There are a total of 40 students in the class after these changes.
Let's use j to represent the number of juniors in the class and s to represent the number of seniors in the class. We know that the ratio of juniors to seniors is 3:2, so we can write:
j/s = 3/2
Multiplying both sides by 2s gives us:
j = (3/2)s
We also know that when six more juniors join and one senior drops, the ratio becomes 4:1, so we can write:
(j + 6) / (s - 1) = 4/1
Multiplying both sides by (s - 1) gives us:
j + 6 = 4(s - 1)
j + 6 = 4s - 4
Substituting j = (3/2)s into this equation gives:
(3/2)s + 6 = 4s - 4
s = 14Therefore, there are 14 seniors in the class. Substituting s = 14 into j = (3/2)s gives:
j = (3/2)(14) = 21
Before the changes, there were 21 juniors and 14 seniors in the class. After six more juniors join and one senior drops, there are:
21 + 6 = 27 juniors
14 - 1 = 13 seniors
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Nan and Santo are designing a coffee table using 4 inch tiles. Nan uses 30 tiles and Sato uses half as many how many . how many total tiles did they use?
Based on the fact that Nan and Santo are designing a coffee table using 4 inch tiles, they used a total of 45 tiles.
What is the total tilesNan uses 30 tiles, and Santo uses half as many as Nan.
To know the amount of tiles Santo uses, one need to divide the number of tiles Nan uses by 2:
30 tiles / 2 = 15 tiles
Hence, Santo uses 15 tiles.
To be able to calculate the total number of tiles used, one need to add the number of tiles used by Nan and Santo and so:
30 tiles + 15 tiles = 45 tiles
Therefore, in the case above, they used a total of 45 tiles.
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Write a polynomial of least degree with real coefficients and with roots 2 and –7–16.
Write your answer using the variable x and in standard form with a leading coefficient of 1.
The polynomial of least degree with real coefficients and with roots 2, -7, and -16 is:
x³ + 21x² + 66x - 224How to write the polynomialTo find a polynomial with roots 2, -7, and -16, we can use the fact that if a number r is a root of a polynomial, then (x - r) is a factor of the polynomial.
Given roots: 2, -7, and -16
To find the polynomial, we can start by setting up the factors:
(x - 2), (x - (-7)), and (x - (-16)), which simplifies to (x + 7) and (x + 16).
Multiplying these factors together, we get:
(x - 2)(x + 7)(x + 16)
Expanding this expression:
(x² + 7x - 2x - 14)(x + 16)
Simplifying:
(x² + 5x - 14)(x + 16)
x³ + 16x² + 5x² + 80x - 14x - 224
x³ + 21x² + 66x - 224
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a) b) QUESTION I Use a protractor to measure the size of angle 9 and a ruler to measure the length of AB and BC in each case. (a) A 8 B What do you notice? (b) Complete the statements: The line drawn from the centre of a circle will be The line drawn from the centre of a circle O the chord. 0 B A to the chord.
(a) When measuring angle 9 and the lengths of AB and BC, one observation is required.
(b) The line drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord, while the line drawn from the center of a circle to an endpoint of a chord bisects the chord.
a) In the given diagram, we have angle AOB (angle 9) and line segments AB and BC.
To measure angle 9, we use a protractor. We align the baseline of the protractor with the line segment OA, ensuring that the center point of the protractor coincides with the vertex O. Then, we read the degree measurement where the line segment OB intersects the protractor. Let's assume the measurement of angle 9 is x degrees.
To measure the length of line segments AB and BC, we use a ruler. We align the ruler with the line segments and read the length in units (e.g., centimeters or inches).
b) The statements to complete are:
(i) The line drawn from the center of a circle to the midpoint of a chord will be perpendicular to the chord.
(ii) The line drawn from the center of a circle to the endpoint of a chord will bisect the chord.
In a circle, the center of the circle lies on the perpendicular bisector of any chord. This means that if we draw a line from the center of the circle to the midpoint of a chord, it will be perpendicular to the chord (statement i). Similarly, if we draw a line from the center of the circle to one of the endpoints of the chord, it will bisect the chord into two equal parts (statement ii).
These properties hold true for any chord in a circle, and they are based on the geometric properties of circles and perpendicular bisectors.
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An airline offers coach and first-class tickets for a flight to Aruba. For the airline to be profitable, it must sell a minimum of 20 first-class tickets and a minimum of 45 coach tickets. The flight must have at least 100 travelers aboard in order for the plane to take off. But the plane has a maximum capacity of 180 travelers. The airline makes a profit of $225 for each coach ticket and $200 for each first-class ticket. Answer ALL of the following questions: (a.) Create a system of constraints that model the given situation. (b.) Graph the system of constraints and shade the feasible region. (c.) Find and label all of the corner points of the feasible region. (d.) Is it feasible for them to sell 150 coach tickets and 25 first class tickets? Why or why not? Mathematically justify your decision. (e.) What is the maximum number of coach tickets they can sell? What is the maximum number of first class tickets they can sell? How can you tell this from the graph? (f.) How many of each ticket should be sold in order to maximize profits? What is the maximum profit they can achieve? (g.) What is the minimum profit they can achieve from the flight
The Range of the total number of travelers constraint (x + y ≥ 100), it is feasible.the maximum number of first-class tickets they can sell is 100 (at the point (100, 80)).The maximum profit they can achieve is $38,500 when they sell 100 first-class tickets and 80 coach tickets.
(a) Let's create a system of constraints based on the given situation:
1. First-class ticket constraint: First-class ticket sales must be at least 20.
x ≥ 20
2. Coach ticket constraint: Coach ticket sales must be at least 45.
y ≥ 45
3. Total number of travelers constraint: The plane must have at least 100 travelers.
x + y ≥ 100
4. Maximum capacity constraint: The plane has a maximum capacity of 180 travelers.
x + y ≤ 180
(b) To graph the system of constraints, we plot the inequalities on a coordinate plane:
The first-class ticket constraint (x ≥ 20) is a vertical line passing through (20, 0).
The coach ticket constraint (y ≥ 45) is a horizontal line passing through (0, 45).
The total number of travelers constraint (x + y ≥ 100) is a line passing through (100, 0) and (0, 100).
The maximum capacity constraint (x + y ≤ 180) is a line passing through (180, 0) and (0, 180).
(c) The feasible region is the shaded area where all constraints are satisfied.
|
180 +--------------------------------------------------+
| /|\
| / | \
| / | \
| / | \
| / | \
| / | \
| / | \
100 +--------------------------------------------------+
|_________________________________________________
20 45 100 180
The corner points of the feasible region are (20, 45), (20, 100), (80, 100), and (100, 80).
(d) To determine if it is feasible to sell 150 coach tickets and 25 first-class tickets, we check if these values satisfy the constraints.
150 coach tickets and 25 first-class tickets would give us a total of 175 travelers (150 + 25). Since this is within the range of the total number of travelers constraint (x + y ≥ 100), it is feasible.
(e) From the graph, we can observe that the maximum number of coach tickets they can sell is 80 (at the point (80, 100)), and the maximum number of first-class tickets they can sell is 100 (at the point (100, 80)).
(f) To maximize profits, we need to determine the corner point within the feasible region that gives the highest profit. We calculate the profit for each corner point and select the maximum.
For (20, 45): Profit = (225 * 45) + (200 * 20) = 10,125 + 4,000 = $14,125
For (20, 100): Profit = (225 * 100) + (200 * 20) = 22,500 + 4,000 = $26,500
For (80, 100): Profit = (225 * 100) + (200 * 80) = 22,500 + 16,000 = $38,500
For (100, 80): Profit = (225 * 80) + (200 * 100) = 18,000 + 20,000 = $38,000
The maximum profit they can achieve is $38,500 when they sell 100 first-class tickets and 80 coach tickets.
(g) The minimum profit they can achieve from the flight is $0 if they don't sell any tickets or if the number of travelers is below the minimum required for profitability.
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use this equation to calculate the profit after selling 300 cappuccinos (cappuccinos=c). p=xc-500
The profit after selling 300 cappuccinos is $1000.
The equation p = xc - 500 can be used to calculate the profit after selling c cappuccinos. In this equation, p is the profit, x is the price per cappuccino, and c is the number of cappuccinos sold.
To calculate the profit after selling 300 cappuccinos, we need to know the price per cappuccino. Let's assume that the price per cappuccino is $5.00. Plugging these values into the equation, we get:
p = 5.00 * 300 - 500
= 1500 - 500
= 1000
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Find The Length of the hypothesis in the triangle below, Write the answer to 2 decimal places
The length of the hypotenuse in this problem is given as follows:
h = 10.6 m.
What is the Pythagorean Theorem?The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side, is equals to the sum of the squares of the lengths of the other two sides.
Hence the equation for the theorem is given as follows:
c² = a² + b².
In which:
c > a and c > b is the length of the hypotenuse.a and b are the lengths of the other two sides (the legs) of the right-angled triangle.The side lengths for this problem are given as follows:
7m and 8m.
Hence the hypotenuse is obtained as follows:
h² = 7² + 8²
[tex]h = \sqrt{7^2 + 8^2}[/tex]
h = 10.6 m.
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What is the product of -2r^5y^2 and 4ry^3
Answer:
-8r^6y^5.
Step-by-step explanation:
In order to find the product, we multiply the coefficients and then add the powers of the same variable.
Coefficient of -2r^5y^2: -2
Coefficient of 4ry^3: 4
Power of r: 5 + 1 = 6
Power of y: 2 + 3 = 5
Product: -2 * 4 = -8
r^6 * y^5 = -8r^6y^5
Therefore, the product of -2r^5y^2 and 4ry^3 is -8r^6y^5.
Subtract
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Your answer should be a polynomial in standard form.
We can see here that the polynomial will be = [tex]-10b^{2} + 12b -16[/tex]
How we arrived at the solution?The given question means:
[tex](b^{2} + 8b - 9) - (11b^{2} -4b+7)[/tex]
Removing brackets, we will have:
[tex]b^{2} + 8b - 9 - 11b^{2} + 4b-7[/tex]
Collecting like terms, we have:
[tex]b^{2} - 11b^{2} + 8b + 4b - 9 -7[/tex]
[tex]-10b^{2} + 12b -16[/tex]
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The complete question is:
Subtract [tex]11b^{2} -4b+7[/tex] from [tex]b^{2} + 8b - 9[/tex]
4
(Graphing Proportional Relationships LC)
The table shows a proportional relationship.
x 12 8 24
y 3 26
Describe what the graph of the proportional relationship would look like.
O Aline passes through the point (0, 0) and continues through the point (3, 12).
O Aline passes through the point (0, 0) and continues through the point (2,8).
A line passes through the point (0, 0) and continues through the point (6, 24).
A line passes through the point (0, 0) and continues through the point (12, 3).
Question 2(Multiple Choice Worth 2 points)
(Graphing Proportional Relationships MG)
The correct answer is option D: A line passes through the point (0, 0) and continues through the point (12, 3)
For the first question:
The table shows a proportional relationship between x and y. To describe what the graph of this proportional relationship would look like, we can examine the given data points.
The data points are (12, 3), (8, 26), and (24, ?). We can observe that as x increases, y also increases. This indicates a positive correlation between the variables. Additionally, we can see that the ratio of y to x remains constant for each data point.
Now, let's analyze the answer options:
A. A line passes through the point (0, 0) and continues through the point (3, 12).
This answer option does not align with the given data because there is no data point with an x-value of 3 and a corresponding y-value of 12.
B. A line passes through the point (0, 0) and continues through the point (2, 8).
This answer option aligns with the given data because (2, 8) is a valid data point from the table, and the line passes through the origin (0, 0).
C. A line passes through the point (0, 0) and continues through the point (6, 24).
This answer option does not align with the given data because there is no data point with an x-value of 6 and a corresponding y-value of 24.
D. A line passes through the point (0, 0) and continues through the point (12, 3).
This answer option aligns with the given data because (12, 3) is a valid data point from the table, and the line passes through the origin (0, 0).
Based on the analysis, the correct answer is option D: A line passes through the point (0, 0) and continues through the point (12, 3). This accurately represents the graph of the proportional relationship given in the table.
For the second question, I'm sorry, but you didn't provide the multiple-choice options or the complete question. Could you please provide the question and options so that I can assist you further?
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Cole and Tisha are about to get married, so they are apartment hunting this week. They compiled a list of 17 apartments to check out, 10 of which are on the ground floor.
If the couple randomly chooses 14 of the apartments to check out in the morning, what is the probability that exactly 9 of the chosen apartments are on the ground floor?
The probability that exactly 9 of the chosen apartments are on the ground floor is approximately 0.3088 or 30.88%.
The number of favourable outcomes and the total number of possible outcomes must be determined in order to determine the likelihood that precisely 9 of the chosen flats are on the ground floor.
The total number of outcomes is the number of ways to select 14 of the 17 available flats. You can compute this using the following combination formula:
C(n, r) = (n!/(n-r)!)
In this instance, the total number of flats is 17 and the number of units that were selected is 14. In other words, C(17, 14) = 17! / (14!(17-14)!) = 17! / (14!* 3!) = (17 * 16 * 15) / (3 * 2 * 1) = 680.
Let's now count the number of successful results. From the 10 flats on the ground floor that are available, we must select 9 and from the 7
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At West High School, 10% of the students participate in sports. A student wants to simulate the act of randomly selecting 20 students and counting the number of students in the sample who participate in sports. The student assigns the digits to the outcomes.
0 = student participates in sports
1–9 = student does not participate in sports
Here is a portion of a random number table.
Beginning at line 1, carry out one trial of this simulation. Use additional lines as needed. How many students in this random sample of 20 students participate in sports?
1
2
3
4
Note that the proportion of trials where 0 students participate in sports is 1 out of 5, or 1/5,which can also be expressed as 0.2 or 20%.
What is the explanation for this ?
To determine the proportion of trials where 0 students participate in sports, we need to carry out the 5 trials and count the number of times we encounter 0 students participating in sports.
Here are the results of the 5 trials -
Trial 1 - 2 students participate in sports
Trial 2 - 1 student participates in sports
Trial 3 - 4 students participate in sports
Trial 4 - 3 students participate in sports
Trial 5 - 0 students participate in sports
Out of the 5 trials,only 1 trial results in 0 students participating in sports.
Therefore, the proportion of trials where 0 students participate in sports is 1 out of 5, or 1/5,which can also be expressed as 0.2 or 20%.
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Please help me important question in image
In the diagram, the correct option is
angle DAB = angle DCB
What is CPCTCCPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." It is a geometric postulate used in proving the congruence of triangles.
According to CPCTC, if two triangles are proven to be congruent (having all corresponding angles and sides congruent), then their corresponding parts, including angles and sides, are also congruent.
The two triangles in the figure are congruent by the SAS congruence theorem and hence applying the CPCTC we have that
angle DAB = angle DCB
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The area of a rectangle is 3x2 - 12x square yards. If the width is 3x yards, what is the length of the rectangle?
Answer:
x - 4 yards.
Step-by-step explanation:
Given:
Area of rectangle = 3x^2 - 12x square yards
Width = 3x yards
To find:
Length of rectangle
Solution:
The area of a rectangle is equal to the product of its length and width.
Area of rectangle = Length * Width
Substituting the given values, we get:
3x^2 - 12x = Length * 3x
Length =( 3x^2 - 12x )3x
Length = x-4
Therefore, the length of the rectangle is x - 4 yards.
Answer: The length of the rectangle is x - 4 yards.
Step-by-step explanation:
We can create an equation to solve this word problem, where the variable L = length.
3x × L = [tex]3x^2 - 12x[/tex]
We need to solve for the variable L, to find the length of the rectangle.
First lets factor the right side of the equation to make it easier to divide with.
3x × L = [tex]3x^2 - 12x[/tex]
We can factor out 3x from the right side of the equation.
3x × L = 3x(x - 4)
Now we need to get the variable L by itself (isolating the variable). In order to do that, we can divide both sides by 3x.
3x × L = 3x(x - 4)
/3x /3x
L = x - 4
The length of the rectangle is x - 4 yards.
