To find the point P on the y-axis that is on both planes, set x = 0 and z = 0 for each plane and solve for y. The unique point P is (0, 1, 0). To find a unit vector u parallel to both planes, find the normal vectors of the planes and normalize one of them. The unit vector u with a positive first coordinate is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)). The vector equation for the line of intersection of the two planes is r(t) = (0, 1, 0) + t(5, -10, -2).
Explanation:To find the unique point P on the y-axis that is on both planes, we need to find the values of x, y, and z that satisfy the equations of both planes.
For the first plane, 2x + y + 5z = 1, if we set x = 0 and z = 0, we can solve for y:
0 + y + 0 = 1
y = 1
Therefore, the point P on the y-axis that is on the first plane is (0, 1, 0).
For the second plane, 2x + 5z = 0, since there is no y term, any point on the y-axis will satisfy this equation. So, we can choose any value for y and set x = 0 and z = 0. Let's choose y = 1:
2(0) + 5(0) = 0
Therefore, the point P on the y-axis that is on the second plane is (0, 1, 0).
Since both points have the same x and z coordinates, the unique point P on the y-axis that is on both planes is (0, 1, 0).
To find a unit vector u with a positive first coordinate that is parallel to both planes, we can find the normal vectors of the planes and then normalize one of them.
For the first plane, the normal vector is (2, 1, 5). For the second plane, the normal vector is (2, 0, 5). Both normal vectors are parallel to the planes.
Let's normalize the first vector:
||u|| = sqrt(2^2 + 1^2 + 5^2) = sqrt(30)
So, the unit vector u with a positive first coordinate that is parallel to both planes is (2/sqrt(30), 1/sqrt(30), 5/sqrt(30)).
To find a vector equation for the line of intersection of the two planes, we can take the cross product of the normal vectors of the planes to get a vector that is perpendicular to both planes.
Let's calculate the cross product:
(2, 1, 5) x (2, 0, 5) = (5, -10, -2)
Now, we can use one of the points on the line of intersection, which is (0, 1, 0), and the direction vector (5, -10, -2) to form the vector equation:
r(t) = (0, 1, 0) + t(5, -10, -2)
Katie wants to buy 6 pencils from the school store.Each pencil costs 15¢.How much money will she get back if she pays with a five dollar bill?
Answer: The amount she will get is $4.10
Step-by-step explanation:
My reason is because .15 times 6 since she is buying 6 pencils, it is .90. The you subtract the decimal 5.00-0.90 and you get 4.10. Hope this helps. PLease mark brainliest!!!!!
Determine c so that f(x) is continuous on the entire real line if f(x)= x^2 when x<=3 and c/x when x>3 ...?
**!!!STUCK ON 2 QUESTIONS!!!!
The time is takes to mow the lawn at a large park m(x) varies inversely with the numbers of workers assigned to the job x. It takes 90 minutes to complete the job when 3 workers are assigned to it.
which equation can be used to find the time to complete the job when x workers are assigned to it?
A.) m(x)= 270/x
B.) m(x) = 270x
C.) m(x) = 30/x
D.) m(x) = 30x
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
What is H(x) when x =2?
A.) 0.5
B.) 6.25
C.) 12.5
D.) 24
Can you also provide how you got the answer as well :)
Answer:
1. Which equation can be used to find the time to complete the job when x workers are assigned to it?
A.) m(x) = 270/x
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
What is H(x) when x =2?
B.) 6.25
Step-by-step explanation:
1. We know when x=3 (workers), m(x)= 90 min (time is takes to mow the lawn at a large park) and we know m(x) varies inversely with x, this means when m(x) increases so x decreases. This is obtained with a division over x.
Now, we can replace to know what is the correct answer:
[tex]m(x)=\frac{y}{x} (1) \\90=\frac{y}{3} \\90*3=y[/tex]
[tex]y=270[/tex]
The answer is A.
[tex]m(x)=\frac{270}{x}[/tex]
We can confirm if we replace x=3 in the equation before. Our answer should be 90.
[tex]m(x)=\frac{270}{3}=90[/tex]
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
We know H(x) varies inversely with x, this means when H(x) increases so x decreases. This is obtained with a division over x. Now, we need to find the constant in the numerator.
Now, we can replace in the eq (1) to know the constant in the numerator:
[tex]H(x)=\frac{y}{x}\\50=\frac{y}{0.25} \\50*0.25=y[/tex]
[tex]y=12.5[/tex]
Now, we can replace in the eq (1) to find H(x) when x=2
[tex]m(x)=\frac{y}{x}[/tex]
[tex]m(x)=\frac{12.5}{2}[/tex]
[tex]m(x)=6.25[/tex]
The answer is B.) 6.25
i am the only number between 41 and 49 that has 5 as a factor. what number am i?
Why is a plane a undefined term
Solve the differential equation using the method of undetermined coefficients
a) 4y"- 4y'-3y= cos2x
Which of the following is not a valid postulate or theorem for proving triangles congruent?
A. SSA
B. ASA
C. AAS
D. SAS
The valid postulates and theorems for proving triangles congruent are ASA, AAS, and SAS. The SSA postulate is not a valid postulate.
Explanation:The valid postulates and theorems for proving triangles congruent are ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SAS (Side-Angle-Side).
The SSA (Side-Side-Angle) postulate, also known as the ambiguous case of the Law of Sines, is not a valid postulate for proving triangles congruent because it can result in two different triangles or no triangle at all.
Thus, the correct answer is A. SSA.
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AB=21; AD=28. What is the value of AC+BD?
The value of AC + BD will be 70.
What is mean by Rectangle?A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.
Given that;
In rectangle ABCD,
AB = 21
AD = 28
Now,
Since, ABCD is a rectangle.
Hence, We get;
⇒ AC = BD
Here, In rectangle ABCD,
AB = 21
AD = 28
So, By using definition of Pythagoras theorem, we get;
⇒ BD² = AB² + AD²
⇒ BD² = 21² + 28²
⇒ BD² = 441 + 784
⇒ BD² = 1225
⇒ BD = 35
Thus, The value of BD + AC is find as;
⇒ AC + BD = 35 + 35
= 70
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explai n how a number and its reciprocal are related
Hi! Please help... a woman drops a front door key to her husband from their apartment window several stories above the ground. the function h=-16t^2 +64 gives the height h of the key in feet, t seconds after she releases it.
A/ how long does it take the key to reach the ground?
B/ what are the reasonable domain and range for the function h? ...?
Answer:
It would take 2 seconds to the key to reach the ground.
The domain the function is: 0≤t≤2 or in interval notation [0,2].
The range of the function is: 0≤h≤64 or in interval notation [0,64].
Step-by-step explanation:
The provided function is: [tex]h=-16t^2 +64[/tex]
Where h represents the height and t represents the time.
Part(A) how long does it take the key to reach the ground?
When key will be at ground, the height will be 0.
Substitute the value of h=0 in above equation and solve for t.
[tex]-16t^2 +64=0[/tex]
[tex]-16(t^2 -4)=0[/tex]
[tex](t^2 -2^2)=0[/tex]
[tex](t+2)(t-2)=0[/tex]
[tex]t+2=0\ or\ t-2=0[/tex]
[tex]t=-2\ or\ t=2[/tex]
Ignore the negative value of t.
Hence, it would take 2 seconds to the key to reach the ground.
Part (B) what are the reasonable domain and range for the function h?
Time and Distance should be a positive number.
Thus, the value of t and h should be greater or equal to 0.
From the part (a) we know that the after 2 second the key will hit the ground so the value of t can be greater or equal to 0 but should be less or equal to 2.
Thus, the domain the function is: 0≤t≤2 or in interval notation [0,2].
The maximum height of the key will be at t=0.
Thus the maximum value of h can be 64 and minimum value can be 0.
The range of the function is: 0≤h≤64 or in interval notation [0,64].
Which of the following options is an equivalent function to f(x) = 3(2)3x?
f(x) = 3(8)x
f(x) = 24x
f(x) = 27(8)x
f(x) = 3(8x)
The correct option is A: [tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]
To determine which option is equivalent to the given function f(x) = [tex]3(2)^{3x} \),[/tex] we will simplify the given function and compare it to each option.
The given function is:
[tex]\[ f(x) = 3(2)^{3x} \][/tex]
We can rewrite the exponent part using properties of exponents:
[tex]\[ (2)^{3x} = (2^3)^x = 8^x \][/tex]
So the function becomes:
[tex]\[ f(x) = 3 \cdot 8^x \][/tex]
Now, let's compare this to each of the given options:
Option A: [tex]\( f(x) = 3(8)^x \)[/tex]
This is exactly the same as our simplified function. Therefore, Option A is equivalent to [tex]\( f(x) = 3(2)^{3x} \).[/tex]
Option B: [tex]\( f(x) = 24^x \)[/tex]
This is not equivalent because it implies a different base raised to the power of x.
Option C:[tex]\( f(x) = 3(8x) \)[/tex]
This is not equivalent because it multiplies by 27 instead of 3.
Option D: [tex]\( f(x) = 3(8x) \)[/tex]
This is not equivalent because it implies multiplying 8 by x, not raising 8 to the power of x.
Therefore, the correct answer is:
[tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]
The complete question is here:
Which of the following options is an equivalent function to f(x) = 3(2)3x?
A. f(x) = 3(8)x
B. f(x) = 24x
C. f(x) = 27(8)x
D. f(x) = 3(8x)
Tim is setting up the course for a 9 mile walk.he places a sign every 0.15 mile along the path. how many signs will tim place
page 128 slopes and intercepts
4 questions that I don't understand how to do. Help please?
Which answer shows these decimals written in order from greatest to least? 46.95
48.1
46.581
The des moines register recently reported the ratings of high school sportsmanship as complied by the iowo high school athletic association. For each school, the participants and coaches were rated by referees, where 1= superior, and 5= unsatisfactory. A regression analysis of the averages scores given to football players and coaches is shown below.
interpret the value of the correlation between the ratings of coaches and participants.
The product of 9 and a number is 12 less than three times that number.
What is the number?
Which fraction is NOT equivalent to 4/12?
A. 1/3 B. 12/30 C. 8/24 D. 16/48
the circular blade on a saw has a diameter of 7.5 inches and rotates at 2400 revolutions per minute.
A) find the angular speed in radians per sec
B) Find the linear speed of the saw teeth ( in feet per sec) as the contact the wood being cut.
Formulas--> Linear speed= arc length/time= s/t
Angular speed= central angle/time ...?
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how to write four hundred thousand in numbers?
keisha is reading a 325 page book at a rate of 25 pages per day. use a point slipe equation to determine whether she will finish reading the book in 10 days.
Using the point slope equation, it is determined that Keisha will not finish reading the 325 pages book in 10 days if she is reading at a rate of 25 pages per day.
Explanation:The subject of this question is Mathematics and it involves solving a problem using a point slope equation.
Let's denote the total number of pages in the book as 'y' and the number of pages that Keisha can read per day as 'm'. We know that y=325 and m=25. The point slope equation in this case would be y=mx, where 'x' represents the number of days. In this case, we are trying to find out whether x=10.
Substituting the known values into the equation we get: 325=25*10. This simplifies to 325=250.
Since 325 does not equal 250, we can conclude that Keisha will not complete reading the book in 10 days.
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Solve log base(x − 1) 16 = 4.
Answer:
x = 3
Step-by-step explanation:
Given:[tex]\log _{x-1}\left(16\right)=4[/tex]
We have to solve for x.
Consider the given expression [tex]\log _{x-1}\left(16\right)=4[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}[/tex]
[tex]\log _{x-1}\left(16\right)=\frac{\ln \left(16\right)}{\ln \left(x-1\right)}[/tex]
Thus, the expression becomes,
[tex]\frac{\ln \left(16\right)}{\ln \left(x-1\right)}=4[/tex]
Multiply both side by [tex]\ln \left(x-1\right)[/tex]
[tex]\frac{\ln \left(16\right)}{\ln \left(x-1\right)}\ln \left(x-1\right)=4\ln \left(x-1\right)[/tex]
Simplify, we get,
[tex]\ln \left(16\right)=4\ln \left(x-1\right)[/tex]
Divide both side by 4, we have,
[tex]\frac{4\ln \left(x-1\right)}{4}=\frac{\ln \left(16\right)}{4}[/tex]
[tex]\frac{\ln \left(16\right)}{4}:\quad \ln \left(2\right)[/tex]
Thus, [tex]\ln \left(x-1\right)=\ln(2)[/tex]
[tex]\mathrm{When\:the\:logs\:have\:the\:same\:base:\:\:}\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\quad \Rightarrow \quad f\left(x\right)=g\left(x\right)[/tex]
thus, x - 1 = 2
simplify for x, we have,
x = 3
Watermelons cost $0.35 per pound. Joseph’s watermelon costs between $4.20 and $5.25. Which compound inequality correctly represents the possible weights of his watermelon?
Answer: 12 < x < 15
Step-by-step explanation:
The sensory somatic nervous system controls all _________ responses of the body.
unconscious
voluntary
sequenced
repeated
Answer:
The sensory somatic nervous system controls all voluntary responses of the body.
Step-by-step explanation:
The sensory somatic nervous system controls all voluntary responses of the body.
The movements includes muscles and organs movements and reflex movements. Here the sensory neurons carry impulses to the brain and the spinal cord.
which correctly describes the root of the following cubic equation? x^3-3x^2+4x-12=0
Answer:
The equation has one real root and two complex roots.
Step-by-step explanation:
The given equation is
[tex]x^3-3x^2+4x-12=0[/tex]
The above equation is true form x=3, therefore (x-3) is a factor of above equation.
Use long division or synthetic division method to divide the equation by (x-3).
[tex](x-3)(x^2+4)=0[/tex]
Equate each factor equal to zero.
[tex]x-3=0[/tex]
[tex]x=3[/tex]
Therefore 3 is a real root of the equation.
[tex]x^2+4=0[/tex]
[tex]x^2=-4[/tex]
[tex]x=\sqrt{-4}[/tex]
[tex]x=\pm 2i[/tex]
2i and -2i are complex roots of the equation.
Therefore the equation has one real root and two complex roots.
At the beginning of April, Owl Corporation has a balance of $12,000 in the Retained Earnings account. During the month of April, Owl had the following external transactions.
1. Issue common stock for cash, $8,000.
2. Provide services to customers on account, $6,100.
3. Provide services to customers in exchange for cash, $2,400.
4. Purchase equipment and pay cash, $5,600.
5. Pay rent for April, $1,000.
6. Pay workers' salaries for April, $2,500.
7. Pay dividends to stockholders, $1,100.
Using the external transactions above, compute the balance of Retained Earnings at April 30 ...?
Name the property of 11xn=1
The ratio of sues age to her fathers age is 2:7 in three years their ages will total 60, how old is sue and her dad right now
Please solve the eqution
Cos4x + cos2x - 2 =0
cos4x = cos2x
We know that:
cos2x = 1-2cos^2 x
==> cos4x = 1-2cos^2 (2x)
Now substitute:
==> 1-2cos^2 (2x) = cos2x
==> 2cos^2 (2x) + cos2x - 1 = 0
Now factor:
==> (2cos2x -1)(cos2x + 1) = 0
==> 2cos2x -1 = 0 ==> cos2x =1/2 ==> 2x= pi/3
==> x1= pi/6 , 7pi/6
==> x1= pi/6 + 2npi
==> x2= 7pi/6 + 2npi
==> cos2x = -1 ==> 2x= pi ==> x3 = pi/2 + 2npi.
==> x= { pi/6+2npi, 7pi/6+2npi, pi/2+2npi}
The solutions to the equation cos(4x) + cos(2x) - 2 = 0 are x = nπ and x = (2n + 1)π/2, where n is an integer.
Here, we have to solve the equation cos(4x) + cos(2x) - 2 = 0, we can use trigonometric identities to simplify it.
We know the following trigonometric identities:
[tex]cos(2x) = 2cos^2(x) - 1\\cos(4x) = 2cos^2(2x) - 1[/tex]
Now, let's rewrite the equation using these identities:
[tex]2cos^2(2x) - 1 + 2cos^2(x) - 1 - 2 = 0[/tex]
Combine like terms:
[tex]2cos^2(2x) + 2cos^2(x) - 4 = 0[/tex]
Divide the entire equation by 2:
[tex]cos^2(2x) + cos^2(x) - 2 = 0[/tex]
Now, let's simplify further:
We can rewrite [tex]cos^2(2x)[/tex] as [tex](1 - sin^2(2x))[/tex]using the Pythagorean identity: [tex]sin^2(\theta) + cos^2(\theta) = 1[/tex]
So, the equation becomes:
[tex](1 - sin^2(2x)) + cos^2(x) - 2 = 0[/tex]
Rearrange the terms:
[tex]cos^2(x) - sin^2(2x) - 1 = 0[/tex]
Now, use the double-angle identity: [tex]sin^2(2x) = 2sin(x)cos(x)[/tex]
[tex]cos^2(x) - 2sin(x)cos(x) - 1 = 0[/tex]
Now, we can rewrite [tex]cos^2(x) as (1 - sin^2(x))[/tex] using the Pythagorean identity:
[tex]sin^2(\theta) + cos^2(\theta) = 1[/tex]
[tex](1 - sin^2(x)) - 2sin(x)cos(x) - 1 = 0[/tex]
Now, simplify further:
[tex]1 - sin^2(x) - 2sin(x)cos(x) - 1 = 0[/tex]
We can see that [tex](1 - sin^2(x))[/tex] cancels out:
-2sin(x)cos(x) = 0
Divide both sides by -2:
sin(x)cos(x) = 0
Now, there are two possible solutions for this equation:
sin(x) = 0
cos(x) = 0
Let's solve each case:
sin(x) = 0
This occurs when x is an integer multiple of π (π, 2π, 3π, ...). So the solutions are x = nπ, where n is an integer.
cos(x) = 0
This occurs when x is an odd multiple of π/2 (π/2, 3π/2, 5π/2, ...). So the solutions are x = (2n + 1)π/2, where n is an integer.
Therefore, the solutions to the equation cos(4x) + cos(2x) - 2 = 0 are x = nπ and x = (2n + 1)π/2, where n is an integer.
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Solve the system of equations and choose the correct ordered pair.
3x + 2y = 12
6x + 3y = 21
A. (4, 3)
B. (4, 0)
C. (2, 3)
D. (3, 2)
Answer:
C.
Step-by-step explanation:
We are going to solve it with substitution, that is, we are going to clear one variable and then replace it in the other equation. Then,
[tex]3x + 2y = 12[/tex]
[tex]3x = 12-2y[/tex]
[tex]x = \frac{12-2y}{3}[/tex]
[tex]x = \frac{12}{3}-\frac{2y}{3}[/tex]
[tex]x = 4-\frac{2y}{3}.[/tex]
Now, we replace that x value in the equation 6x + 3y = 21:
[tex]6(4-\frac{2y}{3}) + 3y = 21[/tex]
[tex]24-\frac{12y}{3} + 3y = 21[/tex]
[tex]24-4y + 3y = 21[/tex]
[tex]24-y = 21[/tex]
[tex]-y = 21-24[/tex]
[tex]-y = -3[/tex]
[tex]y = 3.[/tex]
Finally, we replace the y value found to find x.
[tex]x = 4-\frac{2*3}{3}[/tex]
[tex]x = 4-2[/tex]
[tex]x = 2.[/tex]
So, the solution is (x,y)=(2,3). Then, the answer is C.