Answer:
The sequence of functions [tex]\{x^{n}\}_{n\in \mathbb{N}}[/tex] converges to the function
[tex]f(x)=\begin{cases}0&0\leq x<1\\1&x=1\end{cases}[/tex].
Step-by-step explanation:
The limit [tex]\lim_{n\to \infty }c^{n}[/tex] exists and converges to zero whenever [tex]\lvert c \rvert <1 [/tex]. But, if [tex]c=1[/tex] the sequence [tex]\{c^{n}\}[/tex] is constant and all its terms are equal to [tex]1[/tex], then converges to [tex]1[/tex]. Using this result, consider the sequence of functions [tex]\{f_{n}\}[/tex] defined on the interval [tex][0,1][/tex] by [tex]f_{n}(x)=x^{n}[/tex]. Then, for all [tex]0\leq x<1[/tex] we have that [tex]\lim_{n\to \infty}x^{n}=0[/tex]. Now, if [tex]x=1[/tex], then [tex]\lim_{n\to \infty }x^{n}=1[/tex]. Therefore, the limit function of the sequence of functions is
[tex]f(x)=\begin{cases}0&0\leq x<1\\1&x=1\end{cases}[/tex].
To show that the convergence is not uniform consider [tex]0<\varepsilon<1[/tex]. For any [tex]n>1[/tex] choose [tex]x\in (0,1)[/tex] such that [tex]\varepsilon^{1/n}<x<1[/tex]. Then
[tex]\varepsilon <x^{n}=\lvert f(x)-f_{n}(x)\rvert[/tex]
This implies that the convergence is not uniform.
Larry Calanan has earnings of S518 in a week. He is single and claims 2 withholding allowances. His deductions include FICA, Medicare, federal withholding, state disability insurance, state withholding, union dues of $15, and charitable contributions of $21. Find his net pay.
Answer:
$482
Step-by-step explanation:
Data provided:
Total earning per week = $518
Medicare, federal withholding, state disability insurance, state withholding, union dues = $15
charitable contributions = $21
Now,
The total deductions = $15 + $21 = $36
also,
Net pay = Total income - Total deductions
thus,
Net pay = $518 - $36
or
Net Pat = $482
how many distinguishable a arrangements are there of the letters in "REPRESENTATION"?
Answer: There are 1,816,214,400 ways for arrangements.
Step-by-step explanation:
Since we have given that
"REPRESENTATION"
Here, number of letters = 14
There are 2 R's, 3 E's, 2 T's, 2 N's
So, number of permutations would be
[tex]\dfrac{14!}{2!\times 3!\times 2!\times 2!}\\\\=1,816,214,400[/tex]
Hence, there are 1,816,214,400 ways for arrangements.
Write a meaningful negation for the following statement.
The shirt I’m wearing to my interview is orange.
Answer:
The negation will be : The shirt I am wearing to my interview is not orange.
Step-by-step explanation:
The negation of the statement means adding not, or nor to the statement.
The given statement is :
The shirt I’m wearing to my interview is orange.
The negation will be : The shirt I am wearing to my interview is not orange.
Convert 120 kW to W? MW?
Answer: 120000 W and 0.12 MW
Step-by-step explanation:
The expression 120 kW uses a metric prefix "k" (kilo) which is the same as multiply by 1000. So you can replace k by 1000 to convert the expression to the unit W.
120 kW= 120(1000) W= 120000 W.
To convert 120kW to MW, where the prefix M (mega) is equivalent to 1000000, you can use a conversion factor like (1 MW / 1000 kW) and multiply the expression by it.
Notice that (1 MW / 1000 kW) = 1, so the expression remains unaltered.
Then,
120 kW (1 MW / 1000 kW) = 0.12 MW
The population of a suburb grows at a rate proportional to the population. Suppose the population doubles in size from 3000 to 6000 in a 6- month period and continues at the current rate of growth. Find the particular solution to the differential equation with the initial condition P(0)=3000.
The particular solution to the differential equation with the initial condition P(0) = 3000 is given by P = 3000e^((ln(2)/6)t), where P represents the population at time t. The growth rate is proportional to the population, and the constant of proportionality is ln(2)/6. This equation can be derived by solving the differential equation and using the initial condition.
Explanation:In this problem, we are given that the population of a suburb grows at a rate proportional to the population. We are also given that the population doubles in size from 3000 to 6000 in a 6-month period. We need to find the particular solution to the differential equation with the initial condition P(0)=3000.
Let's denote the population at time t as P(t). Since the growth rate is proportional to the population, we can write the differential equation as dP/dt = kP, where k is the proportionality constant.
Integrating both sides of the equation, we get: ∫dP/P = ∫kdt. This gives us ln|P| = kt + C, where C is the constant of integration.
Using the initial condition P(0) = 3000, we can substitute t = 0 and P = 3000 into the equation to get: ln|3000| = 0 + C. Solving for C, we find C = ln|3000|.
Substituting C = ln|3000| back into the equation, we have ln|P| = kt + ln|3000|. Simplifying, we get ln|P| - ln|3000| = kt.
Since ln|P| - ln|3000| = ln|(P/3000)|, we can write the equation as ln|(P/3000)| = kt.
Taking the exponential of both sides, we get |(P/3000)| = e^(kt).
Since the population cannot be negative, we can remove the absolute value sign and write the equation as (P/3000) = e^(kt).
Substituting the doubling in size from 3000 to 6000 in a 6-month period, we have (6000/3000) = e^(k(6)).
Simplifying, we get 2 = e^(6k).
Taking the natural logarithm of both sides, we have ln(2) = ln(e^(6k)).
Using the property ln(a^b) = bln(a), we can rewrite the equation as ln(2) = 6kln(e).
Since ln(e) = 1, we have ln(2) = 6k.
Solving for k, we get k = ln(2)/6.
Substituting k = ln(2)/6 back into the equation, we have (P/3000) = e^((ln(2)/6)t).
Multiplying both sides by 3000, we get P = 3000e^((ln(2)/6)t).
This is the particular solution to the differential equation with the initial condition P(0) = 3000.
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9x = 99y
y = 2
x = ?
Answer:
x = 22
Step-by-step explanation:
9x = 99y
y = 2
9x = 99 * 2
99 * 2 = 198
9x = 198
--- ----
9 9
x = 22
Hey!
------------------------------------------------
Solution:
9x = 99y
~Substitute
9x = 99(2)
~Simplify
9x = 198
~Divide 9 to both sides
9x/9 = 198/9
~Simplify
x = 22
------------------------------------------------
Answer:
x = 22
------------------------------------------------
Hope This Helped! Good Luck!
After a statistics class, Dr. Johnson asked all 51 of her students the degree to which they felt statistics were going to be useful in their eventual career. She had all of her students respond to the following question, "Statistics will be very useful in my future profession," using the following scale: 1 = not at all useful and 7 = very useful. She found that the median response was a 5.5. Given that the median of 5.5 was based on all of her students, the 5.5 would be called __________.
Answer:
The 5.5 would be called a parameter.
Step-by-step explanation:
A parameter in statistics is a data/number/quantity that gives you information about an entire population. Given that Dr. Johnson asked ALL of her students to respond the question and the median of 5.5 was based on ALL of her students, we can say that this number would be a parameter for the population (in this case, Dr. Johnson's class)
(CO 3) On average, the parts from a supplier have a mean of 97.5 inches and a standard deviation of 6.1 inches. Find the probability that a randomly selected part from this supplier will have a value between 87.5 and 107.5 inches. Is this consistent with the Empirical Rule of 68%-95%-99.7%? Probability is 0.90, which is consistent with the Empirical Rule Probability is 0.95, which is inconsistent with the Empirical Rule Probability is 0.90, which is inconsistent with the Empirical Rule Probability is 0.05, which is consistent with the Empirical Rule Flag this Question
Answer:
C. Probability is 0.90, which is inconsistent with the Empirical Rule.
Step-by-step explanation:
We have been given that on average, the parts from a supplier have a mean of 97.5 inches and a standard deviation of 6.1 inches.
First of all, we will find z-score corresponding to 87.5 and 107.5 respectively as:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{87.5-97.5}{6.1}[/tex]
[tex]z=\frac{-10}{6.1}[/tex]
[tex]z=-1.6393[/tex]
[tex]z\approx-1.64[/tex]
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{107.5-97.5}{6.1}[/tex]
[tex]z=\frac{10}{6.1}[/tex]
[tex]z=1.6393[/tex]
[tex]z\approx 1.64[/tex]
Now, we need to find the probability [tex]P(-1.64<z<1.64)[/tex].
Using property [tex]P(a<z<b)=P(z<b)-P(z<a)[/tex], we will get:
[tex]P(-1.64<z<1.64)=P(z<1.64)-P(z<-1.64)[/tex]
From normal distribution table, we will get:
[tex]P(-1.64<z<1.64)=0.94950-0.05050 [/tex]
[tex]P(-1.64<z<1.64)=0.899[/tex]
[tex]P(-1.64<z<1.64)\approx 0.90[/tex]
Since the probability is 0.90, which is inconsistent with the Empirical Rule, therefore, option C is the correct choice.
By vector methods, find the cosine of the angle between the lines (x - 1)/(3) = (y - 0.5)/(2) = z and x = y = z
Answer:
The angle between the lines [tex]\frac{x-1}{3}= \frac{y-0.5}{2}=\frac{z-0}{1}[/tex] and [tex]\frac{x-0}{1}= \frac{y-0}{1}=\frac{z-0}{1}[/tex] is [tex]\sqrt{\frac{6}{7}}[/tex]
Step-by-step explanation:
The equation of a line with direction vector [tex]\vec{d}=(l,m.n)[/tex] that passes through the point [tex](x_{1},y_{1},z_{1})[/tex] is given by the formula
[tex]\frac{x-x_{1}}{l}= \frac{y-x_{1}}{m}=\frac{z-z_{1}}{n},[/tex] where l,m, and n are non-zero real numbers.
This is called the symmetric equations of the line.
The angle between two lines [tex]\frac{x-x_{1}}{l_{1} }= \frac{y-y_{1}}{m_{1} }=\frac{z-z_{1}}{n_{1}}[/tex] and [tex]\frac{x-x_{2}}{l_{2} }= \frac{y-y_{2}}{m_{2} }=\frac{z-z_{2}}{n_{2}}[/tex] equal the angle subtended by direction vectors, [tex]d_{1}[/tex] and [tex]d_{2}[/tex] of the lines
[tex]cos (\theta)=\frac{\vec{d_{1}}\cdot\vec{d_{2}}}{|\vec{d_{1}}|\cdot|\vec{d_{2}}|}=\frac{l_{1} \cdot\l_{2}+m_{1} \cdot\ m_{2}+n_{1} \cdot\ n_{2}}{\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2}} \cdot \sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}}[/tex]
Given that
[tex]\frac{x-1}{3}= \frac{y-0.5}{2}=\frac{z-0}{1}[/tex] and [tex]\frac{x-0}{1}= \frac{y-0}{1}=\frac{z-0}{1}[/tex]
[tex]l_{1}=3, m_{1}=2,n_{1}=1\\ l_{2}=1, m_{2}=1,n_{2}=1[/tex]
We can use the formula above to find the cosine of the angle between the lines
[tex]cos(\theta)=\frac{3 \cdot 1+2 \cdot 1 +1 \cdot 1}{\sqrt{3^{2}+2^{2}+1^{2}} \cdot \sqrt{1^{2}+1^{2}+1^{2}}} = \sqrt{\frac{6}{7}}[/tex]
20 units Humulin R insulin in 100 mL of normal saline (NS) to infuse for 20 hours. (Round to the nearest tenth if applicable) a. How many units per hour will be infused? ________ b. How many milliliters per hour will be infused? ________
Answer:
a) 1 unit per hour will be infused.
b) 5mL per hour will be infused.
Step-by-step explanation:
Each question can be solved as a rule of three with direct measures, that means we have a cross multiplication.
a. How many units per hour will be infused?
20 units are going to be infused in 20 hours. How many units are going to be infused each hour?
1 hour - x units
20 hours - 20 units
[tex]20x = 20[/tex]
[tex]x = \frac{20}{20}[/tex]
[tex]x = 1[/tex]
1 unit per hour will be infused.
b. How many milliliters per hour will be infused?
100mL are going to be infused in 20 hours. How many mL are going to be infused each hour?
1 hour - x mL
20 hours - 100 mL
[tex]20x = 100[/tex]
[tex]x = \frac{100}{20}[/tex]
[tex]x = 5[/tex]
5mL per hour will be infused.
In an arithmetic sequence, the nth term an is given by the formula an=a1+(n−1)d, where a1 is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1, where r is the common ratio. Use these formulas to determine the indicated term in the given sequence.
The 10th term of 40,10, 5/2, 5/8, ....
Answer:
The 10th term of given sequence is [tex]\frac{5}{32768}[/tex].
Step-by-step explanation:
The given sequence is
[tex]40,10, \frac{5}{2}, \frac{5}{8}, ....[/tex]
The given sequence is a geometric sequence because it have common ratio.
[tex]r=\frac{10}{40}=\frac{\frac{5}{2}}{10}=\frac{\frac{5}{8}}{\frac{5}{2}}=\frac{1}{4}[/tex]
In the given sequence the first term of the sequence is 40.
[tex]a_1=40[/tex]
The nth term of a GP is
[tex]a_n=a_1r^{n-1}[/tex]
where, [tex]a_1[/tex] is first term and r is common ratio.
Substitute [tex]a_1=40[/tex] and [tex]r=\frac{1}{4}[/tex] in the above formula.
[tex]a_n=40(\frac{1}{4})^{n-1}[/tex]
Substitute n=10 , to find the 10th term.
[tex]a_{10}=40(\frac{1}{4})^{10-1}[/tex]
[tex]a_{10}=\frac{5}{32768}[/tex]
Therefore the 10th term of given sequence is [tex]\frac{5}{32768}[/tex].
Input/Output Relationship: Assume that the amount of learning you acquire can be summarized by the following relationship (or equation) and that your motivation to learn and the quality of instruction are both rated on a scale from 1 to 10: Amount of learning acquired = 0.2(number of books read) + 0.25(hours spent studying) + 0.15(quality of instruction) + 0.4(motivation to learn). If your motivation to learn rises from 7 to 9, by how much will the amount of learning acquired rise as a result? Show your work.
Answer:
8%
Step-by-step explanation:
Motivation to learn represents a 40% of the learning acquired and its rated from 1 to 10. An increase from 7 to 9 represents an increase (of motivation to learn of 20%) But since this quality represent 40% of the total, the real increase in learning is 0.2*0.4=0.08 or 8%.
In high school, some students have been confused to believe that 22/7 is already the actual value of π or an acceptable approximation. Show that 355/113 is a better approximation in terms of absolute and relative errors.
Answer:
The absolute and relative error of 355/113 compared with π is less than when π is compared with 22/7, that's why 355/113 is a better approximation for the actual value of π.
Step-by-step explanation:
The absolute error is the difference between a value measured and the real value.
abs = π - approximation of π
The relative error indicates how large the absolute error is when compared with the actual value of π.
Now, let's calculate the absolute an relative error for each approximation of π, for simplicity the calculations will be rounded to 4 decimal digits.
rel = abs / π
For 22/7abs = π - 22/7
abs = -0.0013
rel = (π - 22/7) / π
rel = -0.0402 %
For 355/113abs = π - 355/113
abs = -2.6676 x10-7
rel = (π - 355/113) / π
rel = -8.4914 x10-6 %
You can see that both the value of the absolute and relative error for the 355/113 approximation are smaller numbers, in conclusion, 355/113 is a better approximation for π.
Is the following statement true of false? Givejustification if
true or a counterexample if false.
Any two quadrilaterals are similar if
theircorresponding angles are congruent.
Answer:
No , any two quadrilaterals may not be similar if their corresponding angles are congruent.
Step-by-step explanation:
We need to check that whether two quadrilaterals are similar if
their corresponding angles are congruent.
A quadrilateral is a polygon having two sides .
Two figures are said to be similar if they have same shape .
Two angles are said to be congruent if they have same measure .
Consider two quadrilaterals : rectangle and square
Each angle of square and rectangle is equal to [tex]90^{\circ}[/tex] . So, their corresponding angles are congruent .
But square and rectangle are not similar as they have different shape .
In words, the scalar product of two vectors can be thought of as the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a. It is used to calculate the product of vector quantities when only the parallel components of each vector contribute (e.g., Work = Force • Displacement). Let ~a = h9, 6.75, 0i and ~b = h2.97, 6.075, 0i. Calculate ~a • ~b.
Answer:
67.73625
Step-by-step explanation:
The dot (scalar) product is also the sum of products of corresponding vector components.
~a • ~b = 9·2.97 +6.75·6.075 +0·0 = 27.73 +41.00625 = 67.73625
The scalar product or dot product of two vectors, ~a and ~b, in Cartesian form is calculated by multiplying the matching components of the two vectors and then adding them. Performing these steps will give us 67.73625, which represents the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.
Explanation:In physics, the scalar product, also known as dot product, of two vectors like ~a = h9, 6.75, 0i and ~b = h2.97, 6.075, 0i can be determined using their magnitudes and the cosine of the angle between them. However, in the given question, the vectors are in Cartesian form (i,j,k coordinate system), and we can calculate their dot product directly. The dot product is calculated by multiplying the respective i, j, and k components of the two vectors and then adding them. Let's do this step by step:
The i component of our dot product is 9*2.97 = 26.73.The j component of our dot product is 6.75*6.075 = 41.00625The k component of our dot product becomes 0*0 = 0 as the k component is non-existent for both vectorsSo, the scalar product ~a • ~b = 26.73 + 41.00625 + 0 = 67.73625
This scalar product gives the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.
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Solve the initial value problem: y'(x)=(4y(x)+25)^(1/2) ,y(1)=6. you can't really tell, but the '1/2' is the exponent
Answer:
[tex]y(x)=x^2+5x[/tex]
Step-by-step explanation:
Given: [tex]y'=\sqrt{4y+25}[/tex]
Initial value: y(1)=6
Let [tex]y'=\dfrac{dy}{dx}[/tex]
[tex]\dfrac{dy}{dx}=\sqrt{4y+25}[/tex]
Variable separable
[tex]\dfrac{dy}{\sqrt{4y+25}}=dx[/tex]
Integrate both sides
[tex]\int \dfrac{dy}{\sqrt{4y+25}}=\int dx[/tex]
[tex]\sqrt{4y+25}=2x+C[/tex]
Initial condition, y(1)=6
[tex]\sqrt{4\cdot 6+25}=2\cdot 1+C[/tex]
[tex]C=5[/tex]
Put C into equation
Solution:
[tex]\sqrt{4y+25}=2x+5[/tex]
or
[tex]4y+25=(2x+5)^2[/tex]
[tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex]
[tex]y(x)=x^2+5x[/tex]
Hence, The solution is [tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex] or [tex]y(x)=x^2+5x[/tex]
Multi step equation and check 12x+3-1=35
Answer:
11/4
Step-by-step explanation:
Step 1: Simplify both sides of the equation.
12x+3−1=35
12x+3+−1=35
(12x)+(3+−1)=35(Combine Like Terms)
12x+2=35
12x+2=35
Step 2: Subtract 2 from both sides.
12x+2−2=35−2
12x=33
Step 3: Divide both sides by 12.
12x / 12 = 33 / 12x =
11 / 4
Your waiter at a restaurant suggests you leave a tip of $10 on a $50 bill. What percentage is that?
Answer:
20%
Step-by-step explanation:
To find what percentage is $10 out of $50, we divide 10 by 50:
[tex] \frac{10}{50}=0.2[/tex]
If we want to get the result in percentage form, we simply multiply it by 100%:
[tex]0.2\cdot 100\%=20\%[/tex]
So a tip of $10 on a $50 bill is a tip of 20%.
8 BASIC LEVEL 1. On weekends, a movie ticket costs $10.50. Form an inequality and solve it to find the maximum number of tickets Kate can buy with $205
Answer:
10.50x ≤ 205
The maximum number of tickets, x, would be 19.
Step-by-step explanation:
Given,
The cost of one ticket = $ 10.50,
The cost of x tickets = 10.50x dollars,
Since, the total cost can not exceed $ 205,
⇒ 10.50x ≤ 205
∵ 10.50 > 0 thus, when we multiply both sides by 1/10.50 the inequality sign will not change,
⇒ x ≤ [tex]\frac{205}{10.50}[/tex] ≈ 19.52
Hence, the maximum number of tickets would be 19.
A good-quality measuring tape can be off by 0.49 cm over a distance of 23 m. What is the percent uncertainty?33% Part (a) If 44 ± 3 beats are counted (in exactly 30 s) what is the percent uncertainty in the measurement of their heartbeats per minute?
Step-by-step explanation:
First:
(Error/Total )x 100%
(0.49/23) x100%= 2.13%
Second:
(Absolute uncertainty/mean) x100%
(3/44)x100%= 6.81%
To calculate percent uncertainty, divide the uncertainty by the measured value and multiply by 100%. The percent uncertainty for the measuring tape is 0.0213%. The percent uncertainty for heartbeats (converted to per minute) is 6.82%.
Explanation:The subject of this question is determining the percent uncertainty of measurements. Percent uncertainty is calculated by dividing the uncertainty by the measurement and multiplying by 100%.
1. For the measuring tape, the uncertainty is 0.49 cm which first needs to be converted to meters (0.0049 m) since the measurement is given in meters. The percent uncertainty of the measuring tape is therefore (0.0049 m / 23m) * 100% = 0.0213%, a small percentage reflecting a high level of accuracy.
2. For counting heartbeats, the uncertainty is 3 beats and the measurement is 44 beats. Since it's measured in 30 seconds, we should double it to get beats per minute. Therefore, the uncertainty becomes 6 beats and the measurement becomes 88 beats. The percent uncertainty is therefore (6 beats / 88 beats) * 100% = 6.82%, a more considerable uncertainty.
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Solve these linear equations in the form y=yn+yp with yn=y(0)e^at.
a. y'-4y=-8
b. y'+4y=8
which one has a steady state?
Answer:
a) [tex]y(t) = y_{0}e^{4t} + 2[/tex]. It does not have a steady state
b) [tex]y(t) = y_{0}e^{-4t} + 2[/tex]. It has a steady state.
Step-by-step explanation:
a) [tex]y' -4y = -8[/tex]
The first step is finding [tex]y_{n}(t)[/tex]. So:
[tex]y' - 4y = 0[/tex]
We have to find the eigenvalues of this differential equation, which are the roots of this equation:
[tex]r - 4 = 0[/tex]
[tex]r = 4[/tex]
So:
[tex]y_{n}(t) = y_{0}e^{4t}[/tex]
Since this differential equation has a positive eigenvalue, it does not have a steady state.
Now as for the particular solution.
Since the differential equation is equaled to a constant, the particular solution is going to have the following format:
[tex]y_{p}(t) = C[/tex]
So
[tex](y_{p})' -4(y_{p}) = -8[/tex]
[tex](C)' - 4C = -8[/tex]
C is a constant, so (C)' = 0.
[tex]-4C = -8[/tex]
[tex]4C = 8[/tex]
[tex]C = 2[/tex]
The solution in the form is
[tex]y(t) = y_{n}(t) + y_{p}(t)[/tex]
[tex]y(t) = y_{0}e^{4t} + 2[/tex]
b) [tex]y' +4y = 8[/tex]
The first step is finding [tex]y_{n}(t)[/tex]. So:
[tex]y' + 4y = 0[/tex]
We have to find the eigenvalues of this differential equation, which are the roots of this equation:
[tex]r + 4 = [/tex]
[tex]r = -4[/tex]
So:
[tex]y_{n}(t) = y_{0}e^{-4t}[/tex]
Since this differential equation does not have a positive eigenvalue, it has a steady state.
Now as for the particular solution.
Since the differential equation is equaled to a constant, the particular solution is going to have the following format:
[tex]y_{p}(t) = C[/tex]
So
[tex](y_{p})' +4(y_{p}) = 8[/tex]
[tex](C)' + 4C = 8[/tex]
C is a constant, so (C)' = 0.
[tex]4C = 8[/tex]
[tex]C = 2[/tex]
The solution in the form is
[tex]y(t) = y_{n}(t) + y_{p}(t)[/tex]
[tex]y(t) = y_{0}e^{-4t} + 2[/tex]
A marketing research company desires to know the mean consumption of milk per week among males over age 32. A sample of 710 males over age 32 was drawn and the mean milk consumption was 4.6 liters. Assume that the population standard deviation is known to be 0.8 liters. Construct the 98% confidence interval for the mean consumption of milk among males over age 32. Round your answers to one decimal place.
Answer:
(4.5, 4.7)
Step-by-step explanation:
Hi!
Lets call X to the consumption of milk per week among males over age 32. X has a normal distribution with mean μ and standard deviation σ.
[tex]X \sim N(\mu, \sigma)[/tex]
When you know the population standard deviation σ of X , and the sample mean is [tex]\hat X[/tex], the variable q has distribution N(0,1):
[tex]q = \frac{\hat X - \mu}{\sigma} \sim N(0,1)[/tex]
Then you have:
[tex]P(-k < q <k ) = P(\hat X -\frac{\sigma}{\sqrt{N} }<\mu<\hat X +\frac{\sigma}{\sqrt{N} })=C[/tex]
This defines a C - level confidence interval. For each C the value of k is well known. In this case C = 0.98, then k = 2.326
Then the confidence interval is:
[tex](4.6 - 2.326*\frac{0.8}{\sqrt{710}}, 4.6 + 2.326*\frac{0.8}{\sqrt{710}})\\ (4.5, 4.7)[/tex]
Let P(x) denote the statement "2x+5 > 10." Which of the following is true?
P(0)
P(3)
P(2)
P(1)
Answer: P(3) is True
Step-by-step explanation:
The given statement is an inequality denoted as P(x). To find out which of the options is true you have to evaluate each given value of X in the inequality and perform the arithmetic operations, then you have to see if the expression makes sense.
For P(0): Replace X=0 in 2x+5>10
2(0)+5>10
0+5>10
5>10 is false because 5 is not greater than 10
For P(3): Replace X=3 in 2x+5>10
2(3)+5>10
6+5>10
11>10 is true because 11 is greater than 10
For P(2): Replace X=2 in 2x+5>10
2(2)+5>10
4+5>10
9>10 is false
For P(1): Replace X=1 in 2x+5>10
2(1)+5>10
2+5>10
7>10 is false
Vanessa walks from her house to a bus stop that is 400 yards away. If Vanessa is 22 yards from her house, how far is she from the bus stop? yards Preview 400 − 22 = 400-22= 378. If Vanessa is 163.4 yards from her house, how far is she from the bus stop? yards Preview 400 − 163.4 = 400-163.4= 236.6. Let the variable x x represent Vanessa's varying distance from her house (in yards). As Vanessa walks from her house to the bus stop, the value of x x varies from to . How many values does the variable x x assume as Vanessa walks from her house to the bus stop? Preview
Answer:
[tex]0\le x\le 400[/tex]
x can take infinitely many values
Step-by-step explanation:
Vanessa walks from her house to a bus stop that is 400 yards away.
If Vanessa is 22 yards from her house, how far is she from the bus stop? Preview: 400 − 22 = 400 - 22 = 378 yards. If Vanessa is 163.4 yards from her house, how far is she from the bus stop? Preview: 400 − 163.4 = 400 - 163.4 = 236.6 yards.Let the variable x represent Vanessa's varying distance from her house (in yards). Then 400 - x yards is how far Vanessa is from the bus stop.
The variable x can take any value from 0 to 400 (0 when Vanessa is at home and 400 when Vanessa is at bus station), so
[tex]0\le x\le 400[/tex]
x can take infinitely many values, because there are infinitely many real numbers between 0 and 400.
Vanessa is 378 yards from the bus stop when she is 22 yards from her house and 236.6 yards away when she is 163.4 yards from her house. The variable x denoting Vanessa's distance from home assumes infinitely many values as she walks to the bus stop.
Explanation:When Vanessa is 22 yards from her house, the distance remaining to reach the bus stop is simply the total distance to the bus stop minus her current position from the house. So, it's 400 yards - 22 yards = 378 yards. Similarly, if Vanessa is 163.4 yards from her house, the remaining distance to the bus stop is 400 yards - 163.4 yards = 236.6 yards.
As Vanessa walks from her house to the bus stop, variable x represents her varying distance from her house. The value of x starts at 0 when she is at her house and increases up to 400 yards as she reaches the bus stop. The variable x can assume infinitely many values, as it can represent any real number between 0 and 400, indicating her position at any given moment along her path.
A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0,0,0),(1,3,0)(0,0,0),(1,3,0), and (0,3,2)(0,3,2). By what angle does the tower now deviate from the vertical?
Answer:
The tower deviates [tex]64^\circ36'[/tex] from the vertical.
Step-by-step explanation:
Having 3 point of our new plane we can construct vectors on the plane by substacting 2 of them:
[tex]v_1=(0,3,2)-(0,0,0)\\v_2=(1,3,0)-(0,0,0)[/tex]
These vectors are on the plane, so a cross product between them will give us a vector perpendicular to the plane:
[tex](0,3,2)\times(1,3,0)=\left[\begin{array}{ccc}i&j&k\\0&3&2\\1&3&0\end{array}\right] =(-6,2,-3)[/tex]
Asuming that the aliens used our conventions, the original plane was perpendicular to the z axis, so that a perpendicular vector to that plane was
(0,0,1)
We know that a dot product between 2 vectors |V.W| = |V| |W| cos(α), where α is the angle between them. If we use the vector perpendicular to this plane, and the one perpendicular to the original plane, α will represent the deviation angle of our new plane.
[tex]\|(-6,2,-3)\|=7\\\|(0,0,1)\|=1[/tex]
[tex](-6,2,-3)\odot(0,0,1)= 7 cos(\alpha )\\\\ -3=7 cos(\alpha ) \\ \alpha=arc\ cos \frac{-3}{7} =115.37 ^\circ\\[/tex]
Since this angle is greater than 90 degrees it means that the vector we calculated as perpendicular to the plane points towards negative z (this can be seen by the -3 z component)
To fix this we can calculate a new perpendicular vector, or simply compare ir with the vector (0,0,-1). The latter is easier:
[tex](-6,2,-3)\odot(0,0,-1)= 7 cos(\alpha )\\\\ 3=7 cos(\alpha ) \\ \alpha=arc\ cos \frac{3}{7} =64.6^\circ =64^\circ36'[/tex]
A company's sales decreased by 8% this year, to $9015. What were their sales last year? (Round your answer to the nearest penny.)
My answer was 9,736.2, but the correct answer was 9798.91. Could someone explain how did they get to 9798.91?
Answer:
There are two unknow values, your 8% which is the decrease and the main value which is "the sales last year".
It is correct, the number you gave as an answer $9798,91. Let's get the explanation by a rule of three.
Step-by-step explanation:
The 100 % was "the sales last year", the 8% are what the sales decreased but you don't have that number, you have the result in the substraction, $9015. So this are the step by step, hope you understand!
-1.8-3.9=
A. -2.1
B.5.7
C.2.1
D.-5.7
In order to get the answer to this question you will have to use KCC (Keep, Change, Change) and then solve.
[tex]-1.8 - 3.9=[/tex]
Using KCC:
[tex]-1.8-3.9=-1.8+-3.9[/tex]
[tex]-1.8 + -3.9 = -5.7[/tex]
[tex]= -5.7[/tex]
Therefore your answer is option D "-5.7."
Hope this helps.
Answer:
D "-5.7."
Step-by-step explanation:
Experience raising New Jersey Red chickens revealed the mean weight of the chickens at
five months is 4.35 pounds. The weights follow the normal distribution. In an effort to increase
their weight, a special additive is added to the chicken feed. The subsequent
weights of a sample of five-month-old chickens were (in pounds):
4.41 4.37 4.33 4.35 4.30 4.39 4.36 4.38 4.40 4.39
At the .01 level, has the special additive increased the mean weight of the chickens? Estimate
the p-value.
Answer:
p-value = 0.1277
Step-by-step explanation:
p-value is the probability value tell us how likely it is to get a result like this if the Null Hypothesis is true.
Firstly we find the mean and standard deviation of the given data set.
⇒ Mean = [tex]\frac{4.41 +4.37+ 4.33+ 4.35 +4.30 +4.39 +4.36+ 4.38+ 4.40+ 4.39}{10}[/tex]
⇒ Mean = 4.368
[tex]Standard deviation(\sigma) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}} }[/tex]
where, [tex]\bar{x}[/tex] is mean of the distribution.
⇒ Standard Deviation = 0.034
Applying t- test:
Let out hypothesis is:
H₀: μ = 4.35
H₁: μ ≠ 4.35
Now,
Here, μ = Population Mean = 4.35
[tex]\bar{x}[/tex]= Sample Mean = 4.368
σ = Standard Deviation = 0.034
n = 10
[tex]t=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}} }[/tex]
Putting all values we get, t = 1.6777 with (10 -1) = 9 degree of freedom.
Then the p-value at 99% level of significance.
⇒ p-value = 0.1277
To test whether the special additive has increased the mean weight of the chickens, we can use a t-test. We will calculate the t-value and the p-value and compare the p-value with the significance level of 0.01.
Explanation:To test whether the special additive has increased the mean weight of the chickens, we can use a t-test. We can set up the null hypothesis as follows:
H0: μ = 4.35
And the alternative hypothesis as:
H1: μ > 4.35
We will calculate the t-value and the p-value.
t-value = (mean of the sample - mean of the population) / (standard deviation of the sample / sqrt(sample size))
p-value = P(T > t)
In this case, we have to compare the p-value with the significance level of 0.01.
If the p-value is less than 0.01, we reject the null hypothesis and conclude that the special additive has increased the mean weight of the chickens.
Your friend borrows $100 from you and promises to pay you back $109 in 8 months. What annual percentage rate (A.P.R.) are you charging your friend? Round to the nearest tenth of a percent and write the answer as a decimal
Answer:
[tex]12.0\%=0.12[/tex]
Step-by-step explanation:
We have been given that your friend borrows $100 from you and promises to pay you back $109 in 8 months.
We will use simple interest formula to solve our given problem.
[tex]A=P(1+rt)[/tex], where,
A = Amount after t years,
P = Principal amount,
r = Annual interest rate in decimal form,
t = Time in years.
Convert 8 months to year:
[tex]\frac{8}{12}\text{ year}=\frac{2}{3}\text{ year}[/tex]
[tex]108=100(1+r*\frac{2}{3})[/tex]
[tex]108=100+r*\frac{2}{3}\times 100[/tex]
[tex]108-100+r*\frac{200}{3}[/tex]
[tex]108-100=100-100+r*\frac{200}{3}[/tex]
[tex]8=r*\frac{200}{3}[/tex]
[tex]8\times \frac{3}{200}=r*\frac{200}{3}\times \frac{3}{200}[/tex]
[tex]\frac{24}{200}=r[/tex]
[tex]r=\frac{24}{200}[/tex]
[tex]r=0.12[/tex]
Convert to percent:
[tex]0.12\times 100\%=12\%[/tex]
Therefore, you are charging 12% APR to you friend.
The data were gathered by the following procedure, reported in the study. "Between January and June 1998, parents of children aged 2-16 years [...] that were seen as outpatients in a university pediatric ophthalmology clinic completed a questionnaire on the child’s light exposure both at present and before the age of 2 years." Was this study observational, or was it a controlled experiment? Explain.
Answer:
This was an observational study.
Step-by-step explanation:
Given is that parents of children completed a questionnaire on the child’s light exposure both at present and before the age of 2 years.
This was an observational study since there is no treatment or control group.
We know that treatment, control groups or treatment groups are not specific to randomized control experiments.
The study where parents of pediatric ophthalmology patients completed questionnaires about light exposure is an observational study because data were collected without manipulating any variables.
Explanation:The study described in which parents filled out questionnaires about their children's light exposure is an example of an observational study, not a controlled experiment. In an observational study, researchers collect data without manipulating any variables. In this case, the researchers gathered information on light exposure by asking parents to complete a questionnaire, but they did not control or alter the children's light exposure themselves.
Unlike in an observational study, a controlled experiment involves actively manipulating one variable (the independent variable) to determine if it causes a change in another variable (the dependent variable), often comparing against a control group in a systematic way. An example of a controlled experiment includes the trial of Jonas Salk's polio vaccine, in which one group received the vaccine and another group received a placebo.