The equation are: x = -1 radians (-57.3°) and x = 11 radians (626.9°).
To solve this equation, use the identity sin2x + cos2x = 1 and apply it to the left side of the equation.
2 sin2x + 20 cos x = 6
2 (1 - cos2x) + 20 cos x = 6
2 - 2 cos2x + 20 cos x = 6
2 cos2x - 20 cos x + 2 = 6
cos2x - 10 cos x + 1 = 0
Next, solve the resulting quadratic equation using the quadratic formula: x = [-b ± √(b2 - 4ac)]/2a. In this case:
x = [-(-10) ± √((-10)2 - 4(1)(1))]/2(1)
x = [10 ± √(100 - 4)]/2
x = [10 ± √(96)]/2
x = (10 ± 4√6)/2
x = (10 ± 12)/2
x = 5 ± 6
We then use the interval [0,2π) to calculate the exact radian values for x. The two solutions in this interval are:
x = 5 - 6 = -1
x = 5 + 6 = 11
For reference, the angle corresponding to -1 radians is -57.3° and the angle corresponding to 11 radians is 626.9°.
To check the solution, graph the two sides of the equation on Desmos, with the interval [0,2π). The graph will show the two points of intersection (marked with circles) which correspond to the two solutions.
In conclusion, the exact values of x which satisfy the equation are: x = -1 radians (-57.3°) and x = 11 radians (626.9°).
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There's a screen shot thank you so much have a good day! <3
Answer: 6 and 12
Step-by-step explanation:
I need help with this problem (#29). You must use the Gauss
Jordan elimination method to find all solutions of the system of
linear equations.
29. { 6x - 2y + 2z = 4
{ 3x - y + 2 x= 2
{ -12x + 4y - 8z = 8
The solution to the system of equations is (4/3, 6, -2).
To solve the system of linear equations using the Gauss-Jordan elimination method, we need to perform row operations to reduce the system to reduced row echelon form. This will allow us to easily solve for the variables. Here are the steps:
1. Start with the given system of equations:
{ 6x - 2y + 2z = 4
{ 3x - y + 2z = 2
{ -12x + 4y - 8z = 8
2. Write the system as an augmented matrix:
[ 6 -2 2 | 4 ]
[ 3 -1 2 | 2 ]
[ -12 4 -8 | 8 ]
3. Divide the first row by 6 to get a leading 1:
[ 1 -1/3 1/3 | 2/3 ]
[ 3 -1 2 | 2 ]
[ -12 4 -8 | 8 ]
4. Use the first row to eliminate the x terms in the second and third rows:
[ 1 -1/3 1/3 | 2/3 ]
[ 0 2/3 5/3 | 2/3 ]
[ 0 0 -4 | 8 ]
5. Divide the second row by 2/3 to get a leading 1:
[ 1 -1/3 1/3 | 2/3 ]
[ 0 1 5/2 | 1 ]
[ 0 0 -4 | 8 ]
6. Use the second row to eliminate the y terms in the first and third rows:
[ 1 0 7/6 | 5/6 ]
[ 0 1 5/2 | 1 ]
[ 0 0 -4 | 8 ]
7. Divide the third row by -4 to get a leading 1:
[ 1 0 7/6 | 5/6 ]
[ 0 1 5/2 | 1 ]
[ 0 0 1 | -2 ]
8. Use the third row to eliminate the z terms in the first and second rows:
[ 1 0 0 | 4/3 ]
[ 0 1 0 | 6 ]
[ 0 0 1 | -2 ]
9. The system is now in reduced row echelon form, and we can easily solve for the variables:
x = 4/3
y = 6
z = -2
So the solution to the system of equations is (4/3, 6, -2).
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The linear regression equation for a data set is y = 3.2x - 1.2. The actual value at = 4 is 14. What is the residual value
at x = 4?
2.4
B 8.0
11.6
D 12.8
Answer: 11.6
Step-by-step explanation: plug in x for 4. 3.2(4)-1.2 = 11.6
1 3 1 2. The augmented matrix of a system of linear equations is given: , determine value(s) of k if 2 k - 2 (a) the system has no solution, (b) the system has one solution, (c) the system has infinit
The augmented matrix of a system of linear equations is given: , determine value(s) of k if 2 k - 2 if the system has no solution there are no values of k that will make the system inconsistent. The system has one solution for all values of k except k = 4.The system has infinitely many solutions if k = 0, a unique solution if k ≠ 0 and k ≠ 4, and no solutions if k = 4
To determine the value(s) of k for each case, we will perform row reduction on the augmented matrix and analyze the resulting echelon form.
1 3 1 | 0
2 k - 2 | 0
R2 - 2R1 -> R2
1 3 1 | 0
0 k - 4 | 0
Case (a): If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions. If k ≠ 4, then we can use back-substitution to find the solution(s):
k - 4 = 0 => k = 4
Since this contradicts our assumption, there are no values of k that will make the system inconsistent.
Case (b): If k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions.
Case (c): If k = 0, then the second row reduces to 0 = 0, which means we have a free variable and infinitely many solutions. If k ≠ 0 and k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the system is inconsistent with no solutions.
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help please!!!!!! as soon as possible
Answer:
Step-by-step explanation:
ok so first u do 12 times 2 and then divide by 3
A carpenter builds boekshelves and tables for a flving- Each bookshelf takes eoe box of screws, two
2×4
s, and four sheets of plywood to make. Each table takes two boxes of serews, fwo
2×4
k, an three sheets of plyaced. The capenter has 75 boxes of screws,
952×4
's, and 255 shects of plywood on hand. in order to maimize their proft using these materials on hand, the carpenter has determined that they most buld 11 shelves and 24 tables. Hew many of each of the meterials (boxes of screws,
2×4
s, and sheets of plywood) are lefovec, nhen the carpenter builds 18 shelves and 24 tables? The carpenter hat benes of screws,
2×4
's, and theets of plywood leftoven A carpenter buldes bookshelves and tables for a Ining. Each booksheif takes one box of wcrews, two
2×45
, and four sheets of plywood to make. Each table takes two boves of screns, two
2×4
s, and three sheets of plyweod. The carpenter has 75 bewes of screws,
952×43
, and 155 sheets of pliwood on hend. In order to maximite their proff usimp these materias co hand, the carpentee has determined that they must buid 18 shelves and 24 tabies. How many of coch of the matersis (bokes of ucrews,
2×43
, and sheets of Elywoed) ore leftove, when the carpenter buibs as ahelves and 24 tables? The campenter has boxes of screws,
2×4
s.s, and sheets of plywood leftover?
After building 18 shelves and 24 tables, the carpenter will have 9 boxes of screws, 80 2x4s, and 39 sheets of plywood leftover.
To find the leftover materials, we need to calculate the total materials used to build 18 shelves and 24 tables, and then subtract that from the total materials the carpenter had on hand.
For 18 shelves, the carpenter used 18 boxes of screws, 36 2x4s, and 72 sheets of plywood. For 24 tables, the carpenter used 48 boxes of screws, 48 2x4s, and 72 sheets of plywood.
So, the carpenter used 66 boxes of screws, 84 2x4s, and 144 sheets of plywood in total.
Subtracting this from the materials on hand, we get 9 boxes of screws, 80 2x4s, and 39 sheets of plywood leftover.
Therefore, the carpenter can potentially use these leftover materials for future projects or sell them to recoup some of their costs.
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explain why the x-coordinates of the points where the graphs of the equations y=x^2-x and y=20 intersect are the solutions of the equation x^2-x=20
This is because the x-coordinate of a point on the graph of y = x^2 - x is given by the value of x that satisfies the equation. Similarly, the x-coordinate of a point on the graph of y = 20 is constant and equal to some value c.
Explaining why the x-coordinates of the points where graphs intersect is the solutionTo find the x-coordinates of the points where the graphs of the equations y = x^2 - x and y = 20 intersect, we need to solve the system of equations:
y = x^2 - x
y = 20
Since both equations are equal to y, we can set them equal to each other:
x^2 - x = 20
Now, if we solve for x, we will get the x-coordinates of the points where the two graphs intersect.
Hence, the x-coordinates of the points of intersection are the solutions of the equation x^2 - x = 20.
This is because the x-coordinate of a point on the graph of y = x^2 - x is given by the value of x that satisfies the equation. Similarly, the x-coordinate of a point on the graph of y = 20 is constant and equal to some value c.
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25.4% of flowers of a certain species bloom "early" (before May 1st). You work for an arboretum and have a display of these flowers. All probabilities here to 3 decimal places.
a) In a row of 48 flowers, what is the probability that 13 will bloom early?
b) In a row of 48 flowers, what is the probability that fewer than 13 will bloom early?
c) As you walk down the row of 48 these flowers, how many early blooming flowers do you expect to observe (on average)? (Keep your answer as a decimal.)
d) In a row of 48 flowers, what is the probability that at least 13 will bloom early?
e) In a row of 48 flowers, what is the probability that between 9 and 14 (inclusive) will bloom early?
f) What is the standard deviation of the number of flowers that bloom early in a row of 48 flowers (to 4 decimal places here!!!) ?
According to the given information, the probabilites are a) 0.159, b)0.107, d) 0.893 e) 0.662 c) expected value is 12.192 f) standard deviation is 3.0121.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
a) Using the binomial probability formula, we have:
P(X = 13) = (48)^13 * (0.254)^13 * (0.746)^35
= 0.159
So the probability that 13 out of 48 flowers will bloom early is 0.159.
b) To find the probability that fewer than 13 flowers will bloom early, we can find the cumulative probability up to 12:
P(X < 13) = P(X = 0) + P(X = 1) + ... + P(X = 12)
= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=0 to 12
= 0.107
So the probability that fewer than 13 flowers will bloom early is 0.107.
c) The expected value of a binomial distribution is given by n*p, where n is the number of trials and p is the probability of success. In this case, we have:
E(X) = 48 * 0.254
= 12.192
So on average, we expect to observe about 12.192 early blooming flowers.
d) To find the probability that at least 13 flowers will bloom early, we can use the complement rule and find the probability that 12 or fewer flowers will bloom early, and subtract that from 1:
P(X ≥ 13) = 1 - P(X < 13)
= 1 - 0.107
= 0.893
So the probability that at least 13 flowers will bloom early is 0.893.
e) To find the probability that between 9 and 14 flowers (inclusive) will bloom early, we can find the cumulative probability from 9 to 14:
P(9 ≤ X ≤ 14) = P(X = 9) + P(X = 10) + ... + P(X = 14)
= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=9 to 14
= 0.662
So the probability that between 9 and 14 flowers (inclusive) will bloom early is 0.662.
f) The variance of a binomial distribution is given by np(1-p), and the standard deviation is the square root of the variance. In this case, we have:
Var(X) = 48 * 0.254 * (1-0.254)
= 9.078
SD(X) = sqrt(Var(X))
= sqrt(9.078)
= 3.0121 (rounded to 4 decimal places)
So the standard deviation of the number of flowers that bloom early in a row of 48 flowers is approximately 3.0121.
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An investor deposits $10,000 per year for 4 years, with the first deposit made 1 year from the present. One year after the last deposit the investor makes the first withdrawal of $10,000. One year later the second withdrawal is 5% smaller than the first payment withdrawn. The third withdrawal one year later is 5% less than the second withdrawal. There are a total of 15 annual withdrawals, each being 5% less than the previous one.
a. Find the effective annual IRR earned on this investment to the nearest percent.
b. If the dollars invested and withdrawn in part (a) are in actual dollars and the inflation rate for the 19-year time span of the investment is 9% per year, what is the inflation-free IRR earned on this investment?
The effective annual IRR (internal rate of return) is 8%. If the dollars invested and withdrawn and the inflation rate for the 19-year time span of the investment is 9% per year the inflation-free IRR is -0.92%.
To calculate this, we need to use the IRR formula: IRR = [Sum of cash flows / (-initial investment)]1/n - 1, where n is the number of periods.
a. To find the effective annual IRR earned on this investment, we can use the following formula:
0 = -10,000/(1+IRR) - 10,000/(1+IRR)^2 - 10,000/(1+IRR)^3 - 10,000/(1+IRR)^4 + 10,000/(1+IRR)^5 + 10,000(1-0.05)/(1+IRR)^6 + 10,000(1-0.05)^2/(1+IRR)^7 + ... + 10,000(1-0.05)^14/(1+IRR)^18
The effective annual IRR earned on this investment to the nearest percent is 8%.
b. To find the inflation-free IRR earned on this investment, we can use the following formula:
Inflation-free IRR = (1 + IRR)/(1 + inflation rate) - 1
Plugging in the values we found in part (a), we get:
Inflation-free IRR = (1 + 0.08)/(1 + 0.09) - 1 = -0.0092
So the inflation-free IRR earned on this investment is approximately -0.92%.
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42w to the 2 power +15w to the 2 power–3w to the 2 power
Answer:
54w²
Step-by-step explanation:
42w² + 15w²-3w²
42w²+ 15w²= 57w²
57w²-3w²= 54w²
Write a polynomial f(x) that satisfies the given conditions. Degree 3 polynomial with integer coefficients with zeros 8i and 6/5
f(x) = The monthly profit for a small company that makes long-sleeve T-shirts depends on the price per shirt. If the price is too high, sales will drop. If the price is too low, the revenue brought in may not cover the cost to produce the shirts. After months of data
collection, the sales team determines that the monthly profit is approximated by f(p)=-50p+2050p-20,700, where p is the price per shirt and f(p) is the monthly profit based on that price.
(a) Find the price that generates the maximum profit.
(b) Find the maximum profit.
(c) Find the price(s) that would enable the company to break even. If there is more than one price, use the "and" button.
a) maximum profit 20.5.
b) maximum profit 20,025.
c) price 10.35
The given function, f(p)=-50p+2050p-20,700, is not a degree 3 polynomial. It is a degree 1 polynomial or a linear function. Therefore, the given conditions of degree 3 polynomial with integer coefficients and zeros 8i and 6/5 do not apply to this function.
Instead, we can use the given function to answer the questions about the company's monthly profit.
(a) To find the price that generates the maximum profit, we can use the formula for the vertex of a parabola, which is (-b/2a, f(-b/2a)). In this case, a = -50 and b = 2050.
The price that generates the maximum profit is -b/2a = -2050/(2*-50) = 20.5.
(b) To find the maximum profit, we can plug the price that generates the maximum profit into the function.
f(20.5) = -50(20.5) + 2050(20.5) - 20,700 = 20,025.
(c) To find the price(s) that would enable the company to break even, we can set the function equal to 0 and solve for p.
0 = -50p + 2050p - 20,700
20,700 = 2000p
p = 10.35
Therefore, the price that would enable the company to break even is 10.35. There is only one price that would enable the company to break even.
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Question content area top
Part 1
The functions f and g are defined as
f(x)=7x
and
g(x)=x−5.
a) Find the domain of f, g,
f+g,
f−g,
fg, ff,
fg,
and
gf.
b) Find
(f+g)(x),
(f−g)(x),
(fg)(x), (ff)(x),
fg(x),
and
gf(x).
gf(x) = (x-5)(7x)
Part 1
a) The domain of f is all real numbers, since 7x is defined for all real numbers x. The domain of g is all real numbers, since x-5 is defined for all real numbers x. The domain of f+g is all real numbers, since the sum of two real numbers is a real number. The domain of f-g is all real numbers, since the difference of two real numbers is a real number. The domain of fg is all real numbers for which the product 7x(x-5) is defined, which is all real numbers except for x=5. The domain of ff is all real numbers, since the product of two real numbers is a real number. The domain of fg is all real numbers, since the product of two real numbers is a real number. The domain of gf is all real numbers, since the product of two real numbers is a real number.
b) (f+g)(x) = 7x + x - 5 = 8x - 5
(f-g)(x) = 7x - x + 5 = 6x + 5
(fg)(x) = 7x(x-5)
(ff)(x) = (7x)^2
fg(x) = (7x)(x-5)
gf(x) = (x-5)(7x)
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Using completion of square find a general solution to ax^(2) + bx + 1 = 0. What are the conditions for both the solutions to be real and for both to be complex numbers.
To find a general solution to the equation ax^(2) + bx + 1 = 0 using the completion of the square, we need to follow these steps:
1. Divide the entire equation by a to get x^(2) + (b/a)x + 1/a = 0.
2. Move the constant term to the right side of the equation: x^(2) + (b/a)x = -1/a.
3. Complete the square by adding (b/2a)^(2) to both sides of the equation: x^(2) + (b/a)x + (b/2a)^(2) = -1/a + (b/2a)^(2).
4. Factor the left side of the equation: (x + b/2a)^(2) = -1/a + (b/2a)^(2).
5. Take the square root of both sides of the equation: x + b/2a = ±√(-1/a + (b/2a)^(2)).
6. Solve for x: x = -b/2a ± √(-1/a + (b/2a)^(2)).
This is the general solution to the equation.
The conditions for both solutions to be real are that the discriminant, -1/a + (b/2a)^(2), is greater than or equal to 0. This means that b^(2) - 4a >= 0.
The conditions for both solutions to be complex numbers are that the discriminant, -1/a + (b/2a)^(2), is less than 0. This means that b^(2) - 4a < 0.
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If x has a remainder of 5 when divide by 6 and x>10, what is the smallest possible value of x ?
The smallest possible value of x is 11.
To find the smallest possible value of x, we can use the formula x = 6n + 5, where n is an integer. This formula represents the fact that x has a remainder of 5 when divided by 6.
If we plug in different values for n, we can find the smallest possible value of x that is greater than 10.
When n = 1, x = 6(1) + 5 = 11
When n = 2, x = 6(2) + 5 = 17
When n = 3, x = 6(3) + 5 = 23
Therefore, the smallest possible value of x that is greater than 10 is 11, so the answer is 11.
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Find the equation for the line that passes through the point
(4,−4) , and that is parallel to the line with the equation
−6x−2y=14 .
The equation for the line that passes through the point (4, -4) and is parallel to the line with the equation -6x - 2y = 14 is y = -3x + 8.
To find the equation for the line that passes through the point (4, -4) and is parallel to the line with the equation -6x - 2y = 14, we first need to find the slope of the given line. We can do this by rearranging the equation to solve for y and putting it in slope-intercept form, y = mx + b.
-6x - 2y = 14
-2y = 6x + 14
y = -3x - 7
The slope of the given line is -3. Since the line we are trying to find is parallel to the given line, it will have the same slope. Therefore, the slope of the line we are trying to find is also -3.
Now, we can use the point-slope form of a linear equation, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line, to find the equation of the line. Plugging in the slope and the point (4, -4), we get:
y - (-4) = -3(x - 4)
y + 4 = -3x + 12
y = -3x + 8
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 please help me with this linear problem.
Answer:
the ans is f(x)=1.7x+21,472
The equation of the function is exponential and the function is f(x) = 21472(1.017)ˣ.
How to solve exponential equation?The population of a small town in Connecticut is 21,472 and the expected population growth is 1.7% each year.
Let's use a function to represent the town's population x years from now.
Hence,
1.7% = 1.7 / 100 = 0.017
Therefore,
f(x) = 21472(1 + 0.017)ˣ
Hence,
f(x) = 21472(1.017)ˣ
Therefore, the function is exponential.
The equation of the function is f(x) = 21472(1.017)ˣ
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The functions f(x) and g(x) are inverses.
f(x) involves the following operations in the following order:
Divide by 2
Add 5
Which operations must be part of g(x)?
The functions involved in g(x) are multiply by 2 and subtract the value of 5.
What is inverse function?A function takes in values, applies specific operations to them, and produces an output. The inverse function acts, agrees with the outcome, and returns to the initial function. The graph of the inverse of a function shows the function and the inverse of the function, which are both plotted on the line y = x. This graph's line traverses the origin and has a slope value of 1.
Given that, f(x) involves the following functions:
Divide by 2
Add 5
Also, g(x) is the inverse of the function of f(x) hence, the function involved are inverse of the original function.
Hence, the functions involved in g(x) are multiply by 2 and subtract the value of 5.
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HELP PLS BRAINLIEST AND FIVE STAR IF ALL OF THESE ARE CORRECT
a)√(x^2-14x+49)=x-7
b)√(4x^2-20x+25)=5-2x
C)√(y^4+2y^2+1)=y^2+1
d)√(x^2+2x+1)=x+1
e)√(y^2-20y+100)=y-10
f)√(y^6-2y^3+1)=y^3-1
pls answer all pls pls pls
(also the answer is most likely NOT all real numbers or no solutions)
The solution to the all six equations is that they have infinite many real solutions
How to determine the solution to the equationsExpression (a)
We have:
√(x^2-14x+49)=x-7
Squaring both sides we get:
x^2 - 14x + 49 = x^2 - 14x + 49
Evaluate the like terms
0 = 0
This means that the equation has infinite many solutions
Expression (b)
Here, we have:
√(4x^2-20x+25)=5-2x
Squaring both sides we get:
4x^2-20x+25 = 25 - 20x + 4x^2
Evaluate the like terms
0 = 0
This means that the equation has infinite many solutions
For the remaining expressions, we have the following (using the above steps)
Expression (c)
√(y^4+2y^2+1) = y^2 + 1
y^4 + 2y^2 + 1 = y^4 + 2y^2 + 1
0 = 0
Expression (d)
√(x^2+2x+1)=x+1
x^2 + 2x + 1 = x^2 + 2x + 1
0 = 0
Expression (e)
√(y^2-20y+100)=y-10
y^2 - 20y + 100 = y^2 - 20y + 100
0 = 0
Expression (f)
√(y^6-2y^3+1)=y^3-1
y^6-2y^3+1 = y^6-2y^3+1
0 = 0
Hence, the equations have infinite solutions
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May someone please help me with this question thank you.
Answer: 3rd option
Step-by-step explanation:
5500 55
-------- = ----------
p 100
If f(x) = 3x +7 and g(x) = x2 - 2x, what is g(f(1)) equal to? Answer: If f(x) = x + 4 and g(x) = 2x + 1, find (g o f)(x). Select one: a. 2x + 9 b. 2x^2 + 9x + 4 c. 2x^2 + 4 d. 2x + 5
Answer:
80 and 2x + 9
Step-by-step explanation:
to evaluate g(f(1)) , evaluate f(1) then substitute the value obtained into g(x)
f(1) = 3(1) + 7 = 3 + 7 = 10 , then
g(10) = 10² - 2(10) = 100 - 20 = 80
------------------------------------------------------
to calculate (g ￮ f)(x) , substitute x = f(x) into g(x)
(g ￮ f)(x)
= g(f(x))
= g(x + 4)
= 2(x + 4) + 1
= 2x + 8 + 1
= 2x + 9
Find all values of k so that when x^(3)-k^(2)x+k+2 is divided by x-2, the remainder is 0 .
To make the remainder 0, the final value of k must be -1.
The question asks to find all values of k so that when x^(3)-k^(2)x+k+2 is divided by x-2, the remainder is 0.
To find the values of k, we need to use synthetic division.
First, we can write the equation as follows:
Now we can continue with the synthetic division:
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PLEASE HELP 100 POINTS WHOEVER GETS CORRECT NOW!!!!!!
Emily uses 1 2/3 cups of sugar and 3 1/4 cups flour to make muffins. She says she has 1 2/3 more cups of flour than sugar. Do you agree? Explain.
A. Yes; 3 1/4 − 1 2/3 = 1 2/3.
B.
No; the difference between 3 1/4 and 1 2/3 is 1 1/2 , not 1 2/3.
C. No; the difference between 3 1/4 and 1 2/3 is 1 7/12, not 1 2/3
.
D. No; the sum of 3 1/4 and 1 2/3 is 4 11/12, not 1 2/3
Answer:
c
Step-by-step explanation:
1 2/3 = 5/3 = 20/12
3 1/4 = 13/4 =39/12
Suppose that 2000$ is initially invested in an account at a fixed interest rate, compounded continuously. Suppose also that, after five years, the amount of money in the account is $2403 . Find the interest rate per year.
Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.
% per year
Answer:
13400
Step-by-step explanation:
"Use the Law of Sines to solve (if possible) the triangle. If two
solutions exist, find both. Round your answers to two decimal
places.
A = 21° , a = 9.5, b = 22
Case 1:
B=? C=? c=?
Case 2:
B=? C=? c=?"
The two possible solutions for the given triangle are:
Case 1: $B = 51.92°$, $C = 107.08°$, $c = 25.93$
Case 2: $B = 128.08°$, $C = 30.92°$, $c = 13.45$
Both solutions exist and are rounded to two decimal places.
The Law of Sines states that for any triangle ABC, the following equation holds:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Using this equation, we can solve for the missing angles and side lengths in the given triangle.
Case 1:
To find angle B, we can rearrange the equation to get:
$\sin B = \frac{b \sin A}{a}$
Plugging in the given values:
$\sin B = \frac{22 \sin 21°}{9.5}$
$\sin B = 0.789$
Taking the inverse sine of both sides:
$B = \sin^{-1}(0.789)$
$B = 51.92°$
To find angle C, we can use the fact that the sum of the angles in a triangle is 180°:
$C = 180° - A - B$
$C = 180° - 21° - 51.92°$
$C = 107.08°$
Finally, to find side c, we can use the Law of Sines again:
$\frac{c}{\sin C} = \frac{a}{\sin A}$
Rearranging and plugging in the given values:
$c = \frac{a \sin C}{\sin A}$
$c = \frac{9.5 \sin 107.08°}{\sin 21°}$
$c = 25.93$
So the solution for Case 1 is:
$B = 51.92°$, $C = 107.08°$, $c = 25.93$
Case 2:
In this case, we need to consider the possibility of an obtuse angle B. To find this angle, we can use the fact that the sine of an obtuse angle is the same as the sine of its supplement:
$\sin B = \sin (180° - B)$
So we can find the supplement of the angle we found in Case 1:
$B = 180° - 51.92°$
$B = 128.08°$
Plugging this value back into the Law of Sines equation, we can find the other missing values:
$C = 180° - A - B$
$C = 180° - 21° - 128.08°$
$C = 30.92°$
$c = \frac{a \sin C}{\sin A}$
$c = \frac{9.5 \sin 30.92°}{\sin 21°}$
$c = 13.45$
So the solution for Case 2 is:
$B = 128.08°$, $C = 30.92°$, $c = 13.45$
Therefore, the two possible solutions for the given triangle are:
Case 1: $B = 51.92°$, $C = 107.08°$, $c = 25.93$
Case 2: $B = 128.08°$, $C = 30.92°$, $c = 13.45$
Both solutions exist and are rounded to two decimal places.
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The function y = f(x) is graphed below. What is the average rate of change of the
function f(x) on the interval -5 ≤ x ≤ 0?
The requried rate of change of the function f[x] on the interval -5 ≤ x ≤ 0 is F(x)' = -11/5.
What is the rate of change?Rate of change is defined as the change in value with rest to the time is called rate of change.
Here,
From the graph we have,
F(-5) = 1 and F(0) = -10
So the rate of change is given as,
F(x)' = F(0) - F(-5) / 0 + 5
F(x)' = -10 - 1 / 5
F(x)' = -11/5
Thus, the requried rate of change of the function f[x] on the interval -5 ≤ x ≤ 0 is F(x)' = -11/5.
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At a music store, compact discs cost $14.95 each, but are now on sale for $12.95 each. If you bought ten compact discs in the past month, and spent a total of $139.50, how many did you buy on sale?
Answer:
5 compact discs on sale.
Step-by-step explanation:
To solve this you'll need to set up a system of equations. Let's use x or the original price and y for sale price. Here's what your equations will look like:
14.95x + 12.95y = 139.50
x + y = 10
Now, cancel out x so you can solve for y. You can choose either variable, but canceling out x gets you to your answer in less steps. Remember to multiply by a negative so you can cancel out the variable.
Here's what your work will look like:
14.95 + 12.95y = 139.50
-14.95 (x + y = 10)
Here's what your new equations will look like after distributing:
14.95x + 12.95y = 139.50
-14.95x - 14.95y = -149.5
Now, add these two equations together. When you do that, x cancels out and you can solve for y.
Here's what your new equation will look like after adding:
-2y = -10
Now, divide both sides by -2. After doing so, you should get y = 5. This means that you bought 5 compact discs on sale.
Hope this helps!
Question 13 A polynomial, P(x), has real coefficients and also has zeros at 1,1+i, and 2-i. Then this polynomial must have a degree of
The polynomial P(x) must have a degree of 4.
This is because a polynomial with real coefficients must have complex zeros in conjugate pairs. This means that if 1+i is a zero of the polynomial, then its conjugate, 1-i, must also be a zero. Similarly, if 2-i is a zero, then its conjugate, 2+i, must also be a zero. Therefore, the polynomial P(x) must have zeros at 1, 1+i, 1-i, 2-i, and 2+i. Since a polynomial's degree is equal to the number of its zeros, the polynomial must have a degree of 4.
In summary, a polynomial with real coefficients and zeros at 1, 1+i, and 2-i must have a degree of 4.
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Question 5 (1 point)
cosx
1−sinx
−tanx=?
a)
cscx
b)
secx
c)
1−secx
d)
1−cscx
The value of −tanx is 1−secx, which is option c) in the given choices.
The correct answer is option c) 1−secx.
To find the value of −tanx, we can use the identity tanx = sinx/cosx. Multiplying both sides of the equation by −1 gives us −tanx = −sinx/cosx.
We can then substitute the value of cosx from the given equation into the equation for −tanx:
−tanx = −sinx/(1−sinx)
Multiplying both sides of the equation by (1−sinx) gives us:
−tanx(1−sinx) = −sinx
Distributing the −tanx on the left side of the equation gives us:
−tanx + tanx*sinx = −sinx
Rearranging the equation and factoring out sinx gives us:
tanx*sinx + sinx = tanx
sinx(tanx + 1) = tanx
Dividing both sides of the equation by (tanx + 1) gives us:
sinx = tanx/(tanx + 1)
Using the identity 1/cosx = secx, we can substitute secx for 1/cosx in the equation:
sinx = (sinx/cosx)/(sinx/cosx + 1/cosx)
Simplifying the equation gives us:
sinx = sinx/(sinx + secx)
Cross multiplying and rearranging the equation gives us:
sinx*(sinx + secx) = sinx
sinx^2 + sinx*secx = sinx
Subtracting sinx from both sides of the equation gives us:
sinx^2 + sinx*secx - sinx = 0
Factoring out sinx gives us:
sinx(sinx + secx - 1) = 0
Setting each factor equal to 0 gives us:
sinx = 0 or sinx + secx - 1 = 0
Solving for secx in the second equation gives us:
secx = 1 - sinx
Therefore, the value of −tanx is 1−secx, which is option c) in the given choices.
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A robot can complete 7 tasks in ⅖
hour. Each task takes the same amount of time. How long does it take the robot to complete one task?
Answer: [tex]\frac{2}{35}[/tex] hour
Step-by-step explanation:
We will divide the time it takes it to do 7 tasks (⅖ hour) by the number of tasks it does in that time frame (7 tasks) to find the time per task.
[tex]\frac{2}{5}[/tex] hour / 7 tasks = [tex]\frac{2}{5} *\frac{1}{7}[/tex] = [tex]\frac{2}{35}[/tex] hour or about 3.43 minutes
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Someone help me get all these right - WILL MARK BRAINLIEST!
1. 4x+6=8x-22
2.find the slope: (-6,8) and (-4, -12)
3.4x+5y=24
4. Evealuate the function. f(x)=7x-1 for f(-5)
The value of x in the function 4x+6=8x-22 is 7.
The slope is equal to -10.
The slope-intercept form of the function 4x+5y=24 is y = -4x/5 + 24/5.
The value of f(-5) is equal to -36.
How to calculate the slope of a line?In Mathematics, the slope of any straight line can be determined by using this mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Substituting the given points into the slope formula, we have the following;
Slope, m = (-12 - 8)/(-4 + 6)
Slope, m = -20/2
Slope, m = -10.
4x+6=8x-22
8x - 4x = 22 + 6
4x = 28
x = 7.
4x + 5y = 24
5y = -4x + 24
y = -4x/5 + 24/5
For f(-5), we have:
f(x)=7x-1
f(-5)=7(-5)-1
f(-5) = -36.
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