The distance from the given city to the South Pole is 125 degrees of latitude.
To determine the distance from the given city to various locations, we need to consider the latitude and longitude coordinates.
Distance to the North Pole:
The North Pole is located at 90°N latitude. Since the given city is at 35°S latitude, we need to calculate the difference in latitude between the city and the North Pole.
Distance to the North Pole = 90° - 35° = 55°
The distance from the given city to the North Pole is 55 degrees of latitude.
Distance to the Equator:
The Equator is located at 0° latitude. To determine the distance from the city to the Equator, we need to calculate the absolute value of the latitude of the city.
Distance to the Equator = |35°| = 35°
The distance from the given city to the Equator is 35 degrees of latitude.
Distance to the South Pole:
The South Pole is located at 90°S latitude. Since the given city is at 35°S latitude, we need to calculate the difference in latitude between the city and the South Pole.
Distance to the South Pole = 35° - (-90°) = 35° + 90° = 125°
The distance from the given city to the South Pole is 125 degrees of latitude.
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.
x < 5
–6x – 5 < 10 – x
–6x + 15 < 10 – 5x
A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
Answer:
Third option
Step-by-step explanation:
-3(2x - 5) < 5(2 - x)
-6x + 15 < 10 - 5x <-- Third option
15 < 10 + x
5 < x
x > 5
There only appears to be one option. The solution to the inequality is x>5, not x<5.
Choose the justification for each step of the given equation. -6=-2/3(x+12)+1/3x
The steps used in the solution of the equation -6 = -2/3(x + 12) + 1/3x are based on the principles of the Distributive Property, combining like terms, addition property of equality, symmetric property, subtraction property of equality, and multiplication property of equality.
Let's analyze the steps of the solution for the given equation -6 = -2/3(x + 12) + 1/3x:
Step 1: Distributive Property
The equation begins with the Distributive Property, which states that you can distribute a factor to each term inside parentheses. In this case, we distribute -2/3 to (x + 12), resulting in -2/3 * x and -2/3 * 12.
Step 2: Simplification
We simplify the expression -2/3 * 12 to -8, as multiplying -2/3 by 12 gives us -24, and simplifying the fraction -24/3 yields -8.
Step 3: Combine Like Terms
We combine the like terms -2/3x and -8. The equation becomes -2/3x - 8 + 1/3x.
Step 4: Combine Like Terms
We combine the like terms -2/3x and 1/3x by adding their coefficients. The sum of -2/3x and 1/3x is -1/3x.
Step 5: Addition Property of Equality
We add -1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 - 1/3x = -1/3x.
Step 6: Symmetric Property
Since the equation has a form of -1/3x = -6 - 1/3x, we can rearrange the terms using the Symmetric Property.
Step 7: Addition Property of Equality
We add 1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 = 0.
Step 8: Subtraction Property of Equality
We subtract 0 from both sides of the equation to simplify it further. The equation remains -6 = 0.
Step 9: Multiplication Property of Equality
We multiply both sides of the equation by any non-zero number to check for consistency. In this case, there is no need for multiplication as the equation is already in its simplified form.
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In order to compute a sample mean by hand, first the data values must be added up. Then the sum is divided by the
sample size.
x = xx
Given the data set below, compute the summation, identify the sample size, and calculate the sample mean.
19
10
15
17
16
a.) Ex =
b.) n =
c.) x =
a) The summation (Ex) of the given data set, we add up all the values
Ex = 77
b) n = 5
c) x = 15.4
a) To compute the summation (Ex) of the given data set, we add up all the values:
19 + 10 + 15 + 17 + 16 = 77
b) The sample size (n) is the total number of data points in the set. In this case, there are 5 data points, so:
n = 5
c) To calculate the sample mean (x), we divide the summation (Ex) by the sample size (n):
x = Ex / n
x = 77 / 5
x = 15.4
Therefore, the answers are:
a) Ex = 77
b) n = 5
c) x = 15.4
The summation is 77, the sample size is 5, and the sample mean is 15.4.
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2.3.5 Quiz: Cross-Sections of Geometric Solids
OA. Triangle
OB. Circle
OC. Trapezoid
OD. Rectangle
The cross section of the geometric solid is (d) rectangle
How to determine the cross section of the geometric solidFrom the question, we have the following parameters that can be used in our computation:
The geometric solid
Also, we can see that
The geometric solid is a cylinder
And the cylinder is divided vertically
The resulting shape from the division is a rectangle
This means that the cross section of the geometric solid is (d) rectangle
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The shape of the cross-section for the geometric solid given in the diagram is a rectangle.
The cross section of the geometric solid represents the shape which extends beyond the actual geometric solid which is a cylinder.
A rectangle has opposite side being equal. This means that the width and and length are of different length.
Therefore, the shape of the cross-section is a rectangle.
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4x-5 2x+7 Find the value of x
answers should be from
27
37
47
57
If the mean of a negatively skewed distribution is 122, which of these values could be the median of the distribution
118 be the median of a positively skewed distribution with a mean of 122. Option D.
To determine which of the given values could be the median of a positively skewed distribution with a mean of 122, we need to consider the relationship between the mean, median, and skewness of a distribution.
In a positively skewed distribution, the tail of the distribution is stretched towards higher values, meaning that there are more extreme values on the right side. Consequently, the median, which represents the value that divides the distribution into two equal halves, will typically be less than the mean in a positively skewed distribution.
Let's examine the given values in relation to the mean:
A. 122: This value could be the median if the distribution is perfectly symmetrical, but since the distribution is positively skewed, the median is expected to be less than the mean. Thus, 122 is less likely to be the median.
B. 126: This value is higher than the mean, and since the distribution is positively skewed, it is unlikely to be the median. The median is expected to be lower than the mean.
C. 130: Similar to option B, this value is higher than the mean and is unlikely to be the median. The median is expected to be lower than the mean.
D. 118: This value is lower than the mean, which is consistent with a positively skewed distribution. In such a distribution, the median is expected to be less than the mean, so 118 is a plausible value for the median.
In summary, among the given options, (118) is the most likely value to be the median of a positively skewed distribution with a mean of 122. So Option D is correct.
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Note the complete question is
If the mean of a positively skewed distribution is 122, which of these values could be the median of the distribution?
A. 122
B. 126
C. 130
D. 118
Evaluate the given expression for x=5
x² + 3x - 2
(5)² + 3 × 5- 2
25 + 15 - 2
40 - 2
38...
Please answer ASAP I will brainlist
The resulting matrix after the rows are interchanged is given as follows:
[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]
How to obtain the resulting matrix?The matrix for this problem is defined as follows:
[tex]\left[\begin{array}{cccc}8&-2&1&7\\2&9&4&5\\1&4&-4&9\end{array}\right][/tex]
The row 1 is given as follows:
[8 -2 1 7].
The row 2 is given as follows:
[2 9 4 5].
Interchanging the rows means that the elements of the row 1 in the matrix is exchanged with the elements of row 2, hence the resulting matrix is given as follows:
[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]
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(a) Un ángulo mide 47°. ¿Cuál es la medida de su complemento?
(b) Un ángulo mide 149°. ¿Cuál es la medida de su suplemento?
El supplemento y el complemento de cada ángulo son, respectivamente:
Caso A: m ∠ A' = 43°
Caso B: m ∠ A' = 31°
¿Cómo determinar el complemento y el suplemento de un ángulo?De acuerdo con la geometría, la suma de un ángulo y su complemento es igual a 90° and la suma de un ángulo y su suplemento es igual a 180°. Matemáticamente hablando, cada situación es descrita por las siguientes formulas:
Ángulo y su complemento
m ∠ A + m ∠ A' = 90°
Ángulo y su suplemento
m ∠ A + m ∠ A' = 90°
Donde:
m ∠ A - Ángulom ∠ A' - Complemento / Suplemento.Ahora procedemos a determinar cada ángulo faltante:
Caso A: Complemento
47° + m ∠ A' = 90°
m ∠ A' = 43°
Caso B: Suplemento
149° + m ∠ A' = 180°
m ∠ A' = 31°
ObservaciónEl enunciado se encuentra escrito en español y la respuesta está escrita en el mismo idioma.
The statement is written in Spanish and its answer is written in the same language.
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what is the value of m
Answer:
114°--------------------------
Angle G is central angle and angle E is inscribed angle, both with same endpoints.
According to the inscribed angle theorem the inscribed angle is half of the central angle.
Hence the central angle G measures:
m∠G = 2(m∠E)m∠G = 2(57°)m∠G = 114°0.5(x-4)=4x-3(x-1)+37/5
Answer:
Multiply to remove the fraction, then set it equal to 0 and solve.
Exact Form: x = −124/5
Decimal Form: x = −24.8
Mixed Number Form: x = −24 4/5
Please give the brainliest, really appreciated. Thank you
I WILL GIVE BRAINLIEST
Step-by-step explanation:
In a randomized block design blocked by gender, treatments should be assigned randomly within each gender block. The correct assignment maintains a distribution of one treatment for each gender. Looking at the given options, only one meets this criterion:
OA: (1f, 2f), B: (1m, 2m). C: (3f, 3m). D: (4f, 4m)
Each treatment group A, B, C, and D contains one male and one female, making the distribution of treatments blocked by gender.
For a recent year, 52.7 million people participated in recreational boating. Sixteen years later, that number increased to 57.3
million. Determine the percent increase. Round to one decimal place.
The percent increase was approximately
%.
The percent increase in recreational boating participation over the sixteen-year period is approximately 8.72%. This means that the number of participants increased by around 8.72% from 52.7 million to 57.3 million.
To determine the percent increase in recreational boating participation over the sixteen-year period, we can use the following formula:
Percent Increase = ((New Value - Old Value) / Old Value) * 100
Using the given information, we have an old value of 52.7 million and a new value of 57.3 million.
Percent Increase = ((57.3 million - 52.7 million) / 52.7 million) * 100
= (4.6 million / 52.7 million) * 100
= 0.0872 * 100
= 8.72%
This increase indicates a positive trend in recreational boating, reflecting a growing interest in this activity over time. Factors such as improved accessibility, marketing efforts, and increasing disposable income may have contributed to this upward trend.
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please help quick
Which of the following are solutions to the quadratic equation? Check all that
apply.
The solutions to the quadratic equation 2x² + 6x - 10 = x² + 6 are -8 and 2.
What are the solutions to the quadratic equation?Given the quadratic equation in the question:
2x² + 6x - 10 = x² + 6
First, reorder the quadratic equation in standard form:
2x² + 6x - 10 = x² + 6
2x² - x² + 6x - 10 - 6 = x² - x² + 6 - 6
2x² - x² + 6x - 10 - 6 = 0
x² + 6x - 10 - 6 = 0
x² + 6x - 16 = 0
Next, factor the equation using the AC method:
( x - 2 )( x + 8 ) = 0
Equate each factor to 0 and solve for x:
( x - 2 ) = 0
x - 2 = 0
x = 2
( x + 8 ) = 0
x + 8 = 0
x = -8
Therefore, the solutions are -8 and 2.
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a spinner with 10 equally sized slices 4 yellow, 4 red, 2 blue. probability that the dial stops on yellow?
The probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).
Therefore, there is a 40% chance that the spinner will stop on yellow.
To find the probability that the spinner stops on yellow, we need to determine the number of favorable outcomes (yellow) and the total number of possible outcomes.
The spinner has 10 equally sized slices, with 4 yellow, 4 red, and 2 blue.
The number of favorable outcomes (yellow) is 4 because there are 4 yellow slices.
The total number of possible outcomes is 10 because there are 10 slices in total.
Therefore, the probability of the spinner stopping on yellow can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 4 / 10
Simplifying this fraction, we get:
Probability = 2 / 5
So, the probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).
Therefore, there is a 40% chance that the spinner will stop on yellow.
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PLSPLSS PLSSMSL HELP ME
Answer:
1. x = 18
2. x = 31
Step-by-step explanation:
1. Angles in a quadrilateral sum to 360°.
360 -(60 +130 +70) = 360 -260 = 100
So, x +82 = 100. x = 100 -82 = 18
2. Angles in a triangle sum to 180°. 180 -(29 +75) = 180 -104 = 76.
So, -17 +3x = 76. 3x = 93. x = 93/3 = 31
If sin(x+y)= 1/2(sin x) + square root of 3/2(cos x), what is the value of y
Answer:
y = π/3
Step-by-step explanation:
To find the value of y, we can use the trigonometric identity for the sum of angles:
sin(x + y) = sin x * cos y + cos x * sin y
Comparing this with the given equation:
sin(x + y) = 1/2(sin x) + √3/2(cos x)
We can equate the corresponding terms:
sin x * cos y = 1/2(sin x) ----(1)
cos x * sin y = √3/2(cos x) ----(2)
From equation (1), we can see that cos y = 1/2.
From equation (2), we can see that sin y = √3/2.
To determine the values of y, we can use the trigonometric values of cosine and sine in the first quadrant of the unit circle.
In the first quadrant, cos y is positive, so cos y = 1/2 corresponds to y = π/3 (60 degrees).
Similarly, sin y is positive, so sin y = √3/2 corresponds to y = π/3 (60 degrees).
Therefore, the value of y is y = π/3 (or 60 degrees).
Given the following equation of a line x+6y= 3, determine the slope of a line that is perpendicular.
The slope of a line that is perpendicular to the given line x + 6y = 3 is 6.
To determine the slope of a line that is perpendicular to the given line, we need to find the negative reciprocal of the slope of the given line.
The equation of the given line is x + 6y = 3.
To find the slope of the given line, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope:
x + 6y = 3
6y = -x + 3
y = (-1/6)x + 1/2
From the equation y = (-1/6)x + 1/2, we can see that the slope of the given line is -1/6.
To find the slope of a line that is perpendicular, we take the negative reciprocal of -1/6.
The negative reciprocal of -1/6 can be found by flipping the fraction and changing its sign:
Negative reciprocal of -1/6 = -1 / (-1/6) = -1 * (-6/1) = 6
Therefore, the slope of a line that is perpendicular to the given line x + 6y = 3 is 6.
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Which is equivalent to 4/9 1/2x*?
92x
9 1/8x
Answer:
B. [tex] 9^{\frac{1}{8}x} [/tex]
Step-by-step explanation:
[tex] \sqrt[4]{9}^{\frac{1}{2}x} = [/tex]
[tex] = ({9}^{\frac{1}{4}})^{\frac{1}{2}x} [/tex]
[tex] = 9^{\frac{1}{4} \times \frac{1}{2}x} [/tex]
[tex] = 9^{\frac{1}{8}x} [/tex]
What 2 numbers can multiply to -40 and add up to 6
Answer: 10 and -4
Step-by-step explanation: 10 + - 4 = 6 and 10 x -4 = -40
Pls help I need help on this
Answer:
[tex]5a2b2/4[/tex]
Step-by-step explanation:
Solve it for me please
1a.) The amount that the eldest son received would be =GHç 1,360
b .) The amount received by the daughter would be =G Hç 2176
c.) The difference between the amount the two sons received would be =GHç1,904
How to calculate the amount received by the eldest son?For 1a.)
The amount that the land is worth= $8,600
The amount received for various purposes= $1,800
The remaining amount shared to the sons= 8,600-1800= $6,800
The percentage amount received by the eldest son= 20% of 6800
That is;
= 20/100×6800/1
= 136000/100
= $1,360
The remaining amount= 6800-1360= $5,440
For 1b.)
The ratio that the remaining amount was shared between the other son and the daughter = 3 : 2 respectively.
The total ratio= 3+2=5
For daughter= 2/5× 5440
= 10880/5 = 2176
The other son= 5440-2176 = 3264
For 1c.)
The difference between the amount the two sons received would be =3264-1,360 = GHç1,904.
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(08.01 MC)
The function h(x) is a continuous quadratic function with a domain of all real numbers. The table
x h(x)
-6 12
-57
-4 4
-3 3
-24
-1 7
What are the vertex and range of h(x)?
The vertex of h(x) is (-3, 3), and the range is y ≥ 3.
To find the vertex of the quadratic function h(x), we can use the formula x = -b/2a, where the quadratic function is in the form [tex]ax^2 + bx + c[/tex].
From the given table, we can observe that the x-values of the vertex correspond to the minimum points of the function.
The minimum point occurs between -4 and -3, which suggests that the x-coordinate of the vertex is -3. Therefore, x = -3.
To find the corresponding y-coordinate of the vertex, we look at the corresponding h(x) value in the table, which is 3. Hence, the vertex of the function h(x) is (-3, 3).
To determine the range of h(x), we need to consider the y-values attained by the function.
From the table, we see that the lowest y-value is 3 (the y-coordinate of the vertex), and there are no other y-values lower than 3. Therefore, the range of h(x) is all real numbers greater than or equal to 3.
The vertex of h(x) is (-3, 3), and the range is y ≥ 3.
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The vertex of the quadratic function is (-4, 12).
The range of h(x) is [3, ∞).
To find the vertex and range of the quadratic function h(x) based on the given table, we can use the properties of quadratic functions.
The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be determined using the formula:
x = -b / (2a)
The domain of h(x) is all real numbers, we can assume that the quadratic function is of the form h(x) = ax² + bx + c.
Looking at the table, we can see that the x-values are increasing from left to right.
Additionally, the y-values (h(x)) are increasing from -6 to -4, then decreasing from -4 to -1.
This indicates that the vertex of the quadratic function lies between x = -4 and x = -3.
To find the exact x-coordinate of the vertex, we can use the formula mentioned earlier:
x = -b / (2a)
Based on the table, we can choose two points (-4, 4) and (-3, 3).
The difference in x-coordinates is 1, so we can assume that a = 1.
Plugging in the values of (-4, 4) and a = 1 into the formula, we can solve for b:
-4 = -b / (2 × 1)
-4 = -b / 2
-8 = -b
b = 8
The equation of the quadratic function h(x) can be written as h(x) = x² + 8x + c.
Now, let's find the y-coordinate of the vertex.
We can substitute the x-coordinate of the vertex, which we found as -4, into the equation:
h(-4) = (-4)² + 8(-4) + c
12 = 16 - 32 + c
12 = -16 + c
c = 28
The equation of the quadratic function h(x) is h(x) = x² + 8x + 28.
The range of the quadratic function can be determined by observing the y-values in the table.
From the table, we can see that the minimum y-value is 3.
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Given the information in the diagram, which theorem best justifies why lines j and k must be parallel?
Given the information in the diagram, the theorem that best justifies why lines j and k must be parallel include the following: D. converse alternate exterior angles theorem.
What are parallel lines?In Mathematics and Geometry, parallel lines are two (2) lines that are always the same (equal) distance apart and never meet or intersect.
In Mathematics and Geometry, the alternate exterior angle theorem states that when two (2) parallel lines are cut through by a transversal, the alternate exterior angles that are formed lie outside the two (2) parallel lines, are located on opposite sides of the transversal, and are congruent angles.
Since the alternate exterior angles are congruent, we can logically deduce the following based on the converse alternate exterior angles theorem;
93° ≅ 93° (lines j and k are parallel lines).
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Complete Question:
Given the information in the diagram, which theorem best justifies why lines j and k must be parallel?
alternate interior angles theorem
alternate exterior angles theorem
converse alternate interior angles theorem
converse alternate exterior angles theorem
whats the answer pls
Answer:
Step-by-step explanation:
Write the correct measurement for A-F (Example 2.2, 5.9, 9)
Answer:
9
Step-by-step explanation:
Question 11 of 26
Given the diagram below, what is cos(45)?
Triangle not drawn to scale
A. √2
O B.
√3
C. 3-√2
45⁰
OD.
Answer:
chemical reaction that releases heat energy to the surroundings is known as endothermis reaction
MLMS4 Day 3 (1 hour) SBA Task: Project November 2023 Which sides of the rectangle (that formed the sides of the cylinder) has the same length as the circumference of the circles? Which sides of the rectangle has the same length of the height of the cylinder. What do you notice between your model and the practical calculation.
The sides of the rectangle that have the same length as the circumference of the circles are the sides parallel to the bases of the cylinder. The sides of the rectangle that have the same length as the height of the cylinder are the sides perpendicular to the bases. While the model provides a simplified representation, practical calculations might have slight differences due to real-world factors.
In a cylinder, the two circular bases are connected by a curved surface, forming a three-dimensional shape. The rectangular shape that wraps around the curved surface of the cylinder is called the lateral surface or the lateral area.
To determine which sides of the rectangle have the same length as the circumference of the circles, we need to understand the geometry of a cylinder. The circumference of a circle is calculated using the formula:
Circumference = 2πr,
where r is the radius of the circle. In a cylinder, the bases are identical circles, so the circumference of each base is equal. Therefore, the sides of the rectangle that are parallel to the bases have the same length as the circumference of the circles.
Now, let's consider the height of the cylinder. The height is the distance between the two bases and is perpendicular to the bases. In the rectangular representation of the cylinder, the sides that are perpendicular to the bases represent the height. Hence, the sides of the rectangle that are perpendicular to the bases have the same length as the height of the cylinder.
When comparing the model (rectangular representation) with practical calculations, we may notice some differences. The model provides a simplified representation of the cylinder, assuming that the lateral surface is perfectly wrapped around the curved surface. However, in practical calculations, there might be slight variations due to factors like material thickness, manufacturing processes, or measuring precision. These variations can result in minor deviations between the model and the practical calculations.
It's important to consider that the model is an approximation and serves as a visual aid to understand the basic properties of the cylinder. In real-life applications or engineering calculations, precise measurements and considerations of tolerances are crucial.
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Two models R1 and R2 are given for revenue (in millions of dollars) for a corporation. Both models are estimates of revenues from 2030 through 2035, with t = 0 corresponding to 2030.
R1 = 7.23 + 0.25t + 0.03t^2
R2 = 7.23 + 0.1t + 0.01t^2
How much more total revenue (in millions of dollars) does that model project over the six-year period ending at t = 5? (Round your answer to three decimal places.)
Step-by-step explanation:
To find the difference in total revenue projected by the two models over the six-year period ending at t = 5, we need to calculate the revenue for each model from t = 0 to t = 5 and subtract the results.
For R1:
R1 = 7.23 + 0.25t + 0.03t^2
Substituting t = 5:
R1(5) = 7.23 + 0.25(5) + 0.03(5^2)
R1(5) = 7.23 + 1.25 + 0.75
R1(5) = 9.23 + 0.75
R1(5) = 9.98 million dollars
For R2:
R2 = 7.23 + 0.1t + 0.01t^2
Substituting t = 5:
R2(5) = 7.23 + 0.1(5) + 0.01(5^2)
R2(5) = 7.23 + 0.5 + 0.25
R2(5) = 7.73 + 0.25
R2(5) = 7.98 million dollars
To find the difference, we subtract R2(5) from R1(5):
Difference = R1(5) - R2(5)
Difference = 9.98 - 7.98
Difference = 2 million dollars
Therefore, the model R1 projects 2 million dollars more in total revenue than R2 over the six-year period ending at t = 5.
The average student loan debt for college graduates is $25,200. Suppose that that distribution is normal and that the standard deviation is $11,200. Let X = the student loan debt of a randomly selected college graduate. Round all probabilities to 4 decimal places and all dollar answers to the nearest dollar.
a. What is the distribution of X? X - N
b Find the probability that the college graduate has between $27,250 and $43,650 in student loan debt
c. The middle 20% of college graduates loan debt lies between what two numbers? Low: $ High: $
a) The distribution of X, the student loan debt of a randomly selected college graduate, is normal with a mean of $25,200 and a standard deviation of $11,200. b) The probability is approximately 7.28%.
c) The middle lies between approximately $22,164 and $28,536.
How to Find Probability?a. The distribution of X, the student loan debt of a randomly selected college graduate, is a normal distribution (bell-shaped curve) with a mean (μ) of $25,200 and a standard deviation (σ) of $11,200. We can represent this as X ~ N(25200, 11200).
b. To find the probability that the college graduate has between $27,250 and $43,650 in student loan debt, we need to calculate the z-scores for these two values and then find the area under the normal curve between those z-scores.
First, we calculate the z-score for $27,250:
z1 = (X1 - μ) / σ = (27250 - 25200) / 11200 ≈ 1.8304
Next, we calculate the z-score for $43,650:
z2 = (X2 - μ) / σ = (43650 - 25200) / 11200 ≈ 1.6518
Now, we need to find the area under the normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find this area.
Using a standard normal distribution table or a calculator, the probability is approximately P(1.6518 ≤ Z ≤ 1.8304) ≈ 0.0728.
c. To find the middle 20% of college graduates' loan debt, we need to find the range of values that contain the central 20% of the distribution. This range corresponds to the values between the lower and upper percentiles.
The lower percentile is the 40th percentile (50% - 20%/2 = 40%) and the upper percentile is the 60th percentile (50% + 20%/2 = 60%).
Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these percentiles:
For the lower percentile (40th percentile):
z_lower = invNorm(0.40) ≈ -0.2533
For the upper percentile (60th percentile):
z_upper = invNorm(0.60) ≈ 0.2533
Now, we can convert these z-scores back to the corresponding loan debt values:
Lower debt value:
X_lower = μ + z_lower * σ = 25200 + (-0.2533) * 11200 ≈ $22,164
Upper debt value:
X_upper = μ + z_upper * σ = 25200 + 0.2533 * 11200 ≈ $28,536
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