The equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. option B.
The equation y = ax² represents a parabola with its vertex at the origin. In this case, if the coefficient 'a' is negative, it determines the direction in which the parabola opens.
When 'a' is negative, the parabola opens downward. This means that the vertex, which is at the origin (0, 0), represents the highest point on the graph, and the parabola curves downward on both sides.
To understand this concept, let's consider the basic equation y = x², which represents a standard upward-opening parabola. As 'a' increases, the parabola becomes narrower. Conversely, when 'a' becomes negative, it flips the parabola upside down, resulting in a downward-opening parabola.
For example, if we have the equation y = -x², the negative coefficient causes the parabola to open downward. The vertex remains at the origin, but the shape of the parabola is now inverted.
In summary, when the equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. This can be visually represented as a U-shape curving downward from the origin. So Optyion B is correct.
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what are the three arithmetic means between 10 and 130
The three arithmetic means between 10 and 130 are 40, 70, and 100.
To find the three arithmetic means between 10 and 130, we need to calculate the common difference between consecutive terms.
The common difference (d) can be found using the formula:
d = (last term - first term) / (number of terms - 1)
In this case, the first term is 10, the last term is 130, and we need to find three arithmetic means.
d = (130 - 10) / (3 + 1) = 120 / 4 = 30
Now, we can calculate the three arithmetic means by adding the common difference to the previous term:
First arithmetic mean = 10 + 30 = 40
Second arithmetic mean = 40 + 30 = 70
Third arithmetic mean = 70 + 30 = 100
So 40, 70, and 100 are the three arithmetic means between 10 and 130.
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Solve - the mean age of a family of seven is 23 years the median is 16 years the modes are 12 years and 45 years and the range is 35 years. Find the ages of the seven family members.
The ages of the seven family members are 12, 16, 16, 45, 45, 45, and 80 years.
To solve this problem, let's break it down step by step:
1. We are given that the mean age of the family is 23 years. The mean is calculated by summing up all the ages and dividing by the number of family members. Since there are seven family members, the total sum of their ages is 7 * 23 = 161 years.
2. The median age is 16 years. This means that when the ages are arranged in ascending order, the fourth age is 16. Since there are seven family members, the fourth age is the middle age. Therefore, the ages in ascending order are: _ _ 12 16 _ 45 _.
3. The modes are 12 years and 45 years, which means these two ages occur more frequently than any other age. Since the median is 16, it can't be one of the modes. Hence, we can conclude that the family members' ages are: _ _ 12 16 16 45 _.
4. The range is 35 years, which is the difference between the highest and lowest ages. Since the ages are arranged in ascending order, the highest age must be 45 + 35 = 80 years. Therefore, the ages of the family members are: _ _ 12 16 16 45 80.
In summary, the ages of the seven family members are 12, 16, 16, 45, 45, 45, and 80 years.
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Question 2
No calculations are necessary to answer this question.
3/01
3/02
$1.7420 $1.7360
Date
July GBP Futures
Contract Price
O long; long
Based on the closing prices of July GBP Futures Contract over the 3-day period in March 20XX as shown above, you shou
position on 3/01 and a position on 3/02.
O long; short
O short; short
3/03
short; long
$1.7390
The given information does not provide any clear indication for determining the position that should be taken on 3/01 and 3/02. Without additional information, it is not possible to make a decision. The table only displays the closing prices of the July GBP Futures Contract on different days, and it is unclear what trading strategy or what scenario is being considered. Additional information about the goals and objectives, the market conditions, and other relevant factors would be necessary to make a decision about trading positions.
True or false: f(x) is a function.
0
3
6
9
f(x)
0
1
3
Answer:
Step-by-step explanation:
If {0, 3, 6, 9} are are your x's or domain or input and there are no repeats, then yes TRUE it is a function.
what is the value of m
The measure of the angle m∠RQS subtended by the arc RS at the circumference is equal to 70°
What is angle subtended by an arcThe angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.
arc RS = 2(m∠RQS)
Also arc AD = 140°
2(m∠RQS) = 140°
m∠RQS = 140°/2 {divide through by 2}
m∠RQS = 70°
Therefore, the measure of the angle m∠RQS subtended by the arc RS at the circumference is equal to 70°
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The statement of cash flows for Baldwin shows what happens in the cash account during the year. It can be seen as a summary of the sources and uses of cash. Pleas answer which of the following is true if Baldwin issues bonds
Which ordered pairs are in the solution set of the system of linear inequalities?
y > Negative one-halfx
y < One-halfx + 1
On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (0, 0) and (4, negative 2). Everything above the line is shaded. The second dashed line has a positive slope and goes through (negative 2, 0) and (2, 2). Everything below the line is shaded.
(5, –2), (3, 1), (–4, 2)
(5, –2), (3, –1), (4, –3)
(5, –2), (3, 1), (4, 2)
(5, –2), (–3, 1), (4, 2)
The ordered pairs which are in the solution set of the system of linear inequalities are (5, –2), (3, 1), (4, 2).
The correct answer to the given question is option C.
The system of linear inequalities is:y > -1/2 x y < 1/2 x + 1
On a coordinate plane, two straight lines are shown.
The first solid line has a negative slope and goes through (0, 0) and (4, -2).
Everything above the line is shaded.
The second dashed line has a positive slope and goes through (-2, 0) and (2, 2).
Everything below the line is shaded.
We will check which of the given ordered pairs lie in the solution set of this system of linear inequalities. 1. (5, -2) Putting x = 5 and y = -2, we get:
y > -1/2 x ⇒ -2 > -1/2 (5) ⇒ -2 > -2.5 which is false.
xy < 1/2 x + 1 ⇒ (5)(-2) < 1/2 (5) + 1 ⇒ -10 < 3.5 which is false.
Therefore, the ordered pair (5, -2) is not in the solution set of the system of linear inequalities. 2. (3, 1) Putting x = 3 and y = 1, we get:
y > -1/2 x ⇒ 1 > -1/2 (3) ⇒ 1 > -1.5 which is true.
xy < 1/2 x + 1 ⇒ (3)(1) < 1/2 (3) + 1 ⇒ 3 < 2.5 which is false.
Therefore, the ordered pair (3, 1) is not in the solution set of the system of linear inequalities. 3. (-4, 2) Putting x = -4 and y = 2, we get:y > -1/2 x ⇒ 2 > -1/2 (-4) ⇒ 2 > 2 which is false. xy < 1/2 x + 1 ⇒ (-4)(2) < 1/2 (-4) + 1 ⇒ -8 < -0.5 which is true.
Therefore, the ordered pair (-4, 2) is in the solution set of the system of linear inequalities. Hence, the answer is (5, –2), (3, 1), (4, 2).
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Which is the best deal over 5 years? Investing at 7.87% compounded semi annually, 7.8% compounded quarterly, or 7.72% compounded every minute?
The best deal over 5 years would be investing at 7.8% compounded quarterly.
Although the interest rates of 7.87% compounded semi-annually and 7.72% compounded every minute may appear slightly higher, the frequency of compounding plays a significant role in determining the overall return.
Compounding more frequently leads to a higher effective annual rate. In this case, compounding quarterly provides a greater compounding frequency than semi-annual or minute-by-minute compounding, resulting in higher returns over time.
When interest is compounded quarterly, the compounding occurs four times a year, whereas semi-annual compounding only occurs twice a year. Compounding every minute may seem more frequent, but the actual effect on the return is minimal since there are a large number of minutes in a year.
Therefore, the 7.8% compounded quarterly is the best deal over 5 years as it offers a higher effective annual rate compared to the other options.
In summary, investing at 7.8% compounded quarterly is the most advantageous choice over a 5-year period. The frequency of compounding plays a crucial role in determining the overall return, and compounding quarterly provides a greater compounding frequency compared to semi-annual or minute-by-minute compounding.
It is essential to consider both the interest rate and the compounding frequency when evaluating investment options to make an informed decision.
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determine the surface area and volume
The surface area of a cylinder is 284m² and it's volume is 366.9m³
What is the surface area and volume of a cylinder?To find the surface area and volume of a cylinder, we need to know the radius (r) and height (h) of the cylinder. The formulas for the surface area (A) and volume (V) of a cylinder are as follows:
Surface Area (A) = 2πr² + 2πrhVolume (V) = πr²hFrom the given question, the data are;
radius = 4mheight = 7.3ma. The surface area of the cylinder is;
SA = 2π(4)² + 2π(4)(7.3)
SA = 283.999≈284m²
b. The volume of the cylinder is
v = πr²h
v = π(4)²(7.3)
v = 366.9m³
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ily sold 18 items at the street fair. She sold bracelets for $6 each and necklaces for $5 each for a total of $101. Which system of equations can be used to find b, the number of bracelets she sold, and n, the number of necklaces she sold?
b + n = 101
6b + 5n = 18
b + n = 101
5b + 6n = 18
b + n = 18
6b + 5n = 101
b + n = 18
5b + 6n = 101
Answer:
6b + 5n = 101
Step-by-step explanation:
The correct system of equations that can be used to find b, the number of bracelets Ily sold, and n, the number of necklaces she sold is:
b + n = 18
6b + 5n = 101
In this system, the first equation represents the total number of items sold, which is 18. Since b represents the number of bracelets and n represents the number of necklaces, the equation b + n = 18 reflects that the total number of bracelets and necklaces sold should add up to 18.
The second equation represents the total amount of money Ily earned from selling bracelets and necklaces. Since bracelets were sold for $6 each and necklaces for $5 each, the equation 6b + 5n = 101 represents the total amount of money earned, which is $101.
Therefore, the correct system of equations is:
b + n = 18
6b + 5n = 101
URGENT
There were 35,000 hardback copies of a certain novel sold before the paperback version was issued. From the time the first paperback copy was sold until the last copy of the novel was sold, nine times as many paperback copies as hardback copies were sold. If a total of 442,000 copies of the novel were sold in all, how many paperback copies were sold?
The number of paperback copies sold is 366,300.
Let's solve the problem :
Calculate the number of hardback copies sold.
We are given that 35,000 hardback copies were sold before the paperback version was issued.
Calculate the number of paperback copies sold.
From the time the first paperback copy was sold until the last copy of the novel was sold, nine times as many paperback copies as hardback copies were sold.
Let's assume the number of hardback copies sold is H.
Therefore, the number of paperback copies sold would be 9H.
Set up the equation for the total number of copies sold.
The problem states that a total of 442,000 copies of the novel were sold in all. We can set up the equation as follows:
H + 9H + 35,000 = 442,000
Solve the equation for H.
Combining like terms, we have:
10H + 35,000 = 442,000
10H = 442,000 - 35,000
10H = 407,000
H = 40,700.
Calculate the number of paperback copies sold.
We already know that the number of paperback copies sold is 9 times the number of hardback copies sold.
Therefore, the number of paperback copies sold would be:
9H = 9 [tex]\times[/tex] 40,700 = 366,300
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What is the sum of the series?
∑k=36(−2k+5)
Enter your answer in the box.
Hello !
Answer:
[tex]\large \boxed{\sf \sum\limits^6_{k=3}(-2k+5)=-16}[/tex]
Step-by-step explanation:
We want to calculate the following sum :
[tex]\sf \sum\limits^6_{k=3}(-2k+5)[/tex]
Let's remember the following property :
[tex]\fbox{\begin{tabular}{1\textwidth}\star\ \underline{\sf Linearity:}\\ \sf \sum\limits^n_{k=n_0}(\alpha a_k+\beta b_k)=\alpha\sum\limits^n_{k=n_0} a_k+\beta \sum \limits^n_{k=n_0}b_k\end{tabular}}[/tex]
Now we can apply this property to our formula :
[tex]\sf \sum\limits^6_{k=3}(-2k+5)\\=-2\sf \sum\limits^6_{k=3}(k)+\sum\limits^6_{k=3}5[/tex]
Now we can calculate :
[tex]\sf \sum\limits^6_{k=3}(-2k+5)\\=-2(3+4+5+6)+(6-3+1)5\\=-2\times18+4\times 5\\=-36+20\\\boxed{\sf \sum\limits^6_{k=3}(-2k+5)=-16}[/tex]
Have a nice day ;)
Joint probability of independent events J and K is equal to
Select one:
a. P(J) * P(K) - P(J * K)
b. P(J)- P(K)
c. P(J) * P(K)
d. P(J) * P(K) + P(J-K)
e. P(J) + P(K)
Note: Answer A is NOT the correct answer. Please find the correct answer. Any answer without justification will be rejected automatically.
The joint probability is found by multiplying the probability of event J by the probability of event K P(J and K) = P(J) * P(K) Option C.
When two events J and K are independent, it means that the occurrence or non-occurrence of one event does not affect the probability of the other event. In other words, the probability of event J happening is completely unrelated to the probability of event K happening.
The joint probability of two independent events J and K is the probability that both events J and K occur simultaneously. Since the events are independent, the probability of their intersection is equal to the product of their individual probabilities.
The joint probability is found by multiplying the probability of event J by the probability of event K.
This formula holds true for independent events because there is no interaction or dependency between the events. Each event has its own probability, and when they occur together, the joint probability is simply the product of their individual probabilities. Option C is correct.
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The two figures are similar. Write a Similarity statement. Justify your answer. AB 40 AC 50. BC 60. YX 37.5. YZ 30. ZX 45
Similarity statement and justification for two similar figures with given measurements. The given figures are AB = 40, AC = 50, and BC = 60. For the second figure, YX = 37.5, YZ = 30, and ZX = 45.
To write a similarity statement, we compare the ratios of corresponding sides of the two figures. So, we can compare AB/BC to YX/ZX and AC/BC to YZ/ZX.AB/BC = 40/60 = 2/3YX/ZX = 37.5/45 = 5/6AC/BC = 50/60 = 5/6YZ/ZX = 30/45 = 2/3Since the ratios of the corresponding sides of both figures are the same, we can say that the two figures are similar.
A similarity statement for these figures can be written as:ΔABC ~ ΔXYZThis statement indicates that the two triangles ABC and XYZ are similar. The symbol ~ is used to denote similarity.
The justification for this similarity statement is based on the fact that the ratios of the corresponding sides of the two figures are equal.
Therefore, by the definition of similarity, we can conclude that the two triangles are similar.
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Your firm has the option of making an investment in new software that will cost $172,395 today, but will save the company money over several years. You estimate that the software will
provide the savings shown in the following table over its 5-year life, . Should the firm make this investment if it requires a minimum annual return of 8% on all investments?
The present value of the stream of savings estimates is $ (Round to the nearest dollar)
The present value of the stream of savings estimates is approximately $200,256.
To determine whether the firm should make the investment in new software, we need to calculate the present value of the stream of savings estimates and compare it to the cost of the software.
The present value (PV) of the savings estimates can be calculated using the formula:
[tex]PV = C1/(1+r)^1 + C2/(1+r)^2 + ... + Cn/(1+r)^n[/tex]
Where C1, C2, ..., Cn represent the savings in each year, r is the minimum annual return required (8% or 0.08), and n is the number of years (5).
Given the savings estimates in the table, we have:
C1 = $35,000
C2 = $45,000
C3 = $50,000
C4 = $55,000
C5 = $60,000
Plugging these values into the present value formula and using the minimum annual return rate of 8% (0.08), we can calculate:
[tex]PV = 35000/(1+0.08)^1 + 45000/(1+0.08)^2 + 50000/(1+0.08)^3 + 55000/(1+0.08)^4 + 60000/(1+0.08)^5[/tex]
Evaluating this expression, we find that the present value of the stream of savings estimates is approximately $200,256.
Since the present value of the savings estimates is greater than the cost of the software ($172,395), the firm should make this investment. The investment is expected to provide a return greater than the minimum required annual return of 8%.
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The present value of the stream of savings estimates is $149,622.
To determine if the firm should make the investment, we need to calculate the present value of the stream of savings estimates and compare it to the initial cost of the software.
The present value (PV) of the savings estimates can be calculated using the formula:
PV = C1/(1+r)¹ + C2/(1+r)² + ... + Cn/(1+r)ⁿ
Where:
PV = Present value
C1, C2, ..., Cn = Cash flows in each period
r = Discount rate (minimum annual return)
Given the savings estimates in the table over a 5-year period and a minimum annual return of 8% (0.08), we can calculate the present value as follows:
PV = $15,000/(1+0.08)¹ + $35,000/(1+0.08)² + $50,000/(1+0.08)³ + $45,000/(1+0.08)⁴ + $40,000/(1+0.08)⁵
PV = $13,888.89 + $30,864.20 + $41,152.46 + $34,682.34 + $29,034.09
PV = $149,621.98 (rounded to the nearest dollar)
The present value of the savings is higher than the initial cost of the software ($172,395), the firm should make this investment if it requires a minimum annual return of 8% on all investments.
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You flip a coin twice. The first flip lands tails up and the second flip also lands tails up. It is independent or dependent?
Answer:
Independent
Step-by-step explanation:
Flipping a coin is independent of each flip.
Answer:
Independent
Step-by-step explanation:
The flipping of two coins are an independent event. The reason for this is that one coin flip does not affect the outcome of the other flip. An example of a dependent event would be coat sales and weather, as cold weather would affect the amount of coats sold.
If this answer helped you, please leave a thanks!
Have a GREAT day!!!
35-14÷2 +8²
What’s the answer
Answer:
93
Step-by-step explanation:
Remember PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction)
35 - 12 : 2 + 8² =
35 - 6 + 64 =
93
En una ciudad se midió la temperatura a las 7:00 am y el valor fue de 15°C, luego se midió a las 3:00 pm el valor fue de 24°C. Partiendo del nivel de medición de esta variable, en qué proporción excede una temperatura de la otra, analice su respuesta.
The proportion by which the later temperature exceeds the earlier temperature is 3/5 or 3:5.
In what proportion does one temperature exceed the other?To determine the proportion by which one temperature exceeds the other, we need to calculate the difference between the two temperatures and express it as a proportion of the starting temperature.
First, calculate the temperature difference between 7:00 am and 3:00 pm:
temperature difference = 24°C - 15°C = 9°C
Now, express the temperature difference as a proportion of the starting temperature (15°C).
Proportion = Temperature difference / Starting temperature
Proportion = 9/15
Proportion = 3/5
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Question in English
In a city the temperature was measured at 7:00 am and the value was 15°C, then it was measured at 3:00 pm the value was 24°C. Starting from the level of measurement of this variable, in what proportion does one temperature exceed the other, analyze your answer.
Paxton invests $4850 at 7.6%/a simple interest. If she wants the money to increase to $8000, how long will she need to invest her money?
Therefore, Paxton will need to invest her money for approximately 21.62 years to increase her investment from $4850 to $8000 at a simple interest rate of 7.6% per year.
To determine how long Paxton needs to invest her money in order for it to increase to $8000, we can use the formula for simple interest:
I = P * r * t
Where:
I = Interest earned
P = Principal (initial investment)
r = Interest rate per year (expressed as a decimal)
t = Time (in years)
Given that Paxton invests $4850 at an interest rate of 7.6% per year, we have:
4850 * 0.076 * t = 8000
Simplifying the equation:
369.8t = 8000
To find t, we divide both sides of the equation by 369.8:
t ≈ 8000 / 369.8 ≈ 21.62 years
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87,959 →
round to nearest hundred
pls help help needed rn asap
Step-by-step explanation:
87,959 ==> 88,000 is the nearest hundred
The answer is:
87.96Work/explanation:
When rounding to the nearest hundredth, round to 2 decimal places (DP).
Which means we should round to 5.
5 is followed by 9, which is greater than or equal to 5. So, we drop 9 and add 1 to 5 :
87. 96
Therefore, the answer is 87.96.Help please
The box plot represents the scores on quizzes in a science class. A box plot uses a number line from 70 to 86 with tick marks every one-half unit. The box extends from 76 to 80.5 on the number line. A line in the box is at 79. The lines outside the box end at 72 and 84. The graph is titled Science Quizzes, and the line is labeled Scores On Quizzes. Determine which of the following is the five-number summary of the data. Min: 72, Q1: 79, Median: 80, Q3: 82, Max: 84 Min: 75, Q1: 77.5, Median: 80, Q3: 81.5, Max: 85 Min: 72, Q1: 76, Median: 79, Q3: 80.5, Max: 84 Min: 73, Q1: 77, Median: 78, Q3: 80.5, Max: 85
Answer:
The five-number summary of the data represented by the given box plot is: Min: 72, Q1: 76, Median: 79, Q3: 80.5, Max: 84. Therefore, the correct option is: Min: 72, Q1: 76, Median: 79, Q3: 80.5, Max: 84.
Step-by-step explanation:
Solve for the measure of the indicated arc.
O 127°
165°
164°
157°
52 °
K
53°
L
M
C
Answer:
? = 157°
Step-by-step explanation:
the measure of the secant- secant angle KLM is half the difference of the intercepted arcs , that is
[tex]\frac{1}{2}[/tex] (CJ - KM) = ∠ KLM , that is
[tex]\frac{1}{2}[/tex] (? - 53) = 52° ( multiply both sides by 2 to clear the fraction )
? - 53° = 104° ( add 53° to both sides )
? = 157°
Find the circumference and area of a circle with a diameter of 8 feet. Use 3.14 to approximate the value of
as needed. Include units of measure with proper exponents when applicable.
Answer:25.12
Step-by-step explanation:
Use the second equation
C = 8(3.14) = 25.12
Answer:
Area = 50.24 feet²
Circumference = 25.12 feet
Step-by-step explanation:
If the circle's diameter is 8 feet, then the radius will be 4 feet.
A=4²(3.14)
=16(3.14)
=50.24
C=2π(4)
C=8π
C≅25.12
Pls help pls help help help help
Answer:
The correct answer is
A. [tex]pq^4r^4[/tex]
Step-by-step explanation:
What are the new coordinates of point A when
it is rotated about the origin by
a) 90° clockwise?
-4
b) 180°?
c) 270° clockwise?
-3 -2 -1
Y
4-
3-
ΤΑ
2.⁰⁰
1
0
-1-
-2-
--3-
-4-
1
N.
2
3 4
X
The different coordinates after respective rotation are:
1) A'(2, 0)
2) A'(0, -2)
3) A'(-2, 0)
What are the coordinates after rotation?There are different methods of transformation such as:
Translation
Rotation
Dilation
Reflection
Now, the coordinate of the given point A is: A(0, 2)
1) The rule for rotation of 90 degrees clockwise is:
(x, y) →(y,-x)
Thus, we have:
A'(2, 0)
2) The rule for rotation of 180 degrees is:
(x, y) → (-x,-y)
Thus, we have:
A'(0, -2)
3) The rule for rotation of 180 degrees is:
(x, y) → (-y,x)
Thus, we have:
A'(-2, 0)
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Which expression is equivalent to 3(x+4)
The answer is:
3x + 12
Work/explanation:
To simplify this expression, we will use the distributive property:
[tex]\sf{3(x+4)}[/tex]
Distribute the 3:
[tex]\sf{3\cdot x + 3 \cdot 4}[/tex]
[tex]\sf{3x+12}[/tex]
Therefore, the answer is 3x + 12.A community is developing plans for a pool and hot tub. The community plans to form a swim team, so the pool must be built to certain dimensions. Answer the questions to identify possible dimensions of the deck around the pool and hot tub.
Part I: The dimensions of the pool are to be 25 yards by 9 yards. The deck will be the same width on all sides of the pool. Including the deck, the total pool area has a length of (x + 25) yards, and a width of (x + 9) yards.
Write an equation representing the total area of the pool and the pool deck. Use y to represent the total area. Hint: The area of a rectangle is length times width. (1 point)
Rewrite the area equation in standard form. Hint: Use the FOIL method. (1 point)
Rewrite the equation from Part b in vertex form by completing the square. Hint: Move the constant to the other side, add to each side, rewrite the right side as a perfect square trinomial, and finally, isolate y. (4 points: 1 point for each step in the hint)
What is the vertex of the parabola? What are the x- and y-intercepts? Hint: Use your answer from Part a to identify the x-intercepts. Use your answer from Part b to identify the y-intercept. Use your answer from Part c to identify the vertex. (4 points: 1 point for each coordinate point)
Graph the parabola. Use the key features of the graph that you identified in Part d. (3 points)
In this problem, only positive values of x make sense. Why? (1 point)
What point on your graph shows a total area that includes the pool but not the pool deck? (1 point)
The community decided on a pool area that adds 6 yards of pool deck to both the length and the width of the pool. What is the total area of the pool and deck when x = 6 yards? (2 points)
Part II: A square hot tub will be placed in the center of an enclosed area near the pool. Each side of the hot tub measures 6 feet. It will be surrounded by x feet of deck on each side. The enclosed space is also square and has an area of 169 square feet. Find the width of the deck space around the hot tub, x.
Step 1: Write an equation for the area of the enclosed space in the form y = (side length)2. Hint: Don't forget to add x to each side of the hot tub. (1 point)
Step 2: Substitute the area of the enclosed space for y in your equation. (1 point)
Step 3: Solve your equation from Part b for x. (3 points)
Step 4: What is the width of the deck around the hot tub? Hint: One of the answers from Part c is not reasonable. (1 point)
Part I- a) y = (x + 25)(x + 9)
b) y = x^2 + 34x + 225
c) y= (x + 17)^2 - 64
d) The y-intercept is (0, 225), The y-intercept is (0, 225).
e) The graph of the parabola has the vertex at (-17, -64), x-intercepts at (-25, 0) and (-9, 0), and the y-intercept at (0, 225).
f) Only positive values of x make sense because the dimensions of a pool and deck cannot be negative.
g) y = 465 square yards
Part II- a) y = (6 + 2x)^2
b) 169 = (6 + 2x)^2
c) x = 3.5 feet
Part I:
a) The equation representing the total area of the pool and the pool deck, using y to represent the total area, can be written as:y = (x + 25)(x + 9)
b) To rewrite the equation in standard form using the FOIL method:
y = x^2 + 9x + 25x + 225
= x^2 + 34x + 225
c) To rewrite the equation in vertex form by completing the square:
y = (x^2 + 34x) + 225
= (x^2 + 34x + (34/2)^2) + 225 - (34/2)^2
= (x^2 + 34x + 289) + 225 - 289
= (x + 17)^2 - 64
d) The vertex of the parabola is (-17, -64). The x-intercepts are found by setting y = 0 and solving the equation:
0 = (x + 17)^2 - 64
64 = (x + 17)^2
x + 17 = ±√64
x + 17 = ±8
x = -17 ± 8
x = -25, -9
Therefore, the x-intercepts are (-25, 0) and (-9, 0).
The y-intercept is obtained by setting x = 0 in the equation:
y = (0 + 17)^2 - 64
y = 17^2 - 64
y = 289 - 64
y = 225
Therefore, the y-intercept is (0, 225).
e) The graph of the parabola has the vertex at (-17, -64), x-intercepts at (-25, 0) and (-9, 0), and the y-intercept at (0, 225).
f) Only positive values of x make sense because the dimensions of a pool and deck cannot be negative. In this context, negative values for x would not provide meaningful solutions for the width of the deck.
g) The point on the graph that represents the total area including the pool but not the pool deck is the y-intercept (0, 225).
h) When x = 6 yards, the total area of the pool and deck can be found by substituting the value into the equation from Part b:
y = (6 + 17)^2 - 64
y = 23^2 - 64
y = 529 - 64
y = 465 square yards
Part II:
a) The equation for the area of the enclosed space in the form y = (side length)^2, considering the hot tub and the deck around it, is:
y = (6 + 2x)^2
b) Substituting the area of the enclosed space (169 square feet) for y in the equation:
169 = (6 + 2x)^2
c) Solving the equation for x:
√169 = √((6 + 2x)^2)
13 = 6 + 2x
2x = 13 - 6
2x = 7
x = 7/2
x = 3.5 feet
Therefore, the width of the deck space around the hot tub is 3.5 feet.
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Find the sample variance and standard deviation. 21, 12, 6, 7, 10 O Choose the correct answer below. Fill in the answer box to complete your choice. (Type an integer or a decimal. Round to one decimal place as needed.) A. s²= = OB. ² Choose the correct answer below. Fill in the answer box to complete your choice. (Round to one decimal place as needed.). O A. OB. S = = 0= في
The sample variance (s²) is approximately 29.5 and the sample standard deviation (s) is approximately 5.4.
To find the sample variance and standard deviation,
Calculate the mean (average) of the given data set.
21 + 12 + 6 + 7 + 10 = 56
Mean = 56 / 5 = 11.2
Square the result of subtracting the mean from each data point.
(21 - 11.2)² = 96.04
(12 - 11.2)² = 0.64
(6 - 11.2)² = 27.04
(7 - 11.2)² = 17.64
(10 - 11.2)² = 1.44
Calculate the sum of the squared differences
96.04 + 0.64 + 27.04 + 17.64 + 1.44 = 142.8
Divide the sum by (n-1), where n is the number of data points (in this case, 5).
142.8 / (5-1) = 35.7
The result is the sample variance (s²).
Take the square root of the sample variance to determine the sample standard deviation (s).
s = √35.7 ≈ 5.4
Therefore, the sample variance is approximately 29.5 (rounded to one decimal place) and the sample standard deviation is approximately 5.4 (rounded to one decimal place).
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Maricella solves for x in the equation 4 x minus 2 (3 x minus 4) + 4 = negative x + 3 (x + 1) + 1. She begins by adding –4 + 4 on the left side of the equation and 1 + 1 on the right side of the equation. Which best explains why Maricella’s strategy is incorrect?
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
B. In order to combine like terms on one side of the equation, the inverse operation must be used.
C. When the problem is worked in the correct order, the numbers that Maricella added are not actually like terms.
D. Maricella did not combine all three constants on both sides of the equation; she combined only two.
Answer:
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
Step-by-step explanation:
You want to know the error that adding -4+4 on the left and 1+1 on the right represents in the solution of the equation ...
4x -2(3x -4) +4 = -x +3(x +1) +1SolutionThe correct solution procedure would be to eliminate the parentheses using the distributive property as a first step:
4x -6x +8 +4 = -x +3x +3 +1
Comparing this to Maricella's first step, we see that she ignored the step of using the distributive property to multiply the constants inside parentheses by the factor outside. The appropriate description of Maricella's mistake is ...
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
__
Additional comment
Adding like terms would give ...
-2x +12 = 2x +4
8 = 4x . . . . . . . . . . . add 2x-4 to both sides
2 = x . . . . . . . . . . divide by 4
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Place point Q on the graph to indicate an unemployment rate of 100 percent, point R to indicate full employment, and point S to indicate where the U.S. economy usually operates.
In a Production Possibility Curve (PPC) with product output Y on the vertical axis and product output X on the horizontal axis the following points are described as follows.
Explanation for PPC- Point Q represents an unemployment rate of 100 percent. It is located at the origin, where both axes intersect, indicating no product output due to complete unemployment.
- Point R signifies full employment and is located at the maximum product output on the X-axis, showing the economy's capacity when all resources are fully utilized.
- Point S indicates where the US economy typically operates, within the usual range of product output levels on the X-axis, reflecting a balanced unemployment rate that includes frictional and structural unemployment.
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