Answer:
Faster worker takes 4 hours and slower worker takes 12 hours.
Step-by-step explanation:
Let x be the time ( in hours ) taken by faster worker,
So, according to the question,
Time taken by slower worker = (x+8) hours,
Thus, the one day work of faster worker = [tex]\frac{1}{x}[/tex]
Also, the one day work of slower worker = [tex]\frac{1}{x+8}[/tex]
So, the total one day work when they work together = [tex]\frac{1}{x}+\frac{1}{x+8}[/tex]
Given,
They take 3 hours in working together,
So, their combined one day work = [tex]\frac{1}{3}[/tex]
[tex]\implies \frac{1}{x}+\frac{1}{x+8}=\frac{1}{3}[/tex]
[tex]\frac{x+8+x}{x^2+8x}=\frac{1}{3}[/tex] ( Adding fractions )
[tex]3(2x+8)=x^2+8x[/tex] ( Cross multiplication )
[tex]6x+24=x^2+8x[/tex] ( Distributive property )
[tex]x^2+2x-24=0[/tex] ( Subtraction property of equality )
By quadratic formula,
[tex]x=\frac{-2\pm \sqrt{100}}{2}[/tex]
[tex]x=\frac{-2\pm 10}{2}[/tex]
[tex]\implies x=4\text{ or }x=-6[/tex]
Since, hours can not negative,
Hence, time taken by faster worker = x hours = 4 hours,
And, the time taken by slower worker = x + 8 = 12 hours.
To solve this work rate problem, we set up an equation with combined work rates and find that the faster worker takes 3 hours alone, while the slower worker takes 11 hours alone.
Explanation:The question states that two secretaries can stuff envelopes together in 3 hours. The slower worker takes 8 hours more than the faster worker to complete the job alone. To find how long it takes each secretary to complete the job alone, we can set up an equation using the reciprocal of their work rates.
Let x be the time it takes for the faster worker to stuff the envelopes alone. Then, the slower worker will take x + 8 hours. The work rate of the faster worker is 1/x and the slower worker's rate is 1/(x + 8). Working together, their combined work rate is 1/3 per hour (since they complete the task in 3 hours).
The combined work rate equation will be:
1/x + 1/(x + 8) = 1/3
To solve this equation:
Multiply every term by the common denominator, which is 3x(x + 8).This gives us 3(x + 8) + 3x = x(x + 8).Simplify and solve the resulting quadratic equation.This results in x² + 8x - 3x - 24 = 0, which simplifies to x² + 5x - 24 = 0.Factor the quadratic equation to find the values of x.The factors of 24 that add up to 5 are 8 and -3, so (x + 8)(x - 3) = 0.Therefore, x = 3 or x = -8. Since time cannot be negative, we disregard x = -8.The faster worker takes 3 hours and the slower worker takes 3 + 8 = 11 hours.As a result, the faster worker takes 3 hours to complete the job alone, and the slower worker takes 11 hours to complete the job alone.
Sample methods that embody random sampling are often termed probability sampling methods. OTrue False
Answer:
The answer is true.
Step-by-step explanation:
Sample methods that embody random sampling are often termed probability sampling methods.
Yes this is a true statement.
Random sampling means picking up the samples randomly from a whole population with each sample having an equal chance of getting selected.
For example- selecting randomly 10 students from each class of a school, to survey for the food quality in school's cafeteria.
And this is a type of probability sampling methods. Other types are stratified sampling, cluster sampling etc.
Samantha wants to sort her greetin cards into boxes that hold 24 cards each she has 312 greeting cards how many boxes will she need?
Answer:
Samantha will need 13 boxes.
Step-by-step explanation:
Samantha wants to sort her greeting cards in boxes.
Each box can hold 24 cards.
Samantha has 312 greeting cards.
To calculate the number of boxes she need we have to divide total number of greeting cards to 24.
Therefore, = [tex]\frac{312}{24}[/tex]
= 13 boxes
Samantha will need 13 boxes.
A researcher wants to know if the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. If we assume the standard deviation is 3 years, a 95% confidence interval for the average time served is:
Answer:
The interval is : (7.206 , 7.794)
Step-by-step explanation:
The mean is = 7.5
Standard deviation = 3
n = 400
At 95% confidence interval, the z score is 1.96
[tex]7.5+1.96(\frac{3}{\sqrt{400} } )[/tex]
And [tex]7.5-1.96(\frac{3}{\sqrt{400} } )[/tex]
[tex]7.5+0.294[/tex] and [tex]7.5-0.294[/tex]
So, the interval is : (7.206 , 7.794)
1. A dad holds five coins in his hand. He tells his son that if he can guess the amount of money he is holding within 5% error, he can have the money. The son guesses that dad is holding 81 cents. The dad opens his hand and displays 90 cents. Did the son guess close enough to get the money?
The dad had 90 cents.
Multiply the 90 cents by 5%:
90 x 0.05 = 4.5 cents.
Subtract that from 90:
90 - 4.5 = 85.5 cents.
The lowest guess the son could say was 86 cents to be within 5%
Since the son guessed lower than that he did not get the money.
1. A dad holds five coins in his hand. He tells his son that if he can guess the amount of money he is holding within 5% error, he can have the money. The son guesses that dad is holding 81 cents. The dad opens his hand and displays 90 cents. Did the son guess close enough to get the money?
yes
Determine whether each statement is true or false in double-struck R3. (a) Two lines parallel to a third line are parallel. True False (b) Two lines perpendicular to a third line are parallel. True False (c) Two planes parallel to a third plane are parallel. True False (d) Two planes perpendicular to a third plane are parallel. True False (e) Two lines parallel to a plane are parallel. True False (f) Two lines perpendicular to a plane are parallel. True False (g) Two planes parallel to a line are parallel. True False (h) Two planes perpendicular to a line are parallel. True False (i) Two planes either intersect or are parallel. True False (j) Two lines either intersect or are parallel. True False (k) A plane and a line either intersect or are parallel. True False
Answer:
(a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False
(h) True , (i) True , (j) False , (k) True
Step-by-step explanation:
* Lets explain how to solve the problem
(a) Two lines parallel to a third line are parallel (True)
- Their direction vectors are scalar multiplies of the direction of the 3rd
line, then they are scalar multiples of each other so they are parallel
(b) Two lines perpendicular to a third line are parallel (False)
- The x-axis and the y-axis are ⊥ to the z-axis but not parallel to
each other
(c) Two planes parallel to a third plane are parallel (True)
- Their normal vectors parallel to the normal vector of the 3rd plane,
so these two normal vectors are parallel to each other and the
planes are parallel
(d) Two planes perpendicular to a third plane are parallel (False)
- The xy plane and yz plane are not parallel to each other but both
⊥ to xz plane
(e) Two lines parallel to a plane are parallel (False)
- The x-axis and y-axis are not parallel to each other but both parallel
to the plane z = 1
(f) Two lines perpendicular to a plane are parallel (True)
- The direction vectors of the lines parallel to the normal vector of
the plane, then they parallel to each other , so the lines are parallel
(g) Two planes parallel to a line are parallel (False)
- The planes y = 1 and z = 1 are not parallel but both are parallel to
the x-axis
(h) Two planes perpendicular to a line are parallel (True)
- The normal vectors of the 2 planes are parallel to the direction of
line, then they are parallel to each other so the planes are parallel
(i) Two planes either intersect or are parallel (True)
(j) Two lines either intersect or are parallel (False)
- They can be skew
(k) A plane and a line either intersect or are parallel (True)
- They are parallel if the normal vector of the plane and the direction
of the line are ⊥ to each other , otherwise the line intersect the plane
at the angle 90° - Ф
This question is based on the properties of lines and planes. Therefore, (a) True , (b) False , (c) True , (d) False , (e) False , (f) True , (g) False
, (h) True , (i) True , (j) False , (k) True.
We have to choose correct statement and marked true or false.
Lets solve the problem.
(a) Two lines parallel to a third line are parallel. (True)
Reason - The direction vectors are scalar multiple of the direction of the third line, then they are scalar multiple of each other. So, they are parallel.
(b) Two lines perpendicular to a third line are parallel. (False)
Reason- As we know that, x-axis and the y-axis are perpendicular to the z-axis but not parallel to each other.
(c) Two planes parallel to a third plane are parallel (True)
Reason- The normal vectors of planes are parallel to the normal vector of the third plane. So, these two normal vectors are parallel to each other and the planes are parallel.
(d) Two planes perpendicular to a third plane are parallel. (False)
Reason- x-y plane and y-z plane are not parallel to each other. But they are perpendicular to x-z plane.
(e) Two lines parallel to a plane are parallel. (False)
Reason - Both x-axis and y-axis are not parallel to each other. But, parallel to the plane z = 1.
(f) Two lines perpendicular to a plane are parallel. (True)
Reason - The direction vectors of the lines parallel to the normal vector of the plane, then they parallel to each other , so the lines are parallel.
(g) Two planes parallel to a line are parallel. (False)
Reason- The planes y = 1 and z = 1 are not parallel, but they are parallel to the x-axis.
(h) Two planes perpendicular to a line are parallel. (True)
Reason- The normal vectors of the two planes are parallel to the direction of line. So, they are parallel to each other. Hence, they are parallel.
(i) Two planes either intersect or are parallel (True)
(j) Two lines either intersect or are parallel (False)
Reason- They can also be skew.
(k) A plane and a line either intersect or are parallel (True)
Reason- They are parallel, if the normal vector of the plane and the direction of the line are perpendicular to each other, otherwise the line intersect the plane at the angle 90° [tex]\theta[/tex].
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Write equations for the horizontal and vertical lines passing through the point (4, -6)
Answer:
So you have the vertical line passing through is x=4 and the horizontal line passing through is y=-6.
Step-by-step explanation:
In general the horizontal line passing through (a,b) is y=b and the vertical line passing through (a,b) is x=a.
So you have the vertical line passing through is x=4 and the horizontal line passing through is y=-6.
Solve the following simultaneous linear congruences.
a) x ? 1 (mod 3), x ? 2 (mod 4), x ? 3 (mod 5).
b) x ? 4 (mod 10), x ? 8 (mod 12), x ? 6 (mod 18).
a. The moduli are coprime, so you can apply the Chinese remainder theorem directly. Let
[tex]x=4\cdot5+3\cdot5+3\cdot4[/tex]
Taken mod 3, the last two terms vanish, and [tex]20\equiv2\pmod3[/tex] so we need to multiply by the inverse of 2 modulo 3 to end up with a remainder of 1. Since [tex]2\cdot2\equiv4\equiv1\pmod3[/tex], we multiply the first term by 2.[tex]x=4\cdot5\cdot2+3\cdot5+3\cdot4[/tex]
Taken mod 4, the first and last terms vanish, and [tex]15\equiv3\pmod4[/tex]. Multiply by the inverse of 3 modulo 4 (which is 3 because [tex]3\cdot3\equiv9\equiv1\pmod4[/tex]), then by 2 to ensure the proper remainder is left.[tex]x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4[/tex]
Taken mod 5, the first two terms vanish, and [tex]12\equiv2\pmod5[/tex]. Multiply by the inverse of 2 modulo 5 (3, since [tex]3\cdot2\equiv6\equiv1\pmod5[/tex]) and again by 3.[tex]x=4\cdot5\cdot2+3\cdot5\cdot3\cdot2+3\cdot4\cdot3\cdot3[/tex]
[tex]\implies x=238[/tex]
By the CRT, we have
[tex]x\equiv238\pmod{3\cdot4\cdot5}\implies x\equiv-2\pmod{60}\implies\boxed{x\equiv58\pmod{60}}[/tex]
i.e. any number [tex]58+60n[/tex] (where [tex]n[/tex] is an integer) satisifes the system.
b. The moduli are not coprime, so we need to check for possible contradictions. If [tex]x\equiv a\pmod m[/tex] and [tex]x\equiv b\pmod n[/tex], then we need to have [tex]a\equiv b\pmod{\mathrm{gcd}(m,n)}[/tex]. This basically amounts to checking that if [tex]x\equiv a\pmod m[/tex], then we should also have [tex]x\equiv a\pmod{\text{any divisor of }m}[/tex].
[tex]x\equiv4\pmod{10}\implies\begin{cases}x\equiv4\equiv0\pmod2\\x\equiv4\pmod5\end{cases}[/tex]
[tex]x\equiv8\pmod{12}\implies\begin{cases}x\equiv0\pmod2\\x\equiv2\pmod3\end{cases}[/tex]
[tex]x\equiv6\pmod{18}\implies\begin{cases}x\equiv0\pmod2\\x\equiv0\pmod3\end{cases}[/tex]
The last congruence conflicts with the previous one modulo 3, so there is no solution to this system.
Which of the following is NOT a requirement of the Combinations Rule, Subscript n Baseline Upper C Subscript requalsStartFraction n exclamation mark Over r exclamation mark left parenthesis n minus r right parenthesis exclamation mark EndFraction , for items that are all different? Choose the correct answer below. A. That r of the n items are selected (without replacement). B. That there be n different items available. C. That order is not taken into account (consider rearrangements of the same items to be the same). D. That order is taken into account (consider rearrangements of the same items to be different sequences).
Answer:
C. That order is taken into account
Answer:
D. That order is taken into account (consider rearrangements of the same items to be different sequences).
Step-by-step explanation:
[tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
For example:
Combinations of 2 from a set a,b,c: ab=ba, ac = ca, bc = cb, meaning that the order is not important(ab and ba is the same sequence).
aa, bb, cc are not valid, since there is no replacement.
So the answer for this qustion is:
D. That order is taken into account (consider rearrangements of the same items to be different sequences).
If you roll two fair dice (one black die and one white die), in how many ways can you obtain a 1 on the white die? A 1 on the white die can be obtained in different ways. (u) More Enter your answer in the answer box and then click Check Answer. All parts showing Clear All
Answer:
6
Step-by-step explanation:
Sample space of the experiment
first number in the bracket is white die and second number in the bracket is black
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
As it can be seen that the first numbers in the bracket are (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
∴1 on the white die can be obtained in 6 ways
In the case of rolling two dice and trying to obtain a 1 on the white die, there are 6 ways to accomplish this because the black die outcome is irrelevant and it can show any number from 1 to 6 while pairing with a 1 on the white die.
The question asks about the probability of getting a specific result when rolling two fair dice, which is a problem in the realm of simple probability within mathematics.
Specifically, the question is focused on finding the number of ways to obtain a 1 on the white die.
When rolling two dice, there are a total of 6 different possible outcomes for the black die (since a standard die has 6 faces), and 1 specific outcome we're looking for on the white die, which is a 1.
Each outcome on the black die can be paired with a 1 on the white die, resulting in the combinations (1,1), (2,1), (3,1), (4,1), (5,1), and (6,1).
This gives us a total of 6 ways to achieve a 1 on the white die, regardless of what the black die shows.
The Centers for Disease Control reported the percentage of people 18 years of age and older who smoke (CDC website, December 14, 2014). Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .31. a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02 (to the nearest whole number)? Use 95% confidence.
Final answer:
To estimate the proportion of smokers with a margin of error, use the formula n = (Z^2 * p * (1-p)) / E^2, where n is the sample size, Z is the Z-value for the desired confidence level, p is the preliminary estimate of the proportion who smoke, and E is the margin of error. Plugging in the values from the question, the sample size should be 753.
Explanation:
To estimate the proportion of smokers in the population with a margin of error of 0.02 and a 95% confidence level, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n is the sample sizeZ is the Z-value for the desired confidence level (1.96 for 95% confidence)p is the preliminary estimate of the proportion who smoke (0.31)E is the margin of error (0.02)Plugging in the values, we get:
n = (1.96^2 * 0.31 * (1-0.31)) / 0.02^2 = 752.34
Rounding up to the nearest whole number, the sample size should be 753.
We have three coins: one with heads on both faces, the second with tails on both faces, and the third a regular one. We choose one at random, toss it, and the outcome is heads. What is the probability that the opposite face of the tossed coin is tails?
Answer:
Probability: [tex]\frac{1}{2}[/tex] = 0.5 = 50%
Step-by-step explanation:
Based on the question one coin is chosen at random and tossed. That coin then lands and is heads. Since the coin landed on heads we can eliminate the possibility of the coin that was chosen being the coin with double tails.
The following possibilities are that the coin has double heads or is a regular coin with both tails and heads. Seeing as the coin landed on heads, there are only two possible out comes for the other side of the coin
The other side is either Heads or Tails. That gives us a 50% chance of the other side being tails.
[tex]\frac{1}{2}[/tex] = 0.5 = 50%
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Which type of data in an Enterprise System occasionally changes?
Master data
Date and Time data
Organizational data
Transaction data
Answer:
the correct answer is master data
Step-by-step explanation:
Enterprise system is a information system which provides a company with a wide integration and coordination regarding the important business processes and also helps in providing seamless flow of information through out the company.
Master data is a type of data in the enterprise system which is changed only occasionally , as this data includes all the information related to the customers like name, contact etc which helps a firm in analyzing their behavior and conduct high level research.
A ball is thrown vertically upward. After t seconds, its height h (in feet) is given by the function h(t) = 52t - 16t^2 . What is the maximum height that the ball will reach?
Do not round your answer.
Answer: 42.25 feet
Step-by-step explanation:
We know that after "t" seconds, its height "h" in feet is given by this function:
[tex]h(t) = 52t -16t^2[/tex]
The maximum height is the y-coordinate of the vertex of the parabola. Then, we can use the following formula to find the corresponding value of "t" (which is the x-coordinate of the vertex):
[tex]x=t=\frac{-b}{2a}[/tex]
In this case:
[tex]a=-16\\b=52[/tex]
Substituting values, we get :
[tex]t=\frac{-52}{2(-16)}\\\\t=1.625[/tex]
Substituting this value into the function to find the maximum height the ball will reach, we get:
[tex]h(1.625) = 52(1.625) -16(1.625)^2\\\\h(1.625) =42.25\ ft[/tex]
Answer:
42.25 feet
Step-by-step explanation:
The maximum of a quadratic can be found by finding the vertex of the parabola that the quadratic creates visually on a graph.
So first step to find the maximum height is to find the x-coordinate of the vertex.
After you find the x-coordinate of the vertex, you will want to find the y that corresponds by using the given equation, [tex]y=52x-16x^2[/tex]. The y-coordinate we will get will be the maximum height.
Let's start.
The x-coordinate of the vertex is [tex]\frac{-b}{2a}[/tex].
[tex]y=52x-16x^2[/tex] compare to [tex]y=ax^2+bx+c[/tex].
We have that [tex]a=-16,b=52,c=0[/tex].
Let's plug into [tex]\frac{-b}{2a}[/tex] with those values.
[tex]\frac{-b}{2a}[/tex] with [tex]a=-16,b=52,c=0[/tex]
[tex]\frac{-52}{2(-16)}=\frac{52}{32}=\frac{26}{16}=\frac{13}{8}[/tex].
The vertex's x-coordinate is 13/8.
Now to find the corresponding y-coordinate.
[tex]y=52(\frac{13}{8})-16(\frac{13}{8})^2[/tex]
I'm going to just put this in the calculator:
[tex]y=\frac{169}{4} \text{ or } 42.25[/tex]
So the maximum is 42.25 feet.
A flagpole 94.9 ft tall is on the top of a building. From a point on level ground, the angle of elevation of the top of the flagpole is 35.3degrees, while the angle of elevation of the bottom of the flagpole is 26.2degrees. Find the height of the building.
Answer:
214 ft
Step-by-step explanation:
Height of building = 94.9 ft
The angle of elevation of the top of the flagpole = θ₁ = 35.3°
The angle of elevation of the bottom of the flagpole = θ₂ = 26.2°
Let,
Height of building = x
Distance from observation point to base of building = y
[tex]tan 26.2 =\frac{x}{y}\\\Rightarrow y=\frac{x}{tan26.2}[/tex]
[tex]tan 35.3 =\frac{94.9+x}{y}\\\Rightarrow tan 35.3 =\frac{94.9+x}{\frac{x}{tan26.2}}\\\Rightarrow \frac{x}{tan26.2}tan35.3=94.9+x\\\Rightarrow \frac{tan35.3}{tan26.2}x-x=94.9\\\Rightarrow x=\frac{94.9}{\frac{tan35.3}{tan26.2}-1}\\\Rightarrow x=214.84/ ft[/tex]
I have used the exact values from the calculator.
∴ Height of the building is 214.84 ft
Answer:
The height of the building is 214.84 ft.
Step-by-step explanation:
Given information:
The height of the flagpole = 94.9 ft.
The angle of elevation of top = θ[tex]_1[/tex] = [tex]35.3^o[/tex]
The angle of elevation of bottom = θ[tex]_2=26.2^o[/tex]
If the height of building is [tex]x[/tex]
Then,
[tex]tan 26.2=x/y\\y=x/(tan26.2)\\[/tex]
And:
[tex]tan 35.3=(94.9+x)/y\\[/tex]
[tex]94.9+x=y \times tan35.3[/tex]
On putting the value in above equation:
[tex]x=\frac{94.9}{\frac{tan35.3}{tan26.2}-1 }[/tex]
solving the above equation:
[tex]x=214.84 ft.[/tex]
Hence, the height of the building is 214.84 ft.
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7 = -x+3. Consider the line y = Find the equation of the line that is parallel to this line and passes through the point (-5, 6). Find the equation of the line that is perpendicular to this line and passes through the point (-5, 6)
Answer with explanation:
The equation of line is, y= -x +3
→x+y-3=0---------(1)
⇒Equation of line Parallel to Line , ax +by +c=0 is given by, ax + by +K=0.
Equation of Line Parallel to Line 1 is
x+y+k=0
The Line passes through , (-5,6).
→ -5+6+k=0
→ k+1=0
→k= -1
So, equation of Line Parallel to line 1 is
x+y-1=0
⇒Equation of line Perpendicular to Line , ax +by +c=0 is given by, bx - a y +K=0.
Equation of Line Perpendicular to Line 1 is
x-y+k=0
The Line passes through , (-5,6).
→ -5-6+k=0
→ k-11=0
→k= 11
So, equation of Line Parallel to line 1 is
x-y+11=0
(1 point) The random variables X and Y have the joint density: fX,Y(x,y)={2−x−y00
Answer:
. Let fX,Y(x,y) = 10xy^2 for 0 < x < y < 1 be the joint density function of the random pair (X, Y). (a) Obtain the marginal density f(y) of Y. (b) Obtain the conditional density fx|y(x|y) of X given Y = y. (c) Evaluate the conditional expectation of X, given Y=y
Y = y.
Step-by-step explanation:
Find parametric equations for the line. (Use the parameter t.) The line through the origin and the point (2, 6, −1) (x(t), y(t), z(t)) Find the symmetric equations.
The symmetric equations of the line are: x/(-2) = y/(-6) = z/1 and the x(t) = 2t, y(t) = 6t and z(t) = -t are the parametric equations.
What is Parametric Equation?a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
To find the parametric equations for the line through the origin and the point (2, 6, -1), we can use the vector equation of a line:
r(t) = r₀ + tv
where r(t) = (x(t), y(t), z(t)) is the position vector of a point on the line,
r₀ = (0, 0, 0) is the position vector of the origin,
t is a parameter, and v is the direction vector
v = (2, 6, -1) - (0, 0, 0) = (2, 6, -1)
Now r(t) = (0, 0, 0) + t(2, 6, -1) = (2t, 6t, -t)
Therefore, the parametric equations of the line are:
x(t) = 2t
y(t) = 6t
z(t) = -t
To find the symmetric equations of the line, we can eliminate the parameter t from the parametric equations.
t = -z
Substituting this into the x(t) and y(t) equations, we get:
x = 2t = -2z
y = 6t = -6z
Therefore, the symmetric equations of the line are:
x/(-2) = y/(-6) = z/1
Hence, the symmetric equations of the line are: x/(-2) = y/(-6) = z/1
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Final answer:
Parametric equations for a line passing through the origin and a point are found by using the components of the point as coefficients for the parameter t. Symmetric equations are obtained by equating the ratios of each component to their corresponding coefficients in the direction vector.
Explanation:
To find the parametric equations for the line passing through the origin (0,0,0) and the point (2, 6, −1), we can use the position vector of the point (2,6,−1) and multiply it by the parameter t. This gives us the parametric equations:
x(t) = 2ty(t) = 6tz(t) = -tThe symmetric equations of the line can be obtained by eliminating the parameter t. Since t = x/2 = y/6 = z/(−1), the symmetric equations are:
x/2 = y/6 = z/(−1).
2. Using the Euclidean Algorithm, Calculate the Greatest Common Divisor of the num- bers 11655 and 11340.
Only two steps needed:
11,655 = 1*11,340 + 315
11,340 = 36*315 + 0
This shows that [tex]\mathrm{gcd}(11,655,\,11,340)=315[/tex].
The Greatest Common Divisor (GCD) of the numbers 11655 and 11340 is found using the Euclidean Algorithm by first dividing 11655 by 11340 to get a remainder of 315. This remainder becomes the divisor in the next step and the process repeats until the remainder is zero. Hence, the GCD of 11655 and 11340 is 315.
Explanation:To calculate the Greatest Common Divisor (GCD) of the numbers 11655 and 11340 using the Euclidean Algorithm, follow this process:
Divide 11655 by 11340. The quotient is 1 and the remainder is 315. Now, take the divisor 11340 and divide it by the remainder from the previous step (315). The quotient is 36 with a remainder of 0. The remainder is now zero, which means our last non-zero remainder (315) is the Greatest Common Divisor (GCD).
Therefore, the Greatest Common Divisor of the numbers 11655 and 11340 using the Euclidean Algorithm is 315.
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The exact value of 400 comma 000 times 200 is 8000000. (Use scientific notation. Use the multiplication symbol in the math palette as needed.)
Answer:
The scientific notation of 8,000,000 is 8 × 10^6
Step-by-step explanation:
* Lets explain the meaning of the scientific notation
- Scientific notation is a way of writing very large or very small numbers
- A number is written in scientific notation when a number between 1
and 10 is multiplied by a power of 10
- Ex: 650,000,000 can be written in scientific notation as
6.5 × 10^8
- We put a decimal points to make the number between 1 and 10 and
then count how many places from right to left until the decimal point
The decimal point between 6 and 5 to make the number 6.5 and
there are 8 places from the last zero to the decimal point
* Lets solve the problem
∵ The exact value of 400,000 × 200 = 8,000,000
- Put the decimal point before 8 and count how many places from
the last zero to the decimal point
∵ There are six places from last zero to the decimal point
∴ The scientific notation of 8,000,000 is 8 × 10^6
Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (1, 0, 9) and perpendicular to the plane x + 2y + z = 7
Answer:
r=<1,0,9>+t<1,2,1>
and
x=1+t
y=2t
z=9+t
Step-by-step explanation:
A vector perpendicular to the plane :
[tex]ax+by+cz+d=0[/tex]
is given by (a,b,c)
So a vector perpendicular to given plane will have :
(1,2,1)
[tex]The\ parametric\ equation\ of\ a\ line\ through\ (1,0,9)\ and\ parallel\ to\ vector\ (a,b,c) is\ given\ by:\\x=x_0+ta\\y=y_0+tb\\z=z_0+tb\\x=1+t(1)\\x=1+t\\y=0+t(2)\\y=2t\\z=9+t(1)\\z=9+t\\The\ vector\ form\ is:\\r=<1,0,9>+t<1,2,1>[/tex] ..
The vector equation r(t) is (1, 0, 9) + t(1, 2, 1) and the parametric equations are x = 1 + t, y = 2t, z = 9 + t.
To find the vector equation and parametric equations for the line that passes through the point (1, 0, 9) and is perpendicular to the plane given by x + 2y + z = 7, follow these steps :
1. Find the Normal Vector to the Plane :
The normal vector of the plane x + 2y + z = 7 is given by the coefficients of x, y, and z in the plane equation. Therefore, the normal vector n is (1, 2, 1).2. Vector Equation of the Line :
A line passing through point (1, 0, 9) in the direction of normal vector (1, 2, 1) can be written in vector form as :r(t) = (1, 0, 9) + t(1, 2, 1) where t is the parameter.3. Parametric Equations :
Extract the parametric equations from the vector equation :
x(t) = 1 + ty(t) = 0 + 2tz(t) = 9 + tThus, the parametric equations for the line are :
x = 1 + ty = 2tz = 9 + tAn experiment consists of tossing 4 coins successively. The number of sample points in this experiment is
a. 16
b. 8
c. 4
d. 2
The number of sample points in this experiment is 16.
Explanation:The number of sample points in this experiment can be found by multiplying the number of possible outcomes for each coin toss. Since there are 2 possible outcomes for each coin toss, and we have 4 coin tosses, the total number of sample points is 2 x 2 x 2 x 2 = 16.
Therefore, option a, 16, is the correct answer.
The number of sample points for tossing 4 coins successively is a. 16, calculated using the formula 2⁴. Each coin flip has 2 possible outcomes, and for 4 coins, this results in 2⁴ = 16 outcomes.
When tossing 4 coins successively, each coin has 2 possible outcomes: heads (H) or tails (T). The total number of sample points in such an experiment can be calculated as follows:
Step-by-Step Explanation:
Each coin flip is an independent event with 2 possible outcomes.For 4 coin flips, the number of sample points is given by the formula 2n, where n is the number of coins.In this case, n = 4, so the number of sample points is 2⁴ = 16.Therefore, the number of sample points in this experiment is 16.
The claim is that the IQ scores of statistics professors are normally distributed, with a mean greater than 116. A sample of 20 professors had a mean IQ score of 121 with a standard deviation of 11. Find the value of the test statistic.
Answer: t= 2.032
Step-by-step explanation:
Given : Sample size : [tex]n=20[/tex]
Sample mean : [tex]\overline{x}=121[/tex]
Standard deviation : [tex]\sigma= 11[/tex]
Claim : The IQ scores of statistics professors are normally distributed, with a mean greater than 116.
Let [tex]\mu [/tex] be the mean scores of statistics professors.
Then the set of hypothesis for the given situation will be :-
[tex]H_0:\mu\leq116\\\\H_1:\mu>116[/tex]
As the alternative hypothesis is right tailed , thus the test would be right tail test.
Since the sample size is less than 30, therefore the test would be t-test .
The test statistics for the given situation will be :-
[tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
[tex]\Rightarrow\ t=\dfrac{121-116}{\dfrac{11}{\sqrt{20}}}=2.03278907045\approx2.032[/tex]
Hence, the value of the test statistic : t= 2.032
Answer:
It's 2.05.Step-by-step explanation:
To prove a hypothesis, we have to use test statisticians like the z-value which is used in normally distributed data, and this is the case.
To calculate the z-value we use: [tex]z=\frac{x-u}{\frac{o}{\sqrt{n} } }[/tex]; where x is the sample mean, u is the population mean, o is the standard deviation and n the sample size.
Replacing all values:
[tex]z=\frac{121-116}{\frac{11}{\sqrt{20} } } =\frac{5}{2.44}=2.05[/tex]
Therefore the value of the test statistic is 2.05.
(It's important to clarify that the problem isn't asking about the hypothesis, or the probability value, it's just asking for the test parameter, which in this case is just a z-value).
The half-life of Radium-226 is 1590 years. If a sample contains 500 mg, how many mg will remain after 2000 years? Preview mg Give your answer accurate to at least 2 decimal places. Get help: Video Video
Answer:
[tex]a_{n}[/tex]=209.09 mg
Step-by-step explanation:
given: material= radium
half life= 1590 years
initial mass [tex]a_{0}[/tex] =500mg
we know that to calculate the amount left we use
[tex]a_{n}[/tex] = [tex]a_{0}[/tex][tex]\left ( 0.5\right )^{n}[/tex]
[tex]n=\frac{2000}{1590} = 1.2578[/tex]
therefore
[tex]a_{n}[/tex] = [tex]500\times0.5^{1.2578}[/tex]
[tex]a_{n}[/tex]=209.09058407921 mg
[tex]a_{n}[/tex]=209.09 mg amount left after 2000 years
Assume that the ages for first marriages are normally distributed with a mean of 26 years and a standard deviation of 4 years. What is the probability that a person getting married for the first time is in his or her twenties? (Round your answer to 4 decimal places.)
Answer:
0.7743
Step-by-step explanation:
Mean of age = u = 26 years
Standard Deviation = [tex]\sigma[/tex] = 4 years
We need to find the probability that the person getting married is in his or her twenties. This means the age of the person should be between 20 and 30. So, we are to find P( 20 < x < 30), where represents the distribution of age.
Since the data is normally distributed we can use the z distribution to solve this problem. The formula to calculate the z score is:
[tex]z=\frac{x-u}{\sigma}[/tex]
20 converted to z score will be:
[tex]z=\frac{20-26}{4}=-1.5[/tex]
30 converted to z score will be:
[tex]z=\frac{30-26}{4}=1[/tex]
So, now we have to find the probability that the z value lies between -1.5 and 1.
P( 20 < x < 30) = P( -1.5 < z < 1)
P( -1.5 < z < 1 ) = P(z < 1) - P(z<-1.5)
From the z-table:
P(z < 1) = 0.8413
P(z < -1.5) =0.067
So,
P( -1.5 < z < 1 ) = 0.8413 - 0.067 = 0.7743
Thus,
P( 20 < x < 30) = 0.7743
So, we can conclude that the probability that a person getting married for the first time is in his or her twenties is 0.7743
A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 390 babies were born, and 312 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective?
Answer:Yes
Step-by-step explanation:
Given
n=390 x=312
[tex]\hat{p}=\frac{312}{390}=0.8[/tex]
Confidence level=99 %
[tex] Z_{\frac{\alpha }{2}}=2.575[/tex]
Standard error(S.E.)=[tex]\sqrt{\frac{\hat{p}\left ( 1-\hat{p}\right )}{n}}[/tex]
S.E.=[tex]\sqrt{\frac{0.8\times 0.2}{390}}[/tex]
S.E.=0.0202
Confidence interval
[tex]p\pm \left [ z_{\frac{\alpha }{2}}\cdot S.E.\right ][/tex]
[tex]0.8 \pm 0.0521[/tex]
[tex]\left ( 0.7479,0.8521 \right )[/tex]
Since 0.5 does not lie in interval therefore method appear to be effective
please help, its formula manipulation, only respond if u know how to get the answer, thank you
Answer:
Problem 1:
[tex]r=\frac{3V}{2 \pi h^2}[/tex]
Problem 2:
[tex]h=\frac{3V}{b^2}[/tex]
Problem 3:
The radius is [tex]\frac{25}{\pi}[/tex] cm.
Problem 4:
The width is 15 cm.
Step-by-step explanation:
Problem 1:
We want to solve [tex]V=\frac{2\pi rh^2}{3}[/tex] for [tex]r[/tex].
[tex]V=\frac{2\pi rh^2}{3}[/tex]
Multiply both sides by 3:
[tex]3V=2\pi r h^2[/tex]
Rearrange the multiplication using commutative property:
[tex]3V=2\pi h^2 \cdot r[/tex]
We want to get [tex]r[/tex] by itself so divide both sides by what [tex]r[/tex] is being multiplied by which is [tex]2\pi h^2[/tex].
[tex]\frac{3V}{2 \pi h^2}=r[/tex]
[tex]r=\frac{3V}{2 \pi h^2}[/tex]
Problem 2:
We want to solve for [tex]h[/tex] in [tex]V=\frac{b^2h}{3}[/tex].
Multiply both sides by 3:
[tex]3V=b^2h[/tex]
We want [tex]h[/tex] by itself so divide both sides by what [tex]h[/tex] is being multiply by; that is divide both sides by [tex]b^2[/tex].
[tex]\frac{3V}{b^2}=h[/tex]
[tex]h=\frac{3V}{b^2}[/tex]
Problem 3:
The circumference formula for a circle is [tex]2\pi r[/tex]. We are asked to solve for the radius when the circumference is [tex]50[/tex] cm.
[tex]2\pi r=50[/tex]
Divide both sides by what r is being multiply by; that is divide both sides by [tex]2\pi[/tex]:
[tex]r=\frac{50}{2\pi}[/tex]
Reduce fraction:
[tex]r=\frac{25}{\pi}[/tex]
The radius is [tex]\frac{25}{\pi}[/tex] cm.
Problem 4:
The perimeter of a rectangle is [tex]2w+2L[/tex] where [tex]w[/tex] is the width and [tex]L[/tex] is the length.
We are asked to find w, the width, for when L, the length, is 5, and the perimeter is 40.
So we have this equation to solve for w:
[tex]40=2w+2(5)[/tex]
Simplify the 2(5) part:
[tex]40=2w+10[/tex]
Subtract both sides by 10:
[tex]30=2w[/tex]
Divide both sides by 2:
[tex]\frac{30}{2}=w[/tex]
Simplify the fraction:
[tex]15=w[/tex]
The width is 15 cm.
Without using a calculator and with a simple approach, explain how to use reasoning and mental arithmetic to determine which of the following is a better deal: Cereal A: 15oz for $2.95 or Cereal B: 32oz for $5.95
Answer:
Cereal B
Step-by-step explanation:
Given are two different rates for cereals A and B.
as Cereal A: 15oz for $2.95 or Cereal B: 32oz for $5.95
As such we cannot compare unless we make it unit rate for same number of units
Let us find unit oz rates
Cereal A per oz= [tex]\frac{2.95}{15} =0.1967[/tex]dollars
Cereal B per oz = [tex]\frac{5.95}{32} =0.1859[/tex]dollars
Comparing unit rates per ounce,
we find that Cereal B per oz is lower.
Answer is Cereal B.
A BOX OF 7 ITEMS COSTS $20.79. FIND THE COST OF EACH ITEMS,
A.$0.30
B.$6
C.$0.03
D.$3
Answer:
D. $3.
Step-by-step explanation:
We have been given that a box of 7 items costs $20.79. We are asked to find the cost of each item.
To find the cost of each item, we will divide total cost by total number of items.
[tex]\text{Cost of each item}=\frac{\$20.79}{7}[/tex]
[tex]\text{Cost of each item}=\$2.94142857[/tex]
Upon rounding our answer to nearest dollar, we will get:
[tex]\text{Cost of each item}\approx\$3[/tex]
Therefore, the cost of each item will be approximately $3 and option D is the correct choice.
Find parametric equations for the path of a particle that moves along the circle x2 + (y − 1)2 = 4 in the manner described. (Enter your answer as a comma-separated list of equations. Let x and y be in terms of t.) (a) Once around clockwise, starting at (2, 1). 0 ≤ t ≤ 2π
Answer:
[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex]
Step-by-step explanation:
[tex](x-h)^2+(y-k)^2=r^2[/tex] has parametric equations:
[tex](x-h)=r\cos(t) \text{ and } (y-k)=r\sin(t)[/tex].
Let's solve these for x and y respectively.
[tex]x-h=r\cos(t)[/tex] can be solved for x by adding h on both sides:
[tex]x=r\cos(t)+h[/tex].
[tex]y-k=r \sin(t)[/tex] can be solve for y by adding k on both sides:
[tex]y=r\sin(t)+k[/tex].
We can verify this works by plugging these back in for x and y respectively.
Let's do that:
[tex](r\cos(t)+h-h)^2+(r\sin(t)+k-k)^2[/tex]
[tex](r\cos(t))^2+(r\sin(t))^2[/tex]
[tex]r^2\cos^2(t)+r^2\sin^2(t)[/tex]
[tex]r^2(\cos^2(t)+\sin^2(t))[/tex]
[tex]r^2(1)[/tex] By a Pythagorean Identity.
[tex]r^2[/tex] which is what we had on the right hand side.
We have confirmed our parametric equations are correct.
Now here your h=0 while your k=1 and r=2.
So we are going to play with these parametric equations:
[tex]x=2\cos(t)[/tex] and [tex]y=2\sin(t)+1[/tex]
We want to travel clockwise so we need to put -t and instead of t.
If we were going counterclockwise it would be just the t.
[tex]x=2\cos(-t)[/tex] and [tex]y=2\sin(-t)+1[/tex]
Now cosine is even function while sine is an odd function so you could simplify this and say:
[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex].
We want to find [tex]\theta[/tex] such that
[tex]2\cos(t-\theta_1)=2 \text{ while } -2\sin(t-\theta_2)+1=1[/tex] when t=0.
Let's start with the first equation:
[tex]2\cos(t-\theta_1)=2[/tex]
Divide both sides by 2:
[tex]\cos(t-\theta_1)=1[/tex]
We wanted to find [tex]\theta_1[/tex] for when [tex]t=0[/tex]
[tex]\cos(-\theta_1)=1[/tex]
Cosine is an even function:
[tex]\cos(\theta_1)=1[/tex]
This happens when [tex]\theta_1=2n\pi[/tex] where n is an integer.
Let's do the second equation:
[tex]-2\sin(t-\theta_2)+1=1[/tex]
Subtract 2 on both sides:
[tex]-2\sin(t-\theta_2)=0[/tex]
Divide both sides by -2:
[tex]\sin(t-\theta_2)=0[/tex]
Recall we are trying to find what [tex]\theta_2[/tex] is when t=0:
[tex]\sin(0-\theta_2)=0[/tex]
[tex]\sin(-\theta_2)=0[/tex]
Recall sine is an odd function:
[tex]-\sin(\theta_2)=0[/tex]
Divide both sides by -1:
[tex]\sin(\theta_2)=0[/tex]
[tex]\theta_2=n\pi[/tex]
So this means we don't have to shift the cosine parametric equation at all because we can choose n=0 which means [tex]\theta_1=2n\pi=2(0)\pi=0[/tex].
We also don't have to shift the sine parametric equation either since at n=0, we have [tex]\theta_2=n\pi=0(\pi)=0[/tex].
So let's see what our equations look like now:
[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex]
Let's verify these still work in our original equation:
[tex]x^2+(y-1)^2[/tex]
[tex](2\cos(t))^2+(-2\sin(t))^2[/tex]
[tex]2^2\cos^2(t)+(-2)^2\sin^2(t)[/tex]
[tex]4\cos^2(t)+4\sin^2(t)[/tex]
[tex]4(\cos^2(t)+\sin^2(t))[/tex]
[tex]4(1)[/tex]
[tex]4[/tex]
It still works.
Now let's see if we are being moving around the circle once around for values of t between [tex]0[/tex] and [tex]2\pi[/tex].
This first table will be the first half of the rotation.
t 0 pi/4 pi/2 3pi/4 pi
x 2 sqrt(2) 0 -sqrt(2) -2
y 1 -sqrt(2)+1 -1 -sqrt(2)+1 1
Ok this is the fist half of the rotation. Are we moving clockwise from (2,1)?
If we are moving clockwise around a circle with radius 2 and center (0,1) starting at (2,1) our x's should be decreasing and our y's should be decreasing at the beginning we should see a 4th of a circle from the point (x,y)=(2,1) and the point (x,y)=(0,-1).
Now after that 4th, the x's will still decrease until we make half a rotation but the y's will increase as you can see from point (x,y)=(0,-1) to (x,y)=(-2,1). We have now made half a rotation around the circle whose center is (0,1) and radius is 2.
Let's look at the other half of the circle:
t pi 5pi/4 3pi/2 7pi/4 2pi
x -2 -sqrt(2) 0 sqrt(2) 2
y 1 sqrt(2)+1 3 sqrt(2)+1 1
So now for the talk half going clockwise we should see the x's increase since we are moving right for them. The y's increase after the half rotation but decrease after the 3/4th rotation.
We also stopped where we ended at the point (2,1).
The parametric equations for the path of a particle moving along the circle x^2 + (y - 1)^2 = 4 in a clockwise direction, starting at (2, 1), are x = 2 + 2sin(-t) and y = 1 + 2cos(-t).
Explanation:The parametric equations for the path of a particle moving along the circle x2 + (y - 1)2 = 4 in a clockwise direction, starting at (2, 1), can be found using trigonometric functions. From the given equation of the circle, we can determine that the center of the circle is (0, 1) and the radius is 2. Therefore, the parametric equations are:
x = 2 + 2sin(-t) y = 1 + 2cos(-t).
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