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?
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.
Learn more about equation here
https://brainly.com/question/29657992
#SPJ4
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.
To learn more about variance
brainly.com/question/15707019
#SPJ4
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.
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.
To learn more on volumes: https://brainly.com/question/1578538
#SPJ1
17. Identify any congruent figures in the coordinate plane at the right . Explain .
PLEASE ANSWER
Answer:
(XYZ) and (ABC)
(MNQP) and (RUTS)
Step-by-step explanation:
XYZ and ABC are the same triangle. They are just rotated.
Similarly, MNQP and RUTS are the same figures. They are rotated.
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
Learn more about Pythagoras theorem here:
brainly.com/question/343682
#SPJ4
A recent drug survey showed an increase in the use of drugs and alcohol among local high school seniors as compared to the national percent. Suppose that a survey of 100 local seniors and 100 national seniors is conducted to see if the proportion of drug and alcohol use is higher locally than nationally. Locally, 65 seniors reported using drugs or alcohol within the past month, while 61 national seniors reported using them. Conduct a hypothesis test at the 5% level.
NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)
Null hypothesis is rejected and it was concluded that proportion of the use of drugs in local seniors is more compare to national seniors.
As given in the question,
Total number of local seniors be n₁ = 100
Total number of national seniors be n₂ = 100
Let us consider proportion of local seniors be p₁
Number of local seniors using drugs = 65
Null hypothesis : p₁ = p₂
Alternative hypothesis : p₁> p₂
p₁ = 0.65
1 - p₁ = 0.35
And proportion of national seniors be p₂
Number of national seniors using drugs = 61
p₂ = 0.61
1 - p₂ = 0.39
Standard error = √ [p₁(1-p₁)/n₁ + p₂(1-p₂)/n₂]
= √0.65( 0.35)/100 + 0.61(0.39)/100
= √0.002275 + 0.002379
= 0.0682
Z =( p₁ - p₂ ) / √ [p₁(1-p₁)/n₁ + p₂(1-p₂)/n₂]
= 0.65 - 0.61 / 0.0682
= 0.04/ 0.0682
=0.5865
p-value = 0.2788 > 0.05
Null hypothesis rejected at the 5% level.
Therefore, from the given proportion of the distribution it is concluded that local seniors uses drugs in higher rate than the national seniors as null hypothesis rejected .
Learn more about null hypothesis here
brainly.com/question/28920252
#SPJ4
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]
10. a group conducted a poll of 2052 likely voters just prior to an election. the results of the survey indicated that candidate a would receive 46% of the popular vote and candidate b would receive 44% of the popular vote. the margin of error was reported to be 4%. the group reported that the race was too close to call. use the concept of a confidence interval to explain what this means What does it mean to say the race was too close to call?
The difference between the two candidate voted is less than the margin of error, so the result cannot be predicted.
4% of 2052 is 4/100 x 2052 = 82.08
44% of 2052 is 902.88
46% of 2052 is 943.92
The difference between the two candidate voted is less than the margin of error
It increases as the level of confidence increases because the larger the expected proportion of intervals that will contain the parameter the larger the margin of error.
The race was too close to call means it is within the margin of error for one candidate to receive 44% of the popular vote and for the other candidate to receive 46%, the poll cannot predict the winner.
Therefore, The result cannot be predicted as the race was too close to call.
To learn more about confidence interval refer here
https://brainly.com/question/17097944
#SPJ4
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
read more about integrals on exponential integrals on brainly.com/question/29801384
#SPJ4
23.6 of what number is 45 400
Answer:
Below
Step-by-step explanation:
Not very good syntax on your post
I will assume 23.6 % ow what numer is 45,400
23.6 % = .236 in decimal
.236 * x = 45400
x = 45400/ .236 = 192,372.8814
Simplify the algebraic expression by combining like terms
(15wz^2 + 3wz^3) - 5wz^3
(Can you combine like terms if the exponents are different or do they have to be the same number?)
Answer:
15wz² - 2wz³
Step-by-step explanation:
when collecting like terms the terms must have the same variables, with variables having the same exponents.
(15wz² + 3wz³) - 5wz³ ← remove parenthesis
= 15wz² + 3wz³ - 5wz³ ← collect like terms
= 15wz² + (3wz³ - 5wz³)
= 15wz² + (- 2wz³)
= 15wz² - 2wz³
The corporate team-building event will cost $56 if it has 7 attendees. How many attendees can there be, at most, if the budget for the corporate team-building event is $80? Solve using unit rates.
Answer:
10
Step-by-step explanation:
56 ÷ 7 = 8 each.
80 ÷ 8 = 10
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.
Need help with this one
Answer:i believe its 60 with x=30
Step-by-step explanation:
Step-by-step explanation:
M(<BAC)=95° , M(<ABC)=2x , M(<ACD)=145°
let M(<ACB)=y
Then M(<ACB)+M(<ACD)=180° , Because they are in a straight line and straight line can measure 180°.
so , y+145°=180°
y=180°_145°
y=35° , M(<ACB)=35°
The sum.of the interior angle of any triangle is 180° . so now we have 95° , 38° and 2x so the value of x be
95°+35°+2x =180°
130°+2x=180°
2x=180°_130°
2x=50° , divide both sides by 2 to find the value of x
x=25° , 2*25°=50° , M(<ABC)=50°
M(<BAC)=95°, M(<ABC)=50° , M(<ACB) =35°
95°+50°+35°=180°
Find the P-value for a test of the claim that less than 50% of the people following a particular diet will experience increased energy. Of 100 randomly selected subjects who followed the diet, 47 noticed an increase in their energy level.
Answer
the answer is 0.2743 please show work
The p-value for the given information is 0.247.
The p value refers to a number which is calculated from a statistical test and describes how likely a researcher is to have found a particular set of observations if the null hypothesis were true.
According to the provided information,
Sample size, n = 100
Sample proportion p’ = 47/100 = 0.47
The null hypothesis is: H0 - p = 0.5
The alternative hypothesis is: Ha - p > 0.5
Probability to be tested p = 0.5
Standard deviation = √p(1 – p)/n = √0.5(1 - 0.5)/100 = 0.05
z-test statistics is given by
z = (p’ – p)/σ = (0.47 – 0.5)/0.05 = - 0.6
From the table,
P value = 0.247
Learn more about p value:
https://brainly.com/question/13786078
#SPJ4
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)
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.
learn more about of internet here
https://brainly.com/question/3346870
#SPJ4
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.
Learn more about the numeric values at brainly.com/question/28367050
#SPJ1
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
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
Long or short? Put the names of all the students in your statistics class in a hat. Mix up the names, and draw 4 without looking. Let X = the number whose last names have more than six letters.
This is short. And it is does not have a binomial distribution.
Given,
When a variable has only two possible outcomes and each trial is independent of the others, the binomial distribution is utilised.
Here,
Yes, there are only two possible outcomes for each kid. Their last names either have more than six letters, or they do not.
However, as you choose, the names are not put back into the hat, changing the probabilities. As a result, the trials are not independent, and the distribution is not binomial.
Learn more about binomial distribution here;
https://brainly.com/question/20709876
#SPJ4
Mason worked 48 hours last week. His hourly rate is $11.00. He has the following
deductions taken from his pay: Social Security Tax at the rate of 6.2%, Medicare tax at
the rate of 1.45%, federal income tax withheld at the rate of 10%, and 401k (retirement
savings) contributions of 5% gross pay. Match the correct amount to each number
below.
Since the amounts are not provided, we have mathematically computed the correct amounts for the deductions and gross and net pay, respectively, below.
Gross pay = $528Social Security Tax = $32.74 Medicare Tax = $7.66Federal income tax withheld = $52.80401(k) Retirement savings contributions = $26.40Net pay = $408.40.What is the net pay?The net pay is the amount earned after allowed deductions are made to the gross pay or earnings for the period.
Some of the deductions include social security, medicare, federal income tax withholding, and retirement contributions, among others.
Total hours worked for last week = 48 hours
Hourly rate of pay = $11.00
Gross pay for the week = $528 ($11 x 48)
Social Security Tax = $32.74 (6.2% of $528)
Medicare Tax = $7.66 ($528 x 1.45%)
Federal income tax withheld = $52.80 ($528 x 10%)
401(k) Retirement savings contributions = $26.40 ($528 x 5%)
Total deductions = $119.60
Net pay for the week = $408.40 ($528 - $119.60)
Learn more about the gross and net pay at https://brainly.com/question/13793671
#SPJ1
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
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.
To learn more about P-value from the given link https://brainly.com/question/4621112
#SPJ1
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.
To learn more about P-value from the given link brainly.com/question/4621112
#SPJ1
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]
To learn more about logistic model:
https://brainly.com/question/29451694
#SPJ4
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=
State if the three side lengths form an acute, obtuse or right triangle. 6km, 8km, 12km
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
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°