The eigenvalues of L are λ = 4, -1, and 1.
The eigenvectors associated with λ = 1 are of the form v = [ 1, 0, -1 ] where y is any real number.
To find the eigenvalues and eigenvectors of L, we need to solve the equation LM = ML, where M is the matrix representing L with respect to the basis (t2 + 1, t, 1). We can rewrite this equation as (L - λI)M = 0, where λ is an eigenvalue of L and I is the identity matrix.
Let's solve for the eigenvalues first. We have:
(L - λI)M =[tex]\begin{bmatrix}-1 & -\lambda & 0 \\-1 &0 &1 &1 \\ 1& 0 &1 \\-1 & -\lambda &0 \\\end{bmatrix}[/tex]
[tex]\begin{bmatrix} 0&-\lambda &0 \\ 0 & 0& 0\end{bmatrix} = \begin{bmatrix} 0&0 &0 \\ 0 &-\lambda & 0\end{bmatrix}[/tex]
Expanding the matrix product, we get:
[tex]= > [ (-1-\lambda)(-1) + 2(2)(1-\lambda) 0 (-1-\lambda)(1) + 2(1)(1-\lambda) ] \times [ 0 (-\lambda)(0) 0 ][/tex]
Simplifying the expressions, we obtain:
[tex]\begin{bmatrix}\lambda^2-3\lambda-4 & 0 &3\lambda - 2 \\ 0& 0 &0 \\ 2\lambda - 2 & 0 &\lambda-1 \end{bmatrix}[/tex]
To find the eigenvalues, we need to solve the characteristic equation det(L - λI) = 0. We have:
det(L - λI) = (λ² - 3λ - 4)(λ - 1)
= (λ - 4)(λ + 1)(λ - 1)
Simplifying the equations, we get:
-5x + z = 0
-4y = 0
2x - 3z = 0
From the second equation, we get y = 0. Substituting this into the first and third equations, we get:
-5x + z = 0
2x - 3z = 0
Solving for x and z, we obtain:
x = z/5
z = 2x/3
Therefore, the eigenvectors associated with λ = 4 are of the form v = [ x, 0, z ], where x = z/5 and z = 2x/3. We can choose x = 5 and z = 10/3 to obtain a specific eigenvector:
v = [ 5, 0, 10/3 ]
Similarly, we can find the eigenvectors associated with λ = -1 and λ = 1. The eigenvectors associated with λ = -1 are of the form v = [ x, 0, y ], where x = y/5. Choosing y = 5, we obtain the eigenvector:
v = [ 1, 0, 5 ]
The eigenvectors associated with λ = 1 are of the form v = [ x, y, z ], where x + z = 0. Choosing x = 1 and z = -1, we obtain the eigenvector:
v = [ 1, y, -1 ]
We can choose y = 0 to obtain a specific eigenvector:
v = [ 1, 0, -1 ]
To know more about eigenvectors here
https://brainly.com/question/31013028
#SPJ4
A teacher wants to know whether their course helps students on the SAT. The hypothesized population mean for SAT scores is 500. The standard deviation of the population is 77. The sample size is 100. The sample mean for students who took the course is 533. What is the z-score
The z-score is -0.428
What is a z-score?A z-score, also known as a standard score, provides information on how far a data point is from the mean. Technically speaking, however, it's a measurement of how many standard deviations a raw score is from or above the population mean.
You can plot a z-score on a normal distribution curve.
You must be aware of the mean and population standard deviation to use a z-score.
Z-scores allow results to be compared to a "normal" population.
According to the question,
x=500
μ=533
σ=77
z-score=(x-μ)/σ
On substituting the values,
z-score=(500-533)/77
= -0.428
Learn more about z-scores here:
https://brainly.com/question/25638875
#SPJ4
pls help ILL MARK BRAINLIEST
Answer:
a) Height of the ledge is 1.6 m.
b) roots of the equation are: [tex]\bf d = 2[/tex] , [tex]\bf d = -1[/tex]
c) Maximum height reached = 1.8 m.
Step-by-step explanation:
a) The height of the ledge is the same as the height at which Jake is when he hasn't moved any horizontal distance from the ledge yet, that is, when d = 0:
[tex]h = -0.8(0)^2 + 0.8(0) + 1.6[/tex]
⇒ [tex]\bf h = 1.6 \space\ m[/tex]
∴ Height of the ledge is 1.6 m.
b) The x-intercepts occur where y = 0, that is when h = 0:
[tex]-0.8d^2 + 0.8d + 1.6 = 0[/tex]
Divide both sides of the equation by -0.8:
[tex]d^2 - d -2 = 0[/tex]
Factorizing:
⇒ [tex]d^2 -2d + d - 2 = 0[/tex]
⇒ [tex]d(d-2) +1 (d-2) = 0[/tex]
⇒ [tex](d-2)(d+1) = 0[/tex]
⇒ [tex]d - 2 = 0[/tex] or [tex]d + 1 = 0[/tex]
∴ roots are : [tex]\bf d = 2[/tex] , [tex]\bf d = -1[/tex]
c) To find the maximum value of a quadratic equation in the form
y = ax² + bx + c , use the formula:
max = c - (b² / 4a).
Using the formula for [tex]h = -0.8x^2 + 0.8x + 1.6[/tex] :
max h = [tex]1.6 - ( \frac{0.8^2}{4(-0.8)} )[/tex]
= 1.8
∴ Maximum height reached = 1.8 m.
Answer:
a) 1.6 m
b) -1, 2 meters
c) 1.8 m
Step-by-step explanation:
Apparently, technology tools are allowed when solving this problem. They readily show you the x- and y-intercepts and the vertex of the graph.
a) ledge heightThe height of the ledge is the value of h when d=0. It is the constant in the given equation, and the y-intercept of the graph.
The height of the ledge is 1.6 meters.
b) rootsThe x-intercepts are the values of d that make h equal to zero. The graph shows them to be ...
d = -1
d = 2 . . . . meters
c) maximum heightThe maximum height is the "h" coordinate of the vertex (d, h).
The maximum height is 1.8 meters.
__
Additional comment
"x" is the generic independent variable. The horizontal axis of a graph is often called the "x-axis" and places where the graph crosses that axis are called "x-intercepts" even when the independent variable is named something else. Here, the independent variable is "d", not "x".
Similarly, "y" is the generic dependent variable, and the vertical axis of a graph is often called the "y-axis" even when the dependent variable is something else. This is why we refer to h when d=0 as the "y-intercept". It is the point where the graph crosses the vertical axis.
Fill in the blank: In data analytics, a _____ refers to all possible data values in a certain dataset
In data analytics, a population refers to all possible data values in a certain dataset
Data analysis is a systematic computer-aided analysis of data or statistics. [1] It is used to discover, interpret, and convey meaningful patterns in the data. It also includes applying data patterns for effective decision-making.
It may be useful in areas where there is a lot of recorded information. The analysis relies on the simultaneous application of statistics, computer programming, and operations research to quantify performance.
Organizations can apply analytics to business data to describe, predict, and improve business performance. Areas within the analysis include descriptive analysis, diagnostic analysis, predictive analysis, prescriptive analysis, and cognitive analysis in particular. [2] Analysis can be applied in various fields such as marketing, management, finance, online systems, information security, and software services.
Learn more about the population here: https://brainly.com/question/25630111
#SPJ4
which tree diagram shows all the possible outcomes for 2 coin flips?
Answer:
c
Step-by-step explanation:
im smart like that
What is the value of f(16) - f(0) when f(x) = 4x - 8?
16
48
56
64
Answer:
64
Step-by-step explanation:
evaluate by substituting x = 16 and x = 0 into f(x)
f(16) = 4(16) - 8 = 64 - 8 = 56
f(0) = 4(0) - 8 = 0 - 8 = - 8
then
f(16) - f(0) = 56 - (- 8) = 56 + 8 = 64
Answer:
64
Step-by-step explanation:
f(x) = 4x - 8
First find f(16)
f(16) = 4(16) -8 =64-8=56
Then find f(0) = 4)0) -8 = -8
f(16) - f(0) = 56 - (-8) = 56+8 = 64
Graph the following exponential functions and determine the y-intercept, domain, range, and
asymptote of each function.
f(x)=5(2)^x
The graph of the exponential function f(x) = 5(2)ˣ is as shown in the attached file.
How to draw the graph of an exponential Function?We want to draw the graph of the exponential function;
f(x) = 5(2)ˣ
At input of x = 0, we have;
f(x) = 5(2)⁰ = 5
At input of x = 1, we have;
f(x) = 5(2)¹ = 10
At x = -1, we have;
f(x) = 5(2)⁻¹ = 2.5
At x = -2, we have;
f(x) = 5(2)⁻² = 1.25
Read more about Graph of Exponential Function at; https://brainly.com/question/12940982
#SPJ1
Can someone please tell me which one is correct??? The first screenshot is the question and the second screenshot is the options I have. I am in dire need of help
Answer:
43.5 unit^2.
Step-by-step explanation:
Area of the triangle
= 1/2 * base * height
= 1/2 * AB * AC.
AB = √ [(6-1)^2 + (6-8)^2] = √29
AC = √ [(8 - (-7))^2 + (1 - (-5)^2] = √261
So the area = 1/2 * √29 * √261
= 43.5
CAN ANYONE HELP ME WITH QUESTION 6
jeri is three years younger than laura, whose age is x. How old is jeri?
Answer:
x-3
Step-by-step explanation:
Laura's age goes first. You don't know what it is, so you have to leave it as x. You know Jeri is 3 years younger, so you subtract from x.
What is the denominator of the simplified expression? (startfraction 2 n over 6 n + 4 endfraction) (startfraction 3 n + 2 over 3 n minus 2 endfraction) 2 n 3n+2 3n-2
n/3n - 2.
The denominator of the given simplified expression is
The given expression is ( 2n/6n + 4 ) ( 3n + 2/ 3n - 2 )
To find the denominator of the simplified expression :
First, simplify the given expression as below ;
( 2n/6n + 4 ) ( 3n + 2/ 3n - 2 )
( 2n/2₍₃ₙ ₊ ₂₎ ( 3n + 2/ 3n - 2 ) ( here by taking the common term 2 outside the factor )
= (n/1) ( 1/ 3n - 2 ) ( here by simplifying the factors )
= n/3n - 2.
Therefore, ( 2n/6n + 4 ) ( 3n + 2/ 3n - 2 ) = n/3n - 2
Therefore the result of the given simplified expression is = n/3n - 2.
Learn more about the simplified expression at
https://brainly.com/question/723406
#SPJ4
Jacob is cutting a tile in the shape of a parallelogram. Two opposite angles have measures of (6n − 70)° and (2n + 10)°.
What are the two different angle measures of the parallelogram-shaped tile?
20° and 160°
50° and 130°
30° and 150°
70° and 110°
A parallelogram is a quadrilateral in which opposite sides are equal, and the opposite angles sum up [tex]180^{o}[/tex]. Thus, the measures of the two different angles in the question are 70° and 110° i.e option D.
A parallelogram is a quadrilateral in which opposite sides are equal, and the opposite angles sum up [tex]180^{o}[/tex]. It has four straight sides and can be classified as a quadrilateral.
Thus from the given question, we have:
(6n − 70)° + (2n + 10)° = [tex]180^{o}[/tex]
8n - 60 = [tex]180^{o}[/tex]
8n = [tex]180^{o}[/tex] + 60
8n = 240
n = [tex]\frac{240}{8}[/tex]
= [tex]30^{o}[/tex]
n = [tex]30^{o}[/tex]
So that,
i. (6n − 70)° = (6[30] − 70)°
= 180 - 70
= [tex]110^{o}[/tex]
ii. (2n + 10)° = (2[30] + 10)°
= 60 + 10
= [tex]70^{o}[/tex]
Therefore, the measures of the two different angles are 70° and 110°. Option D.
For more clarifications on angles in a parallelogram, visit: https://brainly.com/question/28001957
#SPJ1
Answer:
b, 50° and 130°
Step-by-step explanation:
edge 2023 :)
^ that shi sucks btw we needa boycott the program or smt
please help asap..... Write the equation of the circle in general form. Show all your work.
will mark brainiest if correct.
Answer:
(x + 3)^2 + (y - 2)^2 = 25 .
Step-by-step explanation:
The center is at (-3, 2) and the radius is 5.
General form is
(x - h)^2 + (y - k)^2 = r^2
So here we have:
(x - (-3)^2 + (y - 2)^2 = 5^2
(x + 3)^2 + (y - 2)^2 = 25
Which one of these is not a step used when constructing an inscribed square using
technology? (5 points)
1) create a circle using the center with given point tool.
2) connect the point with a line through the center of the circle.
3) mark the points of intersection between the three circles.
4) create another circle with the same radius as the original.
The correct option is option () Connect the point with a line through the center of the circle.
According to the problem,
a round planar figure whose circumference is made up of points that are equally spaced from a fixed point (the centre).referring to a unit of measurement whose area is equal to a square whose side is the specified unit.A figure that is inscribed in a circle has vertices that are points on the circle. Learn how to draw equilateral triangles, squares, and regular hexagons that are inscribed in circles, as well as how to build figures in circles.So, the correct option is option ()Connect the point with a line through the center of the circle.
#SPJ4
Very much appreciated!!!
Answer:
80
Step-by-step explanation:
The sum of the interior angles of a triangle is 180. Subtract out the 2 angles that you know.
180 - 70 - 30 = 80
Which of these is the equation of a graph
in which the vertex is (4, 2) and
a=-2
HELPPPPPPPP
The equation of the graph in which the vertex (4,2) and a=-2 is
y=-2[tex](x-4)^{2}[/tex]+2.
Given that the point of the vertex is (4,2) and a=-2.
Equation is like a relationship between two or more variables expressed in equal to form. Equations of two variables look like ax+by=c. It may be a linear equation,quadratic equation, cubic equation or many more depending on the power of variable in the equation.
We know that point of vertex look like (h,k) in the equation and equation look like as under:
y=a[tex](x-h)^{2} +k[/tex].-----------1
So in order to find the equation we have to just put h=4 and k=2 and a=2 in equation 1.
y=-2[tex](x-4)^{2} +2[/tex]
Hence the equation of the graph in which the vertex (4,2) and a=-2 is
y=-2[tex](x-4)^{2}[/tex]+2.
Learn more about equation at https://brainly.com/question/2972832
#SPJ1
What is the PERT weighted average based on an optimistic estimate of six days, a most likely estimate of eight days, and a pessimistic time of sixteen days
the PERT weighted average is 9 days
What is PERT weighted average?
PERT can be defined as an estimating technique that uses a weighted average of three numbers to get a final estimate.
The formula for finding the PERT weighted average is given as;
PERT weighted average = (Optimistic + (4 X Most Likely) + Pessimistic)/6
We have;
Optimistic = 6 daysMost likely = 8 daysPessimistic = 16 daysSubstitute the values into the formula
PERT weighted average = [tex]\frac{6 + 4(8) + 16 }{6}[/tex]
PERT weighted average = [tex]\frac{54}{6}[/tex]
PERT weighted average = 9 days
Thus, the PERT weighted average is 9 days
Learn more about PERT analysis here:
https://brainly.com/question/22280115
#SPJ1
find the midpoint of (0, 4) and (2, -2)
Answer:
[tex]\text{Midpoint} = (1, \; 1)[/tex]
Step-by-step explanation:
[tex]M = (x_M, \; y_M)[/tex]
[tex]M = \left(\dfrac{x_1 + x_2}{2}, \; \dfrac{y_1 + y_2}{2}\right)[/tex]
[tex]M = \left(\dfrac{0 + 2}{2}, \; \dfrac{4 + -2}{2}\right)[/tex]
[tex]M = \left(\dfrac{2}{2}, \; \dfrac{2}{2}\right)[/tex]
[tex]M = (1, \; 1)[/tex]
Answer: (1, 1)
Step by Step Explanation:
The midpoint formula is (x + x/2 , y + y/2)
If you plug in the coordinates it looks like this
( 0 + 2/2 , 4 + (-2)/2)
That turns into this:
(2/2 , 2/2)
Which gives you the final answer of (1, 1)
Solve the triangle. Round your answers to the nearest tenth.
The missing information of the triangle is C = 63°, b ≈ 28.084 and c ≈ 25.084. (Correct choice: C)
How to determine the missing sides and angles
In this question we have a triangle with a known side lengths and two known angle measures. First, we find the missing angle by Euclidean geometry:
C = 180° - 94° - 23°
C = 63°
Lastly, we determine the missing sides by law of sines:
[tex]b = 11 \times \frac{\sin 94^{\circ}}{\sin 23^{\circ}}[/tex]
b ≈ 28.084
[tex]c = 11 \times \frac{\sin 63^{\circ}}{\sin 23^{\circ}}[/tex]
c ≈ 25.084
The missing information of the triangle is C = 63°, b ≈ 28.084 and c ≈ 25.084. (Correct choice: C)
To learn more on triangles: https://brainly.com/question/2773823
#SPJ1
A recipe for dessert calls for 1/3 cup of powdered sugar and 2/6 cup of brown
sugar. What is the total amount of sugar needed for the recipe
Answer:
2/3
Step-by-step explanation:
1/3 cup powdered sugar
2/6 cup brown sugar
Reduce 2/6 to 1/3.
1/3 + 1/3 = 2/3
Kylie is writing a persuasive essay for English class. The essay has to present her opinion and evidence for constructing that opinion. The essay has to have a minimum word count of 150 words and a maximum word count of 500 words.
Which compound inequality represents the length, w, of the essay?
500 ≤ w ≤ 150
150 ≥ w
500 ≤ w
150 ≤ w ≤ 500
Answer:
The last answer is correct.
Step-by-step explanation:
The w stands for the number of words that she can have. She has to have at least 150 words so the w has to be equal to 150 or greater. The w also has to be less then or equal to 500
Twenty-one is 25% of what number?
A. 5.25
B. 84
C. 10.5
D. 63
Answer: 84
Step-by-step explanation:
When substituting this question into an algebraic equation you can write it as:
21=.25x
(x clearly represents the number you are trying to find)
When you divide both sides by .25 to isolate x you get x=84
If sin A + sin^3 A = cos^2 A, prove that cos^6 A - 4cos^4 A + 8cos^2 A = 4.
Please provide all steps. Thank You.
Answer:
Step-by-step explanation:
sinA(1+sin^2A) = cos^2A
sinA(2 -cos^2A) = cos^2A
Squaring both sides,
sin^2A(4-4cos^2A +cos^4A) = cos^4A
(1-cos^2A)(4-4cos^2A +cos^4A) = cos^4A
4-4cos^2A +cos^4A-4cos^2A+4cos^4A-cos^6A = cos^4A
4 -cos^6A +4cos^4A -8cos^2A = 0
cos^6A - 4 cos^4A + 8cos^2A = 4
hence proveproven
Answer:
Step-by-step explanation:
sinA+sin
3
A=cos
2
A
⇒sinA[1+sin
2
A]=cos
2
A
⇒sinA[1+1−cos
2
A]=cos
2
A
squaring on both sides.
⇒sin
2
A[4+cos
4
A−4cos
2
A]=cos
4
A
⇒(1−cos
2
A)[4+cos
4
A−9cos
2
A]=cos
4
A
⇒4+cos
4
A−4cos
2
A−64cot
2
A−cos
6
A+9cos
4
A=cos
4
A
⇒
cos
6
A−4cos
4
A+8cos
2
A=4
Hence Prove
A cross section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base. the cross section can be which of these shapes? select three options.
A cross-section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base square, triangle, and, trapezoid.
What is an area of cross-section?A cross-section parallel to the base will be a square.
One perpendicular to the base will be a trapezoid, or if it passes through the vertex of the pyramid, a triangle.
The cross-section of the pyramid is perpendicular to the base and through the vertex will be a triangle.The cross-section has the same shape but it has a smaller base.The cross-section perpendicular to the triangular bottom then it will be a trapezoid.Hence, a cross-section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base square, triangle, and, trapezoid.
Learn more about the Area of cross-section here:
brainly.com/question/83161
#SPJ4
A rose garden is formed by joining a rectangle and a semicircle, as shown below. The rectangle is 29 ft long and 20 ft wide.
Find the area of the garden. Use the value 3.14 for , and do not round your answer. Be sure to include the correct unit in your answer.
Answer:
15.7 square feet
Step-by-step explanation:
To find the area of the garden we find the area of the rectangle and the area of the semicircle.
The area of the rectangle:
A=bh
A=(29)(20)
A=580ft^2
As shown in the picture, the width of the rectangle is 20 ft wide, which means that the diameter of the semicircle is also 20ft wide. The diameter is two times the radius, which means the radius of the semicircle is 10ft.
The area of a circle is represented by the equation: [tex]A=\pi r^{2}[/tex]
A semicircle is half of a circle, therefore the equation to find the area of a semicircle is: [tex]A=\frac{1}{2}\pi r^{2}[/tex]
Plugging our radius of 10ft in we get:
[tex]A=\frac{1}{2}\pi (10)\\ A=5\pi[/tex]
We use 3.14 for our value of [tex]\pi[/tex] to get:
[tex]A=5(3.14)\\A=15.7ft^{2}[/tex]
Therefore, the area of the rose garden is 15.7 square feet.
TIME REMAINING
01:38:18
The function rule T–4, 6(x, y) could be used to describe which translation?
a parallelogram on a coordinate plane that is translated 4 units down and 6 units to the right
a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up
a rhombus on a coordinate plane that is translated 4 units down and 6 units to the left
a rectangle on a coordinate plane that is translated 4 units to the right and 6 units up
The translation that represents T–4, 6(x, y) is (b) a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up
How to determine the translation?The translation rule is given as:
T–4, 6(x, y)
This can be rewritten as
T(x- 4, y + 6)
The above represents a translation 4 units left and 6 units up
Hence, the translation that represents T–4, 6(x, y) is (b) a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up
Read more about transformation translation at:
https://brainly.com/question/8797028
#SPJ1
2. According to Archimedes, the area of any circle is equal to the area of a right triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Verify this formula for yourself using the formula for the circumference of a circle.
Formula [tex]\text { area }=1 / 2 \times \text { base } \times \text { height }=1 / 2 \times 2 \pi r \times r=\pi r^{2}[/tex] for the circumference of a circle.
How did Archimedes find the circumference of a circle?Archimedes stated in his Proposition that the area of a circle is equal to the area of a triangle with a base equal to the circumference and a height equal to the radius: (1/2)(r · 2πr) = πr2. Archimedes arrived at his approximation of the circumference of the circle by increasing the number of sides of the hexagon. Archimedes claimed that the area of any circle is equal to the area of a right triangle, where the radius of the circle is represented by one side and the circumference by the other.
Archimedes demonstrated using a similar method that the area of a circle of diameter D is equal to the area of a right-angled triangle with one side equal to the radius and the other to the circumference of the circle on the right angle.
Formula for the circumference of a circle:
A circle's area is equal to pi times the radius squared (A = r2). Discover how to apply this formula to determine a circle's area given its diameter.
The circumference is diameter x pi, or 2 x radius x pi.
[tex]\text { area }=1 / 2 \times \text { base } \times \text { height }=1 / 2 \times 2 \pi r \times r=\pi r^{2}[/tex].
Learn more about circumference of a circle here:
https://brainly.com/question/15211210
#SPJ4
Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Yeony2022, tysm for the help!
See
The point D is
3 units left from origin
x coordinate=-31 units up from origin
y coordinate=1The point is
(-3,1)Direct variation
Multiply k=2
k(-3,1)2(-3,1)(-6,2)Option A
A rectangle has a side length of two and seven eighths feet and a side width of 5 feet. What is the area of the rectangle?
A. six and five eighths ft2
B. nine and two eighths ft2
C. fourteen and one eighths ft2
D. fourteen and two eighths ft2
Answer:
14 3/8 ft^2
Step-by-step explanation:
Area = length * width
= 2 7/8 * 5
= 23/8 * 5
= 115/8
= 14 3/8 ft^2.
I really need help with this question
Answer:
39
Step-by-step explanation:
I assume the father's age is a 2-digit number. He is not 9 years old or younger, and he is not 100 years old or older.
The sum of the digits of a two-digit number goes up by 1 from one number to the next unless the number ends in 9.
For example, from 17 to 18, the sum of the digits goes from 8 to 9.
If the number ends in 9, for example, 29, the digits add to 11. Then the next number is 30, and now the sum of the digits is 3, which is less than 11.
The father's age now is a number whose sum of digits is not only greater than the sum of the digits next year, but it is 3 times greater.
Let's look at 2-digit number and the next number and the sum of their digits.
19, 20; sums: 10, 2; 10/2 = 5, not 3
29, 30; sums: 11, 3; 11/3 ≠ 3
39, 40; sums: 12, 4; 12/4 = 3
Answer: The father is 39 years old.
Answer: 39 years
Step-by-step explanation:
The father says that adding both of the digits in his age will be 3 times greater than adding both digits one year later. Before we even start answering the problem, this tells us that the father's age has 2 digits in it.
Let's represent the 10's digit with x, and the 1's digit with y. If the father's age was 45, for example, x would be 4 and y would be 5.
Let's first make an equation to represent this situation:
The sum of the digits of the father's age would be represented by x + y, as x and y are the 10's digit and 1's digit respectively.
The father's age next year would be one greater than the current year, which makes y become y + 1. Then, the sum of the digits would be x + y + 1.
Now, we must make the first equation 3 times larger than the second and solve for the sum.
[tex]x + y = 3(x+y+1)\\ x+y=3(x+y)+3\\ 1(x+y)-3(x+y)=3\\ -2(x+y)=3\\ x+y=\frac{3}{-2}=-\frac{3}{2}[/tex]
However, we can neither have a fraction or a negative as a sum of two digits in an age, as digits are always positive integers from 0 to 9.
This flaw came from the assumption that after 1 year, the sum of the digits will be x + y + 1. This isn't always true, as any number with its 1's digit of 9 turns into 0 the next year, and not 10. Instead, the x value increases by 1.
For example, someone the age 69 becomes 70 the next year, x increases by 1 and y becomes 0.
Let's make a new equation to try and see this.
We know that his current age has to have y = 9 for his new age to be y = 0. Therefore, the current sum of digits is x + 9.
The next year, we know that the 10's place will increase by 1 and the 1's place becomes 0. Therefore the sum of digits is x + 1 + 0, or x + 1.
Since the current sum of digits is 3 times greater than the sum next year, let's multiply the second equation by 3 to get equality. We just need to calculate the 10's digit, as the 1's digit is already 9.
[tex]x+9 = 3(x+1)\\ x+9=3x+3\\ 9=2x+3\\ 6=2x\\ x=3[/tex]
Since the 10's digit (or x) is 3 and the 1's digit (or y) is 9, the father's age is 39, and he will turn 40 next year.
please im in a rush i need help
Answer:
28.5 ft²
Step-by-step explanation:
shaded area = area of triangle minus area of rectangle
shaded area = (1/2)bh - LW
base of triangle = 5 ft + 4 ft = 9 ft
height of triangle = 3 ft + 6 ft = 9 ft
shaded area = (1/2)(9 ft)(9 ft) - (4 ft)(3 ft)
shaded area = 40.5 ft² - 12 ft²
shaded area = 28.5 ft²