By Stokes' theorem, the line integral of [tex]\vec F[/tex] over [tex]C[/tex] is equivalent to the surface integral of the curl of [tex]\vec F[/tex] over [tex]S[/tex], where [tex]S[/tex] is the part of the plane [tex]7x+6y+z=1[/tex] in the first octant, with [tex]S[/tex] having positive/upward orientation.
Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=\dfrac{(1-u)(1-v)}7\,\vec\imath+\dfrac{u(1-v)}6\,\vec\jmath+v\,\vec k[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex].
Take the normal vector to [tex]S[/tex] to be
[tex]\vec s_u\times\vec s_v=\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k[/tex]
The curl of [tex]\vec F[/tex] is
[tex]\nabla\times\vec F(x,y,z)=(x-y)\,\vec\imath-y\,\vec\jmath+\vec k[/tex]
Then the line integral is equivalent to
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]
[tex]=\displaystyle\int_0^1\int_0^1\left(\frac{(6-13u)(1-v)}{42}\,\vec\imath-\dfrac{u(1-v)}6\,\vec\jmath+\vec k\right)\cdot\left(\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k\right)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\frac1{252}\int_0^1\int_0^1(12-6v-19u+19uv)(1-v)\,\mathrm du\,\mathrm dv=\boxed{\frac{11}{1512}}[/tex]
First, find the normal vector of the plane, and then find the curl of the vector field to obtain a new vector field to be integrated over the given surface. Then perform the surface integral of this resultant vector field over the surface to get the required output.
Explanation:Using Stokes' Theorem, we first find the normal vector of the plane 7x + 6y + z = 1 by extracting the coefficients, i.e., the vector is (7,6,1). Next, find the curl of the vector field F(x, y, z) = i + (x + yz)j + (xy - sqrt(z ))k. This provides the vector field, let's say G, that you are going to integrate over the surface that C bounds. The curl is calculated as follows:
The curl of F = ∇ x F= ( ∂/∂y [(xy - sqrt(z))] - ∂/∂z [(x + yz)] )i - ∂/∂x [(xy - sqrt(z))]j + ∂/∂x [(x + yz)]k.
Then perform the surface integral of this resultant vector field, G, over the surface bounded by C to get your answer. The surface integral is ∫∫ G·dS.
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Which is most likely to have a mass of 3 grams? A. an apple B. a backpack C. a paper clip D. a cat
Answer:
C. a paper clip
Step-by-step explanation:
An apple generally weighs more than 3 grams, and can go even into oz.
Depending on the type of material the backpack is made of, and also how big it is, the mass can vary, however, it is safe to say the backpack will be more than 3 grams.
A cat definitely weighs more than 3 grams. Any born cat that weighs only 3 grams will not survive.
~
For this case we have the following options:
An apple, it is known that the apple will never weigh 3 grams.
A backpack will not have that weight either.
A cat would never weigh 3 grams or be newborn.
To have an object of 3 grams, the object must be very light.
A paperclip is very likely to weigh 3 grams because it is a very small and extremely light object.
Answer:
Option C
HELP PLEASE MATH!! A company is testing tires for wear and tear. A given tire is said to either pass the test (P) or fail the test (F). Enter the missing outcome possibilities in each box to show the possible results if three tires are tested.
{PPP,( ), PFP, FPP, FFP, FPF,( )FFF}
Answer:
PPF, PFF
Step-by-step explanation:
There are several ways you can list all the possible combinations. A couple of my favorite are a) use a binary counting sequence; b) use a gray code counting sequence.
Using the first method, the binary numbers 000 to 111 can be listed in numerical order as 000, 001, 010, 011, 100, 101, 110, 111. Letting 0=P and 1=F, the ones missing from your list are the ones in italics in my list.
Using the second method, we change the right-most character, then the middle one, and finally the left-most character so there is one change at a time: 000, 001, 011, 010, 110, 111, 101, 100.
After you have a list of all possible combinations, it is a simple matter to compare the given list to the list of possibilities to see which are missing.
The correct outcome possibilities for the missing boxes in the given scenario are:
1. PPP (all three tires pass)
2. PPF (two tires pass, one fails)
3. PFP (two tires pass, one fails)
4. FPP (two tires pass, one fails)
5. FFP (two tires pass, one fails)
6. FPF (two tires pass, one fails)
7. FFF (all three tires fail)
To determine the missing outcomes, we must consider all possible combinations of passes (P) and failures (F) for three tires. Since each tire can either pass or fail, there are [tex]2^3 = 8[/tex] possible outcomes. We already have five of these outcomes listed, and we need to find the remaining two.
The missing outcomes must include all combinations of passes and failures that have not been listed yet. Since we have all combinations with exactly two passes and one fail, and we have the combination with three passes and three fails, the remaining combinations must have exactly one pass and two fails. These combinations are:
- PFF (one tire passes, two fail)
- FPF (one tire passes, two fail)
Therefore, the complete list of outcomes is:
1. PPP (all three tires pass)
2. PPF (two tires pass, one fails)
3. PFP (two tires pass, one fails)
4. FPP (two tires pass, one fails)
5. FFP (two tires pass, one fails)
6. FPF (two tires pass, one fails)
7. PFF (one tire passes, two fail)
8. FFF (all three tires fail)
Each of these outcomes represents a distinct possibility when testing three tires, and together they account for every possible result of the wear and tear test for the three tires.
If 4a+8b+4c=1
What is 16a+16c+32b?
Answer:
4
Step-by-step explanation:
[tex] 16a+32b+16c= 4*(4a+8b+4c)[/tex]
Answer:
If 4a+8b+4c=1, then 16a+16c+32b = 4
Step-by-step explanation:
To answer the question, a rule of three can be raised:
4a + 8b + 4c --------------- 1
16a + 16c + 32b --------- X
X = (16a + 16c + 32b) ÷ (4a + 8b + 4c)
This is equivalent to the division of the two polynomials, 16a + 16c + 32b by 4a + 8b + 4c.
Before dividing we organize the first polynomial,
16a + 16c + 32b = 16a + 32b + 16c
Dividing the two polynimials, we have
16a + 32b + 16c | 4a + 8b + 4c
-16a - 32b - 16c 4
= 0 =
The exact quotient is 4, since the residue is 0
Hope this hepls!
Compare Standard Deviations
Place data sets in order from the smallest standard deviation to the largest standard deviation.
A = {9, 10, 11, 7, 13}
B = {7, 10, 11, 10, 12}
C = {10, 10, 10, 10, 10}
D = {1, 1, 10, 19, 19}
E = {1, 5, 6, 19, 19}
_________________
1 -
2 -
3 -
4 -
5 -
Answer:
1-C, 2-B, 3-A, 4-E, 5-D
Step-by-step explanation:
It is convenient to let a graphing calculator or spreadsheet compute the standard deviations for you. (Some will compute sample standard deviation; some will compute population standard deviation. It makes no difference to the ordering, as long as the same computation is used for all data sets.)
The correct order of the data sets from smallest to largest standard deviation is: C, B, A, E, D.
Explanation:Calculate the standard deviation for each data set. A = 2.2361, B = 1.5492, C = 0, D = 8.8818, E = 7.7277.Arrange the data sets in order from smallest to largest standard deviation: C, B, A, E, D.Therefore, "the correct order is: C, B, A, E, D".Learn more about ordering data sets by standard deviation here:https://brainly.com/question/37740206
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what is the equation of the graphed line written in standard form?x-4y=4. x+4y=4. y=1/4x-1. y=-1/4x-1
Answer:
y=1/4x-1
Step-by-step explanation:
First step to determine the equation of a line is is to determine its slope.
We see the line passes through points (4,0) and (0,-1), that means its slope is:
S = (0 - -1) / (4 - 0) = 1/4
Since there's only one choice with a slope of 1/4, the choice is easy :-)
But we can also verify the equation by checking if it validates the given points. So, what's the value of y if x = 0?
y = (1/4)0 - 1 = 0 - 1 = -1 Validated.
And when x = 4?
y = (1/4)4 - 1 = 1 - 1 = 0 Validated too.
Answer:
[tex]x-4y=4[/tex]
Step-by-step explanation:
The given function passes through: (4,0) and (0,-1).
The equation is of the form;
y=mx+b
where b=-1 is the y-intercept.
The equation now becomes:
y=mx-1
We substitute the point (4,0) into the function to obtain;
0=m(4)-1
0+1=4m
1=4m
[tex]m=\frac{1}{4}[/tex]
Therefore the equation is:
[tex]y=\frac{1}{4}x-1[/tex]
Multiply through by 4 to get;
[tex]4y=x-4[/tex]
The equation in standard form is;
[tex]x-4y=4[/tex]
In the equation 3x^2-10x=8 please solve for x and show work ty !
Answer:
x=4,-2/3
Step-by-step explanation:
To solve for x, we first subtract 8 from both sides.
So 3x^2-10x-8=0
Using the quadratic formula, we get
X=(10+sqrt(100+96))/6 or x = (10-sqrt(100+96))/6
So x=(10+14)/6 or x=(10-14)/6
So x=4 or x=-2/3.
Answer:
x=4 or x=-2/3.
Step-by-step explanation:
Write a polynomial that represents the area of a rectangle with side lengths of 7x-2 and 3x-5
Answer:
21x^2 - 41x + 10
Step-by-step explanation:
The area of a rectange is length x width.
L = 7x-2
W = 3x-5
So, you would do (7x-2)x(3x-5).
To do this, you can do FOIL.
F (first times first) - (7x)(3x)=21x^2
O (outside times outside) - (7x)(-5)= -35x
I (inside times inside) - (-2)(3x)= -6x
L (last times last) - (-2)(-5) = 10
So, it is 21x^2 - 35x - 6x +10
Then you combine like terms giving you:
21x^2 - 41x + 10
Answer: 21x^2 - 41x + 10
Step-by-step explanation: If the lengths are even then the number would be angle.
50 POINTS
Which of the following can be determined from the table above?
A.
Events B and C are independent.
B.
Events D and A are independent.
C.
Events C and A are independent.
D.
Events E and B are independent.
Answer: Option 'B' is correct.
Step-by-step explanation:
Since we have given that
P(A) = 0.5
P(B) = 0.5
P(C) = 0.15
P(D) = 0.7
P(E) = 0.15
We need to find the independent events.
When we consider,
P(A∩D) = P(Event A and Event D) = 0.35
and
P(A) × P(D) = 0.7 × 0.5 = 0.35
We get that
P(A∩D)= P(A) × P(D) = 0.35
All other options are not satisfying the conditions of "Independent events".
Hence, Option 'B' is correct.
Answer:
option b would be the correct answer
Step-by-step explanation:
140 is decreased to 273
Is there more to the question,
Zach and Roger spent a total of 69 hours building a treehouse. Roger worked 9 hours less than twice the number of hours Zach worked. Which system of equations can be used to determine the number of hours Zach worked, x, and the number of hours Roger worked, y?
A.
x + y = 69
2x + y = 9
B.
x - y = 69
y = 2x + 9
C.
x + y = 69
y = 9x - 2
D.
x + y = 69
y = 2x - 9
Answer:
It is D
Step-by-step explanation:
Find the circumference of a circle:
Whose radius is 3 ½ in. (22/7)
Whose diameter is 3.6 (3.14)
Answer:
1. 22 in
2. 11.304 units
Step-by-step explanation:
Formulas for the circumference are
C = 2πr
C = πd
___
1. Put the given numbers in the appropriate formula:
C = 2·(22/7)·(7/2 in) = 22 in
__
2. Put the given numbers in the appropriate formula:
C = 3.14·3.6 = 11.304 . . . units
find the x-intercepts for the parabola defined by the equation below.
y= 2^2 + 2x - 4
a. -4,0 and 2,0
b. -2,0 and 1,0
c. 0,-2 and 0,1
d. 0,-4 and 0,2
Answer:
b.
Step-by-step explanation:
x-intercepts are found by factoring. We will use standard factoring here since this one is straightforeward and has real zeros as its solutions.
In our equation,
a = 2
b = 2
c = -4
The rules are to take a * c and then find the factors that number, determine which combination of those factors will give you the linear term (the term with the single x on it), and rearrange those signs accordingly. Let's start with that:
Our a * c is 2 * -4 = -8.
We need the factors of |-8|: 1,8 and 2,4
Some combination of those factors needs to give us a +2x. 2,4 will work as long as the 4 is positive and the 2 is negative.
Now we put them back into the equation, the absolute value of the larger number first:
[tex]2x^2+4x-2x-4=0[/tex]
Now group the terms in sets of 2 without moving any of them around:
[tex](2x^2+4x)-(2x-4)=0[/tex]
In each set of parenthesis, pull out what is common to both terms. In the first set, the 2x is common, and in the second set, the 2 is common:
[tex]2x(x+2)-2(x+2)[/tex]
Now what is common between both terms is the (x + 2), so pull that out, grouping what is remaining in its own set of parenthesis:
[tex](x+2)(2x-2)=0[/tex]
To find the zeros, remember that the Zero Product Property tells us that for that equation above to equal zero, one of those factors has to equal zero, so:
x + 2 = 0 or 2x - 2 = 0. Solve both for x:
x = -2 so the coordinate is (-2, 0)
2x - 2 = 0 and
2x = 2 so
x = 1 so the coordinate is (1, 0)
write a polynomial function in standard form with the given roots: -4i
Answer:
[tex]f(x)=x^2+16[/tex]
Step-by-step explanation:
By the conjugate rule, if -4i is a root, then so is +4i. So we have 2 roots, thus, we have a second degree polynomial (namely, a quadratic). If
x = -4i, then
x + 4i is a root.
If
x = 4i, then
x - 4i is a root.
Having (x - 4i)(x + 4i) as roots, we can now FOIL them together to get a polynomial of least degree.
FOILing gives us
[tex]x^2+4ix-4ix-16i^2[/tex]
Notice that the +4ix and the -4ix cancel each other out, leaving you with
[tex]x^2-16i^2[/tex]
Since
[tex]i^2=-1[/tex]
we can make the substitution:
[tex]x^2-16(-1)[/tex]
which simplifies to
[tex]x^2+16[/tex]
In function notation form:
[tex]f(x)=x^2+16[/tex]
Final answer:
The polynomial function in standard form with the root -4i is f(x) = x² + 16. We derive this by realizing that the conjugate, 4i, is also a root and multiplying the factors (x + 4i)(x - 4i), simplifying to the quadratic polynomial.
Explanation:
To write a polynomial function with the given root -4i, we must take into consideration that complex roots come in conjugate pairs for polynomials with real coefficients. Hence, if -4i is a root, then its conjugate 4i must also be a root. The factors of the polynomial corresponding to these roots are (x + 4i) and (x - 4i).
We can find the polynomial by multiplying these two factors:
(x + 4i)(x - 4i)
Applying the difference of squares, we get:
x² - (4i)²
x² - (-16)
x² + 16
The polynomial function in standard form with the given root -4i is:
f(x) = x² + 16
the researchers have also determined that the current rate of the rise in water level is twice the 1880 to 2009 rate. assuming that this new rate began in 2009, you can use the function g(x) = 3.2(x - 2009) + 206, which models the total rise in water level in mm since 1880 for any year x, beginning in 2009.
1. what is the domain of g(x)?
2. write a simplified function, g(x)
3. according to the model, what will be the total rise in water level by 2025?
4. when will the total rise in water level be equal to about half a meter?
please help and thank you!!
Answer:
x ≥ 20093.2x -6222.8257.2 mmyear 2100Step-by-step explanation:
1. The problem statement tells you the function applies for year values (x) 2009 and later. The domain is real numbers greater than or equal to 2009.
__
2. We can use the distributive property to eliminate parentheses:
3.2(x -2009) +206 = 3.2x -6428.8 +206
= 3.2x -6222.8
__
3. Put 2025 in the equation and do the arithmetic
g(2025) = 3.2·2025 -6222.8 = 257.2 . . . . mm
__
4. Put this value of level rise in the equation and solve for x.
g(x) = 3.2x -6222.8
500 = 3.2x -6222.8 . . put 500 mm where g(x) is in the equation
6722.8 = 3.2x . . . . . . . add 6222.8
x = 6722.8/3.2 = 2100.875
The water rise will be equal to about half a meter late in the year 2100.
Which of these is a key feature of an experimental study? A. The treatment in the experiment should be simple enough for each individual in the experimental group to understand. B. The treatment in the experiment must vary for each individual in the experimental group. C. The treatment in the experiment must be applied to each of the individuals in the experimental group. D. The treatment in the experiment should be short so that each individual is tested quickly.
Answer:
I believe it is "E"
Step-by-step explanation:
E-series can eat my memes
Kill me please I'm Swedish ree
Answer: The key feature in the experimental study is C. The treatment in the experiment must be applied to each of the individuals in the experimental group. This is because it is made sure that the variables and conditions in different correspondents are applied so that actual results may be concluded.
Step-by-step explanation:
which fraction is closer to 0 than 1
Answer:
5/12
Step-by-step explanation:
Since 1/2 is the same distance from zero and one, we can use it to judge whether a fraction is closer to 0 or 1.
Since 5/8 is greater than 4/8 which is 1/2, it is closer to 1.
Since 8/10 is greater than 5/10 which is 1/2, it is closer to 1.
Since 5/12 is less than 6/12 which is 1/2, it is closer to 0.
So our answer is C. 5/12
The value close 0 than 1 is 0.41667.
FractionA fraction is a part of a whole number. for example, [tex]\bold{\dfrac{1}{4}}[/tex] represent a quarter of a number. Similarly, when we solve the fraction meaning [tex]\bold{\dfrac{1}{4}}[/tex] in the decimal form we divide the numerator, therefore, 1 in this case with the denominator as 4 in this case. the value thereafter we get is known as the decimal form of a fraction.
for example, = 0.25.
DecimalAlso, to convert a decimal to a fraction we simply divide the number by a multiple of 10. the number will be depending upon the number of numbers are after the decimal.
for example, 0.25 will be written as [tex]\bold{\dfrac{25}{100}}[/tex] while 0.5 will be written as [tex]\bold{\dfrac{5}{10}}[/tex].
Now, solving the question,
Bring every fraction to decimal form,
[tex]\bold{\dfrac{5}{8}= 0.625}[/tex]
[tex]\bold{\dfrac{8}{10} = 0.8}[/tex]
[tex]\bold{\dfrac{5}{12} = 0.41\overline6}[/tex]
[tex]\bold{\dfrac{7}{14} = 0.5}[/tex]
Hence, the value close 0 than 1 is 0.41667.
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If you vertically stretch the cubic function, F(x)=x^3, what is the equation of the new function?
A. J(x)=(1/3x)^3
B. G(x)=3x^3
C. H(x)=(3x)^3
D. K(x)=1/3x^3
Answer:
B. g(x) = 3x^3.
Step-by-step explanation:
In general a f(x) stretches vertically the graph of f(x) by a factor a.
The new function after a vertical stretch of the cubic function F(x) = x^3 is G(x) = 3x³, which multiplies the output of the original function by 3.
Explanation:When you apply a vertical stretch to the cubic function F(x) = x³, you multiply the output of the function, not the input. A vertical stretch by a factor of 3 would mean that each output value is tripled. Therefore, the correct equation for the new function would be G(x) = 3x³.
Option A is incorrect because (1/3x)³ represents a horizontal stretch by a factor of 3. Option C, (3x)³, represents a horizontal shrink by a factor of 1/3, and results in steeper slopes than the original function, which is the opposite effect of a vertical stretch. Option D, 1/3x³, would be a vertical compression by a factor of 1/3, not a stretch.
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Plz help ASAP!! Explain your answer! I will mark at brainliest!!!
Answer:
A. 16.12 ft B. 9.64 ft
Step-by-step explanation:
For both of these, you need Pythagorean's Theorem because we have right triangles. In part A, we are given 2 legs and are asked to find the length of the hypotenuse:
[tex]14^2+8^2=PR^2[/tex] and
[tex]196+64=PR^2[/tex]
and PR = 16.12 ft
In part B, we are given the length of the hypotenuse and side PQ remains the same:
[tex]14^2+QR^2=17^2[/tex] and
[tex]196+QR^2=289[/tex]
[tex]QR^2=289-196[/tex] so
QR = 9.64 ft
A box contains 8 red balls, 5 brown balls, 4 purple balls, and 3 green balls. What is the probability that a purple ball will be selected from the box after a red ball is taken out and not replaced?
Write the probability as a percent. Round to nearest tenth of a percent as needed.
Answer:
21.1 percent
Step-by-step explanation:
Total balls after -1 red ball=7+5+4+3=19
Prob of purple ball= purple ball/total balls
= 4/19
=0.2105...
percentage:21.1%
The probability of selecting a red ball and then a purple one without replacement from a box containing 8 red, 5 brown, 4 purple, and 3 green balls is approximately 8.4%.
Explanation:In probability, there are two events of interest here: selecting a red ball and then selecting a purple ball. Since the total number of balls changes after picking the red ball, these are dependent events. The probability of the first event (selecting a red ball) is 8 out of 20 (total balls). Then, with a red ball removed and not replaced, the probability of the second event (selecting a purple ball) is 4 out of 19. To find the overall probability, we multiply the probabilities of these two events.
To convert the probability to a percentage, multiply the result by 100. By doing this, the probability rounds to approximately 8.4%.
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Which of the following is a zero for the function f(x) = (x + 3)(x − 7)(x + 5)? x = −7 x = −3 x = 3 x = 5
Answer:
x = −3
Step-by-step explanation:
A zero of a function is a value that makes one of the factors be zero.
To make x+3 = 0, x = -3.
To make x-7 = 0, x = 7.
To make x+5 = 0, x = -5.
Of these values that make the factors zero, only x = -3 is on your choices list.
Answer: The correct option is
(B) x = - 3.
Step-by-step explanation: We are given to select the correct zero of the following polynomial function :
[tex]f(x)=(x+3)(x-7)(x+5)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
The zeroes of the function f(x) are given by
[tex]f(x)=0\\\\\Rightarrow (x+3)(x-7)(x+5)=0\\\\\Rightarrow x+3=0,~~x-7=0,~~x+5=0\\\\\Rightarrow x=-3,7,-5.[/tex]
Therefore, the zeroes of the given function are
x = -3, x = 7 and x = -5.
Thus, the correct option is
(B) x = -3.
[tex]4(x - 8) {}^{3} - 18 = 846[/tex]
need help solving it
Answer:
x=14
Step-by-step explanation:
4(x - 8)³ - 18 = 846
4(x - 8)³ = 864
(x - 8)³ =216
∛(x - 8)³ =∛216
x-8 = 6
x= 6+8
x=14
Answer:
x=14
Step-by-step explanation:
look this solution :
Melinda walked 9/12 mile each day for 5 days. What was the total distance , in miles, she walked in the 5 days?
3.75 miles or 3 3/4 would be your answer
PLEASE HELP ME!!!! I need to find m angel 1
Answer:
55 degrees.
Step-by-step explanation:
We can use the postulate if two angles are congruent in both triangles than the triangle is congruent.
That means we can plug in 80 and 45 for triangle QRS.
Then, we can find angle 1 by subtracting the sum of those numbers from 180.
80 + 45 = 125
180 - 125 = 55
What is the volume of the given prism? Round the answer to the nearest tenth of a centimeter. The figure is not drawn to scale.
NEED HELP ASAP!!!!!!!!!!!!!!!!!
Answer:
541.8 cm³
Step-by-step explanation:
Volume of a prism is the height times the area of the base.
V = hA
Area of a rectangle is width times length, so:
V = hwl
Given h = 8.8 cm, w = 4.7 cm, and l = 13.1 cm:
V = (8.8 cm) (4.7 cm) (13.1 cm)
V = 541.8 cm³
Answer:
[tex]541.8cm^{3}[/tex]
Step-by-step explanation:
V = Bh or V = lwh
substitute given measurements, then simplify.[tex]13.1cm * 4.7cm * 8.8cm = 541.8cm^{3}[/tex]
Solve the equation on the interval
Answer:
The solutions of the equations are π/3 , 2π/3 , 4π/3 , 5π/3
Step-by-step explanation:
* Lets revise the four quadrant before solving the equation
# First quadrant the measure of all angles is between 0 and π/2
the measure of any angle is α
∴ All the angles are acute
∴ All the trigonometry functions of α are positive
# Second quadrant the measure of all angles is between π/2 and π
the measure of any angle is π - α
∴ All the angles are obtuse
∴ The value of sin(π - α) only is positive ⇒ sin(π - α) = sinα
# Third quadrant the measure of all angles is between π and 3π/2
the measure of any angle is π + α
∴ All the angles are reflex
∴ The value of tan(π + α) only is positive ⇒ tan(π + α) = tanα
# Fourth quadrant the measure of all angles is between 3π/2 and 2π
the measure of any angle is 2π - α
∴ All the angles are reflex
∴ The value of cos(2π - α) only is positive ⇒ cos(2π - α) = cosα
* Now lets solve the equation
∵ 4 sin²Ф - 3 = 0 ⇒ the domain is 0 ≤ Ф ≤ 2π
- Add 3 for both sides
∴ 4 sin²Ф = 3 ⇒ divide the both sides by 4
∴ sin²Ф = 3/4 ⇒ take square root for both sides
∴ √(sin²Ф) = √(3/4)
∴ sinФ = √3/2 OR sinФ = -√3/2
- When the value of sinФ is positive
∴ The angle Ф is on the first or second quadrant
- When the value of sinФ is negative
∴ The angle Ф is on the third or fourth quadrant
- We have four values of Ф because 0 ≤ Ф ≤ 2π
- Lets find the measure of the acute angle α
∵ sinα = √3/2
∴ α = sin^-1(√3/2) = π/3
- If Ф is on the first quadrant
∴ Ф = α = π/3
- If Ф is on the second quadrant
∴ Ф = π - α = π - π/3 = 2π/3
- If Ф is on the third quadrant
∴ Ф = π + α = π + π/3 = 4π/3
- If Ф is on the fourth quadrant
∴ Ф = 2π - α = 2π - π/3 = 5π/3
* The solutions of the equations are π/3 , 2π/3 , 4π/3 , 5π/3
Use the graph to determine the domain and range of the piecewise defined function.
Domain:
(the picture is a b c d choices)
Answer:
Domain: -6≤x<0 or 0<x≤2
Range: 1<x≤6
Step-by-step explanation:
Domain = The set of starting numbers - the x values.
Range = The set of numbers it becomes - the y values.
For the domain, it starts at -6, on the left and ends at 0; however, it doesn't include 0. Then is starts at 0, exclusive, and ends at 2.
For the range, it starts at 1, but doesn't include 1, and ends at 6, inclusive.
The domain of the given piecewise-defined function is -6≤x≤0 or 0. The domain of a function refers to the set of all possible input values or x-values of the function, while the range refers to the set of all possible output values or y-values of the function. In a piecewise-defined function, each piece has its own domain and range.
The domain of a function refers to the set of all possible input values or x-values of the function, while the range refers to the set of all possible output values or y-values of the function. In a piecewise-defined function, each piece has its own domain and range.
To determine the domain of the given piecewise-defined function, we need to identify the intervals on the x-axis where each piece of the function is defined. From the graph, we can see that the function is defined from -6 to 0, and also from 0 to 2. So the domain is -6≤x≤0 or 0
The range of the function is the set of all y-values that correspond to the x-values in the domain. Looking at the graph, we see that the function takes on values between 1 and 6, inclusive. So the range is 1≤y≤6.
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State whether the given equation or function is linear. Write yes or no. Explain your reasoning.
5x^4 + 5y = 9
Yes, the equation is in linear form. It is in the form xy = C.
Yes, the equation is linear. 0
No, the equation is not linear. It is not in the form Ax + By = C.
No, the equation is not linear. It is in the form x + y = c.
Answer:
Option C is correct.
Step-by-step explanation:
The given equation or function is linear if the variables x and y have degree zero or 1.
The given equation is 5x^4+5y =9
here x has degree =4 and y has degree = 1, but for linear equation both x and y variables must have degree zero or 1.
So, Option C is correct.
Show why, for linear functions, a vertical translation is equivalent to a horizontal translation. For a linear function, what horizontal translation is equivalent to a vertical translation of 3 units up?
Explanation:
A) For the function ...
f(x) = mx + b
the vertical translation by k makes the function ...
g(x) = f(x) + k = mx + b + k
This can be rewritten as ...
g(x) = m(x +k/m) +b = f(x+k/m)
That is, the vertical translation by k is equivalent to a horizontal translation by -k/m, where m is the slope of the linear function.
___
B) For a vertical translation of 3, the equivalent horizontal translation is ...
-k/m for k=3
= -3/m . . . . . where m is the slope of the function
_____
Please note that there is no equivalent for m=0.
Find the value of x in the isosceles trapezoid below?
Answer:
x = 31
Step-by-step explanation:
The sides of an isosceles trapezoid are the same length, so ...
5x -32 = 2x +61
3x = 93 . . . . . . . . add 32-2x
x = 31 . . . . . . . . . . divide by 3
A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?
A.
-3x + 4y = 3
B.
-1.5x − 3.5y = -31.5
C.
2x + y = 20
D.
-2.25x + y = -9.75
This is a PLATO math question, will give 15 pts to the best answer
Answer:
B. -1.5x − 3.5y = -31.5
Step-by-step explanation:
You forgot to provide the reference image which is essential to answer the question, but I managed to find it... and attach it to my answer.
In the given equation for AB, if we place the y term on the left and x term on the right, we see the slope of that line is 7/3 (y = (7x - 21.5)/3 ==> 7/3x).
We see on the image that the line PQ is perpendicular to AB. That means that its slope is -3/7.
If we quickly check the slopes of each of the possible answers...
A. -3x + 4y = 3 ----> 4y = 3x - 3 ==> y = (3x-3)/4 => slope = 3/4
Not what we're looking for.
B. -1.5x - 3.5y = -31.5 ==> 3.5y = -1.5x + 31 ===> y = (-1.5x +31)/3.5
that gives us a slope of -1.5/3.5... We can simplify it... -3(0.5)/7(0.5) = -3/7
Exactly as predicted.
Since we have the point P (7,6), we can enter it in the equation to verify:
-1.5x - 3.5y = -31.5
-1.5 (7) - 3.5 (6) = -10.5 - 21 = -31.5 --- Verified
C. 2x + y = 20 ==> y = 20 - 2x ===> slope is -2, not what we want.
D. -2.25x + y = -9.75 ==> y = 2.25x - 9.75 ==> slop is 2.25 cannot be it.
Answer: B. -1.5x − 3.5y = -31.5
Step-by-step explanation: