"B'C", after a series of transformations. Select from the drop-down menus to correctly complete the statements. C A A A ABC is X Choose. B to becomeA ABC'. Then A A'BC' is C C Choose.. to become A A"B"C" . Because the transformations are Choose... the pre- image and image are Choose..
The pre-image ABC in order to become the reflection at A'B'C' is reflected across the x-axis.
The pre-image ABC in order to become A"B"C" both are CONGRUENT, because the image and pre-image are BOTH RIGID.
**Please note that the rules for the transformation from ABC to A"B"C" were not provided**
i need help with this pls
Hello!
What is an x-intercept:
⇒ value of x when the value of y equals '0'
[tex]8x + 5y=25\\8x+5(0)=25\\8x=25\\\\x=\dfrac{25}{8}[/tex]
x-intercept is 25/8
What is a y-intercept:
⇒ value of y when the value of x equals '0'
[tex]8x+5y=25\\8(0)+5y=25\\5y=25\\y=5[/tex]
y-intercept is 5
Hope that helps!
A model of Spaceship Earth, a major tourist attraction at Epcot Center in Florida, is a sphere whose diameter is approximately 5 inches.The volume of the model sphere is approximately ___ cubic inches.Use 3.14 for pi. Round only your final answer to the nearest hundredth.
The Volume of the model sphere is given as
[tex]V_{\text{sphere}}=\frac{4}{3}\pi r^3[/tex]Given that the diameter is 5inches, the radius will be
[tex]\begin{gathered} \text{diameter,d}=2\times radius \\ r=\frac{d}{2}=\frac{5}{2}\text{inches} \\ r=2.5\text{inches} \end{gathered}[/tex]substituting r in the formula will give
[tex]\begin{gathered} V_{\text{sphere}}=\frac{4}{3}\times3.14\times(2.5)^3 \\ =\frac{4}{3}\times3.14\times76.765625 \\ =\frac{964.17625}{3} \\ =321.3922\text{cubic inches} \end{gathered}[/tex][tex]V_{\text{sphere}}=321.39\text{cubic inches}[/tex]Hence, the volume of the model sphere is approximately 321.39 cubic inches
Type the correct answer in type the answer in the box.Consider functions fand g.f(x)=(x+1)^3g(x) = 3sqrtx + 1Evaluate the function composition.(fg)(-64) = _
f(x) = (x + 1)^3
g(x) = cubic root x + 1
f(g)(x) =
[tex]\begin{gathered} f(g(x))\text{ = (3}\sqrt[]{x}+2)^3 \\ f(g(x))\text{ = x + 6(3}\sqrt[]{x})^2\text{ + 12(3}\sqrt[]{x})\text{ + 8} \\ f(g(-64))\text{ = -64 + 6(3}\sqrt[]{-64})^2\text{ + 12(3}\sqrt[]{-64})\text{ + 8} \\ f(g(-64))\text{ = -64 + 6(16) - 48 + 8} \\ f(g(-64))\text{ = -64 + 96 - 48 + 8} \\ f(g(-64))\text{ = 104 - 112} \\ f(g(-64))\text{ = -8} \end{gathered}[/tex]result
f(g(-64)) = -8
The two plots below show the heights of some sixth graders and some seventh graders:Sixth Graders+52 53 54 55 56 57Height (inches)Seventh Graders52 53 54 55 56 57Height (inches)The mean absolute deviation (MAD) for the first set of data is 1.2 and the MAD for the second set of data is 1.0. Approximately how manytimes the variability in the heights of the seventh graders is the variability in the heights of the sixth graders? (Round all values to the tenthsplace.)
From the information given,
MAD for the heights of some sixth graders = 1.2
MAD for the heights of some seventh graders = 1
Number of times = MAD of data1/MAD of data 2 = 1.2/1
The answer is 1.2
Henry purchased compact disks for $112. The compact discs cost $16 each.Which of the following equations could you use to find how many compact disks, x, Henry purchased?$16 = $112xO $112 = $16xXO $112=16$112 = x - $16
Problem
Henry purchased compact disks for $112. The compact discs cost $16 each.
Which of the following equations could you use to find how many compact disks, x, Henry purchased?
Solution
We can set up the following notation:
x= number of disks
total cost = (number of disks)* (unitary price)
Total cost = $112
number of disks =x
unitary price = $16
Replacing we got:
112 = x*16
For this case the correct equation would be given by:
$112 = $16x
A bird is flying above a tree. You are standing 40 feet away from the tree. The angle of elevation to the top of the tree is 32°, and the angle of elevation to the bird is 42°. What is the distance from the bird to the top of the tree?
A bird is flying above a tree. You are standing 40 feet away from the tree. The angle of elevation to the top of the tree is 32°, and the angle of elevation to the bird is 42°. What is the distance from the bird to the top of the tree?
Let
A -----> you
B-----> a bird
C-----> top of the tree
see the following image
step 1
In the right triangle ABD
tan(42)=BD/AD
substitute given values
tan(42)=BD/40
BD=40*tan(42)
BD=36 ft
step 2
In the right triangle ACD
tan(32)=CD/AD
CD=AD*tan(32)
CD=40*tan(32)
CD=25 ft
step 3
Find the difference BD-CD
36-25=11 ft
therefore
the answer is 11 ftWhen do you know you can stop the long division process when converting a fraction to a decimal number?
Division stops after a certain number of steps as the remainder becomes zero and When division continues as there is a remainder after every step.
What is Division?a division is a process of splitting a specific amount into equal parts.
We can stop the long division process when converting a fraction to a decimal number.
There can be two situations in converting fractions to decimals:
When division stops after a certain number of steps as the remainder becomes zero.
When division continues as there is a remainder after every step.
Hence When division stops after a certain number of steps as the remainder becomes zero and When division continues as there is a remainder after every step.
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can somebody please help me with this question
Answer:
U'(12, 15)
Step-by-step explanation:
Given point U(4, 5) is part of figure STUVW that is dilated by a factor of 3 about the origin, you want the coordinates of U'.
DilationDilation about the origin multiplies each coordinate value by the scale factor:
U' = 3U
U' = 3(4, 5) = (12, 15)
The coordinates of U' are (12, 15).
customer account “numbers” for a certain company consists of 4 letters followed by 2 single digit numbers. how many different account numbers are possible if repetitions of letters and digits are allowed?
A manufacturer knows that their items have a lengths that are skewed right, with a mean of 17.6 inches, andstandard deviation of 3.3 inches.If 37 items are chosen at random, what is the probability that their mean length is greater than 18.9 inches?(Round answer to four decimal places)Question Help: D VideoSubmit Question
The probability is 0.9999
Explanation:Given that the mean is 17.6 inches
standard deviation is 3.3/37 = 0.089
We have:
z(18.9)
[tex]z=\frac{x-\mu}{\sigma}[/tex][tex]\frac{18.9-17.6}{0.089}=14.6067[/tex]Now, we have
P(x > 18.9) = P(z > 14.6067)
= 0.9999
z(0.394) = 0.6532
Part A find the value of each variable
x=14, y=15
x=14, y=-5
x=24, y=-10
x=24, y-5
Part B
the angle measures?
100°
50°
80°
130°
Answer:
Part A:
Option 1
3x + 8 = 5x - 20 (vertically opposite angles)
2x = 28
x = 14
3x + 8 + 5x + 4y = 180° (adjacent angles on a straight line)
8x + 4y = 172
Substitute x = 14 into equation to find y.
8(14) + 4y = 172
4y = 60
y = 15
Hence, x = 14 and y = 15.
Part B:
Option 4
Unlabeled angle = 5x + 4y (vertically opposite angles)
Substitute x and y to find angle measure.
5(14) + 4(15) = 130°
Hence, measure of unlabeled angle is 130°.
MATHEMATICAL PHRASES_____5. the quatient of y and seven is eight_____6. thrice a number is eighteen_____7. the sum of twice a number and five is nine_____8. a number decreased by seven is twenty-eight
Explanation
Part A: The quotient of y and seven is eight.
Answer: y/7 =8
Part B: Thrice a number is eighteen
Answer: 3x = 18
Part C: The sum of twice a number and five is nine
Answer: 2x+5 =9
Part D: A number decreased by seven is twenty- eight
Answer: x-7 =28
The line of best fit is given as y = 9 - 3x. Find the value of y when x = -3.091827
Solution
We are given the equation
[tex]y=9-3x[/tex]We want to find y when x = -3. We only need to put x = -3
[tex]\begin{gathered} y=9-3x \\ y=9-3(-3) \\ y=9+9 \\ y=18 \end{gathered}[/tex]Therefore, the answer is
[tex]\begin{equation*} 18 \end{equation*}[/tex]2.2.30QuesticFind a quadratic function that includes the set of values below.(0,6), (2,8), (3,0)The equation of the parabola is y=0
The form of quadratic is:
[tex]y=ax^2+bx+c[/tex]Since (0,6) is given, we know c = 6, thus we have:
[tex]\begin{gathered} y=ax^2+bx+c \\ ax^2+bx+6 \end{gathered}[/tex]Point 2 is (2,8), replace x and y and find equation:
[tex]\begin{gathered} y=ax^2+bx+6 \\ 8=a(2)^2+b(2)+6 \\ 8-6=4a+2b \\ 4a+2b=2 \end{gathered}[/tex]Putting point 2 (3,0), we have:
[tex]\begin{gathered} y=ax^2+bx+6 \\ 0=a(3)^2+b(3)+6 \\ 9a+3b=-6 \end{gathered}[/tex]Solving the 2 simulatenous equations for a and b, we get:
a = -3
b = 7
Now u have all the values, a, b, and c.
Just put it in the general form of parabola :
[tex]y=-3x^2+7x+6[/tex]y = -3x^2 + 7x + 6
On a recent trip, Carol's car used 7/8 of a tank of gasoline. Which decimal and percentrepresents this amount?
The fraction of gasoline used is:
[tex]\frac{7}{8}[/tex]We need to convert this fraction to decimal and to percentage to find the answer.
Converting 7/8 to decimal:
For this we have to do the division between 7 and 8:
Thus, the decimal form of 7/8 is 0.875
Converting 7/8 to percentage:
we use the result that we previously get of 7/8 as a decimal: 0.875, and to convert it to percentage we multiply it by 100:
[tex]0.875\times100=87.5[/tex]7/8 to percentage is equal to 87.5%
Answer: 0.875 and 87.5%
(4b) If you know that you can drive 230 miles withthat much gas, how many miles per gallon doesyour bus get?Round your answer to the nearest tenth of a mile.The bus gets miles per gallon,
Given the following question:
Part A:
75 dollars to spend on gaS
Gas per gallon costs $2.80
[tex]\begin{gathered} 75\div2.80=26.7857143 \\ 26.7857143 \\ 8>5 \\ 26.8\text{ gallons of gas} \end{gathered}[/tex]With 75 dollars you can buy "26.8 gallons of gas."
Part B:
We know we can travel 230 miles with 26.8 gallons of gas, now we have to find out how many miles can we travel PER gallon of gas.
[tex]\begin{gathered} 230\div26.8=10.4477612 \\ 10.4477612 \\ 4<5 \\ 10.4\text{ miles per gallons} \end{gathered}[/tex]"10.4 miles" per gallon
Function g is a transformation of the parent function f(x) = x2. The graph of fis reflected across the x-axis, and then translated left 4 units anddown 2 units to form the graph of gWrite the equation for g in the form y = ax2 + bx + cO A. y = -x2 + 8x + 14O B. y = -x2 - 8x - 18O C. y = x2 - 8x + 18O D. y = -x2 - 8x + 14
The parent function is given as:
f(x) = x²
y = x²
The graph is reflected across the x - axis
The x axis remains the same but the y axis is negated
g(x) = -x²
It is translated 4 units left
The function g(x) becomes
g(x) = -(x - 4)²
It is the translated 2 units down
g(x) = -(x - 4)² - 2
Simplifying the above equation:
g(x) = - (x² - 8x + 16) - 2
g(x) = -x² + 8x - 16 - 2
g(x) = -x² + 8x - 18
A city has a population of 250,000 people. Suppose that each year the population grows by 6%. What will the population be after 5 years?Use the calculator provided and round your answer to the nearest whole number.
The population after 5 years is approximately 337,465 people
Explanation:The population of the city is 250,000
Annual growth rate is 6%
The population after t years is:
[tex]P=P_oe^{rt}[/tex]After 5 years, t = 5 and the population becomes:
[tex]\begin{gathered} P=250000e^{\frac{6}{100}\times5} \\ \\ =250000e^{0.3} \\ \\ \approx337465 \end{gathered}[/tex]The population after 5 years is approximately 337,465 people
Colleen truman earns 5% commission on all sales in june her sales totaled 54,000$ how much did she earn in commission?
The amount earned as commission in the month of June is $2700
How to determine the amount earned as commission?From the question, we have the following parameters that can be used in our computation:
Commission percentage = 5%
Total sales in the month of June = $54000
The amount earned as commission in the month of June is then calculated as
Amount = Commission percentage x Total sales in the month of June
Substitute the known values in the above equation
So, we have
Amount = 5% x 54000
Evaluate
Amount = 2700
Hence, the commission amount is $2700
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Two motorcycle dealers sell the same motorcycle for the same original price. Dealer A advertises that the motorcycle is on sale for 7.5% off the original price. Dealer B advertises that it is reducing the motorcycle’s price by $599. When Bonnie compares the sale prices of the motorcycles in both dealers, she concludes that the sale prices are equal.
Let p represent the motorcycle’s original price.
Which equation models this situation?
0.075p = p − 599
0.925p = p + 599
0.075(p−599) = p
0.925p =
AND NO SPAM I WILL REPORT YOU AND BAN YOU IMMEADIATLEY AND HELP THIS IS DUE TODAY!! SO FAST PLS
Considering the definition of an equation, the equation 0.925p= p - 599 models the following situation: the sale prices of the motorcycles in both dealers are equal.
Definition of equationAn equation is the equality existing between two algebraic expressions connected through the equals sign in which one or more unknown values appear in addition to certain known data.
The solution of a equation means determining the value that satisfies it and the equality is true. To solve an equation, keep in mind:
When a value that is adding, when passing to the other member of the equation, it will subtract.If a value you are subtracting goes to the other side of the equation by adding.When a value you are dividing goes to another side of the equation, it will multiply whatever is on the other side.If a value is multiplying it passes to the other side of the equation, it will pass by dividing everything on the other side.Equation in this caseBeing "p" the motorcycle’s original price, you know that:
Dealer A advertises that the motorcycle is on sale for 7.5% off the original price → Sale price A= 100%×p - 7.5%×p → Sale price A= (100% - 7.5%)× p → Sale price A= 92.5%p → Sale Price A= 0.925p (expressed as decimals)Dealer B advertises that it is reducing the motorcycle’s price by $599. → Sale price B=p - 599If the sale prices are equal, the equation in this case is:
Sale price A= Sale price B
0.925p= p - 599
Finally, the equation 0.925p= p - 599 represent the situation.
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The sides of a scalene triangle have measures that are consecutive even integers. If the perimeter of this
triangle is 60 inches, what is the length of the longest side of the triangle?
Answer: 22
Step-by-step explanation:
Let the sides have lengths [tex]x-4, x-2, x[/tex]. Since the perimeter is 60,
[tex]x-4+x-2+x=60\\\\3x-6=60\\\\3x=66\\\\x=22[/tex]
So, the length of the longest side is 22 units.
The table and graph show the population or Oregon
From the given table, it is found that:
a. The average population decline was of 2,250 deer a year.
b. The population would have reached 225 thousand deer during the years of 2017 and 2018.
What is the average rate of change of a function?The average rate of change of a function is given by the change in the output divided by the change in the input of the function. Hence, over an interval [a,b], the average rate of change is given as follows:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
In 1984 and 2018, the populations are given as follows:
1984: 250 thousand.2018: 173.5 thousand.Hence the rate of change is given as follows:
r = (173.5 - 250)/34 = -2.25.
(2.25 thousand = 2,250).
For item b, the following linear function is built:
y = 230.5 - 1.45x.
The amount would be of 225 when:
230.5 - 1.45x = 225
1.45x = 5.5.
x = 5.5/1.45
x = 3.79.
Hence between the years of 2017 and 2018.
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Find the variance and standard deviation of the set of data to the nearest tenth (one decimal place){4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9}
Hello there. To solve this question, we'll need to remember how to find the variance and standard deviation of a data set.
First, given a data set with values:
[tex]\mleft\lbrace x_1,x_2,x_{3,\ldots}\mright\rbrace[/tex]The variance can be calculated by the formula:
[tex]\sum ^{}_{}\frac{(x_i-\mu)^2}{n}[/tex]In this case, μ is the arithmetic mean of the data set and x_i is the i-th element of the data set.
The data set given is:
{4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9}
To calculate the mean, we add every term and divide by the number of terms:
[tex]\frac{4+5+5+5+5+5+6+6+6+6+7+7+7+7+8+9}{16}=\frac{98}{16}=6.125[/tex]Now, we calculate the square of the difference between every term and mean.
Notice we have some repeated terms, we rewrite the sum like this:
[tex]\frac{(4-6.125)^2+5\cdot(5-6.125)^2+4\cdot(6-6.125)^2+4\cdot(7-6.125)^2+(8-6.125)^2+(9-6.125)^2}{16}[/tex]Adding the terms and calculating the squares, we have:
[tex]\begin{gathered} \frac{(-2.125)^2+5\cdot(-1.125)^2+4\cdot(-0.125)^2+4\cdot(0.875)^2+(1.875)^2+(2.875)^2}{16} \\ \\ \frac{4.515625+6.328125+0.0625+3.0625+3.515625+8.265625}{16} \\ \\ \frac{25.75}{16}\text{ = }1.609 \end{gathered}[/tex]This is the variance of the values of this data set. We can round it up to the nearest tenth as 1.6
The standard deviation is the square root of the variance.
Then, we calculate:
[tex]\sigma\text{ = }\sqrt[]{1.6}=1.268[/tex]Again, rounding it to the nearest tenth, we have 1.3;
The final answer is: The variance of this data set is 1.6 and its standard deviation is 1.3.
Solving the second question:
We apply the same formula. First, find the mean of the values:
[tex]\mu\text{ = }\frac{4.3+6.4+2.9+3.1+8.7+2.8+3.6+1.9+7.2}{9}=4.54[/tex]Now, as every term is different from each other, we apply the formula and get the following:
[tex]\frac{(4.3-4.54)^2+(6.4-4.54)^2+(2.9-4.54)^2+(3.1-4.54)^2+(8.7-4.54)^2+(2.8-4.54)^2+(3.6-4.54)^2+(1.9-4.54)^2+(7.2-4.54)^2}{9}[/tex]Calculating the difference, squaring it, adding the values and dividing by 9, we get:
Variance is approximately equal to 4.84
The Standard deviation is the square root of the variance, namely 2.2