Answer:
120/s^6
Step-by-step explanation:
There is an easy formula for this...
L(t^n)=n!/(s^(n+1))
Your n=5 here
L(t^5)=5!/(s^6)
L(t^5)=120/s^6
[tex]\mathcal{L}\{t^n\}=\dfrac{n!}{s^{n+1}}[/tex]
So
[tex]\mathcal{L}\{f(t)\}=\dfrac{5!}{s^{5+1}}=\dfrac{120}{s^6}[/tex]
The conversion factor relating feet to meters is 1 ft=0.305 m. Keep in mind that when using conversion factors, you want to make sure that like units cancel leaving you with the units you need. You have been told that a certain house is 164 m2 in area. How much is this in square feet?
Answer:
1763 ft²
Step-by-step explanation:
Using the given conversion factor, ...
(164 m²)(1 ft/(.305 m))² = 165/.093025 ft² ≈ 1763 ft²
_____
The exact conversion factor is 1/0.3048, so the area is closer to 1765 ft². For a 4-significant digit answer, you need to use a conversion factor accurate to 4 significant digits.
To convert square meters to square feet, you must square the feet to meter conversion factor, resulting in approximately 10.764 sq ft/sq m. You then multiply this by the square meter measurement to get the equivalent in square feet. Therefore, the house's area, which was provided as 164 square meters, translates to approximately 1765.736 square feet.
Explanation:The measure of the area in square feet can be derived using the conversion factor for feet to meters, 1 ft = 0.305 m. However, when you deal with areas, you must square the conversion factor. We then apply the conversion factor to the known area in reference, which in our case is 164 square meters.
So our conversion factor becomes (1/0.305)² sq ft/sq m = 10.764 sq ft/sq m.
To use the conversion factor, we multiply it by the metric unit measurement, like this: 164 m²*(10.764 ft²/m²) = 1765.736 square feet.
So, the house's area is approximately 1765.736 square feet.
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You work as a cashier for a bookstore and earn $6 per hour. You also baby sit and earn $6 per hour. You want to earn at least $60 per week, but would like to work no more than 12 hours per week.
Which system of inequalities, along with y ≥ 0 and x ≥ 0, would you use to solve the real-world problem?
Final answer:
To solve the problem, use the system of inequalities: x + y ≥ 0, x ≥ 0, y ≥ 0, 6x + 6y ≥ 60, and x + y ≤ 12.
Explanation:
To solve the given real-world problem, the system of inequalities you would use is:
x + y ≥ 0x ≥ 0y ≥ 06x + 6y ≥ 60x + y ≤ 12These inequalities represent the conditions that need to be met: x and y (representing the number of hours worked as a cashier and as a babysitter, respectively) must be greater than or equal to 0, and the total income from both jobs (6x + 6y) must be greater than or equal to $60, and the total number of hours worked (x + y) must be less than or equal to 12.
The number of geese is modeled by the function G(t) that satisfies the differential equation dG dt equals the product of G divided by 5 and the quantity 350 minus G where t is the time in years and G(0) = 100 . What is the goose population when the population is increasing most rapidly?
Answer:
175
Step-by-step explanation:
The rate of change of the goose population is a function of the population:
G'(x) = (x/5)(350 -x)
This function describes a downward-opening parabola with zeros at x=0 and x=350. The value of x halfway between these zeros, at x = 175, is where the maximum value of G'(x), hence the maximum rate of change, is located.
The goose population is increasing most rapidly when it is 175.
Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 200 and 18, respectively. The distribution of the population is unknown. The mean and the standard error of the distribution of the sample mean are
Answer: The mean and the standard error of the distribution of the sample mean are 200 and 2.
Step-by-step explanation:
Given: Sample size : n= 81
Mean of infinite population : [tex]\mu=200[/tex]
We know that the mean of the distribution of the sample mean is same as the mean of the population.
i.e. [tex]\mu_x=\mu=200[/tex]
The standard error of the distribution of the sample mean is given by :-
[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex]
[tex]\Rightarrow\ S.E.=\dfrac{18}{\sqrt{81}}=\dfrac{18}{9}=2[/tex]
Hence, the mean and the standard error of the distribution of the sample mean are 200 and 2.
The mean of the sample mean distribution for a random sample of size 81 from a population with a mean of 200 and a standard deviation of 18 is 200, and the standard error is 2.
Explanation:The question is about determining the mean and the standard error of the distribution of the sample mean. According to the Central Limit Theorem, regardless of the distribution of the original population, the distribution of the sample mean tends to form a normal distribution as the sample size increases. In this case, the mean of the sample mean distribution is the same as the population mean, which is 200.
The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size. So, the standard error in this scenario would be 18/√81 = 2.
Therefore, in a random sample of size 81 taken from an infinite population with a mean of 200 and a standard deviation of 18, the mean of the sample mean distribution is 200, and the standard error is 2.
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Use the table above to answer the question.
Ed Employee had $70,000 in taxable income. What was his tax?
Tax = $______
Answer:
$16,479
Step-by-step explanation:
The table tells you Ed's tax is ...
14,138.50 + 0.31×(70,000 -62,450) = 16,479 . . . . dollars
Suppose that the distribution of touchdown passes (in football) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 touchdowns.
What is the probability that the 49 touchdowns traveled an average of less than 245 feet? Please explain how you derived your answer.
Answer: 0.2420
Step-by-step explanation:
Given: Mean : [tex]\mu = 250 \text{ feet}[/tex]
Standard deviation : [tex]\sigma =50\text{ inch}[/tex]
Sample size : [tex]n=49[/tex]
The formula to calculate z is given by :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 245
[tex]z=\dfrac{245-250}{\dfrac{50}{\sqrt{49}}}=-0.7[/tex]
The P Value =[tex]P(Z<245)=P(z<-0.7)=0.2419637\approx0.2420[/tex]
Hence, the probability that the 49 touchdowns traveled an average of less than 245 feet= 0.2420
The probability that the 49 touchdowns traveled an average of less than 245 feet is approximately 0.2438, or 24.38%.
Step 1:
To find the probability that the 49 touchdowns traveled an average of less than 245 feet, we can use the Central Limit Theorem (CLT) since we have a large enough sample size (49) to assume that the sample mean follows a normal distribution.
The CLT states that the distribution of sample means of a sufficiently large sample size will be approximately normal, regardless of the distribution of the original population, as long as the sample size is large enough.
Given:
- Population mean mu = 250 feet
- Population standard deviation [tex](\( \sigma \))[/tex] = 50 feet
- Sample size n = 49
Step 2:
The standard error of the sample mean SE is given by:
[tex]\[SE = \frac{\sigma}{\sqrt{n}}\][/tex]
Substituting the given values:
[tex]\[SE = \frac{50}{\sqrt{49}} = \frac{50}{7} \approx 7.14\][/tex]
Step 3:
Now, we can calculate the z-score for the sample mean of 245 feet using the formula:
[tex]\[z = \frac{\bar{x} - \mu}{SE}\][/tex]
Where:
- [tex]\( \bar{x} \)[/tex] is the sample mean
- [tex]\( \mu \)[/tex] is the population mean
- [tex]\( SE \)[/tex] is the standard error of the sample mean
Step 4:
Substituting the given values:
[tex]\[z = \frac{245 - 250}{7.14} \approx -0.6993\][/tex]
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to this z-score.
The probability that the sample mean is less than 245 feet can be found by finding the area to the left of the z-score on the standard normal distribution curve.
From the standard normal distribution table, we find that the probability corresponding to a z-score of -0.6993 is approximately 0.2438.
Therefore, the probability that the 49 touchdowns travelled an average of less than 245 feet is approximately 0.2438, or 24.38%.
Fewer young people are driving. In 1983, 87% of 19-year-olds had a driver’s license. Twenty-five years later that percentage had dropped to 75% (University of Michigan Transportation Research Institute website, April 7, 2012). Suppose these results are based on a random sample of 1200 19-year-olds in 1983 and again in 2008.
a. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 1983?
b. At 95% confidence, what is the margin of error and the interval estimate of the number of 19-year-old drivers in 2008?
c. Is the margin of error the same in parts (a) and (b)? Why or why not?
Answer: the answer to this question is B
Step-by-step explanation: Hope This Helps
Formula to find the margin of error :
[tex]E=z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex] , where n= sample size , [tex]\hat{p}[/tex] is the sample proportion and z*= critical z-value.
Let p be the proportion of 19-year-olds had a driver’s license.
A) As per given , In 1983
[tex]\hat{p}=0.87[/tex]
n= 1200
Critical value for 95% confidence level is 1.96 (By z-table)
So ,Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.87(1-0.87)}{1200}}\approx0.019[/tex]
Interval : [tex](\hat{p}-E , \ \hat{p}+E)=(0.87-0.019 ,\ 0.87+0.019)[/tex]
[tex]=(0.851,\ 0.889)[/tex]
B) In 2008 ,
[tex]\hat{p}=0.75[/tex]
Margin of error : [tex]E=(1.96)\sqrt{\dfrac{0.75(1-0.75)}{1200}}\approx0.0245[/tex]
Interval : [tex](\hat{p}-E, \hat{p}+E)=(0.75-0.0245,\ 0.75+0.0245)[/tex]
[tex]=(0.7255,\ 0.7745)[/tex]
c. The margin of error is not the same in parts (a) and (b) because the sample proportion of 19-year-olds had a driver’s license are not same in both parts.
An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 161 lb. The new population of pilots has normally distributed weights with a mean of 125 lb and a standard deviation of 28.1 lb.
a)If a pilot is randomly selected, find the probability that his weight is between 120 lb and 161 lb.The probability is approximately?
b. If 36 different pilots are randomly selected, find the probability that their mean weight is between 120 lb and 161 lb. The probability is approximately?
c. When redesigning the ejection seat, which probability is more relevant? . Part (b) because the seat performance for a single pilot is more important. B. Part (b) because the seat performance for a sample of pilots is more important. C. Part (a) because the seat performance for a sample of pilots is more important. D. Part (a) because the seat performance for a single pilot is more important.
Which of the following statements is CORRECT? a. A graph of the SML as applied to individual stocks would show required rates of return on the vertical axis and standard deviations of returns on the horizontal axis. b. An increase in expected inflation, combined with a constant real risk-free rate and a constant market risk premium, would lead to identical increases in the required returns on a riskless asset and on an average stock, other things held constant. c. If two "normal" or "typical" stocks were combined to form a 2-stock portfolio, the portfolio's expected return would be a weighted average of the stocks' expected returns, but the portfolio's standard deviation would probably be greater than the average of the stocks' standard deviations. d. If investors become more risk averse, then (1) the slope of the SML would increase and (2) the required rate of return on low-beta stocks would increase by more than the required return on high-beta stocks. e. The CAPM has been thoroughly tested, and the theory has been confirmed beyond any reasonable doubt.
b. An increase in expected inflation, combined with a constant real risk-free rate and a constant market risk premium, would lead to identical increases in the required returns on a riskless asset and on an average stock, other things held constant.
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a figure has a vertex at (-1,-3). if the figure has a line symmetry about x-axis , what are the coordinates of another vertex of the figure?
a. (3,1)
b. (-1,3)
c. (-3,-1)
d. (1,-3)
Answer:
b. (-1,3)
Step-by-step explanation:
the image of the point (x ; y) by symmetry about x-axis is : ( x ;- y)
so the answer "b" : (-1,3)
Line m is parallel to line n. The measure of angle 2 is 74°. What is the
measure of angle 5?
OA) 74°
O B) 120
OC) 106°
OD) 86°
Answer:
C. 106
Step-by-step explanation:
Angles 2 and 6 are corresponding angles so they're both 74. So you just subtract 74 from 180 to get 106.
Answer:
C. 106 is the answer
Step-by-step explanation:
angle 3 = angle 2 (vertically opp. angle)
angle 3+ angle 5 = 180
74+ angle 5 = 180
angle 5 = 106
The speed of cars on a stretch of road is normally distributed with an average 48 miles per hour with a standard deviation of 5.9 miles per hour. What is the probability that a randomly selected car is violating the speed limit of 50 miles per hour? (a) 0.37 (b) 0.48 (c) 0.21 (d) 0.63
Answer: (a) 0.37
Step-by-step explanation:
Given: The speed of cars on a stretch of road is normally distributed with an average 48 miles per hour with a standard deviation of 5.9 miles per hour.
i.e. Mean : [tex]\mu = 48\text{ miles per hour} [/tex]
Standard deviation : [tex]\sigma = 5.9\text{ miles per hour}[/tex]
The formula to calculate z is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For the probability that a randomly selected car is violating the speed limit of 50 miles per hour (X≥ 50).
For x= 80
[tex]z=\dfrac{50-48}{5.9}=0.338983050847\approx0.34[/tex]
The P Value =[tex]P(z>0.34)=1-P(z<0.34)=1-0.6330717\approx0.3669283\approx0.37[/tex]
Hence, the probability that a randomly selected car is violating the speed limit of 50 miles per hour =0.37
F(x)=x^2-14x+33 enter the quadratic function in factored form
Answer:
[tex]F(x)=(x-11)(x-3)[/tex]
Step-by-step explanation:
we have
[tex]F(x)=x^{2} -14x+33[/tex]
Find the zeros of the function
F(x)=0
[tex]0=x^{2} -14x+33[/tex]
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]-33=x^{2} -14x[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]-33+49=x^{2} -14x+49[/tex]
[tex]16=x^{2} -14x+49[/tex]
Rewrite as perfect squares
[tex]16=(x-7)^{2}[/tex]
square root both sides
[tex](x-7)=(+/-)4[/tex]
[tex]x=(+/-)4+7[/tex]
[tex]x=(+)4+7=11[/tex]
[tex]x=(-)4+7=3[/tex]
so
The factors are
(x-11) and (x-3)
therefore
[tex]F(x)=(x-11)(x-3)[/tex]
your bike lock has 4 digits numbered 0-9. Find the total number of possible combinations for the lock.
Answer:
10,000
Step-by-step explanation:
With a bike lock, you can usually use the same number more than one... you can have them the same (7777) if you want.
So, you have 10 possibilities for the first digit, 10 again for the second digit, 10 for the third digit... and also 10 for the last digit. So...
10 * 10 * 10 * 10 = 10,000
These combinations range from 0000 to 9999
Final answer:
The total number of possible combinations for a 4-digit bike lock with each digit ranging from 0-9 is 10,000, as determined by the fundamental counting principle and calculated by multiplying 10 choices for each digit.
Explanation:
To find the total number of possible combinations for a 4-digit bike lock where each digit can range from 0-9, you apply the fundamental counting principle. This principle states that if there are n ways to perform one task and m ways to perform another task, then there are n × m ways to perform both tasks in sequence.
For the bike lock, each of the 4 digits can be chosen in 10 ways (0 through 9). Since the choice of each digit is independent of the others, you multiply the number of choices for each digit together:
Choice 1: 10 waysChoice 2: 10 waysChoice 3: 10 waysChoice 4: 10 waysTherefore, the total number of combinations is 10 × 10 × 10 × 10 = 10,000.
Using composition of functions, determine if the two functions are inverses
of each other. Will Mark Brainliest!
The functions F(x) and G(x) are not inverses of each other.
The correct answer is B. No, because the functions contain different operations.
Given are composition of functions, F(x) = √(x) -6G(x) = (x+6)²
We need to determine if the two functions are inverses of each other.
To determine if the functions F(x) = √(x) - 6 and G(x) = (x + 6)² are inverses of each other using composition of functions, we need to check if their compositions result in the identity function.
Let's calculate the composition:
F(G(x)) = F((x + 6)²) = √((x + 6)²) - 6 = |x + 6| - 6
Now, let's calculate the composition in the reverse order:
G(F(x)) = G(√(x) - 6) = (√(x) - 6 + 6)² = (√(x))² = x
Since F(G(x)) = |x + 6| - 6 and G(F(x)) = x, we can see that they are not equal for all values of x.
Therefore, the functions F(x) and G(x) are not inverses of each other.
The correct answer is B. No, because the functions contain different operations.
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Two functions are inverses if both (f o g)(x) and (g o f)(x) are equal to x. If they are, their composition will yield x, indicating that the two functions are indeed inverses.
Explanation:To determine if two functions are inverses of each other using composition of functions, you should perform the operation (f o g)(x) and (g o f)(x). If f and g are inverse functions, both of these compositions will yield x.
Let's take the example of functions f(x) = 2x + 3 and g(x) = (x - 3) / 2. To check if they are inverses:
Compute (f o g)(x) = f(g(x)) = f((x - 3) / 2) = 2((x - 3) / 2) + 3 = xCompute (g o f)(x) = g(f(x)) = g(2x + 3) = (2x + 3 - 3) / 2 = xSince both (f o g)(x) and (g o f)(x) equals x, so f(x) and g(x) are inverses of each other.
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The mean number of words per minute (WPM) read by sixth graders is 93 with a standard deviation of 22.If 30 sixth graders are randomly selected, what is the probability that the sample mean would be greater than 97.95 WPM? (Round your answer to 4 decimal places)
Answer: 0.1093
Step-by-step explanation:
Given: Mean : [tex]\mu=93[/tex]
Standard deviation : [tex]\sigma = 22[/tex]
Sample size : [tex]n=30[/tex]
The formula to calculate z-score is given by :_
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 97.95, we have
[tex]z=\dfrac{97.95-93}{\dfrac{22}{\sqrt{30}}}\approx1.23[/tex]
The P-value = [tex]P(z>1.23)=1-P(z<1.23)=1-0.8906514=0.1093486\approx0.1093[/tex]
Hence, the probability that the sample mean would be greater than 97.95 WPM =0.1093
Use a proof by contradiction to prove that underroot 3 is irrational.
Let assume that [tex]\sqrt3[/tex] is rational. Therefore we can express it as [tex]\dfrac{a}{b}[/tex] where [tex]a,b\in \mathbb{Z}[/tex] and [tex]\text{gcd}(a,b)=1[/tex].
[tex]\dfrac{a}{b}=\sqrt3\\\dfrac{a^2}{b^2}=3\\a^2=3b^2[/tex]
It means that [tex]3|a^2[/tex] and so also [tex]3|a[/tex].
Therefore [tex]a=3k[/tex] where [tex]k\in\mathbb{Z}[/tex].
[tex](3k)^2=3b^2\\9k^2=3b^2\\b^2=3k^2[/tex]
It means that [tex]3|b^2[/tex] and so also [tex]3|b[/tex].
If both [tex]a[/tex] and [tex]b[/tex] are divisible by 3, then it contradicts our initial assumption that [tex]\text{gcd}(a,b)=1[/tex]. Therefore [tex]\sqrt3[/tex] must be an irrational number.
The volume of water flowing through a pipe varies directly wlth the square of the radius of the pipe. If the water flows at a rate of 80 liters per minute through a pipe with a radlus of 4 cm, at what rate would water flow through a pipe with a radius of 3 cm? (Rigorous) (Competency 007) 11. A) 45 liters per minute B) 6.67 liters per minute C) 60 liters per minute D) 4.5 liters per minute
Answer:
A
Step-by-step explanation:
Volume varies directly with the square of the radius, so:
V = k r²
When V = 80, r = 4.
80 = k (4)²
k = 5
V = 5r²
When r = 3:
V = 5 (3)²
V = 45
The flow is 45 L/min.
find the solution of the following system of equations -5+2y=9 3x+5y=7
The solution to the system of equations is [tex]\(x = -1\) and \(y = 2\)[/tex].
To solve the system of equations:
[tex]\[ \begin{cases} -5x + 2y = 9 \\ 3x + 5y = 7 \end{cases} \][/tex]
We can use either the substitution method or the elimination method. Let's use the elimination method here.
First, let's rewrite the equations in standard form:
Equation 1: [tex]\( -5x + 2y = 9 \)[/tex]
Equation 2: [tex]\( 3x + 5y = 7 \)[/tex]
To eliminate one of the variables, let's multiply Equation 1 by 3 and Equation 2 by 5 to make the coefficients of x the same:
[tex]\[ \begin{cases} -15x + 6y = 27 \\ 15x + 25y = 35 \end{cases} \][/tex]
Now, let's add the two equations:
[tex]\[ (-15x + 6y) + (15x + 25y) = 27 + 35 \]\[ -15x + 6y + 15x + 25y = 62 \]\[ 31y = 62 \][/tex]
Now, let's solve for y:
[tex]\[ y = \frac{62}{31} \][/tex]
y=2
Now that we have found the value of y, let's substitute it back into one of the original equations to find x. Let's use Equation 1:
[tex]\[ -5x + 2(2) = 9 \]\[ -5x + 4 = 9 \]\[ -5x = 9 - 4 \]\[ -5x = 5 \]\[ x = \frac{5}{-5} \][/tex]
[tex]\[ x = -1 \][/tex]
Complete question: Find the solution of the following system of equations
-5x+2y=9
3x+5y=7
\[ x = -\frac{28}{3} \] and \( y = 7 \) are the solutions to the system of equations.
To solve the system of equations:
1. -5 + 2y = 9
2. 3x + 5y = 7
We can start by solving equation 1 for [tex]\( y \):[/tex]
[tex]\[ -5 + 2y = 9 \][/tex]
Add 5 to both sides:
[tex]\[ 2y = 9 + 5 \]\[ 2y = 14 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{14}{2} \]\[ y = 7 \][/tex]
Now that we have the value of [tex]\( y \)[/tex], we can substitute it into equation 2 and solve for [tex]\( x \):[/tex]
[tex]\[ 3x + 5(7) = 7 \]\[ 3x + 35 = 7 \][/tex]
Subtract 35 from both sides:
[tex]\[ 3x = 7 - 35 \]\[ 3x = -28 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{-28}{3} \][/tex]
So, the solution to the system of equations is [tex]\( x = -\frac{28}{3} \) and \( y = 7 \).[/tex]
Determine which statements are true in reals3. (Selectall that apply.)
(a)Two lines parallel to a third line are parallel.
(b) Twolines perpendicular to a third line are parallel.
(c) Twoplanes parallel to a third plane are parallel.
(d) Twoplanes perpendicular to a third plane are parallel.
(e) Twolines parallel to a plane are parallel.
(f) Twolines perpendicular to a plane are parallel.
(g) Twoplanes parallel to a line are parallel.
(h) Twoplanes perpendicular to a line are parallel.
(i) Twoplanes either intersect or are parallel.
(j) Twolines either intersect or are parallel.
(k) A plane and a line either intersector are parallel.
Answer:
(a)Two lines parallel to a third line are parallel.
(c) Two planes parallel to a third plane are parallel.
(f) Two lines perpendicular to a plane are parallel.
(h) Two planes perpendicular to a line are parallel.
(i) Two planes either intersect or are parallel.
(k) A plane and a line either intersect or are parallel
Step-by-step explanation:
(b) Two lines perpendicular to a third line are parallel. -- No. The y-, and z-axes are perpendicular to the x-axis, but are not parallel.
(d) Two planes perpendicular to a third plane are parallel. -- No. The x-y and y-z coordinate planes are both perpendicular to the x-z coordinate plane, but are at right angles to each other.
(e) Two lines parallel to a plane are parallel. -- No. Two intersecting lines in the plane z=0 are both parallel to the plane z=1, but are not parallel to each other.
(g) Two planes parallel to a line are parallel. -- No. Both the x-z plane and the y-z plane are parallel to the line (x, y, z) = (1, 1, z), but those coordinate planes are perpendicular to each other.
(j) Two lines either intersect or are parallel. -- No. The lines may be skew, running different directions in parallel panes.
Final answer:
The student's inquiry into the truth of various geometric statements has been addressed, confirming the true relationships and correcting the false ones, based on the principles of Euclidean geometry which govern lines and planes.
Explanation:
When it comes to geometry in the context of Euclidean space, which is the setting for high school mathematics, the rules governing the behavior of lines and planes can be understood through the principles of parallel and perpendicular relationships. Now, let's assess each of the statements given by the student:
(a) True: Two lines parallel to a third line are parallel to each other based on the Transitive Property of parallel lines.(b) True: Two lines perpendicular to a third line are parallel to each other as they both create right angles with the third line, leading to them being parallel.(c) True: Two planes parallel to a third plane are parallel to each other, by the definition of parallel planes.(d) False: Two planes perpendicular to a third plane need not be parallel as they can intersect along a line.(e) True: Two lines parallel to a plane are parallel to each other as they never intersect with the plane or each other.(f) False: Two lines perpendicular to a plane are not necessarily parallel; they can intersect each other at a point.(g) False: Two planes parallel to a line are not necessarily parallel to each other; they could intersect along lines that are both parallel to the given line.(h) True: Two planes perpendicular to a line are parallel to each other as the line is a line of intersection for the planes, and they do not intersect each other anywhere else.(i) True: Two planes either intersect or are parallel, this is a foundational concept in Euclidean geometry.(j) True: Two lines either intersect or are parallel, as there is no other possibility for their relationship in Euclidean space.(k) True: A plane and a line either intersect or are parallel.These principles form the basis for understanding the complex relations of geometric shapes and objects which are important for most geometrical problems and real-world applications.
Solve the equation for x. If a solution is extraneous, be sure to identify it in your final answer.
Square root of x-2+8=x
ANSWER
Extraneous solution: x=6
Real solution: x=11
EXPLANATION
The given expression is
[tex] \sqrt{x - 2} + 8 = x[/tex]
Add -8 to both sides:
[tex]\sqrt{x - 2} + 8 + - 8= x + - 8[/tex]
[tex] \implies\sqrt{x - 2} = x - 8[/tex]
Square both sides.
[tex]\implies(\sqrt{x - 2} )^{2} =( x - 8)^{2} [/tex]
[tex]x - 2=( x - 8)^{2} [/tex]
We expand the to get
[tex]x - 2 = {x}^{2} - 16x + 64[/tex]
Write in standard quadratic form.
[tex] {x}^{2} - 16x - x + 64 + 2 = 0[/tex]
[tex] {x}^{2} - 17x + 66 = 0[/tex]
Factor to get:
[tex](x - 6)(x - 11) = 0[/tex]
[tex]x = 6 \: or \: \: x = 11[/tex]
We check for extraneous solutions by substituting each value of x into the original equation.
When x=6
[tex]\sqrt{6 - 2} + 8 = 6[/tex]
[tex]\sqrt{4} + 8 =6[/tex]
[tex]2 + 8 = 10 \ne8[/tex]
Hence x=6 is an extraneous solution.
When x=11
[tex]\sqrt{11- 2} + 8 = 11[/tex]
[tex]\sqrt{9} + 8 = 11[/tex]
[tex]3 + 8 = 11[/tex]
This statement is true.
Hence x=11 is the only solution.
a theater has two screens and shows its movies continuously. a 30 minute documentary is shown on one. a 120 minute film is shown on the other. If both showings start at noon, how many minutes will pass before both movies begin again at the same time?
I need help with math work!!
Answer:
(g-h)=x^2+5-8-x
=x^2-x-3
(g-h)(-9) means x= -9
x^2-x-3
=(-9)^2-(-9)-3
=81+9-3
=90-3
=87
x^2+y^2=25 Find the distance of point (x,y) from origin.
Answer:
5 units
Step-by-step explanation:
This is a circle with a center of (0, 0). The square root of 25 represents the radius of the circle which is 5. The radius represents the distance that the outside of the circle is from the center.
A coin is tossed 30 times it lands 12 times on heads and 18 times on tails what is experimental probability of the coin landing on tails?
Answer:
3/5
Step-by-step explanation:
Total tossed : 30
# of times landed on tails : 18
Experimental probability of tails = 18/30 = 3/5
Which of the following is the major negative aspect of crossover designs for research studies? A. Prohibitive cost B. Residual effects C-Subject drepout D. Incomplete randomization E. Large sample size required
Answer:
D. Incomplete randomization
Step-by-step explanation:
let f(x) = -2x/(x^2-x-5) There are 2 numbers that are not in the domain of f. Give the larger value to 2 decimal places.
Answer:
Step-by-step explanation:
The 2 numbers that are not in the domain of the function are the 2 numbers that cause the denominator of the function to equal 0. In order to find those 2 numbers, we have to factor the quadratic that is in the denominator. When you factor, you get x = 2.79 and x = -1.79
Those are the values of x that cause the denominator to equal 0, which of course is NEVER allowed in math!
A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?
Answer:
The probability of the sample mean foot length less than 23 cm is 0.120
Step-by-step explanation:
* Lets explain the information in the problem
- The eighth-graders asked to measure the length of their right foot at
the beginning of the school year, as part of a science project
- The foot length is approximately Normally distributed, with a mean of
23.4 cm
∴ μ = 23.4 cm
- The standard deviation of 1.7
∴ σ = 1.7 cm
- 25 eighth-graders from this population are randomly selected
∴ n = 25
- To find the probability of the sample mean foot length less than 23
∴ The sample mean x = 23, find the standard deviation σx
- The rule to find σx is σx = σ/√n
∵ σ = 1.7 and n = 25
∴ σx = 1.7/√25 = 1.7/5 = 0.34
- Now lets find the z-score using the rule z-score = (x - μ)/σx
∵ x = 23 , μ = 23.4 , σx = 0.34
∴ z-score = (23 - 23.4)/0.34 = -1.17647 ≅ -1.18
- Use the table of the normal distribution to find P(x < 23)
- We will search in the raw of -1.1 and look to the column of 0.08
∴ P(X < 23) = 0.119 ≅ 0.120
* The probability of the sample mean foot length less than 23 cm is 0.120
Six customers enter a three-floor restaurant. Each customer decides on which floor to have dinner. Assume that the decisions of different customers are independent, and that for each customer, each floor is equally likely. Find the probability that exactly one customer dines on the first floor.
The probability that exactly one customer dines on the first floor is:
0.26337
Step-by-step explanation:We need to use the binomial theorem to find the probability.
The probability of k success in n experiments is given by:
[tex]P(X=k)=n_C_k\cdot p^k\cdot (1-p)^{n-k}[/tex]
where p is the probability of success.
Here p=1/3
( It represents the probability of choosing first floor)
k=1 ( since only one customer has to chose first floor)
n=6 since there are a total of 6 customers.
This means that:
[tex]P(X=1)=6_C_1\times (\dfrac{1}{3})^1\times (1-\dfrac{1}{3})^{6-1}\\\\\\P(X=1)=6\times (\dfrac{1}{3})\times (\dfrac{2}{3})^5\\\\\\P(X=1)=0.26337[/tex]
Using the binomial distribution, it is found that there is a 0.2634 = 26.34% probability that exactly one customer dines on the first floor.
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For each customer, there are only two possible outcomes, either they dine on the first floor, or they do not. The probability of a customer dining on the first floor is independent of any other customer, which means that the binomial probability distribution is used to solve this question.
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Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [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]
And p is the probability of a success on a single trial.
----------------
Six customers, thus [tex]n = 6[/tex].They are equally as likely to dine on any of the three floors, thus [tex]p = \frac{1}{3} = 0.3333[/tex].----------------
The probability that exactly one customer dines on the first floor is P(X = 1), thus:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 1) = C_{6,1}.(0.3333)^{1}.(0.6667)^{5} = 0.2634[/tex]
0.2634 = 26.34% probability that exactly one customer dines on the first floor.
A similar problem is given at https://brainly.com/question/13036444
Write the standard equation of a circle with center (-4 0) and radius 3 brainly