The function f(x) is not a straight line and therefore cannot be represented by a linear function like g(x).
The function g(x)=2x-3 is a linear function with a slope of 2 and a y-intercept of -3. It is not directly related to the function f(x) represented by the graph.
The graph of f(x) shows a non-linear function with a maximum value at x=2 and a minimum value at x=6. The function f(x) is not a straight line and therefore cannot be represented by a linear function like g(x).
While the functions f(x) and g(x) may have different shapes and characteristics, they can still be compared and analyzed in various ways. For example, one could find the x- and y-intercepts of both functions, determine their rates of change or slopes, or find their zeros or critical points. However, it's important to note that the relationship between two functions depends on the specific context and the questions being asked.
Learn more about linear function here
https://brainly.com/question/30177575
#SPJ11
What is the relationship between the two functions f(x) and g(x) based on the graph shown? A graph is shown for a function f(x) . The equation g(x)=2x-3 represents a second function.
Find the monthly return of amazon and nasdaq. run a liner regression to estimate beta of amazon. (3) use corrl (amazon return, nasdaq return) x stdev(standard deviation of amazon return)/ stdev (standard deviation of nasdaq return) to find beta of amazon. (4) use the security market line or capital asset pricing model to find the required rate of return of amazon, assuming market return is 12% and risk free rate is 3%.
To find the monthly return of Amazon and Nasdaq, we need to calculate the percentage change in their prices over the month. Once we have these values, we can calculate the beta of Amazon using a linear regression analysis.
Beta measures the sensitivity of a stock's return to changes in the market, with a beta of 1 indicating that a stock moves in line with the market. To find the beta of Amazon, we can use the formula: corrl (Amazon return, Nasdaq return) x stdev (standard deviation of Amazon return) / stdev (standard deviation of Nasdaq return). This will give us a measure of Amazon's volatility relative to the market.
Once we have the beta of Amazon, we can use the security market line or capital asset pricing model to find the required rate of return of Amazon. This involves taking the risk-free rate (which is typically the yield on a government bond) and adding a premium based on the level of risk associated with the stock. Assuming a market return of 12% and a risk-free rate of 3%, we can use the formula: required rate of return = risk-free rate + beta x (market return - risk-free rate).
In summary, to find the monthly return of Amazon and Nasdaq, we need to calculate the percentage change in their prices over the month. We can then use a linear regression analysis to estimate the beta of Amazon and use this to calculate the required rate of return based on the security market line or capital asset pricing model.
To find the monthly return of Amazon and Nasdaq, first calculate the percentage change in their prices for each month. Then, run a linear regression using Nasdaq returns as the independent variable and Amazon returns as the dependent variable to estimate Amazon's beta.
Next, use the formula: CORREL(Amazon return, Nasdaq return) x STDEV(Amazon return) / STDEV(Nasdaq return) to find Amazon's beta. This represents the stock's sensitivity to market movements.
Finally, use the Capital Asset Pricing Model (CAPM) to find Amazon's required rate of return: Risk-Free Rate + Beta x (Market Return - Risk-Free Rate). With a 3% risk-free rate and a 12% market return, the calculation is: 3% + Beta x (12% - 3%). This result represents the expected return for investing in Amazon, considering its risk relative to the market.
To learn more about Nasdaq: brainly.com/question/484147
#SPJ11
Consider the initial value problem dx/dt = [-2 -2 4 2] x, x(0) = [3 9] (a) Find the eigenvalues and eigenvectors for the coefficient matrix. lambda_1 = v_1 vector = [], and lambda_2 = v_2 vector = [] (b) Solve the initial value problem. Give your solution in real form. x(t) = []. Use the phase plotter pplane9.m in MATLAB to answer the following question. An ellipse with counterclockwise orientation 1. Describe the trajectory.
(a) Eigenvalues: lambda_1 = -4, lambda_2 = 3. Eigenvectors: v_1 = [1 2]T, v_2 = [-1 2]T.
(b) Solution: x(t) = 2 e^(-4t) [1 2]T + 5 e^(3t) [-1 2]T. Trajectory is an ellipse with counterclockwise orientation reflecting the exponential decay and growth along the eigenvectors.
(a) To find the eigenvalues and eigenvectors, we need to solve the characteristic equation:
det(A - λI) = 0
where A is the coefficient matrix and I is the identity matrix. We have:
A =
[ -2 -2 ]
[ 4 2 ]
and
A - λI =
[ -2-λ -2 ]
[ 4 2-λ ]
Taking the determinant, we get:
det(A - λI) = (-2-λ)(2-λ) - (-2)(4) = λ^2 + λ - 12 = (λ + 4)(λ - 3)
So the eigenvalues are λ1 = -4 and λ2 = 3.
To find the eigenvector corresponding to λ1, we solve the system:
(A - λ1I) v1 = 0
Substituting λ1 = -4 and solving, we get:
v1 = [1 2]T
To find the eigenvector corresponding to λ2, we solve the system:
(A - λ2I) v2 = 0
Substituting λ2 = 3 and solving, we get:
v2 = [-1 2]T
Therefore, lambda_1 = -4, v_1 = [1 2]T and lambda_2 = 3, v_2 = [-1 2]T.
(b) Using the eigenvectors, we can write the solution as:
x(t) = c1 e^(-4t) [1 2]T + c2 e^(3t) [-1 2]T
where c1 and c2 are constants determined by the initial conditions.
To find c1 and c2, we use the initial condition x(0) = [3 9]. Substituting t = 0, we get:
[3 9] = c1 [1 2]T + c2 [-1 2]T
Solving for c1 and c2, we get:
c1 = 2
c2 = 5
Substituting these values into the solution, we get:
x(t) = 2 e^(-4t) [1 2]T + 5 e^(3t) [-1 2]T
To express the solution in real form, we use Euler's formula:
e^(ix) = cos(x) + i sin(x)
Substituting x = -4t and x = 3t, we get:
x(t) = 2 [cos(4t) + 2i sin(4t)] [1 2]T + 5 [cos(3t) + 2i sin(3t)] [-1 2]T
Expanding and simplifying, we get:
x(t) = [2 cos(4t) - 5 cos(3t) - 4 sin(4t) + 15 sin(3t), 4 cos(4t) - 10 cos(3t) + 8 sin(4t) + 10 sin(3t)]
(c) The phase plotter pplane9.m can be used to plot the trajectory of the system. Since the system has two dimensions, we can plot the solution as a curve in the plane. The ellipse with counterclockwise orientation corresponds to the trajectory of the system.
From the eigenvalues, we know that the solution will decay exponentially along the direction of the eigenvector v1 and grow exponentially along the direction of v2. The orientation of the ellipse reflects this behavior: it is elongated along the direction of the eigenvector v2, which corresponds to the growth direction. As time increases, the ellipse stretches out in the v2 direction and contracts in the v1 direction. The trajectory spirals towards the equilibrium point at the origin, as the exponential decay and growth eventually dominate the solution.
Learn more about Eigenvalues:
https://brainly.com/question/15245981
#SPJ11
Find the GCF of 75m + 85
The GCF of the expression A = 75m + 85 is k = 5
Given data ,
Let the GCF of the expression be represented as k
Now , the expression is
A = 75m + 85
On simplifying , we get
Taking the common factors of 75 and 85 , we get
A = 5 ( 15m + 17 )
Hence , the GCF is k = 5
To learn more about GCF click :
https://brainly.com/question/13257989
#SPJ1
Find the P-value for the test value.
t = 1.619, n = 23, right-tailed
0.0299
0.1196
0.0598
0.0704
The P-value represents the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.
In this case, the null hypothesis is not provided, so we cannot determine the exact P-value.
However, we can use the t-distribution table or a calculator to estimate the P-value for a one-sample t-test with a sample size of 23 and a right-tailed alternative hypothesis.
Using a t-distribution table with 22 degrees of freedom (n-1), we find that the t-critical value for a 0.05 level of significance (alpha) is 1.717. Since the observed t-value of 1.619 is less than the critical value, we can conclude that the P-value is greater than 0.05.
Using a calculator or software, we can obtain a more accurate estimate of the P-value. For a one-sample t-test with the given parameters, the P-value is approximately 0.1196. This means that if the null hypothesis is true, we would expect to obtain a t-value as extreme or more extreme than 1.619 in about 11.96% of all possible samples of size 23.
Therefore, the answer is 0.1196.
To know more about probability visit:
https://brainly.com/question/30034780
#SPJ11
factorise 6x+9xcubed
The factored form of expression 6x + 9x³ is 3x(2 + 3x²).
The given expression is 6x + 9x³
Six times of x plus nine times of x cube
x is the variable in the expression
To factorize 6x + 9x³, we can take out the greatest common factor, which is 3x:
6x + 9x³
= 3x(2 + 3x²)
Hence, the factored form of 6x + 9x³ is 3x(2 + 3x²).
To learn more on Expressions click:
https://brainly.com/question/14083225
#SPJ1
A tent with sides of 3 ft has a rope of 5 ft going from the tent to the tent post. How far away are the posts placed in the ground?
The posts are placed 4 feet away from the tent in the ground by using pythagoras theorem
The sides of the tent form a right triangle, with one side of length 3 ft and the other side of length x ft (the distance between the post and the tent). The rope forms the hypotenuse of the triangle, with a length of 5 ft.
Using the Pythagorean theorem, we can write:
5² = 3² + x²
Simplifying and solving for x, we get:
25 = 9 + x²
x² = 16
x = ±4
Since we're looking for a distance, we take the positive root, x = 4.
Therefore, the posts are placed 4 feet away from the tent in the ground.
To learn more on trigonometry click:
https://brainly.com/question/25122835
#SPJ1
For the following quadratic function, find the axis of symmetry, the vertex and the y-intercept. y= x^2 + 16x+ 24
A. axis of symmetry: x = -16; vertex (-16,24); y-intercept: -16
B. axis of symmetry: x = 8; vertex: (8,194); y-intercept: 24
C. axis of symmetry: x = 16; vertex: (16,536); y-intercept: 16
D. axis of symmetry: x = -8; vertex: (-8,-40); y-intercept: 24
Explain how u got the answer
For the given quadratic equation y = x² + 16x + 24 the axis of symmetry x = -8, and the vertex is (-8, -40) and the y-intercept is 24. So, the correct option is D.
Given quadratic equation y = x² + 16x + 24
From the above equation a = 1; b = 16; c = 24. [co-efficients]
The formula for axis of symmetry x = -b/2a = -16/2(1) = -16/2 = -8.
The formula for vertex is ((-b/2a), f(-b/2a)) = (-8, ((-8)² + 16(-8) + 24))
= (-8, (64-128+24))
= (-8, -40)
The formula for y-intercept = f(0) = (0)² + 16(0) + 24 = 24
From the above analysis,
axis of symmetry = -8vertex = (-8, -40)y-intercept = 24So the option D is the correct answer.
To know more about quadratic equations,
https://brainly.com/question/28038123
#SPJ1
14 let z be a standard normal variable. find p(-1.63 < z < 0.25).
the probability of having a standard normal variable z between -1.63 and 0.25 is approximately 0.5471.
Using a standard normal table or a calculator, we can find the area under the standard normal curve between -1.63 and 0.25.
The standard normal distribution is a continuous probability distribution with mean 0 and standard deviation 1.
Therefore,
P(-1.63 < z < 0.25) = P(z < 0.25) - P(z < -1.63)
Using a standard normal table or a calculator, we can find that:
P(z < 0.25) = 0.5987
P(z < -1.63) = 0.0516
Therefore,
P(-1.63 < z < 0.25) = P(z < 0.25) - P(z < -1.63)
= 0.5987 - 0.0516
= 0.5471
what is probability?
Probability is a measure of the likelihood of an event occurring. It is a numerical value that ranges from 0 (indicating the event is impossible) to 1 (indicating the event is certain).
To learn more about probability visit:
brainly.com/question/30034780
#SPJ11
Carrie is finding the set of even numbers within the set of prime numbers.of the sets described, which is the universal set?all real numbersall integersprime numberseven numbers
Carrie is finding the set of even numbers within the set of prime numbers. The universal set is all integers.
The universal set is the set of all possible elements in a particular context. In this case, Carrie is finding the set of even numbers within the set of prime numbers. The set of prime numbers is a subset of the set of all integers, which means that the set of all integers is the universal set. This is because all prime numbers are integers, but not all integers are prime numbers. Therefore, the universal set in this context is all integers.
Learn more about integers here
https://brainly.com/question/929808
#SPJ11
A beam of light in air is incident upon a stack of four flat transparent materials with indices of refration 1.20,1.40, 1.32, 1.28. If the angle of incidence for the beam on the first of the four materials is 60 degrees, what angle does the beam make with the normal when it emerges into the air after passing through the entire stack?
When the beam of light emerges into the air after passing through the entire stack, it makes an angle of approximately 29.4° with the normal.
We need to apply Snell's Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction for two different media.
Step 1: Calculate the angle of refraction in the first material using Snell's Law.
n1 * sin(i1) = n2 * sin(r1)
1 * sin(60°) = 1.20 * sin(r1)
sin(r1) = 0.5/1.20
r1 ≈ 25.4°
Step 2: Repeat the process for the remaining materials, using the previous angle of refraction as the new angle of incidence. Calculate the final angle of refraction in the last material.
For the second material:
1.20 * sin(25.4°) = 1.40 * sin(r2)
r2 ≈ 21.4°
For the third material:
1.40 * sin(21.4°) = 1.32 * sin(r3)
r3 ≈ 22.9°
For the fourth material:
1.32 * sin(22.9°) = 1.28 * sin(r4)
r4 ≈ 23.3°
Step 3: Calculate the angle of emergence in air.
1.28 * sin(23.3°) = 1 * sin(e)
e ≈ 29.4°
When the beam of light emerges into the air after passing through the entire stack, it makes an angle of approximately 29.4° with the normal.
Visit here to learn more about beam of light:
brainly.com/question/9054772
#SPJ11
1+1=
2+2=
3+3=
4+4=
4163123 x 418247 =
please help
Answer:
2
4
6
8
1.7412137e+12
Step-by-step explanation:
the first 4 are basic addition and the last one I put into calculator and it came up with that
what is the sample size needed to create a 90% confidence interval estimate for a population mean with a width of no more than 1 unit given that the population standard deviation is 10? a. 17 b. 33 c. 1083 d. 27
The answer is d. 27 is not a correct answer as it is much smaller than the required sample size of 271.
To create a 90% confidence interval estimate for a population mean with a width of no more than 1 unit, we need to find the sample size required to achieve this level of precision.
Using the formula for the margin of error of a confidence interval, we have
Margin of Error = z* (σ / sqrt(n))
where z* is the critical value from the standard normal distribution corresponding to a 90% confidence level, σ is the population standard deviation (given as 10), and n is the sample size.
We want the margin of error to be no more than 1 unit, so we can write
1 = z* (σ / sqrt(n))
Solving for n, we get
n = (z* σ / 1)^2
Using a standard normal distribution table or calculator, we can find that the critical value z* corresponding to a 90% confidence level is approximately 1.645. Substituting this value and the given population standard deviation into the formula for n, we get
n = (1.645 * 10 / 1)^2 = 270.25
Rounding up to the nearest integer, we get n = 271. Therefore, the answer is d. 27 is not a correct answer as it is much smaller than the required sample size of 271.
To learn more about Margin of Error click here, brainly.com/question/29419047
#SPJ11
a recent study of 3,000 of a college's recent alumni found that 1,725 of them were homeowners. find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n
The mean of the sampling distribution is equal to the population proportion, and the standard deviation can be calculated using the formula sqrt(p*(1-p)/n), where p is the population proportion and n is the sample size.
Explanation: The sample proportion of homeowners can be calculated as the ratio of the number of homeowners to the sample size:
p^- = 1725 / 3000 = 0.575
This sample proportion provides an estimate of the population proportion of homeowners among recent alumni of the college. Therefore, we can conclude that the population proportion is estimated to be 0.575.
The mean of the sampling distribution is equal to the population proportion, which is 0.575. This means that if we were to take an infinite number of random samples of size n from the population, the average of the sample proportions would be equal to the population proportion.
The standard deviation of the sampling distribution can be calculated using the formula:
σ = sqrt(p*(1-p)/n)
Where p is the population proportion and n is the sample size. Substituting the values, we get:
σ = sqrt(0.575*(1-0.575)/3000) = 0.015 (rounded to three decimal places)
This means that if we were to take an infinite number of random samples of size n from the population, the sample proportions would be distributed around the population proportion with a standard deviation of 0.015.
To learn more about mean click here, brainly.com/question/31101410
#SPJ11
Given the table of values below from a quadratic function, write an equation of that function
X | -6 | -5 | -4 | -3 | -2|
f(x) | 2 | -1 | -2 | -1 | 2 |
Answer:
(x+4)²-2
Step-by-step explanation:
The lowest point of the curve is -2 at x=(-4)
=> The vertex of the upwards facing parabola is (-4,-2)
So, the vertex has shifted from (0,0) (vertex of x²) to (-4, -2)
Using function transformation (translation),
f(x) = (x+4)²-2
<
G
X
Jen has two ropes. The blue rope is 13 feet long and the green rope is 21 feet long.
Calculate how much longer the green rope is than the blue rope.
O A. 21 - 13
= 8²/ feet longer
OB. 21-13 = 21 - 13 = 8 feet longer
c. 21+13= 21 + 13 = 33 feet longer
D. 21 - 13 = 21 - 13 = 8 feet longer
The green rope is 8 ft longer than the blue rope.
Given that Jen has two ropes. The blue rope is 13 feet long and the green rope is 21 feet long.
We need to calculate how much longer the green rope is than the blue rope.
So,
To find the same we will just subtract the shorter blue rope from the longer green rope,
21-13 = 8
Hence the green rope is 8 ft longer than the blue rope.
Learn more about subtraction click;
https://brainly.com/question/2346316
#SPJ1
Solve 2x 10 = -4
X = ?
Which of the following expressions represents "five more than half of x"?
The expression that represents "five more than half of x" is
We have,
"five more than half of x"
To understand , let's break down the phrase
"five more than half of x". "Half of x" can be represented as 0.5x,
and "five more than" means we add 5 to that result.
Therefore, the expression is 0.5x + 5.
Learn more about Expression here:
https://brainly.com/question/14083225
#SPJ1
Given: KLMN is a trapezoid, KL = MN,
angle 1 = angle 2, LM/KN = 8/9,
Perimeter of KLMN = 132
Find: The length of the legs
The length of each leg of trapezoid KLMN is 6.
The length of the legs of trapezoid KLMN can be found using the given information and the formula for the perimeter of a trapezoid, which is P = a + b1 + b2 + c, where a and c are the lengths of the legs and b1 and b2 are the lengths of the bases.
Since KL = MN and angle 1 = angle 2, trapezoid KLMN is an isosceles trapezoid. Let x be the length of each leg and y be the length of each base. Then we have:
a = c = x
b1 = KN = 9x
b2 = LM = 8x
Using the given perimeter of 132, we can substitute these values into the perimeter formula and solve for x:
132 = x + 9x + 8x + x
132 = 19x + 2x
132 = 21x
x = 6
Therefore, the length of each leg of trapezoid KLMN is 6.
Learn more about trapezoid here
https://brainly.com/question/26687212
#SPJ11
Determine whether the following statements are true and give an explanation or counterexample. a) For linear functions, the slope of any secant line always equals the slope of any tangent line. b) The slope of the secant line passing through the points P and Q is less than the slope of the tangent line at P. c) Consider the graph of the parabola f(x) = x². For x> 0 and h> 0, the secant line through (x, f(x)) and (x+h, f(x + h)) always has a greater slope than the tangent line at (x, f(x)).
a) This statement is false. The slope of a secant line is the average rate of change between two points on a function, while the slope of a tangent line is the instantaneous rate of change at a specific point. In general, the slope of a secant line will not equal the slope of the tangent line unless the function is both continuous and differentiable.
b) This statement is false. The slope of a secant line can be greater than, less than, or equal to the slope of the tangent line depending on the shape of the function.
c) This statement is true. The slope of the secant line between two points on a parabola is given by the difference quotient: (f(x+h) - f(x))/h. Taking the limit as h approaches 0 gives the derivative of the function at x, which is 2x. Therefore, for x > 0, the slope of the tangent line at (x, f(x)) is 2x, while the slope of the secant line is (f(x+h) - f(x))/h = ([(x+h)² - x²])/h = (2x + h). Since h > 0, (2x + h) is greater than 2x, so the slope of the secant line is always greater than the slope of the tangent line.
To know more about secant line visit:
https://brainly.com/question/31124520
#SPJ11
Look at the system of equations below.
Which coordinate plane shows the solution to this system of equations?
Adam has been walking to the post office each day. He walks 3.6 miles in 32 minutes. He decided to walk to the store which is 5.2 miles away. About how long should Adam expect the walk to take?
Adam should expect the walk to the store to take approximately 46 minutes and 13 seconds.
To calculate how long it will take Adam to go to the storeWe can set up the following proportion:
32 minutes x 3.6 miles equals 5.2 miles.
When we solve for x, we obtain the following:
x = (5.2 miles x 32 minutes) / 3.6 miles
x = 46.22 minutes.
So, Adam should expect the walk to the store to take approximately 46 minutes and 13 seconds.
Learn more about proportion here : brainly.com/question/19994681
#SPJ1
What is the slope of the line that contains the points in the table?
O A 1/2
O B. -1/2
O c. -2
O d. 2
An angle measures 32.2° more than the measure of its complementary angle. What is the measure of each angle?
The angle measures 61.1 degrees, and its complementary angle measures 28.9 degrees.
Let x be the measure of the angle.
Then the measure of its complementary angle is 90 - x.
We know that x = (90 - x) + 32.2°, because the angle measures 32.2° more than its complementary angle.
Simplifying this equation, we get:
2x = 90 + 32.2
2x = 122.2
x = 61.1
Therefore, the angle measures 61.1 degrees, and its complementary angle measures 90 - 61.1 = 28.9 degrees.
To learn more on Angles click:
https://brainly.com/question/28451077
#SPJ1
Select the real world situation that can be represented by 2m+3
A. A music download website has a membership fee of $2 and charges $3 for each download.
B. An online company charges a $3 shipping fee and $2 per item for a purchase.
C. Grace bikes 2 more than 3 times the miles Jamie bikes.
D. Bentley swims 3 less than 2 times the lap Dan swims.
Option C is the real-world situation that can be represented by 2m + 3. The equation 2m + 3 represents the number 3 more than twice a certain number (m).
In option A, the equation that represents the situation would be 3d + 2, where d is the number of downloads. In option B, the equation that represents the situation would be 2i + 3, where i is the number of items purchased.
In option D, the equation that represents the situation would be 2l - 3, where l is the number of laps Dan swims.
Therefore, option C is the correct answer as it is the only option that represents the situation where one person bikes 2 more than 3 times the miles another person bikes.
To learn more about real world situation click on,
https://brainly.com/question/16179216
#SPJ1
the students who attend memorial high school have a wide variety of extra-curricular activities to choose from in the after-school program. students are 38% likely to join the dance team; 18% likely to participate in the school play; 42% likely to join the yearbook club; and 64% likely to join the marching band. many students choose to participate in multiple activities. students have equal probabilities of being freshmen, sophomores, juniors, or seniors.what is the probability of the union of being either a freshman or senior?
For a student who participate in a wide variety of extra-curricular memorial high school, the probability of the union of being either a freshman or senior is equals to the 0.50.
We have students who attend memorial high school have a wide variety of extra-curricular activities.
Probability of student who likely to join the dance team = 38% = 0.38
Probability of student who likely to participate in the school play = 18% = 0.18
Probability of students who likely to join the yearbook club = 42% = 0.42
Probability of students who likely to join the marching band = 64% = 0.64
Also, students have equal probabilities of being freshmen, sophomores, juniors, or seniors.
So, the probability ( a student is senior )
= 1/4 = 0.25
The probability ( a student is freshman )
= 1/4 = 0.25
We have to determine the union of being either a freshman or senior. Now, the probability of the union of disjoint or independent events is the sum of their individual probabilities.
So, probability ( a student is freshman or senior) = P( freshman ∪ senior )
= P( freshman) + P( senior)
= 0.25 + 0.25 = 0.50
Hence, required probability is 0.50.
For more information about probability, visit:
https://brainly.com/question/30673594
#SPJ4
Which polynomial factors below yield a product of 3x^4 - 75? A 3x^2 (x^2 + 5) B 3(x^2 + 5)^2 C 3(x^2 − 5)^2 D 3(x^2 + 5)(x^2 − 5)
The equation is [tex]3x^{4}-75[/tex] has a polynomial factors [tex]3(x^{2} -5)(x^{2} +5)[/tex]. Thus, the option d is correct.
A polynomial having coefficients in a certain field or in integers is expressed as a combination of indivisible factors having coefficients in the exact same domain in math and computing algebra, which is known as polynomial factorization.
One of the essential building blocks of algorithms for computers is polynomial factorization.
The given equation is [tex]3x^{4}-75[/tex].
To solve for polynomial factors, take the like terms outside, we get
[tex]3x^{4}-75=3(x^{4} -25)[/tex]
Rewrite the terms [tex]x^4[/tex] as [tex](x^{2}) ^{2}[/tex], and [tex]25[/tex] as [tex]5^{2}[/tex].
[tex]3x^{4} -75=3((x^{2} )^{2} -5^2)[/tex]
By using the algebraic identity [tex]a^{2} -b^{2} =(a+b)(a-b)[/tex], we get
[tex]3x^{4} -75=3(x^{2} -5)(x^{2} +5)[/tex]
Hence, the option d is correct.
Read more about polynomial factorization:
brainly.com/question/11434122
The height and base of a triangle are in the ratio 1 : 3 respectively. if the area of the triangle is 216cm², find the length of the base.
The solution is, the length of the base is 36 cm.
Here, we have,
given that,
The height and base of a triangle are in the ratio 1 : 3 respectively.
if the area of the triangle is 216cm²,
let, the length of the base is 3x
so, we get, The height = x
now, we know that,
area = 1/2 * base * height
so, we have,
216 = 1/2 * x* 3x
or, 432 = 3x^2
or, x^2 = 144
or, x = 12
Hence, The solution is, the length of the base is 36 cm.
To learn more on triangle area click:
brainly.com/question/19305981
#SPJ1
differentiate. f(y) = 1 y2 − 9 y4 (y + 3y3)
The derivate of f(y) is -9y^6 - 33y^4 + 84y^2.
To differentiate the given function f(y), we will use the product rule and the chain rule of differentiation. Let's break down the function into two parts:
f(y) = (1 y^2 - 9 y^4) * (y + 3y^3)
Using the product rule, we can differentiate each part separately:
f'(y) = (1 y^2 - 9 y^4)' * (y + 3y^3) + (1 y^2 - 9 y^4) * (y + 3y^3)'
The derivative of the first part is:
(1 y^2 - 9 y^4)' = 2y - 36y^3
Now we need to differentiate the second part using the chain rule. Let's call the inner function u:
u = y + 3y^3
Using the power rule, the derivative of u with respect to y is:
u' = 1 + 9y^2
Now we can substitute these values back into our original equation:
f'(y) = (2y - 36y^3) * (y + 3y^3) + (1 y^2 - 9 y^4) * (1 + 9y^2)
Simplifying further:
f'(y) = 2y^2 + 6y^4 - 36y^4 - 108y^6 + y^2 + 9y^4 - 9y^6 + 81y^2
Combining like terms:
f'(y) = -9y^6 - 33y^4 + 84y^2
Therefore, the derivative of f(y) is -9y^6 - 33y^4 + 84y^2.
Visit here to learn more about chain rule:
brainly.com/question/28972262
#SPJ11
In the diagram shown, two lines intersect to form four angles. The measures of the angles are given in terms of the variables a, b, c, and d. Find the value of a. Show your work.
The value of a in the given figure is 7.5.
Given is a figure we need to simplify for the values of the angles in it,
Using the vertical angle theorem,
4a = 2b+6
2a = b+3........(i)
Also,
12b+6 + 2b+6 = 180 [linear pair angle]
14b+12 = 180
7b+6 = 90
7b = 84
b = 12
Put b = 12 in equation (i)
2a = 12+3
2a = 15
a = 7.5
Hence, the value of a in the given figure is 7.5.
Learn more about vertical angle theorem, click;
https://brainly.com/question/18868430
#SPJ1
The triangle above has the following measures. a = 10 yd b = 10v3 yd Use the 30-60-90 Triangle Theorem to find the length of the hypotenuse. Include correct units. Show all your work.
We can see that that the size of the hypotenuse is 20.
What is the right triangle?A perfect 90-degree angle (a right angle) distinguishes a right triangle from other triangle types. The junction of the triangle's two perpendicular sides results in the formation of this angle.
Right triangles are important in mathematics and many real-world applications because they have unique properties and relationships between their sides and angles.
We know that;
a = 10 yd
b = 10√3
Then;
Hypotenuse would be obtained from the sine ratio;
Sin 30 = 10/c
c = 10/Sin 30
c = 10/0.5
c = 20
Learn more about right triangle:https://brainly.com/question/6322314
#SPJ1
