[tex]\cfrac{1}{3}(9x+12)=15\implies \cfrac{9x+12}{3}=15\implies 9x+12=45 \\\\\\ 9x=33\implies x=\cfrac{33}{9}\implies x=\cfrac{3}{3}\cdot \cfrac{11}{3}\implies x=\cfrac{11}{3}\implies x=3\frac{2}{3}[/tex]
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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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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7. The size of a television screen is determined by its diagonal measure. If the height of
a screen is 32 inches and the width is 57 inches, what size is the TV considered to be
in inches? Round to the nearest whole number.
Using Pythagoras theorem, the size of the TV will be 65 inches.
What is Pythagorean Theorem?
The Pythagorean Theorem is a fundamental concept in mathematics that describes the relationship between the sides of a right triangle. It states that in any right triangle, the sum of the squares of the lengths of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse).
In equation form, the Pythagorean Theorem can be written as:
a² + b² = c²
where a and b are the lengths of the two legs and c is the hypotenuse.
Now,
We can use the Pythagorean theorem to find the diagonal measure of the television screen.
In this case, the height and width of the screen form the legs of a right triangle, so we can use the following equation:
(diagonal)² = (height)² + (width)²
Substituting the given values, we get:
(diagonal)² = (32)² + (57)²
(diagonal)² = 1024 + 3249
(diagonal)² = 4273
Taking the square root of both sides
diagonal = √(4273) ≈ 65.37
Therefore, the size of the TV screen, as measured diagonally, is approximately 65 inches (rounded to the nearest whole number).
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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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I have 5 mathematics books, 4 astronomy, and 3 physics books. I have always fancied both math and physics but don’t like these books to touch each other. I feel astronomy plays well with both subjects so astronomy can touch either. In how many ways can I arrange the books on a shelf if:
-All the books are different
-I want subjects grouped together
-Math and physics cannot touch
1. How many ways can I arrange the books.
2. What if I let the math and physics books touch?
1.) The total number of ways in which the books can be arranged so that mathematics and physics cannot touch = 478,961,280.
2.) The total number of ways the books can be arranged when math and physics books touch = 479,001,600
What is permutation?Permutation is defined as the expression that shows how objects can be arranged in a definite order.
The total number of mathematics books = 5
The total number of astronomy books = 4
The total number of physics = 3
The total number of books that the student has in possession = 5+4+3 = 12.
The arrangement of the books so that math and physics cannot touch = 12!-8!
= 479001600 - 40320
= 478,961,280
The arrangement of the books so that math and physics can touch = 12! = 479,001,600
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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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help please!!!!!! as soon as possible
Answer:
Step-by-step explanation:
ok so first u do 12 times 2 and then divide by 3
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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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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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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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
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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Cos ^4 x rewrite the expression as an equivalent expression that does not contain powers are trigonometric functions greater than 1.
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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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
50x8+25 take away what equals 225?
Answer:
1.8 i think
Step-by-step explanation:
i used a calculator
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:
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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There's a screen shot thank you so much have a good day! <3
Answer: 6 and 12
Step-by-step explanation:
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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Help me please now pls
Answer: C
Step-by-step explanation: LxWxH
8x8= 64x4=256i
hope this helped :)
Answer:
256
Step-by-step explanation:
start with 8x8 that will equal 64 then 64x4 which equals 256.
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
Use interest tables on page A10 to solve for the total interest
$600 was deposited at 5.5% interest rate compounded daily for 50 days
Interest on $600 after 50 days of depositing is $4.53
What is interest?In finance and economics, interest is the amount paid by a borrower or deposit-taking financial institution to a lender or depositor in excess of the repayment of the principal at a specified rate. It is different from a fee that a borrower can pay to a lender or a third party.
Given,
Principal amount = $600
Interest rate = 5.5%
Compounded daily for 50 days
By the interest table,
$1 compounded daily at 5.5% rate for 50 days = $1.00755
$600 compounded daily at 5.5% rate for 50 days
= $600 × 1.00755
= $604.53
Interest = Amount with interest - Principal
Interest = $604.53 - $600
Interest = $4.53
Hence, $4.53 is the interest on $600 after 50 days of depositing.
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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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If A and B are two independent events in certain sample space S,
then show that A^c and B^c are also independent where "c" denotes
the complement operator
P(A ∩ B) = P(A)P(B) and P(Ac ∩ Bc) = P(Ac)P(Bc).
The complement operator “c” refers to the inverse of an event. Therefore, for two independent events A and B in a sample space S, the inverse events Ac and Bc are also independent. This is because the probability of two independent events occurring is equal to the product of their individual probabilities. Therefore, the probability of two inverse events occurring is equal to the product of their individual probabilities.
Mathematically, this is expressed as follows: P(A ∩ B) = P(A)P(B) and P(Ac ∩ Bc) = P(Ac)P(Bc).
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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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What is the ratio of red to blue squares in its simplest form?
Red Blue
00
The ratio of red to blue squares is given by the division of the number of red squares by the number of blue squares.
How to obtain the ratio?The ratio between two amounts, a and b, is obtained applying a proportion, as the ratio is the division of the amount a by the amount b.
The amounts for this problem are given as follows:
Amount a: number of red squares.Amount b: number of blue squares.Hence the ratio is given by the division of the number of red squares by the number of blue squares.
For example, for 10 red and 20 blue squares, the ratio is given as follows:
r = 10:20 = 1:2.
Missing InformationThe problem is incomplete, hence the general procedure to obtain the ratio is presented.
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Which equation could be used to find the value of x?
cos 49° = X/55
tan 49° = x/55
cos 49° = 55/x
tan 49° = 55/x
PLEASE ANSWER FAST!
tan 49° = x/55
tan Θ = opposite/adjacent
x - opposite
55 cm - adjacent
3.[10points]Suppose{v1,…,vk}is an orthogonal basis of a subspaceVofRnand{w1,…,wℓ}is an orthogonal basis ofV⊥. (a) Show that{v1,…,vk,w1,…,wℓ}is an orthogonal set. (b) Show that{v1,…,vk,w1,…,wℓ}is basis ofRn. (c) Show thatdimV+dimV⊥=n
(a) To show that {v1,…,vk,w1,…,wℓ} is an orthogonal set, we must show that the inner product of any two distinct vectors in this set is 0.
(b) To show that {v1,…,vk,w1,…,wℓ} is a basis of Rn, we must show that any vector in Rn can be written as a linear combination of these vectors.
(c) To show that dim V + dim V⊥ = n, we must show that the number of vectors in {v1,…,vk,w1,…,wℓ} is equal to n.
Let v and w be any two distinct vectors in the set {v1,…,vk,w1,…,wℓ}. By definition, since {v1,…,vk} is an orthogonal basis of a subspace V of Rn and {w1,…,wℓ} is an orthogonal basis of V⊥, we have that v and w are either in V or in V⊥, but not both.
Thus, if v is in V, then we have = 0, because w is in V⊥. Similarly, if w is in V⊥, then = 0, because v is in V. Therefore, = 0 for any two distinct vectors v and w in {v1,…,vk,w1,…,wℓ}, which implies that {v1,…,vk,w1,…,wℓ} is an orthogonal set.
(b) Let x be any vector in Rn. Since {v1,…,vk} is a basis of V, x can be written as a linear combination of the vectors in {v1,…,vk}, that is, x = a1v1 + a2v2 + ... + akvk.
Since {w1,…,wℓ} is a basis of V⊥, x can also be written as a linear combination of the vectors in {w1,…,wℓ}, that is, x = b1w1 + b2w2 + ... + bℓwℓ.
Combining these two equations, we can write x = (a1v1 + a2v2 + ... + akvk) + (b1w1 + b2w2 + ... + bℓwℓ). Thus, any vector in Rn can be written as a linear combination of the vectors in {v1,…,vk,w1,…,wℓ}, which implies that {v1,…,vk,w1,…,wℓ} is a basis of Rn.
By definition, {v1,…,vk} is an orthogonal basis of a subspace V of Rn, so dim V = k. Similarly, {w1,…,wℓ} is an orthogonal basis of V⊥, so dim V⊥ = ℓ. Thus, the number of vectors in {v1,…,vk,w1,…,wℓ} is k + ℓ = n, which implies that dim V + dim V⊥ = n.
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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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