Answer:1/4 (I think)
factor x^2-4x-5=0 steps plsss
x^2-4x-5=0
Using sum and product method
sum= -4
product= -5
factor= -5, 1
x^2-5x+x-5
x(x-5)+1(x-5)
(x-5)(x+1)
an article in neurology (1998, vol. 50, pp. 1246-1252) discussed that monozygotic twins share numerous physical, psychological, and pathological traits. the investigators measured an intelligence score of 10 pairs of twins, and the data follow: pair birth order: 1 (first) birth order: 2 (second) 1 6.08 5.73 2 6.22 5.80 3 7.99 8.42 4 7.44 6.84 5 6.48 6.43 6 7.99 8.76 7 6.32 6.32 8 7.60 7.62 9 6.03 6.59 10 7.52 7.67 (a) State the appropriate hypotheses. (b) Create a Paired- T test at α=0.05 in Minitab and find the P-value. Do you reject the null?
Answer: The resulting P-value is 0.923, which is greater than 0.05. Therefore, we do not reject the null hypotheses.
Step-by-step explanation:
(a) The appropriate hypotheses for this study would be:
Null hypothesis: There is no difference in intelligence scores between first-born and second-born monozygotic twins.
Alternative hypothesis: There is a significant difference in intelligence scores between first-born and second-born monozygotic twins.
(b) In Minitab, we can conduct a Paired-T test using the following steps:
1. Enter the data into two columns, one for first-born twins and one for second-born twins.
2. Select "Stat > T Tests > Paired T" from the menu.
3. In the "Paired-T Test" dialog box, select the two columns of data as the "Pairs" and specify the significance level as 0.05.
4. Click "OK" to run the test.
The P-value obtained from this test is 0.752, which is greater than the specified significance level of 0.05. Therefore, we do not reject the null hypotheses.
Why is it appropriate to use a Paired-T test in this study?
It is appropriate to use a Paired-T test in this study because the data consists of paired samples (first-born and second-born monozygotic twins) and we are comparing the mean difference between the two groups. The Paired-T test is specifically designed to compare the means of paired samples and allows for the correction of any inherent dependency between the two groups.
The P-value of 0.752 indicates that there is a 75.2% chance of obtaining the observed difference in intelligence scores between first-born and second-born twins if the null hypothesis is true.
Since this probability is relatively high, we cannot conclude that there is a significant difference in intelligence scores between the two groups. This means that the intelligence scores of first-born and second-born twins are not significantly different.
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P-value = 0.29, which is higher than 0.05. Therefore, we don't reject the null hypotheses.
a)The proper hypotheses for this investigation would be:
Null hypotheses: First-born and second-born monozygotic twins do not differ in their cognitive ratings.
Appropriate hypotheses: First-born and second-born monozygotic twins' IQ scores differ significantly.
(b)The steps listed below can be used to do a Paired-T test in Minitab:
1. Separate the data into first-born twins and second-born twins columns.
2. "Stat > T Tests > Paired T" can be selected from the menu.
3. Choose the two data columns as the "Pairs" in the "Paired-T Test" dialogue box, and enter 0.05 for the significance level.
4. To launch the test, click "OK".
This test yielded a P-value of 0.752, which is higher than the 0.05 designated threshold of significance. Therefore, we don't reject the null hypotheses.
Why is a Paired-T test useful for this study?
The first-born and second-born monozygotic twins in this study are paired samples, and we are comparing the mean difference between the two groups, therefore a Paired-T test is acceptable. The Paired-T test allows for the correction of any inherent dependencies between the two groups and is specifically created to compare the means of paired samples.
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a school board has a plan to increase participation in the pta. currently only about 16 parents attend meetings. suppose the school board plan results in logistic growth of attendance. the school board believes their plan can eventually lead to an attendance level of 48 parents. in the absence of limiting factors the school board believes its plan can increase participation by 30% each month. let m denote the number of months since the participation plan was put in place, and let p be the number of parents attending pta meetings.
The required logistic model is
[tex]p=\frac{48}{1+2e^{0.11394t} }[/tex]
As given in the question,
a school board has plan to increase the participation in the pta
number of parents attending the meetings is 16
the school board believes that their would be rise in attendance level of
48 parents
the percentage of attendance rise is 30%
a logistic model is
[tex]N = \frac{k}{1+be^{-rt} }[/tex]
where
k is a constant
the constant b is founded by
[tex]b = \frac{k}{N(0)}-1[/tex]
r is the intrinsic exponential growth rate
if there is no limiting factors then logistic model will be
[tex]N=N(0) e^{rt}[/tex]
so the corresponding growth factor for the equation will be
[tex]a=e^t[/tex]
so the variable can be obtained by using
[tex]r=lna[/tex]
A) p is the number of parents attending pta meetings
P(0) = 16
then the limiting value is k
where k = 48
therefore,
the carrying capacity ( limiting value ) k is 48
B) the constant for the logistic model
[tex]b=\frac{k}{p(o)} -1[/tex]
where
the value of k is 48
the value of p(o) is 16
now
[tex]b = \frac{48}{16} -1\\[/tex]
[tex]b= \frac{48-16}{16}[/tex]
[tex]b = \frac{32}{16}[/tex]
b = 2
the constant for b in logistic model is
b = 2
C)
the r value to construct the logistic model because the number of participation increased by 30%
a = 1+0.30
a = 1.30
to find r
r = ln a
r = ln (1.30)
r = 0.11394
D)
now substitute the values k = 48 and b=2 and r = 0.11394 in the equation
[tex]p=\frac{k}{1+be^{-rt} }[/tex]
=> p = [tex]\frac{48}{1+2e^{0.11394t} }[/tex]
this the required logistic model which represents the number of parents attending the PTA after this month
=> p= [tex]\frac{48}{1+2e^{0.11394t} }[/tex]
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Note:- The complete question is,
A school board has a plan to increase participation in the pta. currently only about 16 parents attend meetings. suppose the school board plan results in logistic growth of attendance. the school board believes their plan can eventually lead to an attendance level of 48 parents. in the absence of limiting factors the school board believes its plan can increase participation by 30% each month. let m denote the number of months since the participation plan was put in place, and let p be the number of parents attending pta meetings.
(a) What is the carrying capacity K for a logistic model of P versus m ?
K=
(b) Find the constant b for a logistic model.
b=
(c) Find the r value for a logistic model. Round your answer to three decimal places.
r =
(d) Find a logistic model for P versus m.
P=
A number is rounded off to the nearest thousand. The rounded number is less than the original
number. Which of these can be the original number?
a) 4832
b) 5789
c) 6315
d) 7956
Answer:
c or d or b but probely d
Step-by-step explanation:
because
Let f be a scalar field and F a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field.
(a) curl f
(b) div F
(c) del (upside down traingle) F
(d) curl(curl F)
(e) (grad F)X(divF)
(f) grad(div F)
Please show your work when you answer, thank you!
On solving the provided question, Triple product - A.(BXC) = B.(CXA)=C.(AXB) and AX(BXC) = B(A.C) - C(A.B)
What is vector field?A vector field is a vector association with each point in a subset of space in the field of vector mathematics and physics. For example, a vector field in the plane can be represented as a series of arrows. Each arrow is associated with a point in the plane and each has a specific size and direction.
Triple product -
A.(BXC) = B.(CXA)=C.(AXB)
AX(BXC) = B(A.C) - C(A.B)
Product Rules-
∇(fg) = f(∇g) + g()∇f
∇A.B = AX (∇XB) + BX(∇XA) + (AX∇)B + (B.∇)A
∇(fA) = f(∇.A) + A(∇.f)
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Read the following statement:
If the measure of an angle is more than 90º, then it is an obtuse angle.
Which of the following choices is a counterexample to the statement?
the measure of an angle that is 180º
the measure of an angle that is 150º
the measure of an angle that is 91º
Answer:
the measure of an angle that is [tex]180^{\circ}[/tex]
Step-by-step explanation:
This would be a straight angle, not an obtuse angle.
In a board game, a certain number of points is awarded to a player upon rolling a six sided die (labeled 1 to 6) according to the function f(x)=4x+4, where xx is the value rolled on the die. Find and interpret the given function values and determine an appropriate domain for the function.
The numeric values of the function are given f(1) = 8, meaning when a 1 is rolled on the die, the player is awarded 8 points. This interpretation makes sense in the context of the problem.
What are the numeric values of the function?The function in this problem is defined as follows:
f(x) = 4x+4
The variables are given as x is the number rolled and y is the score.
The numbers that can be rolled are:
1, 2, 3, 4, 5 and 6.
Hence the domain is
{x ∈ N | 1 ≤ x ≤ 4}.
The numeric values are obtained replacing the variable x by the input, as follows:
f(1) = 4 + 4 =8
f(10) = 4(10) + 4= 44-> does not make sense, as a 10 cannot be rolled.
This interpretation does not make sense in the context.
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two adjacent sides of a unit square are trisected and lines drawn between the points as shown. what fraction of the square is hatched?
the square is hatched that is Area of the square adjacent to the third side of the triangle is 85 units square that is 85/100= 17/20
What is Pythagoras theorem?
Pythagoras theorem states that a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.
Given that areas of the squares adjacent to two sides of a right triangle are :
35 sq units
50 sq units
To determine the area of the square adjacent to the third side of the triangle
According to Pythagoras theorem,
Hypotenuse² = Base² + Perpendicular²
As figure is missing assuming given sides are perpendicular sides
Then Area of the square adjacent to the third side of the triangle = 35 + 50
= 85 units square
Now, assuming 50 sq unit is an area on hypotenuse
Then Area of the square adjacent to the third side of the triangle = 35 - 50
= 15 units square
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FILL IN THE BLANK. (n) does not extend beyond the boundaries of a particular organization. group of answer choices internet extranet intranet arpanet
ARPANET does not extend beyond the boundaries of a particular organization.
ARPANET is considered the forerunner of the internet.
ARPANET stands for Advanced Research Projects Agency Network. It was established by the Department of Defense of the United States in 1969, and it used two protocols that were determinant for the invention of the Internet: the packet-switching network and the TCP/IP protocol suite. The scientist that participated in the development of the research were Paul Baran, Lawrence Robles, and Donald Davis. So ARPANET is considered the forerunner of the internet.
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Please what’s the answer
The coefficient of the given term is 2208
What is the binomial formula?The binomial formula is used for binomial expansion and is given by: [tex](x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}[/tex]
Given here (2u-s)²⁴
The 22nd term will be given by ;
T₂₂ = ²⁴C₂₂× (2u)²× (-s)²²
= 552×4× (u)²× (s)²²
= 2208 (u)²(s)²²
Hence, the coefficient of the given term is 2208
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find the area of the parallelogram height 8 length 15
Answer:
Step-by-step explanation:
The area of a parallelogram is given by the formula:
Area = Base * Height
In this case, the base of the parallelogram is 15 and the height is 8, so the area of the parallelogram is given by:
Area = 15 * 8
= 120
Therefore, the area of the parallelogram with a height of 8 and a length of 15 is 120 square units.
Answer:
120 units
area of parallelogram = height x length
a = 8 x 15
a = 120
In Exercises 1, 2, 3, 4, 5, and 6, show that v is an eigenvector of A and find the corresponding eigenvalue. A= [ -1 16 0 ] V = [1 -2]
Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
A*v = [-1 16 0] * [1 -2] = [-1 -32 0]
λv = [-2 -4 0]
Since Av = λv, we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
1. First, we multiply the matrix A with vector v to find the product Av.
2. A*v = [-1 16 0] * [1 -2] = [-1 -32 0]
3. Then, we multiply the eigenvalue λ with vector v which gives the result λv.
4. λv = [-2 -4 0]
5. We compare Av and λv. If they are equal, then v is an eigenvector of A and the corresponding eigenvalue is λ.
6. Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
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Please show your work: Select ALL the transformations that occur to the parent function f(x) to form g(x)=0.8f(x−2)−8
Answer:
Step-by-step explanation:
Vertical shrink by 0.8
horizontal shift by 2 to the right
vertical shift by 8 down
URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
The main similary between a cone and a cylinder: They both have a circular base, area of the base is A = π · r². (Correct choice: B)
What similarity does exist between a cone and a cylinder?In this problem we must compare a cone with a similar cylinder, that is, two solids with similar key measures (same height and same base radius). The area of the base is described by the following formula:
A = π · r²
Where r is the radius of the circular base, in square units.
The volume formula for each solid is shown below:
Cone
V = (1 / 3) · A · h
Cylinder
V = A · h
Where h is the height of the solid, in units.
The main conclusions are described below:
The volume of the cone is a third of the volume of cylinder.Both solids have a circular base. The cone has one circular base and the cylinder has two parallel circular bases.Three cones fit a similar cylinder.Hence, the right choice for this question is B.
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Consider again the electric circuit in Problem 21 of Section 7.6. This circuit is described by the system of differential equations ()=(-4,-)(0) a. Show that the eigenvalues are real and equal if L = 4R2C. b. Suppose that R=1.12, C = 1 F, and L = 4 H. Suppose also that I(0) = 1 A and V(0) = 2 V. Find I(t) and V(t).
The solution of a differential equation to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.
a. When L = 4R2C, the eigenvalues of the system of differential equations are found by solving the characteristic equation, which is -4±2√(4R2C) = -4±8RC. Since R, C, and L are all positive constants, the eigenvalues are real and equal, and they are both equal to -4RC.
b. Setting R = 1.12, C = 1 F, and L = 4 H, we find that the eigenvalues of the system of differential equations are -4·1.12·1 = -4.48. The solution to the system is then given by I(t) = Ae-4.48t + Be4.48t and V(t) = -4RAe-4.48t + 4RB e4.48t.
Substituting the initial conditions I(0) = 1 A and V(0) = 2 V, we get A + B = 1 and -4RA + 4RB = 2. Solving for A and B, we find that A = 3/4 and B = -1/4.
The solution to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.
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Please help
Leo is buying fabric to make curtains. The graph shows the cost, y, of buying x yards of fabric.
What is the cost of buying 7 yards of fabric?
Write the point-slope form of the equation of the line that passes through the points (-8, 2) and (1, -4). Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
Slope: -2/3
Point-Slope Form: y - 2 = -2/3 ( x - (-8))
The equation of line in point slope form is y = -2/3 x - 10/3
Define Equation of line.
A line in geometry is a collection of points that can be stretched infinitely in two different ways.To put it another way, a line is created by continually extending the two end points in any direction. As a result, we can assert that a line lacks end points. However, because it just has length and no width, it is a one-dimensional geometric form. Based on the information provided, we can create multiple forms of equations for lines algebraically, such as point slope form, slope-intercept form, and two point form.Two Point Form Formula,y - y₁ = m (x - x₁)
Given points are,
(-8, 2) and (1, -4)
We know,
Slope formula,
m = (y₂ - y₁) / (x₂ - x₁)
Two Point Form Formula,
y - y₁ = m (x - x₁)
Let, (x₁, y₁) = (-8, 2)
and, (x₂, y₂) = (1, -4)
First find the slope,
m = (-4 - 2) / (1 - (-8))
m = -6 / 9
m = -2/3
Now, point slope equation is,
y - 2 = -2/3(x -(-8) )
y - 2 = -2/3 ( x + 8)
y - 2 = -2/3 x - 2/3 * 8
y - 2 = -2/3 x - 16/3
y = -2/3 x - 16/3 + 2
y = -2/3 x - ( 16/3 - 2 )
y = -2/3 x - ( (16 - 6) / 3)
y = -2/3 x - ( 10/3 )
y = -2/3 x - 10/3
Hence, the equation of line in point slope form is y = -2/3 x - 10/3
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Suppose that a steel company views the production of its continuous caster as a continuous income stream with a monthly rate of flow at time t given by f(t) = 36,000e0.05t (dollars per month). Find the total income from this caster in the first year. (Round your answer to the nearest dollar.)
total income from the continuous caster in the first year is $43200
A serial caster's first year gross income can be found by integrating the function f(t) representing the number of months in a year from t = 0 to t = 12.
To find this integral, we can use the formula for integrals of exponential functions.
∫ f(t) dt = F(t) = 36000/0.05 * (e0.05t - 1)
Inserting the limits of the integration, we get:
∫36000e0.05t dt from t = 0 to t = 12 = 36000/0.05 * (e0.05*12 - 1) = 36000/0.05 * (1.63 - 1) = 36000/0.05 * 0.63 = 43200
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At the bearest wholesales store, 3 bags of cereal cost $21.75. Which equation represents the proportional relationship? y=3x
y=7.25
y=65.25
y=21.75
7. A customer ordered 20 kg of fertilizer that contains 15% nitrogen. To
fulfill the customer's order, how much of the stock 30% nitrogen
fertilizer must be mixed with the 10% nitrogen fertilizer?
Answer:
5 kg of 30% nitrogen an d 15 kg of 10% nitrogen.
Step-by-step explanation:
Meryl stood X meters from the base of a lighthouse. The angle from where she stood to the top of lighthouse was 60 degrees. If the lighthouse was 66.5 meters tall, how far away was Meryl standing from the base of the lighthouse? Round your answer to the nearest whole meter.
The required distance of Meryl from the base of the lighthouse is 38.39 meters.
What is angle ?An angle is the formed when two straight lines meet at one point, it is denoted by θ.
Given that,
The height of the lighthouse = 66.5 meters.
Also, the angle to the top of the lighthouse = 60°.
To find the distance X from the base of the lighthouse,
use tan θ = perpendicular / base
Substitute the values here,
tan θ = 66.5 / X
tan 60 = 66.5 / X
1.732 = 66.5 / X
X = 66.5 / 1.732
X = 38.39 meters.
The distance of Meryl from the base of the lighthouse is 38.39 meters.
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Given the equation 3x-12=18. Which equation are equivalent to the given equation above
3x=30
3x=6
3(x-4)=18
-3x+12=-18
6x-24=34
a line with a slope of 1/10 passes through the point (20,2). what is its equation in slope-intercept form?
-------------------------------------------------------------------------------------------------------------
Answer: [tex]y= \frac{1}{10}x[/tex]
-------------------------------------------------------------------------------------------------------------
Given: [tex]\textsf{Slope = 1/10 and goes through (20, 2)}[/tex]
Find: [tex]\textsf{Slope in slope-intercept form}[/tex]
Solution: First we need to plug into the point slope formula which is [tex](y - y_1) = m(x - x_1)[/tex]. Let us plug in the values first and then solve for the slope-intercept form.
Plug in the values
[tex](y - y_1) = m(x - x_1)[/tex][tex](y - 2) = \frac{1}{10}(x - 20)[/tex]Simplify
[tex]y - 2 = (\frac{1}{10} * x)+(\frac{1}{10}*-20)[/tex][tex]y - 2 = \frac{1}{10}x-2[/tex][tex]y - 2 + 2 = \frac{1}{10}x-2 + 2[/tex][tex]y= \frac{1}{10}x[/tex]Answer: Therefore, the final answer in slope-intercept form for the line with a slope of 1/10 that passes through the point (20, 2) is [tex]y= \frac{1}{10}x[/tex]
exploration of the clinical significance of the findings, as opposed to the statistical significance
While statistical significance indicates the reliability of the study results, clinical significance reflects its impact on clinical practice.
Can you have clinical significance without statistical significance?
Clinical researchers and practitioners should concentrate on clinically relevant improvements rather than statistical significance, which is the emphasis of most research. A research result may be clinically important but not statistically important, and vice versa.
In clinical research, statistically significant study results are frequently considered to be of clinical significance. Clinical significance emphasizes the study's influence on clinical practice, whereas statistical significance highlights the validity of the study's findings.
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Describe or show how to solve for sine, cosine, and tangent values.
Answer:
SOH CAH TOA
The ratios of sine, cosine and tangent.
Step-by-step explanation:
Sine: Opposite / Hypotenuse
Cosine: Adjacent / Hypotenus
Tangent: Opposite / Adjacent
What correctly explains 55 percent of 64
Answer:
15
Step-by-step explanation:
Round 55 percent to 40 percent and 64 to 60. Think of 40 percent as . So, of 60 is 15.
State if the three side lengths form an acute, obtuse or right triangle. 6km, 8km, 12km
organic chemists often purify organic compounds by a method known as fractional crystallization. an experimenter wanted to prepare and purify 4.85 g of aniline. ten 4.85-gram specimens of aniline were prepared and purified to produce acetanilide. the following dry yields were obtained: 3.85, 3.88, 3.90, 3.62, 3.72, 3.80, 3.85, 3.36, 4.01, 3.82 construct a 95% confidence interval for the mean number of grams of acetanilide that can be recovered from 4.85 grams of aniline. assume that the obtained dry yields of acetanilide are i.i.d. and approximately normally distributed with an unknown mean and variance.
As calculated from the given data the confidence interval for μ is (3.65, 3.91).
Let X_1,..,X_nX 1 ,..,X n are n observations
from population with mean μ.
Than sample mean
= n1i=1/n∑X i
Sample variance is defined by,
s = 1/( n-1 )∑(X i - mean) ²
n = sample size = 10
c = 95%
therefore ,
mean = 37.82/10
=3.782
variance = 0.29556/9
=0.03
Data points' variance from the mean is a measure of how they vary. A variance, according to Layman, is a measurement of how widely apart a set of data (numbers) are from their mean (average) value.
Finding the expected difference of deviation from the actual number is what is meant by variance. As a result, variance is influenced by the data set's standard deviation.
Data is more dispersed from its mean the higher the variance value, and less dispersed from mean if the variance value is low or minimal. As a result, it is referred to as a measure of data spread from mean.
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Use vectors to find the interior angles of the triangle with the given vertices. (Enter your answers as a comma-separated list. Enter your answers in terms of degrees. Round your answers to two decimal places.)
(−2, 4), (−3, 8), (6, 8)
Please Help ASAP
Answer:
77.47°
75.96°
26.57°
Step-by-step explanation:
Given vertices of the triangle:
A = (−2, 4)B = (−3, 8)C = (6, 8)Find the vectors from A to B, B to C and A to C:
[tex]\begin{aligned}AB = B - A &=(x_B-x_A,y_B-y_A) \\&=(-3-(-2), 8-4)\\& = (-1, 4) \end{aligned}[/tex]
[tex]\begin{aligned}BC=C-B &=(x_C-x_B,y_C-y_B)\\ &=(6-(-3),8-8)\\&=(9,0)\end{aligned}[/tex]
[tex]\begin{aligned}AC = C - A &=(x_C-x_A,y_C-x_A)\\&= (6-(-2), 8-4) \\&= (8, 4)\end{aligned}[/tex]
Use Pythagoras Theorem to calculate the magnitudes of the vectors:
[tex]|AB| = \sqrt{(-1)^2+4^2}=\sqrt{17}[/tex]
[tex]|BC|=\sqrt{9^2+0^2}=9[/tex]
[tex]|AC| = \sqrt{8^2+4^2}=4\sqrt{5}[/tex]
[tex]\boxed{\begin{minipage}{6 cm}\underline{Dot Product of two vectors}\\\\$a \cdot b=|a||b| \cos \theta$\\\\where:\\ \phantom{ww}$\bullet$ $|a|$ is the magnitude of vector a. \\ \phantom{ww}$\bullet$ $|b|$ is the magnitude of vector b. \\ \phantom{ww}$\bullet$ $\theta$ is the angle between $a$ and $b$. \\ \end{minipage}}[/tex]
Rearrange the dot product formula to make θ the subject:
[tex]\implies \theta=\cos^{-1}\left(\dfrac{a \cdot b}{|a||b|}\right)[/tex]
Use the rearranged dot product formula to find the angles between two pairs of vectors.
[tex]\boxed{\begin{minipage}{4 cm}\underline{Dot Product}\\\\$\textbf{u} \cdot \textbf{v}=u_1v_1+u_2v_2$\\\\where:\\ \phantom{ww}$\bullet$ $\textbf{u}=\left\langle u_1,u_2 \right\rangle$ \\\phantom{ww}$\bullet$ $\textbf{v}= \left\langle v_1,v_2 \right\rangle$ \\ \end{minipage}}[/tex]
Angle A
[tex]\implies A=\cos^{-1}\left(\dfrac{AB \cdot AC}{|AB||AC|}\right)[/tex]
[tex]\implies A=\cos^{-1}\left(\dfrac{-1 \cdot 8+4 \cdot4}{\sqrt{17} \cdot 4 \sqrt{5}}\right)[/tex]
[tex]\implies A=\cos^{-1}\left(\dfrac{8}{4 \sqrt{85}}\right)[/tex]
[tex]\implies A=77.47^{\circ}\; \sf (2 \; d.p.)[/tex]
Angle C
[tex]\implies C=\cos^{-1}\left(\dfrac{BC \cdot AC}{|BC||AC|}\right)[/tex]
[tex]\implies C=\cos^{-1}\left(\dfrac{9 \cdot 8+0 \cdot4}{9 \cdot 4 \sqrt{5}}\right)[/tex]
[tex]\implies C=\cos^{-1}\left(\dfrac{72}{36 \sqrt{5}}\right)[/tex]
[tex]\implies C=26.57^{\circ}\; \sf (2 \; d.p.)[/tex]
Interior angles of a triangle sum to 180°.
[tex]\implies B=180^{\circ}-A-C[/tex]
[tex]\implies B=180^{\circ}-77.47^{\circ}-26.57^{\circ}[/tex]
[tex]\implies B=75.96^{\circ}[/tex]
Therefore, the interior angles of the triangle with the given vertices are:
77.47°75.96°26.57°2. From basis to differential equation In each of the following problems you are given a set of basis functions for the general solution to homogeneous, linear DE with constant coefficients. Work backwards to a determine the DE (in unknown function y y(z)) which they solve. 1. The number of basis functions corresponds to the order of the differential equation (and dimension of the solution space). 2. You can determine all characteristic equation roots and their multiplicities from the basis functions. 3. From the characteristic equation, you can write the DE. (b) (e2e e cos (2z), e r sin(2r))
On solving the provided question, the differential equations we obtained are as follows - [tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex] And [tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]
what is differential equations?
An equation comprising one or more functions and their derivatives is known as a differential equation in mathematics. A function's derivative expresses the rate of change of the function at a specific moment. It is mostly employed in the sciences of physics, engineering, and biology.
Since there are 3 basic function, the order of the differential equation is 3. Since the basic finction e^{x} is repeating thrice, the characteristic roots are 1,1,1,(i.e the characteristic root is 1 with multiplicity 3). Hence the characteristic equation is (D-1)^{3} = 0.
[tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex]
Since there are 3 basic function, the order of the differential equation is According to the basic finctions, the characteristic roots are [tex]2,-1[/tex]±[tex]2i.[/tex] Hence the characteristic equation is (D-2)(D^{2}+2D+5) = 0. which means D^{3}+D-10 = 0.
[tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]
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