Final answer:
The sample mean (μ) and standard deviation (σ) are calculated using the provided z-scores and corresponding sample values, resulting in a mean of 30 and a standard deviation of 10.
Explanation:
To solve for the sample mean and standard deviation using the given z-scores and sample values, we can employ the formula for calculating a z-score:
z = (X - μ) / σ
where X is the sample value, μ is the mean, and σ is the standard deviation. For X = 45 with a z-score of 1.50, the equation is:
1.50 = (45 - μ) / σ
For X = 40 with a z-score of 1.00, the equation becomes:
1.00 = (40 - μ) / σ
By solving these two equations simultaneously, we can find the values of μ and σ.
From the first equation, we have:
1.50σ = 45 - μ
From the second equation, we have:
1.00σ = 40 - μ
If we multiply the second equation by 1.5, it becomes:
1.50σ = 60 - 1.5μ
We can set the expressions for 1.50σ equal to each other:
45 - μ = 60 - 1.5μ
Solving for μ gives us the sample mean:
μ = 60 - 45 = 15 / (1.5 - 1) = 15 / 0.5 = 30
To find the standard deviation σ, we substitute μ = 30 into one of the original equations:
1.50σ = 45 - 30 = 15
Therefore, σ = 15 / 1.50 = 10
So, the sample mean (μ) is 30 and the sample standard deviation (σ) is 10.
Final answer:
The sample mean (μ) is 30 and the standard deviation (σ) is 10.
Explanation:
The question is asking to find the sample mean and standard deviation based on given z-scores for specific values in a normal distribution.
To find these parameters, we'll use the formula for calculating a z-score:
z = (X - μ) / σ, where z is the z-score, X is the value, μ is the mean, and σ is the standard deviation.
For X = 45 and z = 1.5, the equation becomes 1.5 = (45 - μ) / σ. For X = 40 and z = 1.0, the equation is 1.0 = (40 - μ) / σ.These two equations can be solved simultaneously to find the values of μ and σ.
Steps for solving them are:
Rewrite the equations:Therefore, the sample mean (μ) is 30 and the standard deviation (σ) is 10.
One hundred teachers attended a seminar on mathematical problem solving. The attitudes of representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows...... 4; 7; -1; 1; 0; 5; -2; 2; -1; 6; 5; -3
What is the mean change score? (Round your answer to two decimal places.)
What is the standard deviation for this population? (Round your answer to two decimal places.)
What is the median change score? (Round your answer to one decimal place.)
Find the change score that is 2.2 standard deviations below the mean. (Round your answer to one decimal place.)
Answer:
a) 1.92
b) 3.25
c) 1.5
d) -5.23
Step-by-step explanation:
We are given the following in the question:
4, 7, -1, 1, 0, 5, -2, 2, -1, 6, 5, -3
a) mean of score change
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{23}{12} = 1.92[/tex]
b) standard deviation for this population
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
Sum of squares of differences = 126.92
[tex]\sigma = \sqrt{\frac{126.92}{12}} = 3.25[/tex]
c) median change score
[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]
Sorted data: -3, -2, -1, -1, 0, 1, 2, 4, 5, 5, 6, 7
Median =
[tex]\dfrac{6^{th} + 7^{th}}{2} = \dfrac{1+2}{2} = 1.5[/tex]
d) change score that is 2.2 standard deviations below the mean.
[tex]x = \mu - 2.2(\sigma)\\x = 1.92-2.2(3.25)\\x = -5.23[/tex]
Alondra received a 14% hourly raise, but the number of hours worked decreased by 7.5%. If her wage was $10.50 an hour and she worked 40 hours per week before the changes, how much money will she earn in one week after the changes? Is this more or less than her previous weekly earnings?
Answer:
She will earn $442.89 in one week after the changes.
And this is more than her previous weekly earnings.
Step-by-step explanation:
Given:
Alondra received a 14% hourly raise, but the number of hours worked decreased by 7.5%. If her wage was $10.50 an hour and she worked 40 hours per week before the changes.
Now, to find money she will earn in one week after the changes.
Her wage was = $10.50.
She worked per week = 40 hours.
So, her salary before changes:
[tex]10.50\times 40\\\\=\$420.[/tex]
Thus, the salary per week before changes is $420.
Now, to get her salary after 14% hourly raise:
[tex]10.50+14\%\ of\ 10.50\\\\=10.50+\frac{14}{100} \times 10.50\\\\=10.50+1.47\\\\=11.97[/tex]
Salary after hourly raise = $11.97 per hour.
Then, to get the number of hours worked decreased by 7.5%:
[tex]40-7.5\%\ of\ 40\\\\=40-\frac{7.5}{100} \times 40\\\\=40-3\\\\=37.[/tex]
Number of hours per week after hours of worked decreased = 37 hours.
Now, to get the salary after changes:
Salary after hourly raise × number of hours per week after hours of worked decreased
[tex]=11.97\times 37[/tex]
[tex]=\$442.89.[/tex]
Salary after changes in one week = $442.89.
As, the previous salary was $420 in one week.
And after changes this salary is $442.89 in one week which is more than previous.
Therefore, she will earn $442.89 in one week after the changes.
And this is more than her previous weekly earnings.
Explaining How to Compare Water Levels Ericka decided to compare her observation to the average annual trend, which shows the water rising 1.8 mm/year. Remember, she used 6.2 years as her time period. Explain how she would calculate the difference between how much water levels rose on average and how much the water level fell in the part of the river she observed.
Step-by-step explanation:
Below is an attachment containing the solution
Answer: She would multiply the rate by the years to find the average rise in water levels, or 1.8 times 6.2 = 11.16. To find the difference between the water levels, she would subtract -13.64 from 11.16.
Step-by-step explanation:
In the test of hypothesis H 0 : μ = 100 vs Ha: μ ≠ 100, a sample of size 250 yields the standardizedtest statistic z = 1.47. Find the p-value for the test and state your conclusion at α = 0.10
Answer:
The p-value of the test is 0.1416.
The null hypothesis was not rejected concluding that μ = 100.
Step-by-step explanation:
The hypothesis is defined as:
H₀: μ = 100 vs. Hₐ: μ ≠ 100
The test is a two-tailed test.
The test statistic value is z = 1.47.
The significance level of the test is α = 0.10.
The p-value is computed as follows:
[tex]p-value=2\times P(Z<-1.47)=2\times0.0708=0.1416[/tex]
Decision rule:
If the p-value of the test is less than the significance level 0.10, then the null hypothesis is rejected and vice-versa.
The p-value = 0.1416 > α = 0.10.
The p-value is more than the significance level.
The null hypothesis was not rejected.
Conclusion:
The mean value is not different than 100.
To find the p-value for the test, calculate the area under the standard normal curve more extreme than the observed test statistic. The p-value is 0.1416, and we fail to reject the null hypothesis at α = 0.10.
Explanation:To find the p-value for the test, we need to calculate the area under the standard normal curve that is more extreme than the observed test statistic. In this case, the test statistic is z = 1.47. Since it is a two-tailed test, we need to find the probability in both tails.
First, we find the area to the right of 1.47 by subtracting the cumulative probability from the mean to 1.47 from 1: P(Z > 1.47) = 1 - P(Z < 1.47) = 1 - 0.9292 = 0.0708.
Next, we double this probability to get the total p-value for both tails: p-value = 2 * 0.0708 = 0.1416. Since the p-value (0.1416) is greater than the significance level (α = 0.10), we fail to reject the null hypothesis. This means we do not have enough evidence to conclude that the population mean is not equal to 100.
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please help ! i got through half the problem - i just can't figure out how to split the $2 to jay and kay. giving out brainliest !!
Jay, Kay, and Lee are hiking. Lee forgot to bring water. Jay brought 2 gallons of water and Kay brought 1.75 gallons of water. The three of them agree to divide all the water equally among them. Lee gives $2 for the water he receives, which Jay and Kay agree to divide fairly between the two of them. How much money should Jay receive?
Answer:
The answer to your question is Jay will receive $1.07 and Kay will receive $0.93.
Step-by-step explanation:
Data
Jay brought 2 gallons
Kay brought 1.75 gallons
Money = $2
Process
1.- Calculate the total amount of water
Water = 2 + 1.75
= 3.75 gallons
2.- Calculate the money Jay will receive using proportions
3.75 gallons ---------------- $2.00
2 gallons --------------- x
x = (2 x 2) / 3.75
x = 4/3.75
x = $1.07
3.- Calculate the money Kay will receive using proportions
3.75 gallons ----------------$2.00
1.75 gallons --------------- x
x = (1.75 x 2.00) / 3.75
x = 3.5/3.75
x = $0.93
PLLLZ HELP Find Sn if a1 = 20, d = –10, and n = 25.
–2500
–2750
2500
–5000
Option A: [tex]-2500[/tex] is the value of [tex]S_{25[/tex]
Explanation:
It is given that the first term is [tex]a_1=20[/tex]
The common difference is [tex]d=-10[/tex]
We need to determine the sum of 25 terms.
The sum of terms of an arithmetic series can be determined using the formula, [tex]S_n=\frac{n[2a_1+(n-1)d]}{2}[/tex]
Substituting [tex]n=25[/tex] , [tex]a_1=20[/tex] and [tex]d=-10[/tex]
Thus, we have,
[tex]S_{25}=\frac{25[2(20)+(25-1)(-10)]}{2}[/tex]
Simplifying the values, we get,
[tex]S_{25}=\frac{25[40+(24)(-10)]}{2}[/tex]
[tex]S_{25}=\frac{25[40-240]}{2}[/tex]
Subtracting the terms within the bracket, we get,
[tex]S_{25}=\frac{25[-200]}{2}[/tex]
Multiplying the terms in the numerator, we have,
[tex]S_{25}=\frac{-5000}{2}[/tex]
Dividing, we get,
[tex]S_{25}=-2500[/tex]
Thus, the sum of the 25 terms is [tex]-2500[/tex]
Therefore, Option A is the correct answer.
Answer:
-2500 answer
Step-by-step explanation:
What is the area of the largest circular fire that can be made in this fire pit? Use 3.14 for π. Round to the nearest square inch.
!no absurd answers, please! : (
Area of the largest circle fit in the fire pit is 4069.44 square inches.
Solution:
Length of the rectangular pit = 7 feet
Width of the rectangular pit = 6 feet
The diameter of a largest circle inscribed in a rectangle is equal to the smaller side of the rectangle.
Diameter of the largest circle = 6 feet
Radius of the largest circle = 3 feet
Area of the circle = πr²
= 3.14 × 3²
Area of the circle = 28.26 square feet
Let us convert square feet into square inches.
1 square feet = 144 square inches
28.26 square feet = 28.26 × 144
= 4069.44 square inches
Area of the largest circle fit in the fire pit is 4069.44 square inches.
The equation of the piecewise function f(x) is below. What is the value of f(3)
When x is greater than or equal to 0 use the equation x +2
The x value is given as 3 in f(3)
Now replace x with 3 in the equation and solve:
F(3) = 3 + 2 = 5.
The answer is 5
A landscaping company charges a $10 fee to make a house call plus an hourly rats for labor. If the total bill for a job that takes two hours comes to $61.00 write an equation that can be used to solve for the hourly labor rate
Answer:
61 divided by 10 and that should even out how many hours they worked
Step-by-step explanation:
Answer: the equation that can be used to solve for the hourly labor rate is
2x + 10 = 61
Step-by-step explanation:
Let x represent the amount charged by the landscaping company per hour of labour.
The landscaping company charges a $10 fee to make a house call plus an hourly rate for labor. This means that the total charge for y hours is
xy + 10
If the total bill for a job that takes two hours comes to $61.00, the equation that can be used to solve for the hourly labor rate would be
2x + 10 = 61
A consumer organization inspecting new cars found that many had appearance defects. While none had more than 3 defects, 7% had three, 11% had two, and 21% had one defect. Find the expected number of appearance defects in a new car, and the standard deviation.
Answer:
E(X) = 0.64
Sd(X) = 0.933
Step-by-step explanation:
3(0.07) + 2(0.11) + 1(0.21)
E(X²) = 3²(0.07) + 2²(0.11) + 1²(0.21)
= 1.28
Var(X) = 1.28 - 0.64² = 0.8704
Sd = 0.933
If a quadrilateral does not have two pairs of opposite sides that are parallel, then it may be a _____. A. parallelogram B. rhombus C. trapezoid D. square E. rectangle
Answer:
C Trapezoid
Step-by-step explanation:
A trapezoid mostly has only one pair of parallel sides, not two
Answer: Trapezoid
Step-by-step explanation:
An NHL hockey player has 59 goals so far in season what are the possible numbers of additional goals the player can score to make or break the NHL record of 92 goals in a season
Answer:
53 by the fact NFL players use a for wheeler u Dan je e eka a f wjd r
Chelsea collects sand dollars.The number she has collected is greater than 400 and less than 460.The number in hundreds place is one less than the number in the tens place.The number in the ones place is two more than the number in the tens place.How many sand dollars has Chelsea collected?
Answer:
457 sand dollars
Step-by-step explanation:
Let the number=x
The number she has collected is greater than 400 and less than 460
400<x<460
The number in hundreds place is one less than the number in the tens place.
The number in the Hundreds place is 4. Since it is one less, our number for now will be of the form 45* where * is the ones digit.
Also, the number in the ones place is two more than the number in the tens place.
The number in the tens place is 5. If it is two more than 5, the number in the ones place is 7.
Therefore Chelsea has collected 457 sand dollars
An isosceles triangle with each leg measuring 13 is inscribed in a circle. If the altitude to the base of the triangle is 5, find the radius of the circle.
Using the principles of Pythagorean theorem, we can figure out that the radius of the circle inscribed by the given isosceles triangle is 5 units.
Explanation:To solve this problem, we need to apply the principles of the Pythagorean theorem and radius calculation in a circle inscribed by a triangle.
To recall, the Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, if you have a right triangle, you can calculate the hypotenuse c with the formula: c = √a² + b².
In this case, the altitude to the base forms two right-angled triangles within the isosceles triangle. These triangles both have legs of 5 (altitude) and half the base.
We first need to calculate this half base. The half base can be calculated by using Pythagorean theorem where one leg of the right triangle is the altitude (5) and the other leg is half the base, and the hypotenuse is one side of the isosceles triangle (13). Solving this yields a half base of 12.
Now, with the whole base equal to twice this value, or 24, we have a right triangle where the hypotenuse of the triangle (the diameter of the circle) is also the side of the isosceles triangle (13) and one leg is the whole base of the isosceles triangle (24), and the other leg is the altitude from the center of the base to the top of the isosceles triangle which is also the radius of the circle we are looking for.
Applying the Pythagorean theorem here yields a radius of √(13² - 12²) which simplifies to 5. Therefore, the radius of the circle is 5 units.
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Fran is limited to watching television less than 12.6 hours per week. She has already watched 4.2 hours, and each show is 0.7 of an hour long. 0.7x + 4.2 < 12.6 How many more shows can Fran watch this week?
Answer: the number of hpurs that Fran can watch this week is lesser than 12
Step-by-step explanation:
Let x represent the number of hours of television that Fran can watch in a week.
Fran is limited to watching television less than 12.6 hours per week. She has already watched 4.2 hours, and each show is 0.7 of an hour long. The inequality representing the situation is expressed as
0.7x + 4.2 < 12.6
0.7x + 4.2 < 12.6 - 4.2
0.7x < 8.4
x < 8.4/0.7
x < 12
Answer:
12
Step-by-step explanation:
Fran has wached 4.2 hours of tv out of 12.6, if the equasion is curect that means that she has wached 6 shows already.
From here there is two ways to do his one is to look at how meny shows she can watch in 12.6 hours and subtract how meny shows she has wached from it and then convert back to desimals.
The other way to do this( the better way) is to do 12.6-4.2=8.4 then do 8.4 / .7 and you would get 12
A sociologist surveyed 300 people about their level of anxiety on a scale of 1 to 100. Unfortunately, the person inputting the data into the computer accidentally transposed six of the numbers causing the statistics to have errors.What type of error is this?1. Sampling error 2. Non sampling error
Answer:
non sampling error
Step-by-step explanation:
Sampling error in a statistical analysis arising from the unrepresentativeness of the sample taken.
Non Sampling error is a term used in statistics that refers to an error that occurs during data collection, causing the data to differ from the true values.
Answer:
sampling error
Step-by-step explanation:
A social scientist measures the number of minutes (per day) that a small hypothetical population of college students spends online. Student Score Student Score A 58 F 92 B 77 G 99 C 87 H 84 D 87 I 99 E 91 J 22 (a) What is the range of data in this population? min (b) What is the IQR of data in this population? min (c) What is the SIQR of data in this population? min (d) What is the population variance?
Answer:
The range of the data = 99 -22 = 77min
The mean of the dataset is given as
The IQR = 92 - 77 = 15 min
SIQR = IQR / 2 = 15 / 2 = 7.5 min
variance = 55.05 min
Step-by-step explanation:
First we need to arrange the data in ascending or descending order
22 ,58, 77, 84, 87, 87,91, 92, 99, 99 in ascending order
The range of the data is calculated by substracting the numbers at the extreme that is the lowest number subtracted from the highest number
range = 99 - 22 = 77min
The IQR stands for Inter-quartile range Q3 - Q1 where
Q1 is the middle value in the first half of the data set. i.e
Q1 is the middle of 22 ,58, 77, 84, 87 which is 77
Q1 = 77 min
Q3 is the middle value in the second half of the data set. i.e
Q3 is the middle of 87,91, 92, 99, 99 which is 92
Q1 = 92 min
Therefore IQR = Q3 - Q1 = 92min - 77min = 15 min
The SIQR stands for the semi-interquartile range. it is calculated by IQR / 2
SIQR = 15 / 2 = 7.5 min
To calculate the population variance we need to get the mean say X
The mean is the data point at the center. Since the dataset is even, there are two of them. which is 87 and 87
Therefore the mean is X = (87 + 87)/ 2 = 87
The variance = ∑[tex](X-x)^{2} /n[/tex]
where X is the mean = 87
x is a datapoint on the given dataset
n is the datasize = 10
variance = [tex]((22-87)^{2} + (58-87)^{2} + (77-87)^{2} + (84-87)^{2} + (87-87)^{2} + (87-87)^{2} + (91-87)^{2} + (92-87)^{2} + (99-87)^{2} + (99-87)^{2} ) /10[/tex]
variance = [tex]((-65)^{2} + (-29)^{2} + (-10)^{2} + (-3)^{2} + (0)^{2} + (0)^{2} + (4)^{2} + (5)^{2} + (12)^{2} + (12)^{2} ) /10[/tex]
variance = [tex]((4225) + (841) + (100) + (9) + (0) + (0) + (16) + (25) + (144) + (144) ) /10[/tex]
variance = 5505/10
variance = 550.5 min
Answer:
Range =77
IQR=15
SIQR=7.5
Variance=550.5
Step-by-step explanation:
Range:
First find the lowest and the highest number in the data and then subtract high with the low to find the range. 99-22=77.
IQR:
Ascend the data from low to high like this:
22 58 77 84 87 87 91 92 99 99
Then break into two half
22 58 77 84 87 | 87 91 92 99 99
Find the median of all the two half
77 is rhe median in the first half and 92 is the median in the second half.
Then subtract them: 92-77=15 IQR
SIQR:
Divide the IQR/2
Hence 15/2=7.5.
Variance:
Find the mean of the data first i.e. 79.6
variance = 5505/10
variance = 550.5
Plz help
The volume of the rectangular prism is 60x3 + 145x2 + 70x. Factor to find the possible expressions for length, width and height of the prism.
4x(5x + 7)(3x + 2)
x(2x + 7)(10x + 1)
5x(4x + 7)(3x + 2)
5x(7x + 4)(3x + 2)
Option C: [tex]5 x(4 x+7)(3 x+2)[/tex] is the possible expressions for length, width and height of the prism.
Explanation:
The volume of the rectangular prism is [tex]60 x^{3}+145 x^{2}+70 x[/tex]
To determine the length, width and height of the rectangular prism, let us factor the expression.
Thus, factoring 5x from the expression, we have,
[tex]5 x\left(12 x^{2}+29 x+14\right)[/tex]
Let us break the expression [tex]12 x^{2}+29 x+14[/tex] into two groups, we get,
[tex]5x[\left(12 x^{2}+8 x\right)+(21 x+14)][/tex]
Factoring 4x from the term [tex]12 x^{2}+8 x[/tex] , we get,
[tex]5x[4 x(3 x+2)+(21x+14)][/tex]
Similarly, factoring 7x from the term [tex]21 x+14[/tex] , we get,
[tex]5x[4 x(3 x+2)+7(3x+2)][/tex]
Now, let us factor out [tex]3x+2[/tex], we get,
[tex]5 x(4 x+7)(3 x+2)[/tex]
Hence, the possible expressions for length, width and height of the prism is [tex]5 x(4 x+7)(3 x+2)[/tex]
Therefore, Option C is the correct answer.
Solve each problem.
(8 +5i) + (6 - 7i)
Answer:
2(7-i)
Step-by-step explanation:
(8 +5i) + (6 - 7i)
Opening each bracket
8 +5i +6 -7I
8 +6 +5i-7i
14-2i
2(7-i)
Answer:
14 - 2 i
Step-by-step explanation:(8 + (5 * i)) + (6 - (7 * i)) =
Consider the differential equation: y′′−8y′=7x+1. Find the general solution to the corresponding homogeneous equation. In your answer, use c1 and c2 to denote arbitrary constants. Enter c1 as c1 and c2 as c2. yc= Apply the method of undetermined coefficients to find a particular solution. yp=
Answer:
yp = -x/8
Step-by-step explanation:
Given the differential equation: y′′−8y′=7x+1,
The solution of the DE will be the sum of the complementary solution (yc) and the particular integral (yp)
First we will calculate the complimentary solution by solving the homogenous part of the DE first i.e by equating the DE to zero and solving to have;
y′′−8y′=0
The auxiliary equation will give us;
m²-8m = 0
m(m-8) = 0
m = 0 and m-8 = 0
m1 = 0 and m2 = 8
Since the value of the roots are real and different, the complementary solution (yc) will give us
yc = Ae^m1x + Be^m2x
yc = Ae^0+Be^8x
yc = A+Be^8x
To get yp we will differentiate yc twice and substitute the answers into the original DE
yp = Ax+B (using the method of undetermined coefficients
y'p = A
y"p = 0
Substituting the differentials into the general DE to get the constants we have;
0-8A = 7x+1
Comparing coefficients
-8A = 1
A = -1/8
B = 0
yp = -1/8x+0
yp = -x/8 (particular integral)
y = yc+yp
y = A+Be^8x-x/8
One day the appliance store offers a $50 discount on all purchases over $300. The store also has a sale with 15% off of all refrigerators. The 15% discount is applied after the $50 discount. What is the price, in dollars, of a $435 dollar refrigerator after both discounts? Answer the problem. Explain how you would solve the problem (list the steps you would take).
Answer:
$327.25
Step-by-step explanation:
$435 - $50 = 385
15% = 15/100 = 0.15
$385 * 0.15 = 57.75
$385 - $57.75 = $327.25
The price, in dollars, of a $435 dollar refrigerator after both discounts is $327.25.
The calculation is as follows:= $435 - $50
= 385
Since there is 15% discount
So here we have to do 15% discount of $385
i.e.
= $385 - 15% of $385
= $385 - $57.75
= $327.25
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Find a1, d, and the Explicit Rule for this arithmetic sequence: -3, 4, 11, 18, 25, ...
1.)a1 =-3, d=-7, an = -7n - 3
2.)a1 =-3, d = 7, an = 7n - 10
3.)21 = -3, d =-7, an = -7n - 10
4.)a1 = -3, d = 7, an = 7n - 3
Answer:
2
Step-by-step explanation:
Answer: 2.)a1 =-3, d = 7, an = 7n - 10
Step-by-step explanation:
In an arithmetic sequence, consecutive terms differ by a common difference.
The formula for determining the nth term of an arithmetic sequence is expressed as
an = a1 + d(n - 1)
Where
a represents the first term of the sequence.
d represents the common difference.
n represents the number of terms in the sequence.
From the information given,
a1 = - 3
d = 4 - - 3 = 11 - 4 = 18 - 11 = 25 - 18 = 7
Therefore, the Explicit Rule for this arithmetic sequence is
an = - 3 + 7(n - 1)
an = - 3 + 7n - 7
an = 7n - 7 - 3
an = 7n - 10
Given f(x)= x+1 and g(x)=√x+2 determine the following. Write each answer using interval notation.
Determine the Domain of g(f(x))
Domain:
Answer:
g(f(x)) = [tex]\sqrt{f(x)} +2=\sqrt{x+1} +2[/tex]
Domain = [-1,∞)
Step-by-step explanation:
Given f(x) = x+1 and g(x) = √x + 2
g(f(x)) is a composite function.
g(f(x)) = [tex]\sqrt{f(x)} +2=\sqrt{x+1} +2[/tex]
To find the domain of composite function we must get both domains right (the composed function and the first function used).
The domain of f(x) is all the real numbers.
The domain of g(f(x)) is the values of x provide that the square root is greater than or equal zero
So, x+1 ≥ 0
∴ x ≥ -1
So, the domain = [-1,∞)
The domain of the composite function g(f(x)) is x ≥ -2.
Explanation:The domain of a function is the set of all possible input values for which the function is defined. To determine the domain of the composite function g(f(x)), we need to consider two things:
The domain of the inner function f(x)The domain of the outer function g(x)In this case, the domain of f(x) is all real numbers because there are no restrictions on the input values for f(x) = x + 1.
However, the domain of g(x) = √(x + 2) is limited by the requirement that the radicand (the expression inside the square root) must be greater than or equal to zero. So, x + 2 ≥ 0.
Solving this inequality, we get x ≥ -2.
Therefore, the domain of g(f(x)) is x ≥ -2, which can be written in interval notation as (-∞, -2] or [-2, ∞).
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Ramon wants to cut a rectangular board into identical square pieces. If the board is 18 inches by 30 inches, what is the least number of square pieces he can cut without wasting any of the board?
Answer: 15
Step-by-step explanation:
Answer:
he have to kep cutting them until he find out he answer
Step-by-step explanation:
Simplify 3^1/2 * 3^1/2. Show work
[tex]\(3^\frac{1}{2} \cdot 3^\frac{1}{2} = 3\)[/tex].
To simplify [tex]\(3^\frac{1}{2} \cdot 3^\frac{1}{2}\)[/tex], you can use the properties of exponents.
When you multiply two powers with the same base, you add their exponents:
[tex]\[a^m \cdot a^n = a^{m+n}\][/tex]
In this case, both exponents are [tex]\(\frac{1}{2}\)[/tex], so when you multiply them together, you add the exponents:
[tex]\[3^\frac{1}{2} \cdot 3^\frac{1}{2} = 3^{\frac{1}{2} + \frac{1}{2}}\][/tex]
[tex]\[= 3^1\][/tex]
= 3
So, [tex]\(3^\frac{1}{2} \cdot 3^\frac{1}{2} = 3\)[/tex].
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In triangle $ABC$, the measure of angle $A$ is $x$ degrees, the measure of angle $B$ is $2x$ degrees and the measure of angle $C$ is $5x$ degrees. What is the value of $x$? Express your answer as a decimal to the nearest tenth.
Answer:
22.5
Step-by-step explanation:
All of the angles inside the triangle equals 180. Therefore, the equation is x+2x+5x=180. Then, you solve for x. The final equation should look like 8x=180 And that is how we get 22.5
Answer:
22.5
Step-by-step explanation:
The sum of the interior angles in a triangle is 180 degrees, so we have the equation $x+2x+5x=180$, so $x=\boxed{22.5}$.
In a sample of n = 6 scores, the smallest score is X = 3, the largest score is X = 10, and the mean is M = 6. If the largest score is changed from X = 10 to X = 22, then what is the value of the new mean?
Answer:
8
Step-by-step explanation:
Given that in a sample of n = 6 scores, the smallest score is X = 3, the largest score is X = 10
Mean = 6
Since mean = 6 we get sum of all the 6 scores = [tex]6(6) = 36[/tex]
Now II part says 10 is changed to 20
i.e. original sum = 36
Changed value = 10
Adjusted value =26
Add: new value =22
New sum =48
So we have sum = 48
New mean= [tex]\frac{48}{6} =8[/tex]
(This can also be done using the formula
old mean + positive change in one score/6)
Final Answer:
The new mean after changing the largest score from 10 to 22 is 8.
Explanation:
To solve this problem, we will follow these steps:
1. Calculate the total sum of all scores using the original mean.
2. Subtract the original largest score from the total sum.
3. Add the new largest score to the total sum to get the new total sum.
4. Calculate the new mean by dividing the new total sum by the sample size.
Let's go through these steps one by one:
Step 1: Calculate the original total sum of scores.
Given that the mean (M) is 6 for a sample size (n) of 6:
Total sum of scores (original) = Mean × Sample size = M × n = 6 × 6 = 36
Step 2: Subtract the original largest score from the total sum.
Original total sum = 36 (from Step 1)
Original largest score = 10
Total sum after removing the original largest score = 36 - 10 = 26
Step 3: Add the new largest score to the total sum to get the new total sum.
Total sum after removing the original largest score = 26 (from Step 2)
New largest score = 22
New total sum of scores = 26 + 22 = 48
Step 4: Calculate the new mean.
New total sum of scores = 48 (from Step 3)
Sample size (n) = 6
New mean = New total sum of scores ÷ Sample size = 48 ÷ 6 = 8
Therefore, the new mean after changing the largest score from 10 to 22 is 8.
The supreme choice pizza at Pizza Paradise contains 2 different meats and 2 different vegetables. The customer can select any one of 6 types of crust. If there are 4 meats and 9 vegetables to choose from, how many different supreme choice pizzas can be made
There are 1,296 different ways to make supreme choice pizza.
Step-by-step explanation:
Here, the total number of crusts available = 6
The number of crust to be chosen = 1
So, the number of ways that can be done = [tex]^6 C_1 = 6[/tex] ways ...... (1)
Similarly, the total number of meats available = 4
The number of types meats to be chosen = 2
So, the number of ways that can be done = [tex]^4 C_2 = 6[/tex] ways ...... (2)
Similarly, the total number of vegetables available = 9
The number of types vegetables to be chosen = 2
So, the number of ways that can be done = [tex]^9 C_2 = 36[/tex] ways ...... (3)
Now, combining (1), (2) and (3):
The number of ways one can choose 1 crust, 2 meat and 2 vegetables
= 6 ways x 6 ways x 36 ways = 1,296 ways
Hence, there are 1,296 different ways to make supreme choice pizza.
Chris wants to fence three sides of a rectangular exercise yard for his dog. The fourth side of the exercise yard will be a side of the house. He has
120
feet of fencing available. Find the dimensions that will enclose the maximum area.
The fence parallel to the house is ___
feet, the fence perpendicular to the house is ____
feet and the area of the yard is ____
square feet.
Answer:
parallel: 60 ftperpendicular: 30 ftarea: 1800 ft^2Step-by-step explanation:
Let x represent the length of fence parallel to the house. Then the length perpendicular is ...
y = (120 -x)/2
The area of the yard is the product of these dimensions, so is ...
A = xy = x(120-x)/2
This is the equation of a parabola that opens downward and has zeros at x=0 and x=120. The maximum (vertex) is on the line of symmetry, halfway between these zeros, at x=60.
The fence parallel to the house is 60 feet.
The fence perpendicular to the house is (120-60)/2 = 30 feet.
The area of the yard is (60 ft)(30 ft) = 1800 ft^2.
A small pizza has a diameter of 10 inches. A slice had a central angle of π/3 radians. What is the area of the slice?
The area of the slice is 13.0899 inch².
Explanation:
The pizza has an angle of 360°. If each slice has a central angle of π/3 = 60° then the number of slices = [tex]\frac{thetotalangleofthepizza}{theangleofoneslice}[/tex] = [tex]\frac{360}{60}[/tex] = 6 slices. So the pizza has 6 slices.To calculate one slice's area, we calculate the the entire pizza's area and divide it by 6 (number of slices).The circle's area is given by multiplying π with the square of its radius (r²). If the diameter is 10 inches, the radius is half i.e. the radius = 5 inches.The area of the pizza = π × 5 × 5 = 78.5398 inch². The area of the slice = [tex]\frac{78.5398}{6}[/tex] = 13.0899 inch².