Answer:
The formula to find the sum of the first n terms of our sequence is n divided by 2 times the sum of twice the beginning term, a, and the product of d, the common difference, and n minus 1. The n stands for the number of terms we are adding together.
A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position has the same duties and voting rights. There are 9 faculty members and 10 students eligible to serve on the committee. In how many ways can the committee be formed
Answer:
The committee can be formed in 31,752 ways.
Step-by-step explanation:
Every committee position has the same duties and voting rights. This means that the order in which the members are chosen is not important to solve this question. So the combinations formula is used.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In how many ways can the committee be formed
4 faculty members from a set of 9
5 students from a set of 10.
Then
[tex]T = C_{9,4} \times C_{10,5} = \frac{9!}{4!(9-4)!} \times \frac{10!}{5!(10-5)!} = 31752[/tex]
The committee can be formed in 31,752 ways.
HELP ME PLEASE, it’s confusing
The number at the end of the equation shifts the graph up or down.
Changing the 1/2 to a -2 would shift the graph down which would change the y-intercept.
The answer is D.
Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.25cm and a standard deviation of 0.43cm. Using the empirical rule, what percentage of the apples have diameters that are less than 6.82cm? Please do not round your answer.
Answer:
16% of the apples have diameters that are less than 6.82cm
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 7.25cm
Standard deviation = 0.43cm
The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.
Using the empirical rule, what percentage of the apples have diameters that are less than 6.82cm?
6.82 = 7.25 - 1*0.43
So 6.82 is one standard deviation below the mean.
Of the 50% of the diameters below the mean, 68% are within 1 standard deviation of the mean.
So 100-68 = 32% are not within 1 standard deviation of the mean.
0.5*0.32 = 0.16
16% of the apples have diameters that are less than 6.82cm
- + 106 = 315 cubic inches
2. Find the volume of the blue rectangular prism first.
v=-
cubic inches
V=
cubic inches
Total volume =
X
x
9 in.
X
x
Sin
Answer:585 cubic inch
Step-by-step explanation:
volume=15 x 10 x 3 + 9 x 3 x 5
Volume=450 + 135
Volume=585
Are two regular hexagons always congruent?
Answer:
Polygons - Hexagons. So, the sum of the interior angles of a hexagon is 720 degrees. All sides are the same length (congruent) and all interior angles are the same size (congruent).
Step-by-step explanation:
Search google
Final answer:
Two regular hexagons are congruent when they have equal side lengths, making them identical in shape and size. The definition of a regular hexagon as having equal sides and angles, paired with the criteria for congruence, confirms this. However, without equal side lengths, hexagons may only be similar, not congruent.
Explanation:
The question of whether two regular hexagons are always congruent comes down to understanding the definition of both a regular hexagon and congruence in geometry. A regular hexagon is a six-sided polygon where all sides and angles are equal. Congruent figures, on the other hand, are those that are identical in shape and size, meaning their corresponding sides and angles are equal. Thus, two regular hexagons that have the same lengths of sides are indeed congruent because they will have identical shapes and sizes, fulfilling the criteria for congruence.
However, it's important to note that while all regular hexagons will be similar (having the same shape but not necessarily the same size), they are only congruent if their side lengths are equal. This distinction is key in understanding the conditions under which two regular hexagons can be considered congruent. Without the specification of size, two hexagons can still be similar but not congruent.
In summary, two regular hexagons are congruent when they have the same side lengths, hence matching in both shape and size. This specific condition must be met for congruence to hold between any two geometric figures, including regular hexagons.
2. A random sample of 29 employees of a large company has their systolic blood pressure checked. Summary statistics are provided in the table below. Assume that the systolic blood pressure of all U.S. adults follows a normal distribution with a mean of 122 mm Hg and a standard deviation of 20 mm Hg.
(a) Approximately what percent of all U.S. adults have systolic blood pressure greater than 142 mm Hg?
(b) Describe the distribution of systolic blood pressure for the 29 employees of this company that was sampled.
(c) The company CEO wants to know if the mean systolic blood pressure of employees at her company is higher than the national average. State the hypotheses for testing this concern.
(d) The conditions for the hypothesis test in part (c) were satisfied. The hypothesis test resulted in a t-score of 5.495 and a p-value of 3.591 × 10−6. Interpret the p-value in the context of this hypothesis test. What would this p-value lead you to conclude?
(e) Explain in context what it would mean to make a type II error for the hypothesis test in part (c).
In addition to recording the systolic blood pressure of the 29 employees at the company, their ages were also recorded. A linear regression model was fit to these data. Graphical and numerical summaries of this analysis are given below. Use this information to answer the questions that follow. (Image is attached below)
(f) Interpret the slope of the regression line in this context.
(g) Comment on the strength, direction, and form of the relationship between age and systolic blood pressure.
Answer:
Step-by-step explanation:
Hello!
The variable of interest is
X: Systolic blood pressure of a U.S. adult. (mmHg)
X~N(μ;σ²)
μ= 122 mmHg
σ= 20 mmHg
For all calculations you have to work under the standard normal distribution, because it is tabulated. It is easier and faster to standardize or "translate" all values of X into values of Z and look for the corresponding probabilities in the table than to manually calculate them using the density function of the normal distribution.
The standard normal distribution is derived from the normal distribution. Considering a random variable X with normal distribution, mean μ and variance δ², the variable Z =(X-μ)/δ ~N(0;1) is determined.
Any value of any random variable X with normal distribution can be "converted" by subtracting the variable from its mean and dividing it by its standard deviation.
Since the distribution is centered in zero, there are two entries for its table, the left entry shows the cumulated probabilities corresponding to negative values of Z: P(Z≤z)=α and the right entry show the cumulated probabilities corresponding to positive values of Z: P(Z≤z)= 1 - α
a)
You need to calculate the percentage/ proportion of U.S. adults that have a systolic pressure greater than 142 mmHg, symbolically:
P(X>142) = 1 - P(X≤142)
To standardize this value od the variable you have to do the following calculation:
Z =(X-μ)/δ= (142-122)/20= 1
Now you look in the right entry for the cumulative probability until z= 1.00
(Remember, the first column of the table shows you the integer and first digit, the first row of the table shows you the second integer of the z value)
P(Z≤1)= 0.84134
Now you calculate the asked value:
P(X>142) = 1 - P(X≤142)= 1 - P(Z≤1)= 1 - 0.84134= 0.15866
b)
The sample mean is derived from a random variable with a normal distribution it shares that distribution with the exception that its variance is directly affected by the sample size:
X[bar]~N(μ;σ²/n)
μ= 122 mmHg
σ/√n= 20/√29= 3.71 mmHg
c)
The claim is that the mean systolic pressure of the employees is higher than the national average, symbolically μ> 122
The statistical hypotheses are:
H₀: μ≤ 122
H₁: μ> 122
d)
t= 5.495 p-value: 0.000003591
In this example the test statistic depends on the mean and the p-value is 3.591*10⁻⁶. This value indicates that 0.0003591% of the samples with size n=29 taken from a population with mean 122 mmHg, will produce a mean that provides evidence as (or stronger) than the current sample that μ is not at most 122 mmHg.
e)
The type II error is the scenario when you fail to reject the null hypothesis when the hypothesis is false. In this case, it is to fail to reject that the average systolic pressure of the company's employees is at most as the national average.
f)
The variable "X: Age of an employee" was recorded and linear regression of the systolic blood pressure as a function of the employee's ages estimated.
^Y= 97.0771 + 0.9493Xi
0.9493mmHg/years is the modification of the estimated average systolic blood pressure of the company's employees when their age increases one year.
g)
To determine the type of linear regression between the two variables, you have to analyze the slope of the equation:
As you can see in the graphic, the slope of the regression is positive, which means there is a positive regression between the systolic pressure and the age of the employees. I.e. each time the age of the employee increases, his systolic pressure also increases.
To determine the strength of the regression between these two variables you have to analyze the coefficient of determination R²:
The coefficient of determination gives you an idea of how much of the variability of the dependent variable (Y) is due to the explanatory variables under the estimated regression. It takes values between 0 to 1 or 0 to 100% if expressed in percentage. The closer the coefficient is to zero, the weaker the relationship between these two variables.
The closer the coefficient is to 100%, the stronger the relationship between these two variables.
R²= 0.712
71.2% of the variability of the systolic pressure is explained by the age fo the employees under this estimated model ^Y= 97.0771 + 0.9493Xi
The relationship between these variables is strong enough to consider the regression.
I hope this helps!
help, Which equation should be used to find the volume of the figure?
Answer:
V = (10*6*12)/3
V = 720/3
V = 240 cm^3
Step-by-step explanation:
V = LWH/3
L = base length
W = base width
H = pyramid height
Please, I need help on this problem!
Answer:
$676
Step-by-step explanation:
26 percent of 2600 is 676.
2600 x .26
Write an exponential function in the form y = ab that goes through points (0,18)
and (2,288).
Final answer:
The desired exponential function that goes through the points (0,18) and (2,288) is y=18 × 4^x.
Explanation:
To write an exponential function in the form y = ab^x that passes through the points (0,18) and (2,288), we first use the given points to find the values of a and b.
When x = 0, the equation simplifies to y = a because b0 equals 1. Therefore, from the point (0,18), we can conclude that a equals 18.
Next, substituting x = 2 and y = 288 into the equation and using the value of a, we have 288 = 18 × b^2. To find b, divide both sides by 18 to get b^2 = 16, which means b = 4 since b is positive in the growth of an exponential function.
So the desired exponential function is y = 18 × 4^x.
kevin started his new job with a salaryof $26,550. every year, he recieves a 3.1% increase in his salary. write and use a continuous exponential growth to find his salary after 20 years?
its $48892.00
Answer:
His salary after 20 years is $48892
Step-by-step explanation:
Kevin started his new job with a salary of $26,550.
We are also given that every year, he receives a 3.1% increase in his salary
So, rate of increase = 3.1%
Formula : [tex]y(t)=y_0(1+r)^t[/tex]
Where y(t)= Amount after t years
[tex]y_0[/tex] = initial amount = 26550
r = rate of increase in decimals = [tex]\frac{3.1}{100}[/tex]
t = time =20
we are supposed to find his salary after 20 years
Substitute the values in the formula :
Salary after 20 years = [tex]26500(1+\frac{3.1}{100})^{20}=48892.00[/tex]
Hence His salary after 20 years is $48892
A company has a policy of retiring company cars; this policy looks at number of miles driven, purpose of trips, style of car and other features. The distribution of the number of months in service for the fleet of cars is bell-shaped and has a mean of 64 months and a standard deviation of 5 months. Using the empirical rule, what is the approximate percentage of cars that remain in service between 69 and 74 months?
Answer:
By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 64
Standard deviation = 5.
Using the empirical rule, what is the approximate percentage of cars that remain in service between 69 and 74 months?
69 = 64 + 1*5
So 69 is one standard deviation above the mean.
74 = 64 + 2*5
So 74 is two standard deviations above the mean.
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
This means that:
95 - 68 = 27% between 1 and 2 standard deviations of the mean.
The normal distribution is symmetric, which means that 27/2 = 13.5% of those are below the mean(within 1 and 2 standard deviations below the mean) and 13.5% of those are above the mean(between 69 and 74).
So
By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.
1. Which of the following is a binomial?
A 5b2 + 3b - 8
B-2x} + 14x=1
C -9a - 15
D-v3 + 4uv+'lly - 1
Answer:
C -9a - 15
Step-by-step explanation:
bionomial means two terms
Shring's dining room table is round
and has a radius of 45 feet What is
the tabletop's area?
Answer:
it is 6,358.2. hope this helps
Write the equation of the circle with a center (5,4) and a radius of 7.
Answer:(x-5)^2+(y-4)^2=49
Step-by-step explanation:
equation of a circle is:
(x-h)^2+(y-k)^2=r^2
(x-5)^2+(y-4)^2=7^2
(x-5)^2+(y-4)^2=7x7
(x-5)^2+(y-4)^2=49
The equation of the circle of center (5, 4) and radius 7, is:
(x - 5)^2 + (y - 4)^2 = 49
How to find the equation of the circle?
The general equation for a circle of radius R centered at the point (a, b) is:
(x - a)^2 + (y - b)^2 = R^2
In this case, the center is (5, 4) and the radius is 7, so we need to replace:
a = 5
b = 4
R = 7
Then the equation is:
(x - 5)^2 + (y - 4)^2 = 7^2 = 49
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Your small farm encompasses 120 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $3,000 for each acre of tomatoes, $1,400 for each acre of lettuce and $400 for each acre of carrots. You would like to spend at least $480 on fertilizer and your farm laborers can supply up to 600 hours per week. How many acres of each crop should you plant to maximize total profits?
tomatoes acre(s) =
lettuce acre(s) =
carrots acre(s)=
profit $ =
In this event, will you be using all 120 acres of your farm?
Answer:
tomatoes acre(s) = 120lettuce acre(s) = 0carrots acre(s)= 0profit $ = $360,000all 120 acres are usedStep-by-step explanation:
You can write the linear system model as follows. Let t, l, c represent acres of tomatoes, lettuce, and carrots, respectively. The we have ...
t + l + c ≤ 120 . . . . . constraint on available land
5t +4l +2c ≥ 480 . . requirement for spending on fertilizer
4t +2l +2c ≤ 600 . . constraint on available labor
t ≥ 0; l ≥ 0; c ≥ 0 . . requirement for non-negative acres
Then the objective function (profit) is ...
p = 3000t +1400l +400c
The linear programming problem is to maximize p subject to the above constraints.
__
Any of a variety of solvers can find the solution to this problem. That solution is ...
(t, l, c) = (120, 0, 0) and p = 360,000
In summary ...
tomatoes acre(s) = 120 (all available acres are used)
lettuce acre(s) = 0
carrots acre(s) = 0
profit $ = $360,000
_____
Additional comments
This solution suggests that it can be found simply by examining the profit associated with each unit of resource.
profit per acre is maximized for tomatoes, at $3000 per acre
profit per fertilizer dollar is maximized for tomatoes, at $600 per dollar
profit per labor hour is maximized for tomatoes, at $750 per hour
That is, the profit per acre is maximized for tomatoes, regardless of the resource being considered. Thus it make sense to put all of the acreage in tomatoes. At $5 per acre for fertilizer, we use $600 worth of fertilizer. At 4 hours per week per acre, we use 480 hours of labor, so not all available labor is used.
To maximize profits, use linear programming to find the number of acres for each crop. The total farm acres used depends on the constraints.
Explanation:To maximize total profits, the number of acres of each crop should be determined. Let's assign the variables t, l, and c to represent the number of acres of tomatoes, lettuce, and carrots, respectively. We have the following constraints:
The total cost of fertilizer must be at least $480: 5t + 4l + 2c ≥ 480The total hours of labor must not exceed 600: 4t + 2l + 2c ≤ 600We want to maximize total profits: Profit = 3000t + 1400l + 400cThe objective is to find the values of t, l, and c that satisfy the constraints and maximize the profit. This is a linear programming problem that can be solved using methods such as the simplex method or graphical method.
To determine if all 120 acres will be used, we need to check if t + l + c = 120 under the constraints.
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If the tax rate on homes in Topeka, Kansas is 1.8 percent of their value and Bonnie owns a $100,000 house, how much will the real estate taxes add to her monthly mortgage payment?
the real estate taxes add to her monthly mortgage payment?
Answer:
$150
Step-by-step explanation:
Annual Real Estate Tax Rate =1.8%
Value of Bonnie's House = $100,000
Annual Tax= 1.8 % × $100,000 =0.018 × $100,000 = $1,800
Therefore, tax payment on a monthly basis
= $1,800÷12 Months
= $150 per month.
Real estate tax will add $150 per month to Bonnie's mortgage payment.
Final answer:
The real estate taxes on Bonnie's $100,000 house at a 1.8 percent tax rate add $150 to her monthly mortgage payment, after calculating the annual tax and dividing by 12 months.
Explanation:
To calculate the real estate taxes that will add to Bonnie's monthly mortgage payment, we need to apply the tax rate to the value of the home and then divide by 12 to find the monthly amount.
The annual real estate taxes would be calculated as follows:
Therefore, the real estate taxes will add $150 to Bonnie's monthly mortgage payment.
Use the complement to find the probability. Enter your answer in simplified fraction form.
A spinner has 3 equal sections that are white, green, and blue. What is the probability of not landing on blue?
The probability of not landing on blue is
The system of equations y = three-fourths x minus 4 and y = –x + 3 is shown on the graph below.
On a coordinate plane, 2 lines intersect at (4, negative 1).
According to the graph, what is the solution to this system of equations?
(4, –1)
(–1, 4)
(4, 1)
(1, 4)
Answer:
(4, -1)
Step-by-step explanation:
The point of intersection of two lines is also the solution to the system of equations of the two lines
Step-by-step explanation:
The answer to this question is where both of the lines intersect. For this problem, the answer is (4, -1)
Look at the attachment for clarification:
Find the value of g(7) for the function below. g(x)= 7/8x -1/2
g(x) = ⅞x - ½
g(7) = ⅞(7) - ½
= 49/8 - ½
= 49/8 - 4/8
= 45/8
= 5 ⅝
So, the value of g(7) for that function is 5 ⅝
Hope it helpful and useful :)
The value of the given function for g(7) is 45/8.
The given function is [tex]g(x)=\frac{7}{8}x-\frac{1}{2}[/tex].
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
We need to find the value of g(7)
Substitute x=7 in the given function and simplify, we get
[tex]g(7)=\frac{7}{8}(7)-\frac{1}{2}[/tex]
= 49/8 - 1/2
= 49/8 - 4/8
= 45/8
Hence, the value of the given function for g(7) is 45/8.
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find the amount each contestant would carry if the total amount carried (5 cups) was redistributed equally among all the contestants.
Complete Question
The line plot shows the amount of green ooze (in cup) carried by ten contestants. Find the amount each contestant would carry if the total amount carried (5 cups) was redistributed equally among all the contestants.
Answer:
1/2 Cup
Step-by-step explanation:
This is a Line Plot in which:
1/4 had two Xs3/8 had two Xs1/2 had two Xs5/8 had three Xs7/8 had one XsTotal
[tex]=(\frac{1}{4}X2) +(\frac{3}{8}X2) +(\frac{1}{2}X2) +(\frac{5}{8}X3) +(\frac{7}{8}X1) \\\\=\frac{2}{4} +\frac{6}{8} +1 +\frac{15}{8} +\frac{7}{8}\\\\$=5 Cups[/tex]
If this amount was redistributed equally among the ten contestants:
Each contestant would carry (5/10) cups =1/2 cup.
elena lists the ages of all of the Family members who live in our house below what is the mean absolute deviation of all their ages?
Answer:
19
Step-by-step explanation:
We'll begin by calculating the mean of the data. This is illustrated below:
Mean = summation of data / number of data
Data from the question: 9, 13, 43, 55
Mean = (9 + 13 + 43 + 55)/4
Mean = 120/4
Mean = 30.
Now, we can calculate the mean absolute deviation (MAD) as follow:
MAD = summation of the absolute deviation from the mean / number of data.
MAD = [ |9–30| + |13–30| + |43–30| + |55–30| ] /4
MAD = [ 21 + 17 + 13 + 25] /4
MAD = 76/4
MAD = 19.
Therefore, the mea absolute deviation (MAD) of their ages is 19
Answer:
19
Step-by-step explanation:
Kyle and Julie are playing a game where they flip a fair coin four times and try to predict the outcomes. ... Using the sample space of possible outcomes listed below, answer each of the following questions. What is P(A)P(A)P, left parenthesis, A, right parenthesis, the probability that the first flip is heads?
Answer: 1/2
Step-by-step explanation:
144
A. Is not a perfect square
B. Is a perfect square
Use a letter to represent the unknown Twenty packs of fruit snacks come in a box. Each pack weighs 6 ounces. Students eat some. There are 48 ounces of fruit snacks left in the box. How many ounces of fruit snacks did the students eat?
Answer:
The students have eaten 72 ounces of fruit snacks.
X = 72 ounces
Step-by-step explanation:
There are total 20 packs of fruit snacks
The weight of each pack is 6 ounces.
Let us first find the total weight of the snacks.
Total weight = 6*20
Total weight = 120 ounces
Let X represents the ounces of fruit snacks that students have eaten.
Let Y represents the 48 ounces of fruit snacks left in the box
Total weight = X + Y
120 = X + 48
X = 120 - 48
X = 72 ounces
Therefore, the students have eaten 72 ounces of fruit snacks.
Answer:
x = 72 ounces
The number of ounces of fruit snacks the students ate is 72 ounces
Step-by-step explanation:
Given;
Let x represent the number of ounces of fruit the students ate;
The total number of ounces of fruit initially in the box is;
20 packs × 6 ounces/pack = 120 ounces
The remaining ounces of fruit is;
48 ounces
Therefore;
120 - x = 48
Solving for x;
x = 120 - 48
x = 72 ounces
The number of ounces of fruit snacks the students ate is 72 ounces
What is the area of the sector, in square units, determined by an arc with a measure 45° in a circle with radius 6? Round to the nearest tenth
Answer:
14.1 u^2
Step-by-step explanation:
A=pi*6^2*45/360=3.14*36*45/360=3.14*45/10≈14.1
What percentage of the Mountain View polled supported Olunloyo?
Answer: The answer is C (about 47%) because the total amount of Mountain View residents is 0.38, 0.18 people chose Olunloyo, which is about 47 percent of residents.
To find out what percentage of the Mountain View polled supported Olunloyo, divide the number of Olunloyo supporters by the total number of poll participants, and multiply by 100. This reflects the views of the polled sample, not the entire population.
Explanation:To determine what percentage of the Mountain View polled supported Olunloyo, you would first need to know the total number of people who participated in the poll from Mountain View. Then, find out the number of these participants who indicated their support for Olunloyo. Divide the number of supporters by the total number of participants, and multiply the result by 100. This will give you the percentage of Mountain View residents in the poll who supported Olunloyo. Remember that this only applies to the sample of the population that participated in the poll and may not reflect the entire population of Mountain View.
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What is the length of the diagonal?
Answer:
The value of x is equal to 26 cm.
Step-by-step explanation:
We can use the Pythagorean Theorem to solve this problem.
[tex]a^2+b^2=c^2[/tex]
[tex]a=10\\b=24\\c=?[/tex]
[tex]10^2+24^2=c^2[/tex]
[tex]100+576=c^2[/tex]
[tex]676=c^2[/tex]
[tex]\sqrt{676} =\sqrt{c^2}[/tex]
[tex]26=c[/tex]
Students in a statistics class are conducting a survey to estimate the mean number of hours each week that students at their college study. The students collect a random sample of 49 students. The mean of the sample is 12.2 hours. The standard deviation is 1.6 hours. Use the T-distribution Inverse Calculator applet to answer the following question. What is the 95% confidence interval for the number of hours students in their college study?
The 95% confidence interval for the number of hours students at the college study weekly, given a sample size of 49, mean of 12.2 hours, and standard deviation of 1.6, is from 11.62 hours to 12.78 hours.
Explanation:In statistics, the 95% confidence interval for the mean number of hours students study in a week can be calculated using the collected data and the t-distribution. Given that the sample size is 49 (n=49), the sample mean(x) is 12.2 hours, and the standard deviation(s) is 1.6 hours, we need to apply the formula for the confidence interval which is x ± t∗(s/√n). The t-value for 95% confidence with 48 degrees of freedom (49-1) can be found using the t-distribution Inverse Calculator applet, which is about 2.01 (it varies slightly depending on the calculator used).
Now, plug in the given values into the formula: 12.2 ± 2.01 * (1.6 / √49), it simplifies to 12.2 ± 0.58. So, the 95% confidence interval for the number of hours students in the college study each week is from 11.62 hours to 12.78 hours.
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Solve the logarthmic equation. 2 log3(x + 4) - log39 = 2
Answer:
5
Step-by-step explanation:
2 log₃(x + 4) − log₃9 = 2
2 log₃(x + 4) − log₃3² = 2
2 log₃(x + 4) − 2 = 2
2 log₃(x + 4) = 4
log₃(x + 4) = 2
3^(log₃(x + 4)) = 3^2
x + 4 = 9
x = 5
The solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.
To solve the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2, we can use logarithmic properties to simplify the equation and then solve for x.
First, let's apply the properties of logarithms:
2 log₃(x + 4) - log₃(9) = 2
Using the power rule of logarithms, we can rewrite 2 log₃(x + 4) as log₃((x + 4)²):
log₃((x + 4)²) - log₃(9) = 2
Next, we can combine the logarithms using the quotient rule:
log₃((x + 4)² / 9) = 2
Now, we can eliminate the logarithm by rewriting the equation in exponential form:
3² = (x + 4)² / 9
9 = (x + 4)² / 9
Multiplying both sides of the equation by 9:
81 = (x + 4)²
Taking the square root of both sides:
±√81 = ±√((x + 4)²)
±9 = x + 4
Solving for x, we have two possible solutions:
x + 4 = 9
x = 9 - 4
x = 5
or,
x + 4 = -9
x = -9 - 4
x = -13
Therefore, the solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.
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