The amount invested in the account that earns 8% interest is $6500
The amount invested in the account that earns 12% interest is $8500
How much did she invest in each account?a + b = $15,000 equation 1
0.08a + 0.12b = $1540 equation 2
Where:
a = amount invested in the account that earns 8% interest
b = amount invested in the account that earns 12% interest
The elimination method would be used to determine the required values:
Multiply equation 1 by 0.08
0.08a + 0.08b = 1200 equation 3
Subtract equation 3 from equation 2
0.04b = 340
Divide both sides of the equation by 0.04
b = 340 / 0.04
b = $8500
a + 8500 = 15,000
a = $15,000 - $8,500
a = $6500
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a. Use the points (1, 226.9) and (4, 275.2) to write an equation for the line of fit in slope-intercept form, where x is the number of years since 2010 and y is the median price in thousands of dollars. what will be the approximate price by 2025?
The Equation of line is y= 16.1x + 210. 8.
The approximate price by 2025 is 452.3.
What is Slope?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,
m = Δy/Δx where, m is the slope
Given:
The points are (1, 226.9) and (4, 275.2).
So, Slope for line
m = ( 275.2- 226.9)/( 4-1)
m = 48.3/ 3
m= 16.1
Now, using the slope- intercept form
y= mx+ b
y = 16.1x + b
Now, x= 1 and y= 226.9
By, y = 16.1 + x
226.9 = 16.1(1)+ b
b= 210.8
So, the equation of line is y= 16.1x + 210. 8
Now, x is the number of years since 2010 and by 2025
x= 2025 - 2010 = 15
Then, the approximate price by 2025
y= 16.1(15) + 210.8
y= 452.3
Hence, the approximate price by 2025 is 452.3.
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Under certain conditions, the number of feet a car will skid after the brakes are fully applied is given by the function d(x)=0.05x2+x, where x is the speed of the car in miles per hour. For what values of x will the car skid at most 120ft ?
Answer:
x ≤ 40 . . . miles per hour
Step-by-step explanation:
You want to know the values of x for which d(x) = 0.05x² +x is at most 120.
Solutiond(x) ≤ 120 . . . . . . . . the required limit
0.05x² +x ≤ 120 . . . . . substitute the expression for d(x)
x² +20x -2400 ≤ 0 . . . . . multiply by 20, subtract 2400
(x -40)(x +60) ≤ 0 . . . . . . . . factor. Factors are zero for x=40, x=-60.
The product will be negative for x-values between the zeros:
-60 ≤ x ≤ 40
We only care about positive values of x:
0 ≤ x ≤ 40 . . . . . miles per hour
To estimate the number of bass in a lake, a biologist catches and tags 32 bass. Several weeks later, the biologist catches a new sample of 55 bass and finds that 5 are tagged. How many bass are in the lake.
NO LINKS!!!
The model t = 16.708 ln(x/(x-705)) approximates the term of a mortgage of $150,000 at 6% interest rate, where t is the term of the mortgage in years and x is the monthly payment plan in dollars.
a. Approximate in terms (in yr) of a $150,000 mortgage at 6% when the monthly is $897.72 and when the monthly payment is $1526.49 (Round your answers to the nearest whole number).
$897.72 ________ yr
$1526.49_________ yr
b. Approximate the total amounts paid (in dollars) over the term of the mortgage with a monthly payment plan of $897.72 and with a monthly payment plan of $1526.49. (Round your answers to 2 decimal places.)
$897.72 $_____________
$1526.49 $_____________
What is the amount of the total is interest costs (in dollars) in each case? (Round your answers to 2 decimal places)
$897.72 $__________
$1526.49 $__________
c. What is the vertical asymptote for the model? ____________
Interpret its meaning in the context of the problem.
The monthly payment must be (more or less) than $________, and the close to this value is, the (quicker you will be able or longer it will take) to pay off the mortgage.
Answer:
(a) $897.72: 26 yr
$1526.49: 10 yr
(b) See below.
(c) x = 705
more, $705, longer it will take
Step-by-step explanation:
Given equation:
[tex]t=16.708 \ln \left(\dfrac{x}{x-705}\right)[/tex]
where:
t = term of the mortgage (in years)x = monthly payment plan (in dollars)Part (a)[tex]\begin{aligned}x=897.72 \implies t& =16.708 \ln \left(\dfrac{897.72}{897.72-705}\right)\\& =16.708 \ln \left(4.65815691...\right)\\&=16.708(1.53861985...)\\&=25.70726...\\&=26\; \rm years\end{aligned}[/tex]
[tex]\begin{aligned}x=1526.49 \implies t& =16.708 \ln \left(\dfrac{1526.49}{1526.49-705}\right)\\& =16.708 \ln \left(1.85819669...\right)\\& =16.708(0.619606496...)\\&=10.352385...\\&=10 \; \rm years\end{aligned}[/tex]
Part (b)To approximate the total amounts paid (in dollars) over the term of the mortgage, multiply the monthly payment by the term.
Please note I have provided two calculations per monthly payment:
(1) by using the exact term, and (2) using the rounded term from part (a).
[tex]\implies \$897.72 \times 25.7072605... \times 12 =\$276935.06[/tex]
[tex]\implies \$897.72 \times 26 \times 12=\$280088.64[/tex]
[tex]\implies \$1526.49 \times 10.3523853... \times 12=\$189633.75[/tex]
[tex]\implies \$1526.49 \times 10 \times 12 =\$183178.80[/tex]
To calculate the amount of interest costs (in dollars) in each case, subtract $150,000 from the total amounts paid:
[tex]\$897.72\implies 276935.06-150000=\$126935.06[/tex]
[tex]\$897.72 \implies 280088.64-150000=\$130088.64[/tex]
[tex]\$1526.49 \implies 189633.75-150000=\$39633.75[/tex]
[tex]\$1526.49\implies 183178.80-150000=\$33178.80[/tex]
Part (c)The natural logarithm of a negative number cannot be taken.
Therefore, x > 705.
So the vertical asymptote for the model is:
x = 705The monthly payment must be more than $705, and the closer to this value the payment is, the longer it will take to pay off the mortgage.
Select the correct answer from each drop-down menu.
The function f(x) = 500(1+004 models the balance in a savings account.
The savings account had an initial balance of
$500
$515
and compounds
Reset
Next
at an interest rate of
Answer:
Initial balance 500, compounds 4 times at an interest rate of 15%
Step-by-step explanation:
Initial balance 500, compounds 4 times at an interest rate of 15%
Kelvin spends 1/4 of his $100 on a ticket to a play and then buys a shirt for $22.50. How much
of the $100 does Kelvin have left?
A $5.50
B $10.00
C $52.50
D $73.50
need answer right now
Suppose f left parenthesis x right parenthesis equals 1 minus x squared and g left parenthesis x right parenthesis equals 2 x plus 5. Find the value of f(g(-1).
Answer:
f(g(-1)) = -8
Step-by-step explanation:
⭐ The two equations we are given are:
[tex]f(x) = 1-x^2[/tex][tex]g(x) = 2x + 5[/tex]This problem is an example of composite functions.
⭐What are composite functions?
Composite functions are functions inside of a functionYou compute the value of an x-value for one function, and use that value for another function.First, we have to compute the value for the 2nd function you see. The 2nd function is inside f(x), and it is g(-1).
Essentially, we have to find the corresponding y-value for when x = -1 in g(x)
[tex]g(-1) = 2(-1) + 5[/tex]
[tex]g(-1) = 3[/tex]
Next, take this value and substitute it as the x-value for f(x).
[tex]f(3) = 1-x^2[/tex]
Now we have to find the y-value for f(x) for when x = 3.
[tex]f(3) = 1 -3^2[/tex]
⚠️⚠️⚠️!!! CAUTION !!! ⚠️⚠️⚠️
Some people may make the mistake of computing f(3) like this:
[tex]f(3) = 1 - 3^2[/tex]
[tex]f(3) = 1+9\\f(3) = 10[/tex]
This is wrong because only 3 is being squared, not -3. Make sure to read the equations you are given carefully to avoid mistakes like this.
[tex]f(3) = 1-3^2\\f(3) = 1-9\\f(3) = -8[/tex]
∴ f(g(-1)) = -8
Please help me, I need to get this now
Answer:
see the attachment for a graph
Step-by-step explanation:
You seem to want a graph of the inequalities ...
y ≤ -3/2x -2y > 1/2x +6Boundary linesThe equations of the boundary lines are given by replacing the inequality symbol with an equal sign:
y = -3/2x -2y = 1/2x +6These are graphed in the usual way. It is often convenient to find the y-intercept, then use the slope to locate other points on the line.
y = -3/2x -2
The y-intercept is (0, -2). The line has a rise/run of -3/2, so goes down 3 units for each 2 to the right. Another point on the line is (2, -5).
y = 1/2x +6
The y-intercept is (0, 6). The line has a rise/run of 1/2, so goes up 1 unit for each 2 to the right. Another point on the line is (2, 7).
Line typeWhen the "or equal to" symbol (≤ or ≥) is used, the boundary line is solid. When the "or equal to" case is not part of the solution, the boundary line is dashed.
The line with negative slope through (0, -2) is solid; the line with positive slope through (0, 6) is dashed.
ShadingAll you need to determine shading is a variable with a positive coefficient, and the inequality symbol.
y ≤ . . . . . tells you shading is below the solid line
y > . . . . . tells you shading is above the dashed line
We notice x has a positive coefficient in the second inequality, so we could determine shading from ...
> x . . . . . tells you shading is left of the line (where x values are less than those on the line)
Of course, you can rearrange the inequality so a variable of interest has a positive coefficient. For example, we could add 3/2x to the first inequality to get ...
3/2x + y ≤ -2
Then, looking at the x-variable, we see ...
x ≤ . . . . . tells you shading is to the left of the line
Answer:
See attachment.
Step-by-step explanation:
When graphing inequalities:
< or > : draw a dashed line.≤ or ≥ : draw a solid line.< or ≤ : shade under the line.> or ≥ : shade above the line.Treat the inequalities as equations (swap the inequality sign for an equals sign) to find two points on the line to help draw the lines.
Inequality 1
[tex]y \leq -\dfrac{3}{2}x-2[/tex]
[tex]\begin{aligned}x=0 \implies y&=-\dfrac{3}{2}(0)-2\\y&=-2\end{aligned}[/tex]
[tex]\begin{aligned}x=2 \implies y&=-\dfrac{3}{2}(2)-2\\y&=-5\end{aligned}[/tex]
Plot the points (0, -2) and (2, -5).
Draw a solid straight line through the points.
Shade under the line.
Inequality 2
[tex]y > \dfrac{1}{2}x+6[/tex]
[tex]\begin{aligned}x=0 \implies y&=\dfrac{1}{2}(0)+6\\y&=6\end{aligned}[/tex]
[tex]\begin{aligned}x=4 \implies y&=\dfrac{1}{2}(4)+6\\y&=8\end{aligned}[/tex]
Plot the points (0, 6) and (4, 8).
Draw a dashed straight line through the points.
Shade above the line.
The solution to the two inequalities is the overlap of the shaded parts.
Using the definitional formula, compute Ss, variance and the standard deviation for the following population of scores. Scores: 1, 8, 0, 4,2 SS 40 variance 10 standard deviation 3.16
Ss, variance and the standard deviation for the population of scores is 2.83 and 8 respectively
What is standard deviation and variance?
Standard deviation is a measure of the distribution of statistical data, whereas variance is a measure of how data points differ from the mean. The fundamental distinction between the two is that although the variance is expressed in squared units, the standard deviation is expressed in the same units as the data's mean.
Given,
SS = 40
Variance = 10
Standard Deviation = 3.16
Using Population Variance Formula :
= [tex]\frac{1}{n}[/tex]∑( x-xbar)² where i is 1
For the complete solution see the attachment of table
sum ( x - x bar)² = 40
so, variance = 40/5
= 8
standard deviation = [tex]\sqrt{8}[/tex]
= 2.83
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A small town has two local high schools. High School A currently has 700 students and is projected to grow by 65 students each year. High School B currently has 900 students and is projected to grow by 40 students each year. Let A(t) represent the number of students in High School A in t years, and let B(t) represent the number of students in High School B after t years. Write the equation for each function and determine how many students would be in each high school in the year they are projected to have the same number of students. A(t) = B(t) =
Answer:
1. A(t) = 65t + 700
2.B(t) = 40t + 900
3. High School A and High School B will both have 1,220 students in the 8th year
Step-by-step explanation:
1. The equation for the number of students in High School A represents a linear function.
⭐ What is a linear function?
A linear function is a type of equation where every y-value increases by a constant, additive amountOne way to write the equation for a linear function is: [tex]y = mx+b[/tex], where m is the constant, additive amount, and b is the y-intercept, or the initial value.Let's write the equation for High School A in the [tex]y = mx+b[/tex] format, known as slope-intercept form:
The constant, additive amount for High School A is 65 (m)The initial value for High School A is 700 (b)∴ High School A: [tex]y = 65x + 700[/tex]
2. Let's write the equation for High School B in the [tex]y = mx + b[/tex] format, known as slope-intercept form:
The constant, additive amount for High School B is 40 (m)The initial value for High School B is 900 (b)∴ High School B: [tex]y = 40x + 900[/tex]
3. To find at what year High School A and High School B will have the same number of students, we need to solve a system of linear equations.
⭐What is a system of linear equations?
A system of linear equations is two or more linear equations that intersect at one point (x,y)For this problem, let's set both linear equations equal to each other to see at what point will the high school populations be the same.
[tex]A(t) = B(t)[/tex]
[tex]65t + 700 = 40t + 900[/tex]
[tex]25t = 200[/tex]
[tex]t = 8[/tex]
Now we know that in the 8th year, High School A will have the same population as High School B.
We need to find what the population will be in year 8.
Thus, substitute the value of t into one of the functions and solve.
I am choosing to substitute t into A(t), but you can also do B(t).
[tex]A(8) = 65(8) + 700[/tex]
[tex]A(8) = 520 + 700[/tex]
[tex]A(8) = 1,220[/tex]
⚠️!!! CAUTION !!! ⚠️
Some people may stop at this point and write that in the 8th year, both high schools will have a population of 1,220 students.
However, you should also substitute 8 into the other function you didn't substitute it into to make sure that 8 is correct.
[tex]B(8) = 40(8) + 900[/tex]
[tex]B(8) = 320 + 900[/tex]
[tex]B(8) = 1,220[/tex]
∴ In the 8th year, both high schools will have a population of 1,220 students.
how many 3 digit multiple of 11 ends in 2
Answer:
22
There are 22 multiples that end in 22.
Below is the table of values of a function. Write the output when the input is n.
input 1,4,6,n
output 2,8,12,?
Answer:
Step-by-step explanation:
What is the maximum height of the firework? How long is the firework in the air before it explodes?
See attached picture
The maximum height of the firework is; 387.78 ft
The time for which the firework is in the air before it explodes is; 1 second
How to solve projectile equations?
We are given the function;
h = -(500/9)t² + (1000/3)t + 10
where;
h is height after time of t seconds
At height of zero, the firework would either be about to launch or when it has come down.
Thus, let us set h = 0 to find the times.
0 = -(500/9)t² + (1000/3)t + 10
Using quadratic equation calculator, we have;
t ≈ 2 seconds
Now the firework explodes at the highest point which will be the mid point of where the launching started and the endpoint where it stopped.
This means time at midpoint = (0 + 2)/2 = 1 sec
It explodes after 1 second.
Thus maximum height at this time is;
h = -(500/9)(1)² + (1000/3)(1) + 10
h = 387.78 ft
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Rain is flowing into 2 containers at different rates. The figure below shows the volume of water in each container what is the difference in rate of change between the two containers
With the help of Linear relationships The difference in the rate of change between the two containers is 1/5 gallon per minute.
what is Linear relationships?
A statistical word used to express a straight-line relationship between two variables is a linear relationship (or linear association). Graphs and mathematical equations of the form y = mx + b can both be used to represent linear relationships. In everyday life, linear relationships are quite prevalent.
Given:
Volume of water in each container.
To find:
Difference in the rate of change.
Solution:
Take any two points on container 1.
Let the points are (10, 2) and (20, 4).
m = (y2 - y1) / ( x2 - x1 )
= ( 4 - 2 ) / ( 20 - 10 )
= 2/ 10
= 1/ 5
Rate of change for container 1 is .
Take any two points on container 2.
Let the points are (5, 2) and (10, 4).
m = (y2 - y1) / ( x2 - x1 )
= ( 4 - 2 ) / ( 10 - 5)
= 2 / 5
Rate of change for container 2 is 2/5
Difference = (2 /5 ) - ( 1/ 5 )
= 1 / 5
Hence, The difference in the rate of change between the two containers is 1/5 gallon per minute.
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What are the values of t and u?
t = ?°
u= ?°
18) What is the slope of the line that contains points (–6, –6) and (–3, 1)?
The slope of the line is 7/9
How to determine the slope of the lineIt is important to note that the equation of a line is represented as;
y = mx + c
Where;
y is a point on the linem is the slope of the linex is a point on the x - axisc is the intercept of the y-axisThe formula for calculating the slope of a line is expressed as;
Slope, m = y₂ - y₁/x₂ - x₁
Now, let's substitute the values into the formula from the points given we have;
Slope, m =1 -(-6)/ -3 - (-6)
expand the bracket
Slope, m = 1 + 6/ 3 + 6
add the values
Slope, m = 7/9
Hence, the value is 7/9
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8. Sketch a model to represent the equation
3x - 5 = 10. Then, solve the equation.
Answer:
x=5
Step-by-step explanation:
(Not sure how I was supposed to do the model..so I ended up doing it this way..)
3x-5=10
Add 5 on both sides and then you get
3x=15
divide both sides by 3 and then you get
x=5
(Hopefully you can see my model even if it's just a little...)
For the model I drew 3 rectangles and put an x inside each one of them. for the -5, I drew 5 squares with a minus symbol/sign inside. And then I put an equal sign beside and I drew 10 squares with nothing inside but you can draw positive signs in it to show that it's positive..
So basically mines was
3 rectangles with x inside -5 squares with - inside = 10 squares
Marijuana legalization: In a Public Policy Institute of California (PPIC) poll, 53% of 1,706 California adult residents surveyed say that marijuana 10 points should be legal. Based on the results, the 95% confidence interval is (0.506, 0.554). Which of the following is an appropriate interpretation of this confidence interval? A. We are 95% confident that between 50.6% and 55.4% of California residents say that marijuana should be legal. B. We can conclude that 95% of states have 50.6% to 55.4% of adult residents who say that marijuana should be legal. C. We are 95% confident that between 50.6% and 55.4% of all American adults say that marijuana should be legal. D. If we took many samples of adults from around the nation, between 50.6% and 55.4% of them would say that marijuana use should be legal.
The correct interpretation of the confidence interval is: "We are 95% confident that between 50.6% and 55.4% of California adult residents say that marijuana should be legal." (Option A)
The confidence interval represents the range of values within which we are 95% confident that the true population parameter (in this case, the percentage of California adult residents who say that marijuana should be legal) lies. It is specific to the population of California adult residents that was surveyed, and cannot be generalized to other states or to all American adults.
Therefore, Option A is the correct answer that we are 95% confident that between 50.6% and 55.4% of California adult residents say that marijuana should be legal.
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The confidence interval denotes the range of values within which we are 95% certain that the true population parameter (in this case, the proportion of adult Californians who support marijuana legalization) resides. It cannot be extrapolated to adults in other states or to all Americans as a whole because it is specific to the demographic of adult inhabitants of California who were polled.
Accordingly, Option A is the right response, and we are 95% certain that between 50.6% and 55.4% of adult inhabitants of California agree that marijuana should be legal.
Kyra is blocking off several rooms in a hotel for guests coming to her wedding. The hotel can reserve large rooms that can hold 8 people, and small rooms that can hold 6 people. Kyra reserved twice as many large rooms as small rooms, which altogether can accommodate 88 guests. Determine the number of small rooms reserved and the number of large rooms reserved.
The number of small rooms reserved is 4.
The number of large rooms reserved is 8.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
We have,
Small rooms are denoted as S.
Large rooms are denoted as L.
Now,
We will make two equations.
L = 2S _____(1)
8L + 6S = 88 ______(2)
Putting (1) in (2) we get,
8L + 6S = 88
8 x (2S) + 6S = 88
16S + 6S = 88
22S = 88
S = 88/22
S = 8/2
S = 4
Now,
Putting S = 4 in (1) we get,
L = 2 x 4
L = 8
Thus,
The number of small rooms and large rooms reserved is 4 and 8.
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Use the points A(1,5), B(1,-4), C(-3,5), D(-3,-4)
Pls lmk by Thursday
Using the points A(1,5), B(1,-4), C(-3,5), D(-3,-4) the following are matched:
Distance between points B and C = √97.
Distance between points A and B = 9.
Midpoint between points B and C = (-1, 1/2).
Midpoint between C and D = (-3, 1/2).
Slope of the line through B and D = 0.
What is the distance between two points of a line?The distance between two points of a line is given as:
Distance = [tex]\sqrt{c - a)^2 + (d - b)^2}[/tex]
We have,
A(1,5), B(1,-4), C(-3,5) and D(-3,-4).
The distance between A and B.
= [tex]\sqrt{(1 - 1)^2 + (-4 - 5)^2}[/tex]
= √81
= 9
The distance between B and C.
= [tex]\sqrt{(-3 - 1)^2 + (5 + 4)^2}[/tex]
= √(16 + 81)
= √97
The midpoint (x, y) between B and C.
x = (1 - 3)/2 = -1
y = (-4 + 5) / 2 = 1/2
The midpoint (x, y) between C and D.
x = (-3 -3) / 2 = -6/2 = -3
y = (5 - 4) / 2 = 1/2
The slope of the line through points B and D.
= (-4 + 4) / (-3 - 1)
= 0/ -4
= 0
Thus,
The distance between A and B is 9.
The distance between B and C is √97.
The midpoint between B and C is (-1, 1/2).
The midpoint between C and D is (-3, 1/2).
The slope of the line through B and D is 0.
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In response to nutrition concerns raised last year about food served in school cafeterias, the Smallville School District entered into a one-year contract with the Healthy Alternative Meals (HAM) company. Under this contract, the company plans and prepares meals for 2,500 elementary, middle, and high school students, with a focus on good nutrition. The school administration would like to survey the students in the district to estimate the proportion of students who are satisfied with the food under this contract. Two sampling plans for selecting the students to be surveyed are under consideration by the administration. One plan is to take a simple random sample of students in the district and then survey those students. The other plan is to take a stratified random sample of students in the district and then survey those students. (a) Describe a simple random sampling procedure that the administrators could use to select 200 students from the 2,500 students in the district. (b) If a stratified random sampling procedure is used, give one example of an effective variable on which to stratify in this survey. Explain your reasoning>
The students corresponding to those 200 numbers would beasked to participate in the survey.
What is corresponding ?
Having or participating in the same relationship (such as kind, degree, position, correspondence, or function) especially with regard to the same or like wholes (such as geometric figures or sets) corresponding parts of similar triangles. : related, accompanying.
Have given,
a) The administrators could number an alphabetical list of students from 1 to 2,500.Then use a random number generator from a calculator or computer to generate 200 unique random numbers from 1 to2,500.(if you don’t say “unique”, then say “ignoring repeats”)The students corresponding to those 200 numbers would beasked to participate in the survey.
b) One advantage is that stratified random sampling guarantees that each of the school-level strata will have some representation, because it is possible that a simple random sample would miss one or more of the strata completely.(NOTE: It is NOT sufficient to say that a stratified sample “makes it MORELIKELY that all school-levels are represented in the sample”… the student must state that stratifying by school level GUARANTEES or ENSURES that all school levels are represented in the sample.)
The students corresponding to those 200 numbers would beasked to participate in the survey.
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Given a metric space M with metric d, verify that any ε-ball is an open set.
Answer:
Prove that for any x0∈X and any r>0, the open ball Br(xo) is open. My attempt: Let y∈Br(x0). By definition, d(y,x0)<r
Step-by-step explanation:
Prove that for any x0∈X and any r>0, the open ball Br(xo) is open. My attempt: Let y∈Br(x0). By definition, d(y,x0)<r
4 times the difference of x and 2
Answer: 2x-4
Step-by-step explanation:
Answer:
solution
you must multiply a difference of x and 2 which is. >{x-2}
mathematical
4 x {x-2}
What sides are similar to this triangle with the side dimensions of 6cm, 12cm, and 15cm?
Answer: Pythagorean triplets.
Step-by-step explanation:
not sure but this is the triangle that helps at all.
(20points) Let A be a 4 x 4 matrix and let A be a eigenvalue of multiplicity 3. If A - AI has rank 1, is A defective? Explain.
No, A is not defective.
What is the rank nullity theorem?The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its nullity (the dimension of its kernel).
Given here A be a 4 x 4 matrix and let λ be an eigenvalue of multiplicity 3 and rank 1, then by rank nullity theorem we have
Rank(A-λI) + Nullity((A-λI) =4
⇒dimN(N-λI) = Nullity(A-λI)
⇒dimN(N-λI) = 4 - Rank(A-λI)
⇒dimN(N-λI) = 4-1
⇒dimN(N-λI) = 3
Therefore 3 linearly independent eigenvectors belong to eigenvalue λ. Since λ is an eigenvalue of multiplicity 3, this ensures that there is a different eigenvalue different from λ. Along with 3 linearly independent eigenvectors in the eigenspace of λ, this gives us 4 linearly independent eigenvectors in the matrix A. Now we An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Thus A is nondefective
Hence, matrix A is not defective.
Learn more about the rank nullity theorem here:
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let
[tex]A=(\sqrt{2} +\sqrt{3} )(\sqrt{4} +\sqrt{5} )...(\sqrt{2020} +\sqrt{2021} )\\B=(\sqrt{2021} -\sqrt{2020} )(\sqrt{2019} -\sqrt{2018} )...(\sqrt{3} -\sqrt{2} )[/tex]
what is A times B?
Answer:
1
Step-by-step explanation:
A)
[tex]( \sqrt{3 } - \sqrt{2} )( \sqrt{3} + \sqrt{2} )( \sqrt{5 } - \sqrt{4} )( \sqrt{5} + \sqrt{4} )....[/tex]
using difference of two squares
[tex]( \sqrt{a} + \sqrt{b} )(\sqrt{a} - \sqrt{b} ) = a - b[/tex]
we can rewrite the A) to 1×1×1×1.... so the answer would be 1
The career placement office at a large university tracks the number of job applications graduating seniors complete before getting their first job. Data on college graduates suggests that most will send out a large number of applications before getting their first job. Which one of the following histograms best represents the distribution of job applications for graduating seniors?
a.Histogram III
b.Histogram II
c.Histogram IV
d.Histogram I
The histogram that best represents the distribution of job applications for graduating seniors is given as follows:
a. Histogram III.
How to identify the correct histogram?An histogram shows the number of times that each observation appears in a data-set.
Data on college graduates suggests that most will send out a large number of applications before getting their first job, hence the left bins on the histogram should have small numbers, while the right bins on the histogram should have greater numbers.
The only histogram that follows this pattern is Histogram III, hence option A gives the histogram that best represents the distribution of job applications for graduating seniors.
Missing InformationThe histograms are given by the image shown at the end of the answer.
More can be learned about histograms at https://brainly.com/question/25983327
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What is the solution of the system? Use elimination.
3x +9y=33
-10x-6y=-14
a. (-4,5)
b. (4,-1)
c. (-10, 3)
d. (-1,4)
To solve the given system of equations using elimination, we can add the two equations together to eliminate one of the variables. Doing this, we get 3x + 9y + (-10x - 6y) = 33 + (-14), which simplifies to -7x + 3y = 19. We can then solve for y by dividing both sides of the equation by 3, giving us y = 19/3 = 6.33. We can substitute this value for y in either of the original equations to solve for x. For example, if we substitute 6.33 for y in the first equation, we get 3x + 9 * 6.33 = 33, which simplifies to 3x = 3.67. Dividing both sides of the equation by 3, we get x = 1.22. Therefore, the solution of the system is (1.22, 6.33), which is approximately (1, 6). The answer choice that is closest to this solution is (d) (-1, 4). Thus, the solution of the system is (-1, 4).
If a share of a stock jumped from $9 to $25, what was the percent increase of this stock (rounded to the
nearest tenth)?
Answer: 177.8
Step-by-step explanation: To find the percent increase, we have to use this formula:
New value-old value/ old value x 100
So, following this formula, we would do 25-9, which, is 16, divided by 9. That would be 1.777... So now let's round that to 1.78, so 1.78 x 100 = 177.8. I hope this helped.
4x+6y=8
y = -2/3x+1/3
how do i graph this and what is the solution please help it’s urgent