Answers
So first of all let's take:
[tex]x_1=x\text{ and }x_2=y[/tex]Then we get:
[tex]\begin{gathered} \text{Min}z=1.5x+2y \\ x+y\ge300 \\ 2x+y\ge400 \\ 2x+5y\leq750 \\ x,y\ge0 \end{gathered}[/tex]The next step would be operate with the inequalities and the equation so we end up having only the term y at the left side of each:
[tex]\begin{gathered} \text{Min}z=1.5x+2y \\ 1.5x+2y=\text{Min}z \\ 2y=\text{Min}z-1.5x \\ y=\frac{\text{Min}z}{2}-0.75x \end{gathered}[/tex][tex]\begin{gathered} x+y\ge300 \\ y\ge300-x \end{gathered}[/tex][tex]\begin{gathered} 2x+y\ge400 \\ y\ge400-2x \end{gathered}[/tex][tex]\begin{gathered} 2x+5y\leq750 \\ y\leq150-\frac{2}{5}x \end{gathered}[/tex]So now we have the following inequalities and equality:
[tex]\begin{gathered} y=\frac{\text{Min}z}{2}-0.75x \\ y\ge300-x \\ y\ge400-2x \\ y\leq150-\frac{2}{5}x \end{gathered}[/tex]If we take the three inequalities and replace their symbols by "=' we'll have three equations of a line:
[tex]\begin{gathered} y=300-x \\ y=400-2x \\ y=150-\frac{2}{5}x \end{gathered}[/tex]The following step is graphing these three lines and delimitating a zone in the grid that meets the inequalities:
Where the blue area is under the graph of y=150-(2/5)x which means that it meets:
[tex]y\leq150-\frac{2}{5}x[/tex]And it is also above the x-axis, y=400-2x and y=300-x which means that it also meets:
[tex]\begin{gathered} x\ge0 \\ y\ge0 \\ y\ge400-2x \\ y\ge300-x \end{gathered}[/tex]All of this means that the values of x and y that give us the correct minimum of z are given by the coordinates of a point inside the blue area. The next thing to do is take the four possible values for Min(z) and use them to graph four lines using this equation:
[tex]y=\frac{\text{Min}z}{2}-0.75x[/tex]Then we have four equations of a line:
[tex]\begin{gathered} y=\frac{450}{2}-0.75x \\ y=\frac{780}{2}-0.75x \\ y=\frac{647}{2}-0.75x \\ y=\frac{354}{2}-0.75x \end{gathered}[/tex]The line that has more points inside the blue area is the one made with the closest value to Min(z). Then we have the following graph:
As you can see there are two lines that have points inside the blue area. These are:
[tex]\begin{gathered} y=-\frac{3}{4}x+\frac{450}{2} \\ y=-\frac{3}{4}x+\frac{354}{2} \end{gathered}[/tex]That where made using:
[tex]\begin{gathered} \text{Min }z=450 \\ \text{Min }z=354 \end{gathered}[/tex]Taking a closer look you can see that the part of the orange line inside the blue area is larger than that of the red line. Then the value used to make the orange line would be a better aproximation for the Min z. The orange line is -(3/4)x+450/2 which means that the answer to this problem is the first option, 450.
Related Questions
In △ABC, m∠A=45°. The altitude divides side AB into two parts of 20 and 21 units. Find BC.
Answers
Answer:
29 units
Step-by-step explanation:
BC is a side of ACB, which is a 45 45 90 triangle. BC = AB/sqrt2
If In △ABC, m∠A=45°. The altitude divides side AB into two parts of 20 and 21 units. Then BC is 29 units
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
Given,
In △ABC, m∠A=45°. The altitude divides side AB into two parts of 20 and 21 units.
We need to find BC
In triangle ACD,
tan 45° = CD/AD
CD = tan 45° x AD
= 1 x 20= 20 units
In triangle CDB,
tanФ = CD/BD
Ф = tan⁻¹(CD/BD)
= tan⁻¹(20/21)
= 43.6°
so, sin 43.6° = CD/BC
BC = CD/sin 43.6°
= 20/0.689
= 29 units
Hence the length of BC will be 29 units.
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given that f(x)=3x-6, determine f(8)
Answers
According to the given data we have the following function:
f(x)=3x-6
To determine f(8) we would have to plug in into the equation the 8 and then proceed to calculate it, so:
If f(x)=3x-6
Then, f(8)=3(8)-6
f(8)=24-6
f(8)=18
What is the area of the blue shape?
Answers
In rectangle , 38.5 sq units is the area of the blue shape.
What is rectangle?
A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. As a result, it is sometimes referred to as an equiangular quadrilateral. Because the opposite sides of a rectangle are equal and parallel, it can also be referred to as a parallelogram.We can split the shape into two rectangles.
The small rectangle at the top has an area of 14 sq units (2 * 7).
The middle rectangle has an area of 49 sq units ( 7*7).
Since the blue part in the middle rectangle is half of the whole rectangle, the area of the blue part there is 24.5 sq units.
Adding that to 14 sq units will give us 38.5 sq units.
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At Paul's Pet Palace, 3/16 of the animals are dogs and 5/24 of the animals are cats. What fraction of the animals are neither dogs nor cat?
Answers
You have that 3/16 of the animals are dogs and 5/24 of the animals are cat.
To determine the fraction of animals that are neither dogs nor cat, consider the following:
If 3/16 are dogs, then 13/16 are other animals, but from this fraction, 5/24 are cats. Then, the subtraction 13/16 - 5/24 results in the fraction of animals that are neither dogs nor cat:
[tex]\frac{13}{16}-\frac{5}{24}=\frac{312-80}{384}=\frac{232}{384}[/tex]simplify the last fraction:
[tex]\frac{232}{384}=\frac{116}{192}=\frac{58}{96}=\frac{29}{48}[/tex]Hence, 29/48 of the animal are neither dogs nor cats
Using the formula C =5/9(F −32), find C when F is −58∘.? C∘
Answers
ANSWER
C = -50 degree Celcius
STEP-BY-STEP EXPLANATION:
What to find? The value of C in degree Celcius
Given Parameters
F = -58 degree Fahrenheit
The formula is given below
[tex]C=\text{ }\frac{5}{9}(F\text{ - 32)}[/tex]Substitute the value of into the equation
[tex]\begin{gathered} C\text{ = }\frac{5}{9}(-58\text{ - 32)} \\ \text{Solve the expression inside the parenthesis first} \\ C\text{ = }\frac{5}{9}(-90) \\ C\text{ = }\frac{-5\cdot\text{ 90}}{9} \\ C\text{ = }\frac{-450}{9} \\ C=-50^oC \end{gathered}[/tex]Hence, the value of C is -50 degrees
How do you evaluate the following polynomials for a domain value?P(x) = -3x² + 9x find P(-5)
Answers
The given polynomial is expressed as
P(x) = - 3x^2 + 9x
The domain values are the x values. To find P(-5), it means that we would substitute x = - 5 into the polynomial. It becomes
P(- 5) = - 3(- 5)^2 + 9(- 5)
P(- 5) = - 3 * 25 - 45
P(- 5) = - 75 - 45
P(- 5) = - 120
Which one of the following angle measurements is the largest?
Answers
We have
[tex]\pi\approx3.14\text{ radians}[/tex]and
[tex]\pi=180^0[/tex]From these,
[tex]2\text{ radians<3 radians<}\pi<200^o[/tex]The largest measurement is 200 degrees. Thus, option B is correct.
If f(x) = sin(x ^ 5) , find f^ prime (x)
Answers
Solution
Step 1
Write the function.
[tex]f(x)\text{ = sin\lparen x}^5)[/tex]Step 2
Use the chain rule to find f'(x)
[tex]\begin{gathered} f^{\prime}(x)\text{ = }\frac{df}{du}\times\frac{du}{dx} \\ \\ u\text{ = x}^5 \\ \\ \frac{du}{dx}\text{ = 5x}^4 \\ f(x)\text{ = sinu} \\ \\ \frac{df}{du}\text{ = cosu} \end{gathered}[/tex]Step 3
[tex]\begin{gathered} f^{\prime}(x)\text{ = 5x}^4\text{ }\times\text{ cosu} \\ \\ f^{\prime}(x)\text{ = 5x}^4cos(x^5) \end{gathered}[/tex]Step 4
Substitute x = 4 to find f'(4).
[tex]\begin{gathered} f^{\prime}(4)\text{ = 5}\times4^4\times cos(4^5) \\ \\ f^{\prime}(4)=\text{ 1280}\times cos1024 \\ \\ f^{\prime}(x)\text{ = 715.8} \end{gathered}[/tex]Final answer
I need help with this question! I tried to work the question out but the answer I got is not an answer choice.
Answers
Given:
The given expression is
[tex]2\times72-3\times8+6\times5+4[/tex]Required:
We have to find the value of the given expression.
Explanation:
[tex]\begin{gathered} 2\times72-3\times8+6\times5+4 \\ =144-24+30+4 \end{gathered}[/tex][tex]\begin{gathered} =120+30+4 \\ =150+4 \end{gathered}[/tex][tex]=154[/tex]Final answer:
Hence the final answer is
A small town has two local high schools. High School A currently has 900 studentsand is projected to grow by 50 students each year. High School B currently has 500students and is projected to grow by 100 students each year. Let A represent thenumber of students in High School A in t years, and let B represent the number ofstudents in High School B after t years. Graph each function and determine whichhigh school is projected to have more students in 4 years.
Answers
High School A currently has 900 students and is projected to grow by 50 students each year.
We can write an equation using the above information
[tex]A=900+50t[/tex]Where A represents the number of students in High School A in t years.
High School B currently has 500 students and is projected to grow by 100 students each year.
We can write an equation using the above information
[tex]B=500+100t[/tex]Where B represents the number of students in High School B in t years.
Let us graph these two equations
Determine which high school is projected to have more students in 4 years.
Let us substitute t = 4 into both equations
[tex]\begin{gathered} A=900+50t \\ A=900+50(4) \\ A=900+200 \\ A=1100 \end{gathered}[/tex]High school A is projected to have 1100 students in 4 years.
[tex]\begin{gathered} B=500+100t \\ B=500+100(4) \\ B=500+400 \\ B=900 \end{gathered}[/tex]High school B is projected to have 900 students in 4 years.
Therefore, high school A is projected to have more students (1100) as compared to high school B (900) in 4 years.
Riley has $955 in a savings account that earns 15% interest, compounded annually.To the nearest cent, how much interest will she earn in 2 years?
Answers
In order to calculate the interest generated in 2 years, we can use the formula below:
[tex]I=P((1+r)^t-1)[/tex]Where I is the interest generated after t years, P is the principal (initial amount) and r is the interest rate.
So, for P = 955, r = 0.15 and t = 2, we have:
[tex]\begin{gathered} I=955((1+0.15)^2-1) \\ I=955(1.15^2-1) \\ I=955(1.3225-1) \\ I=955\cdot0.3225 \\ I=307.99 \end{gathered}[/tex]Therefore the interest generated is $307.99.
Solve the inequality and how do i graph ?
Answers
The most appropriate choice for linear inequation will be given by-
[tex]m > \frac{1}{2}[/tex] is the correct solution
What is linear inequation?
At first it is important to know about algebraic expressions.
Algebraic expressions consists of variables and numbers connected with addition, subtraction, multiplication and division.
Inequation shows the comparision between two algebraic expressions by connecting the two algebraic expressions by > , < , [tex]\geq, \leq[/tex]
A one degree inequation is known as linear inequation.
Here,
The given inequation is [tex]\frac{1}{2} - \frac{m}{4} < \frac{3}{8}[/tex]
Now,
[tex]\frac{1}{2} - \frac{m}{4} < \frac{3}{8}\\\\\frac{m}{4} > \frac{1}{2} - \frac{3}{8}\\\\\frac{m}{4} > \frac{4 - 3}{8}\\\\\frac{m}{4} > \frac{1}{8}\\\\m > \frac{1}{8} \times 4\\m > \frac{1}{2}[/tex]
The number line has been attached here.
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John owes $25.20 to his mom. He borrowed $27.60 from his dad. Howmuch does he owe in all? Write your answer as a rational number *
Answers
John owes to his Mom and Dad.
From Mom = 25.20
From Dad = 27.60
The total amount he owes is the sum of both. We just add both the amounts.
Total: 25.20 + 27.60 = $52.80
Consider the line y=7x-1Find the equation of the line that is perpendicular to this line and passes through the point −2, 3.Find the equation of the line that is parallel to this line and passes through the point −2, 3.Note that the ALEKS graphing calculator may be helpful in checking your answer.Equation of per pendicular line:Equation of parallel line:
Answers
Algebra / Graphs and Functions / Equations of Parallel and Perpendicular Lines
We have the line:
[tex]y=7x-1.[/tex]We must find the equation:
0. of the perpendicular line,
,1. and the parallel line,
to the given line that passes through the point (-2, 3).
1) Perpendicular line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The slope of the perpendicular line mₚ is given by the equation:
[tex]\begin{gathered} m\cdot m_p=-1, \\ 7\cdot m_p=-1, \\ m_p=-\frac{1}{7}. \end{gathered}[/tex]Replacing mₚ = -1/7 and (x₀, y₀) = (-2, 3) in the equation of the perpendicular line, we get:
[tex]y=-\frac{1}{7}\cdot(x-(-2))+3=-\frac{1}{7}\cdot(x+2)+3=-\frac{1}{7}\cdot x-\frac{2}{7}+3=-\frac{1}{7}\cdot x+\frac{19}{7}.[/tex]2) Parallel line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The parallel line has the same slope as the given line, so we have:
[tex]\begin{gathered} m_p=m, \\ m_p=7. \end{gathered}[/tex]Replacing mₚ = 7 and (x₀, y₀) = (-2, 3) in the equation of the parallel line, we get:
[tex]y=7\cdot(x-(-2))+3=7\cdot(x+2)+3=7x+14+3=7x+17.[/tex]3) Graph
Plotting the equations obtained, we get the following graph:
Answer1) Equation of the perpendicular line:
[tex]y=-\frac{x}{7}+\frac{19}{7}[/tex]2) Equation of the parallel line:
[tex]y=7x+17[/tex]describe and state a situation where you can apply the concepts of point, line, and plane in real-life situation
Answers
Point
A point is an exact position or location on a plane surface.
Real life situation: The end of a knife used for cutting meat in preparation for a meal.
Description: The end of knife is a point whose location is precise. It has ability to pass through surfaces easily due to this fact.
Line
A line is a one-dimensional figure, which has length but no width.
Real-life situation : The edge of a wall or rectangular table.
Description: The edge of a table represent a distance from one end to another end.
Plane:
A plane is a flat, two-dimensional surface that extends indefinitely.
Real-life situation: The surface of the floor or table.
Description: The surface of a floor is such that it can accommodate things on it showing that it is 2-dimensional.
You are taking 2 shirts(white and red) and 3 pairs of pants (black, blue, and gray) on a trip. How many different choices of outfits do you have?
Answers
Consider the line segment porque shown. For which of the following transformations would the image porque be contained entirely in Quadrant II?
Answers
We will have the following:
In order to have PQ entirely in the quadrant II, the transformation must be:
*Translate PQ up 4 units and to the left 3 units. [Option K]
find two vectors each of norm 1 that I perpendicular to vector A={3,2}
Answers
(2√13/13 , 3√13/13) and (-2√13/13 , -3√13/13) are two vectors of norm 1 that are perpendicular to A = (3 , 2) .
What is perpendicular vectors ?In Cartesian coordinates, the given vector can be represented by the line y = -2x/3. The vector is the line segment that connects (0,0) and (3,-2).y = 3x/2 can be used to represent the normal.
If the vector is represented by a line connecting (0,0) to a point (p,q), then,p2 + q2 = 1 because the normal is one length, and q = 3p/2.
As a result,p² + 9p²/4 = 1, 13p²/4 = 1, p = ±√(4/13) = ±2/√13, q = ±3/√13.
After rationalization, one normal vector is (2√13/13 , 3√13/13) and the other is (-2√13/13 , -3√13/13).The two vectors of norm 1 perpendicular to A = (3 , 2) is :
(2√13/13 , 3√13/13) and (-2√13/13 , -3√13/13).
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If RT = 36, RS = 2x + 3 and ST = 7x + 6, find RSand ST.
Answers
We know that RT=36 and that RS=2x+3 and ST=7x+6. We notice that
[tex]RT=RS+ST[/tex]Then, plugging the corresponding values and expressions we have
[tex]36=(2x+3)+(7x+6)[/tex]Solving this equation for x,
[tex]\begin{gathered} 36=(2x+3)+(7x+6) \\ 36=2x+3+7x+6 \\ 36=9x+9 \\ 36-9=9x \\ 27=9x \\ x=\frac{27}{9} \\ x=3 \end{gathered}[/tex]Then te value of x is 3.
Once we have the value of x we are able to find the value of RS and ST, we just have to substitute said value in the expressions. Then
[tex]\begin{gathered} RS=2(3)+3=6+3=9 \\ ST=7(3)+6=21+6=27 \end{gathered}[/tex]Therefore RS=9 and ST=27.
33. Let f(x) = 5x2 - 4 and g(x) = 3x + 1. Find f(x) + g(x):
Answers
The addition of f(x) and g(x) is derived as follows;
5x^2 - 4 + (3x + 1)
5x^2 - 4 + 3x + 1
5x^2 + 3x - 4 + 1
5x^2 +3x -3
The correct answer is option D
i am supposed to find the volume of this pyramid
Answers
For this type of problems we use the formula for the volume of a pyramid:
[tex]\begin{gathered} V=\text{ }\frac{1}{3}A_bh \\ A_b\text{ is the area of the base} \\ h\text{ is the height of the pyramid} \end{gathered}[/tex]Substituting h=12 yd and knowing that the area of a square is side*side we get that:
[tex]\begin{gathered} A_b=\text{ 10yd }\cdot10yd=100yd^2 \\ V=\frac{1}{3}100yd^212yd=100yd^24yd=400yd^3 \end{gathered}[/tex]In a music class of 20 students, there are 12 who play the Guitar (G), 7 who play the Piano (P) and 4 who do not play any of these instruments.1)Represent this situation using a Venn diagram2)What is the probability no randomly selected student plays guitar and piano3)What is the probability that no randomly selected student plays either of these two instruments?4) How likely is it that no randomly selected student plays the piano
Answers
GIVEN:
Total number of students: 20
Number that plays the Guitar: 12
Number that plays the Piano: 7
Number that does not play any of the instruments: 4
Note that included in the number that plays either the guitar and the piano is the number that plays both instruments. Let's call this number x.
Number that play both instruments: To calculate this number, we have to get the number that plays each instrument alone and find the sum equated to the total number of students.
Number that plays the Guitar only: 12 - x
Number that plays the Piano only: 7 - x
Therefore, the total number of students will be:
[tex]\begin{gathered} 12-x+7-x+x+4=20 \\ 23-x=20 \\ \therefore \\ x=23-20 \\ x=3 \end{gathered}[/tex]Hence, there are 3 students that play both instruments.
Therefore, 9 students play the guitar only and 4 students play the piano only.
VENN DIAGRAM:
QUESTION 2:
The formula to calculate the probability of an event E is given to be:
[tex]P(E)=\frac{n(E)}{n(U)}[/tex]where
[tex]\begin{gathered} P(E)=\text{ Probability of event E happening} \\ n(E)=\text{ Number of times event E happened} \\ n(U)=\text{ Number of times all possible events occured} \end{gathered}[/tex]The number of students that play the guitar and piano is 3. This means that the number that does not play the guitar and piano is
[tex]\Rightarrow20-3=17\text{ students}[/tex]Therefore, the probability is given to be:
[tex]P=\frac{17}{20}[/tex]QUESTION 3:
This question requires us to find the probability of students that do not play any of the two instruments. This number is 4.
Therefore, the probability is given to be:
[tex]P=\frac{4}{20}=\frac{1}{5}[/tex]QUESTION 4:
The number of students that play the piano is 7. Hence, the number that doesn't play the piano will be:
[tex]\Rightarrow20-7=13[/tex]Therefore, the probability is:
[tex]P=\frac{13}{20}[/tex]What would the answer be?
Nvm, I got it wrong
Answers
Applying the definition of similar triangles, the measure of ∠DEF = 85°.
What are Similar Triangles?If two triangles are similar, then their corresponding angles are all equal in measure to each other.
In the image given, since E and F are the midpoint of both sides of triangle BCD, then it follows that triangles BCD and EFD are similar triangles.
Therefore, ∠DBC ≅ ∠DEF
m∠DBC = m∠DEF
Substitute
4x + 53 = -6x + 133
4x + 6x = -53 + 133
10x = 80
10x/10 = 80/10 [division property of equality]
x = 8
Measure of ∠DEF = -6x + 133 = -6(8) + 133
Measure of ∠DEF = 85°
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Suppose you have a piece of ribbon that is 6 feet long, and you cut off one piece that's 3 2/3
inches long and a second piece that's 2 5/6 inches long. How much ribbon is left in inches?
inches of ribbon left
Answers
Answer:
65 1/2 inches of ribbon is left, my exact work is shown on a piece of paper below if you need it.
Step-by-step explanation:
1 foot = 12 inches
6 x 12 = 72 inches
72 - (3 2/3 + 2 5/6) = ?
My explanation:
The rib ion originally started at 6 feet
You first cut 3 2/3 off, so your now left with 7/3 or 2 1/3 (simplified)
Now your left with 2 1/3.
Then you cut a second piece off, 2 5/6 inches….so 2 5/6 minus 2 1/3 = 1/2
So you should have about 1/2 of a ribbon left.
(I tried btw. I’m very sleepy rn.)
Higher Order ThinkingIn 448,244, how is the relationship between the first pair of 4s the same as the relationship between the second pair of 4s?4 grade studentLesson place value relationship
Answers
In the number 44,244, we can see two pairs of 4's.
The first pair (to the left) has a higher value than the second pair to the right, but the 4's have something in common: The leftmost 4 is ten times as high as the rightmost 4.
For this reason, we start the number as forty-four thousand and end up with forty-four.
Can 37° 111° and 32° be measurements of a triangle?
Answers
Answer
The angles given can easily be the measurements of one triangle because they sum up to give 180°.
Explanation
The sum of angles in a triangle is known to be 180°
So, for the given angles to be to belong to one triangle, the sum of all the angles must be equal to 180°
So, we check by adding them
37° + 111° + 32° = 180°
Hope this Helps!!!
Teresa is participating in a 4day cross-country bike challenge. She biked it for 61, 67, and 66 miles on the first three days. How many miles does she need to bike on the last day so that her average (mean) is 63 miles per day?
Answers
The number of miles that she need to bike on the last day so that her average (mean) is 63 miles per day is 62 miles.
What is a mean?The mean is the average of a set of numbers. Let the biking of the last day be represented as x. This will be:
(61 + 67 + 66 + x) / 4 = 64
(194 + x) / 4 = 64
Cross Multiply
194 + x = 64 × 4
194 + x = 256
Collect like terms
x = 256 - 194
x = 62
The miles is 62 miles.
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I need help answering this if you can show your work to the be good
Answers
Let:
x = Number of sodas purchased
y = Number of hamburgers purchased
The food truck charges $3 for sodas, so the total cost for sodas will be:
3*x=3x
also, it charges $8 for each hamburger, hence, the total cost for hamburgers will be:
8*y = 8y
Since Jack wants to spend no more than $30, the total cost must be less or equal than $30:
[tex]\begin{gathered} \text{Total cost }\leq\text{ 30} \\ \text{Total cost = total cost for sodas+total cost for hamburgers} \\ 3x+8y\le30 \end{gathered}[/tex]May I please get help with numbers (4), (5), and (6). I have tried multiple times to find the correct answers for each of them but still could not get the accurate or at least correct answers for each of them. I would appreciate it so much if I could get help
Answers
EF =21
4) Let's find out the measure of the line segment EF, using the Trapezoid Midsegment Formula:
[tex]M=\frac{B+b}{2}[/tex]4.1) We can plug into that the lengths of AD and BC:
[tex]M=\frac{18+24}{2}=\frac{42}{2}=21[/tex]Note that the Midsegment is the average of the bases of a trapezoid.
4.3) Hence, the answer is 21
The amount of pollutants that are found in waterways near large cities is normally distributed with mean 9.9 ppm and standard deviation 1.8 ppm. 39 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.
Answers
ANSWER:
a. 9.9, 1.8
b. 9.9, 0.2882
c. 0.5239
d. 0.6368
e. No
f.
Q1 = 9.7069
Q3 = 10.0931
IQR = 0.3862
STEP-BY-STEP EXPLANATION:
a.
X ~ N (9.9, 1.8)
b.
x ~ N (9.9, 1.8/√39)
x ~ N (9.9, 0.2882)
c.
P(X > 9.8)
We calculate the probability as follows:
[tex]\begin{gathered} P\left(X>9.8\right)=1-p\left(\frac{X-9.9}{1.8}<\frac{9.8-9.9}{1.8}\right) \\ \\ P\left(X>9.8\right)=1-p(z<-0.06) \\ \\ P\left(X>9.8\right)=1-0.4761 \\ \\ P\left(X>9.8\right)=0.5239 \end{gathered}[/tex]d.
p (x > 9.8)
We calculate the probability as follows:
[tex]\begin{gathered} P\left(x>9.8\right)=1-p\left(\frac{X-9.9}{\frac{1.8}{\sqrt{39}}}<\frac{9.8-9.9}{\frac{1.8}{\sqrt{39}}}\right) \\ \\ P\left(x>9.8\right)=1-p(z<-0.35) \\ \\ P\left(x>9.8\right)=1-0.3632 \\ \\ P\left(x>9.8\right)=0.6368 \end{gathered}[/tex]e.
No, you don't need to make the assumption
f.
Q1 = 0.25
In this case the value of z = 0.25, so we look for the closest value in the normal table, like this:
Thanks to this, we make the following equation:
[tex]\begin{gathered} -0.67=\frac{x-9.9}{\frac{1.8}{\sqrt{35}}} \\ \\ x-9.9=-0.19311 \\ \\ x=-0.1931+9.9 \\ \\ x=9.7069 \\ \\ Q_1=9.7069 \end{gathered}[/tex]Q3 = 0.75
In this case the value of z = 0.75, so we look for the closest value in the normal table, like this:
Therefore:
[tex]\begin{gathered} -0.67=\frac{x-9.9}{\frac{1.8}{\sqrt{39}}} \\ \\ x-9.9=0.1931 \\ \\ x=0.1931+9.9 \\ \\ x=10.0931 \\ \\ Q_3=10.0931-9.7069 \end{gathered}[/tex]Therefore, the interquartile range would be:
[tex]\begin{gathered} IQR=Q_3-Q_1 \\ \\ IQR=10.0931-9.7069 \\ \\ IQR=0.3862 \end{gathered}[/tex]