Consider the following time series data. t 1 2 3 4 5 yt 7 12 8 15 16 (a) Construct a time series plot. What type of pattern exists in the data
Answers
(a) The time series plot shows an increasing non-linear trend pattern in the data.
(b) The parameters for the line that minimizes MSE for this time series are:
\(b_o = 2.4376\\b_i = 2.434\)
(a)
t 1 2 3 4 5
\(y_t\) 7 12 8 15 16
Plotting the data points on a graph with time (t) on the x-axis and the observed values (\(y_t\)) on the y-axis, we obtain the following time series plot.
From the time series plot, we can observe an increasing trend in the data. The values of \(y_t\) generally rise over time, indicating a non-linear positive trend pattern.
(b) To find the parameters for the line that minimizes the Mean Squared Error (MSE) for the given time series data, we can use simple linear regression analysis.
The equation for simple linear regression is given by:
\(Y_t = b_o + b_i * t\)
\(\sum Y_t = n * b_o + b_i * \sum t\\\sum Y_t * t = b_o * \sum t + b_i * \sum t^2\)
where n is the number of observations.
Let's calculate the required values:
n = 5 (number of observations)
\(\sum Y_t = 5 + 12 + 8 + 15 + 16 = 56\\\sum t = 1 + 2 + 3 + 3 + 4 + 5 = 18\\\sum Y_t * t = (5 * 1) + (12 * 2) + (8 * 3) + (15 * 3) + (16 * 4) = 117\)
Now, we can substitute these values into the equations:
\(n * b_o + b_i * \sum t = \sum Y_t\\5 * b_o + 18 * b_i = 56 ---(1)\\b_o * \sum t + b_i * \sum t^2 = \sum Y_t * t\\18 * b_o + 30 * b_i = 117 ---(2)\)
To solve this system of equations, we can multiply equation (1) by 18 and equation (2) by 5 to eliminate bo:
\(90 * b_o + 324 * b_i = 1008 ---(3)\\90 * b_o + 150 * b_i = 585 ---(4)\)
Subtracting equation (4) from equation (3):
\(174 * b_i = 423\)
Dividing both sides by 174:
\(b_i = 423 / 174 = 2.434\)
Now, substitute the value of \(b_i\) back into equation (1):
\(5 * b_o + 18 * 2.434 = 56\)
Simplifying:
\(5 * b_o + 43.812 = 56\\5 * b_o = 56 - 43.812\\5 * b_o = 12.188\\b_o = 12.188 / 5 = 2.4376\)
Therefore, the parameters for the line that minimizes the MSE for this time series data are approximately:
\(b_o = 2.4376\\b_i = 2.434\)
So, the equation for the line is:
\(Y_t = 2.4376 + 2.434 * t\)
Complete Question:
Consider the following time series data. t 1 2 3 3 4 5 Yt 5 12 8 15 16
(a) Construct a time series plot. What type of pattern exists in the data?
(b) Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series.
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Related Questions
if a is any integer, is a (a plus 1 )even or odd? say which it is (4 pts) and explain why (as a simple proof) (8 pts).
Answers
==================================================
Proof:
We'll break the proof into two cases which I'll label A and B
Case A: 'a' is evenCase B: 'a' is odd-----------
Case A: 'a' is even
k = some integer
a = 2k = some even integer
a+1 = 2k+1
a(a+1) = 2k(2k+1) = 2(2k^2+k) = 2*(some integer)
Since 2 is a factor of that last expression, this shows that a(a+1) is even when 'a' is even.
-----------
Case B: 'a' is odd
k = some integer
a = 2k+1 = some odd integer
a+1 = (2k+1)+1 = 2k+2
a(a+1) = (2k+1)(2k+2) = 2(2k+1)(k+1) = 2(some integer)
This shows that a(a+1) is even when 'a' is odd.
-----------
Therefore, for any integer 'a', the expression a(a+1) is always even.
Some examples:
a = 3, a+1 = 3+1 = 4, a(a+1) = 3*4 = 12 which is evena = 12, a+1 = 12+1 = 13, a(a+1) = 12*13 = 156 which is even-----------
Here's a slightly different way to interpret why the proof works.
a(a+1) consists of factors 'a' and 'a+1'
If 'a' was even, then a(a+1) is automatically even since 2 is a factor of 'a'.If 'a' was odd, then a+1 is even and we arrive at the same conclusion as before.Either way, we'll have 2 as a factor somewhere in a(a+1).
helpp pleaaaaaaaaaseee
Answers
The linear equations would be
4x + 5y = 62 (the equation representing the total cost of Kent)
6x + 2y = 60 (the equation representing the total cost of Rew)
and the solution would be x = 8, y = 6.
What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that describe a relationship between multiple variables. The variables in each equation are related to each other, and the solution of a system of linear equations is the set of values that satisfy all of the equations in the system.
In this case, since Kent and Rew each bought the same amount of boondins and hopts, we can use the same variable names (x for boondins and y for hopts) for both of them. We can use this information to write two equations to model the relationship between the amounts of boondins and hopts that Kent and Rew bought:
4x + 5y = 62 (the equation representing the total cost of Kent)
6x + 2y = 60 (the equation representing the total cost of Rew)
This system of linear equations is called a system of two equations in two variables.
Hence, the linear equations would be
4x + 5y = 62 (the equation representing the total cost of Kent)
6x + 2y = 60 (the equation representing the total cost of Rew)
and the solution would be x = 8, y = 6.
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DL and BM are the heights on sides AB and AD respectively of parallelogram ABCD. If the area of the parallelogram is 1470 cm2 , AB=35cm and AD=49cm, Find the length of BM and DL.
Answers
Answer:
BM = 30 cm, DL = 42 cm
Step-by-step explanation:
The area of the parallelogram = 1470 cm², AB = 35 cm, AD = 49 cm, DL and BM are the heights on sides AB and AD.
area of the parallelogram = base × height = AD × height of AD = AD × BM
⇒ 1470 cm² = 49 cm × BM
BM = 1470 cm² / 49 cm = 30 cm
BM = 30 cm
Also:
area of the parallelogram = base × height = AB × height of AB = AB × DL
⇒ 1470 cm² = 35 cm × DL
DL = 1470 cm² / 35 cm = 42 cm
DL = 42 cm
2
.
Toru has $80 to pay for dog grooming. The fee is $72 plus 7.25% tax.
Does he have enough money? Explain
Answers
We have been given that Toru has $80 to pay for dog grooming. The fee is $72 plus 7.25% tax. We are asked to determine whether Toru has enough money for grooming or not.
The total charges for grooming would be $72 plus 7.25% of $72.
\(\text{Total charges for grooming}=\$72+\$72\times \frac{7.25}{100}\)
\(\text{Total charges for grooming}=\$72+\$72\times0.0725\)
\(\text{Total charges for grooming}=\$72+\$5.22\)
\(\text{Total charges for grooming}=\$77.22\)
Since total cost of grooming is $77.22 and Toru has $80 to pay for grooming, therefore, Toru has enough money for grooming.
Math class, last chance to take it!
Answers
Answer:
\(\sf x=-4\) \(\sf \ and\) \(\sf y=45\)
Step-by-step explanation:
given:
\(\sf{-\frac{1}{2} x+\frac{1}{5} y=11}\)
\(\sf{3 x+\frac{1}{4} y=-\frac{3}{4} }\)
make x subject:
\(\sf{-\frac{1}{2} x+\frac{1}{5} y=11}\)
\(\sf{-\frac{1}{2} x =11-\frac{1}{5}y}\)
\(\sf{- x =22-\frac{2}{5}y}\)
\(x =-22+\frac{2}{5}y}\)
solve using substitution method:
\(\sf{3 x+\frac{1}{4} y=-\frac{3}{4} }\)
\(\sf{3 (-22+\frac{2}{5}y) +\frac{1}{4} y=-\frac{3}{4} }\)
\(\sf-66 +\frac{6}{5} y+\frac{1}{4} y = -\frac{3}{4}\)
\(\sf 1.45y = 65.25\)
\(\sf y=45\)
then x will be:
\(x =-22+\frac{2}{5}y}\)
\(x =-22+\frac{2}{5}(45)}\)
\(\sf x=-4\)
circle $c$ has radius 6 cm. how many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $c$?
Answers
The area of the largest possible inscribed triangle with one side as a diameter of the circle is 72 square centimeters.
The largest possible inscribed triangle with one side as a diameter of the circle has a right angle.
To find the area of the inscribed triangle, we need to calculate the length of the base and the height. Since the diameter of the circle is the base of the triangle, it has a length of 2 times the radius, which is 12 cm.
The height of the triangle is equal to the radius of the circle, which is 6 cm.
To calculate the area of the triangle, we use the formula:
Area = 1/2 * base * height
= 1/2 * 12 cm * 6 cm
= 72 square cm
Therefore, the area of the largest possible inscribed triangle with one side as a diameter of the circle is 72 square centimeters.
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what number does not appear in the first 30 digits of pi?
Answers
Answer:0
Step-by-step explanation:
HELP NOW I BEEN STRESSING OUT
Answers
Answer:
(0,-3)
Step-by-step explanation:
the y intercept is when x =0
Assuming a linear relationship exists, and that the relationship continues unchanged through 1986, what would you predict for the average cost of a gallon of milk in 1986
Answers
Given the average cost of a gallon of milk is the dependent variable, and the year is the independent variable, assuming that a linear relationship exists, you can determine the slope and y-intercept values.
Since the relationship is linear, you can use a simple linear regression equation of the form:
y = mx + b where y represents the average cost of a gallon of milk and x represents the year.
For example, if the average cost of a gallon of milk in 1980 is $2.50, and the average cost in 1985 is $3.25, then the slope would be
(3.25 - 2.50) / (1985 - 1980) = 0.15.
This indicates that for every one year increase, the cost of milk increases by $0.15.
The y-intercept can be found by using either of the years and the slope value.
Substituting the slope and y-intercept values into the equation y = mx + b gives the equation
y = 0.15x + b,
where x is the year and y is the average cost of a gallon of milk.
To predict the average cost of a gallon of milk in 1986,
you would substitute x = 1986 into the equation and solve for y.
Therefore, the predicted average cost of a gallon of milk in 1986 would be:
y = 0.15(1986) + b
y = 298.9 + b
where b is the y-intercept.
Since we do not know the value of b, we cannot determine the exact predicted cost of milk in 1986.
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marsha deposited $9,500 into a savings account 3 years ago. The simple interest rate is 5%. How much money did marsha earn in interest?
Answers
Answer:
$ 10, 925
Step-by-step explanation:
Interest = P( 1 + rt)
9500 { 1 + ( 0.05 x 3 ) }
= 9500 x 1.15
= $ 10,925
Anthony wants to get a new TV. He has been working hard as a busboy at the local restaurant and wants
to treat himself to something nice. Today is his lucky day as he finds a 60" TV in stock that he has been
looking for. Unfortunately, it costs $700, and he does not have enough money saved up in order to buy it.
He recently received a credit card and is trying to decide what to do. Should he keep saving money buy it
Answers
The credit card provides the opportunity for Anthony to purchase the TV by paying at a rate of $50 a month.
The correct responses are;
1. The final price of the TV is $791.62. Anthony has to work for 70 hours to save enough money to buy the TV3. The number of extra hours of work bussing tables when using his credit card is approximately 9.16 hours.Reasons:
The price of the TV, P = $700
The interest rate is r = 18%
The monthly payment, M = $50
Monthly payment formula is given as follows;
\(M = \mathbf{ \dfrac{P \cdot \left(\dfrac{r}{12} \right) \cdot \left(1+\dfrac{r}{12} \right)^n }{\left(1+\dfrac{r}{12} \right)^n - 1}}\)
The time it will take to pay using the credit card is therefore;
\(\displaystyle 50 = \dfrac{700 \times \left(\dfrac{0.18}{12} \right) \cdot \left(1+\dfrac{0.18}{12} \right)^n }{\left(1+\dfrac{0.18}{12} \right)^n - 1} = \frac{10.5 \cdot 1.015^n}{1.015^n-1}\)
Which gives;
50 × (1.015ⁿ - 1) = 10.5 × 1.015ⁿ
Solving for n gives;
\(\displaystyle n = \frac{ln\left(\frac{50}{39.5}\right) }{ln(1.015)} \approx \mathbf{15.832}\)
It will take approximately 15.832 months for Anthony to pay for the TV at $50 per month.
1. The final price of the TV = $50 × 15.832 ≈ $791.6
2. The number of hours of work, t, is given as follows;
\(\displaystyle t = \frac{\$700}{\$10 \ per \ hour} = \mathbf{ 70 \ hours}\)
Anthony has to work for 70 hours to save enough money to purchase the TV.
3. The extra hours of work is given by the hours of work needed to pay for the interest amount for the loan on the card
The interest amount on the card = $791.6 - $700 = $91.6
\(\displaystyle Extra \ hours \ to \ work = \mathbf{ \frac{\$ 91.6}{\$10 \ per \ hour}} = 9.16 \ hours\)
Anthony has to work for extra 9.16 hours bussing tables using his credit card.
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Two containers designed to hold a water our side-by-side, both in the shape of a cylinder. Container a has A diameter of 6 feet and a height of 15 feet. Container b has a diameter of 8 feet and a height of 10 feet. Container is full of water in the water is pumped into container be until container is empty. After the pumping is complete what is the volume of the empty portion of container b, to the nearest 10th of a cubic foot
Answers
Answer:
78.5 ft3
Step-by-step explanation:
First we need to find the volume of both containers.
The volume of a cylinder is given by the equation:
Volume = pi * diameter^2 * height / 4
If cylinder A has a diameter of 6 feet and height of 15 feet, we have:
Volume_A = pi * 6^2 * 15 / 4 = 424.115 ft3
If cylinder B has a diameter of 8 feet and height of 10 feet, we have:
Volume_A = pi * 8^2 * 10 / 4 = 502.6548 ft3
After pumping the water from A to B, the volume of the empty portion of B will be the difference of their volumes:
Volume_B - Volume_A = 502.6548 - 424.115 = 78.5398 ft3
Rounding to the nearest 10th of a cubic foot, we have a difference of 78.5 ft3
Justin gave 2/3 of his pizza to Forest friends who shared the pizza equally. what fraction of the original Pizza did each friend get.
Answers
Since 2 contains 6 one-thirds, we can say that 2 divided by one-third is 6. 2 divided by 1/3 = 6 and 2 x 3 = 6 1/3 and 3 are reciprocals
1/4 divided by 2/8 = 1/4 x 3/2 = 3/8
Please help me I will give brainlest if right
What is the complete factorization of the polynomial below?
x^2 + 4x^2 + 16x + 64
O A. (x-4)(x + 4i)(x + 4i)
B. (x + 4)(x + 4i)(x + 4i)
C. (x + 4)(x + 4i)(x-4i)
O D. (x-4)(x + 4i)(x-4i)
Answers
Given:
Consider the given polynomial is:
\(x^3+4x^2+16x+64\)
To find:
The complete factorization of the given polynomial.
Solution:
We have,
\(x^3+4x^2+16x+64\)
Using grouping method, we get
\(=(x^3+4x^2)+(16x+64)\)
\(=x^2(x+4)+16(x+4)\)
\(=(x+4)(x^2+16)\)
\(=(x+4)(x+4i)(x-4i)\) \([\because x^2+y^2=(x+iy)(x-iy)]\)
Therefore, the correct option is C.
The weather forecaster says that tomorrow there is a 87% chance that it will rain. Which word or phrase describes the probability that it will rain?
O certain
O likely
O impossible
O unlikely
Answers
Answer: likely
Step-by-step explanation:
In ADEF, the midpoint of the side opposite vertex D is M and the centroid is C. If DM is 42, what are DC and CM?
Answers
Answer:
21
Step-by-step explanation:
If it is in the center than DC is 21 and CM is 21
42÷ 2
The endpoints of the diameter of a circle have the coordinates (3,8) and (9,16) .
What are the coordinates of the center of the circle?
Answers
9514 1404 393
Answer:
(6, 12)
Step-by-step explanation:
The center of the circle is the midpoint of the diameter, the average of the end point coordinates:
((3, 8) +(9, 16))/2 = (3+9, 8+16)/2 = (12, 24)/2 = (6, 12)
The center of the circle is (6, 12).
A painting and sculpture cost 1500 in total the painting is 1000 more than the sculpture how much the painting cost?
a.1200
b.1000
c.1250
d.1300
Answers
painting 750+500 = $1250.
Option C
Complete the square to re-write the quadratic function in vertex form.
y=x²+7x+3
Answers
Answer:
To complete the square and rewrite the quadratic function y = x² + 7x + 3 in vertex form, we follow these steps:
Factor out the coefficient of x² from the first two terms:
y = 1(x² + 7x) + 3
Take half of the coefficient of x (which is 7 in this case) and square it. Add this value inside the parentheses, and subtract the same value multiplied by the coefficient of x² (which is 1) outside the parentheses to maintain the same value of the expression:
y = 1(x² + 7x + (7/2)² - (7/2)²) + 3
Simplify inside the parentheses by combining the first three terms using the square of the binomial formula (a + b)² = a² + 2ab + b²:
y = 1(x + 7/2)² - 1/4 + 3
Combine the constant terms to simplify:
y = 1(x + 7/2)² + 11/4
Therefore, the quadratic function y = x² + 7x + 3 can be written in vertex form as y = (x + 7/2)² + 11/4. The vertex is located at the point (-7/2, 11/4).
Hope This Helps!
Tuck is 11 years old, and he has a younger
brother. His brother's age is represented by the
expression 2x - 14, when x represents Tuck's age.
How old is Tuck's brother?
Answers
Answer:
his brother is 8
Step-by-step explanation:
i did math :)
Answer:
Tucks brother is 8 years old.
Step-by-step explanation:
2x = 22 - 14= 8
$1200 are invested for three years at 6% compound interest.
a) How much is the investment worth at the end of three years?
b) How much interest has been earned?
Answers
Answer:
I think its A 1200
Step-by-step explanation:
If f(x) is a linear function, f(-2) = 1, and f(1) = -2, find an equation for f(x) f(x) = __
Answers
To find an equation for the linear function f(x), we can use the two given points (-2, 1) and (1, -2) to determine the slope and y-intercept of the function.
Let's use the point-slope form of a linear equation, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Using the points (-2, 1) and (1, -2), we can calculate the slope: slope (m) = (y₂ - y₁) / (x₂ - x₁) = (-2 - 1) / (1 - (-2)) = (-3) / 3 = -1
Now that we have the slope, we can choose one of the given points and substitute it into the point-slope form to find the equation of the linear function.
Let's use the point (-2, 1):
y - y₁ = m(x - x₁)
y - 1 = -1(x - (-2))
y - 1 = -1(x + 2)
y - 1 = -x - 2
y = -x - 1
Therefore, the equation for the linear function f(x) is f(x) = -x - 1.
In summary, by using the given points and the point-slope form of a linear equation, we determined the slope to be -1. Substituting one of the points into the point-slope form, we found the equation of the linear function f(x) to be f(x) = -x - 1.
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Ethan makes punch by mixing orange juice with
sparkling water. He uses the same ratio of orange juice
to sparkling water each time he makes punch. The table
shows some of the amounts of orange juice and
sparkling water he uses for different amounts of punch.
Orange Juice (fl oz) 7 21 28 ?
Sparkling Water (fl oz) 4 12 16 28
If Ethan uses 28 fl oz of sparkling water, how much orange juice will he use and what will be the total amount of punch?
Use the number pad to enter your answers in the boxes.
Ethan will use ____
fl oz of orange juice, and he will have a total of ___
fl oz of punch.
Answers
Answer:
Ethan will use 49
fl oz of orange juice, and he will have a total of 77
fl oz of punch.
Step-by-step explanation:
He uses a ratio of 7 to 4 of orange juice to sparkling water.
7/4 = ?/28
4 * ? = 7 * 28
Divide both sides by 4.
? = 7 * 7
? = 49
He will use 49 fl oz of orange juice.
The total amount made is: 49 fl oz + 28 fl oz = 77 fl oz
Answer:
Ethan will use 49
fl oz of orange juice, and he will have a total of 77
fl oz of punch.
What is the answer to 9/10x8/20? Plz I need help I forgot how to do fractions
Answers
Answer:
\(\boxed{\frac{9}{25} }\)
Step-by-step explanation:
\(9 \div 10 \times 8 \div 20\)
\(\frac{\frac{9}{10}(8) }{20}\)
\(= \frac{\frac{35}{5} }{20}\)
\(= \frac{9}{25}\)
Hope this helped you!
● Answer:
9/25
● Step-by-step explanation:
Attachment
Archimedes drained the water in his tub.
The amount of water left in the tub (in liters) as a function of time (in minutes) is graphed.
How fast did the water drain?
Choose one answer:
A. 72 liters per minute
B. 74 liters per minute
C. 76 liters per minute
D. 78 liters per minute
Answers
Based on the amount of water in the tub that Archimedes is draining as a function of time, the rate that the water drains is A. 72 liters per minute
How to find the rate of water drainage?The speed that the water drained is the same as the slope of the line. That slope can be found by the formula:
= Change in y / Change in x
Pick two points from the line;
(0, 360) and (5, 0)
The slope is:
= (0 - 360) / (5 - 0)
= -360 / 5
= - 72
This means that every minute, Archimedes would have drained the water by 72 liters.
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Find the value of the following. P(-3,8) Q(x,4) slope=3
Answers
The possible first coordinate x with a slope of 3 is equal to -13/3.
How to calculate the slope of a line?In Mathematics, the slope of any straight line can be determined by using this mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Based on the information provided, we can logically deduce the following data points and slope on the line:
Points on x-axis = (-3, x).Points on y-axis = (8, 4).Slope = 3Substituting the given data points into the slope formula, we have the following;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
3 = (4 - 8)/(x + 3)
3x + 9 = -4
3x = -13
x = -13/3
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Identify the constant terms of the expression 5x + 4y - 8x + 9x - 1
Answers
Answer:
7
Step-by-step explanation:
which of the following are solutions to the equation below x^2+4x-9=x+1 check all that apply a. 2 b. 4 c. -7 d. -3 e. -5 f. -4
Answers
Answer:
a. and e.
Step-by-step explanation:
Hello, there are two ways to handle this kind of question.
... Either you solve the equation ...
\(x^2+4x-9=x+1\\\\x^2+3x-10=0\\\\x^2-2x+5x-10=x(x-2)+5(x-2)=(x+5)(x-2)=0\\\\x= -5 \ \ or \ \ x = 2\)
So the right answers are a. and e.
... Or, you check which answer can be right.
We replace x by 2 and the two expressions are equal.
\(2^2+4*2-9=4+8-9=3 \ and \ 2+1=3\)
So this is a. 2
We replace x by 4 and the two expressions are not equal.
\(4^2+4*4-9=16+16-9=23 \ and \ 4+1=5\)
So this is not b. 4
We replace x by -7 and the two expressions are not equal.
\(7^2-4*7+9=49-28+9=30 \ and \ -7+1=-6\)
So this is not c. -7
We replace x by -3 and the two expressions are not equal.
\(3^2-4*3-9=9-12-9=-12 \ and \ -3+1=-2\)
So this is not d. -3
We replace x by -5 and the two expressions are equal.
\(5^2-4*5-9=25-20-9=-4 \ and \ -5+1=-4\)
So this is e. -5
We replace x by -4 and the two expressions are not equal.
\(4^2-4*4-9=16-16-9=-9 \ and \ -4+1=-3\)
So this is not f. -4
Thank you.
SOMEONE HELP ASAP Ill give brainist and 100 points
Answers
Answer:
x=0
Step-by-step explanation:
the axis of symmetry is 0
Heyo Brother!
It's me Violet! ^w^
I could be wrong.. but, I believe the answer is..
\((0,20)\)
Hopefully, this helps chu!
:3
Have a awesome day~
\(XxVioletXx\)
True or false?
67.1 < 67.09
True or false?
1880.09 < 1880.023
Answers
Answer: true
Answer:
Step-by-step explanation:
67.1< (less than) 67.09
False
1880.09< (less than) 1880.023
False