Using the method of orthogonal polynomials described in Section 7.1.2, fit a third-degree equation to the following data: y (index): 9.8 11.0 13.2 15.1 16.0 (year): 1950 1951 1952 1953 1954 Test the hypothesis that a second-degree equation is adequate. 2. Show that the least squares estimates of B1 and B2 for the model (7.28) are still unbiased even when the true model includes an interaction term B12, that is, E[Y] = Bo + B121 + B2X2 + B120122. Find the least squares estimate of B12. 3. Suppose that the regression curve E[Y] = Bo + BjI + B2x2 has a local maximum at I = Im where I'm is near the origin. If Y is observed at n points 1: (i = 1,2,..., n) in (-a, a), I = 0, and the usual normality assumptions hold, outline a method for finding a confidence interval for Im. Hint: Use the method of Section 6.1.2. (Williams (1959: p. 110])
Answers
Orthogonal polynomials are used to fit a third-degree equation to data points in the query. We must then prove that a second-degree equation is sufficient. We must also demonstrate that least squares regression coefficient estimations are unbiased when the genuine model includes an interaction component. Finally, we must provide a confidence interval for the regression curve's local maximum.
Fitting a third-degree equation using orthogonal polynomials: To fit a third-degree equation to the given data, we can use orthogonal polynomials such as Legendre polynomials. These polynomials form an orthogonal basis for the space of polynomials. By applying the method described in Section 7.1.2, we can calculate the coefficients of the third-degree equation that best fit the data.Testing the hypothesis of a second-degree equation: To test the hypothesis that a second-degree equation is adequate, we can perform an analysis of variance (ANOVA) test. This test compares the fit of the second-degree equation to the third-degree equation by assessing the reduction in sum of squares. If the reduction in sum of squares is statistically significant, it suggests that the third-degree equation provides a significantly better fit.Unbiasedness of least squares estimates with an interaction term: The least squares estimates of the regression coefficients (B1 and B2) remain unbiased even when the true model includes an interaction term (B12). This property holds as long as the usual assumptions of linear regression, such as the errors being normally distributed with zero mean and constant variance, are satisfied.Estimating B12: To estimate the interaction term B12, we can include the interaction term in the regression model and use the least squares method to obtain the estimate. By minimizing the sum of squared residuals, we can find the least squares estimate of B12.Confidence interval for the local maximum: To find a confidence interval for the local maximum of the regression curve at Im, we can utilize the method described in Section 6.1.2, as suggested in the hint. This method involves constructing a confidence interval for the location parameter based on the normality assumptions. By applying appropriate statistical techniques, such as calculating the standard error and using the t-distribution, we can determine a confidence interval that represents the uncertainty around the estimated local maximum.Learn more about Orthogonal polynomials here:
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Related Questions
we assume that with a linear relationship between two variables, for any fixed value of x, the observed ________ follows a normal distribution.
Answers
We assume that with a linear relationship between two variables, for any fixed value of x, the observed residuals follows a normal distribution.
This assumption is based on the Central Limit Theorem, which states that when the sample size is large enough, the distribution of sample means will be approximately normal, regardless of the shape of the underlying population distribution.
In the case of a linear relationship between two variables, we can assume that the residuals (the difference between the observed y values and the predicted values based on the linear regression model) follow a normal distribution with mean 0 and constant variance. This assumption is important because it allows us to use statistical methods that rely on normality, such as hypothesis testing and confidence intervals.
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1) Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE. )
f(x, y) = 5x2 + 5y2; xy = 1
2) Find the extreme values of f subject to both constraints. (If an answer does not exist, enter DNE. )
f(x, y, z) = x + 2y; x + y + z = 6, y2 + z2 = 4
Answers
The maximum and minimum values for the given two cases are
for the first case
the maximum value = 10
minimum value = 10
for the second case
the maximum value = 7
minimum value = 7
first case,given, f(x, y) = 5x2 + 5y2; xy = 1
In order to evaluate the maximum and minimum values of the given above function using Lagrange multipliers,
Here the conversion of the Lagrangian function takes priority:
L(x, y, λ) = f(x , y) - λ(x y - 1)
f(x , y) = 5x² + 5y² and x y = 1.
so, we evaluated the partial derivatives of L concerning x, y and λ:
∂l/∂x = 10x - λy
∂l/∂y = 10y - λx
∂l/∂λ = x y - 1
Staging the partial derivatives = 0 and calculating for x, y and λ
x = y
x y = 1
10x - λy = 0
10y - λx = 0
For the set of equation there are two critical points: (1,-1) and (-1,1).
In order to find if the critical points extend to a maximum or minimum value of f(x , y),
Then,
f(1 , -1) = f(-1 , 1) = 10
the maximum value = 10
minimum value = 10
for second case,Here
f(x,y,z) = x + 2y and x+y+z=6 and y²+z²=4.
so, we evaluated the partial derivatives of L concerning x, y, z, λ1 and λ2:
∂L/∂x = 1 - λ1
∂L/∂y = 2 - λ1 - 2λ2y
∂L/∂λ1 = x + y + z - 6
∂L/∂λ2 = y² + z² - 4
Staging the partial derivatives = 0 and calculating for x, y, z, λ1 and λ2
x = 3
y = 1
z = 2
λ1 = 1/3
λ2 = -1/3
For the set of equation there are critical points correspond to a maximum or minimum values
f(3,1,2) = 7
the maximum value = 7
minimum value = 7
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How are the hours spent on homework per day related to grade level in school?
Grade in School Hours spent on Homework (per day)
4 1
6 1.5
8 2.5
10 3
12 3.5
a. The higher the grade level in school, the more hours spent on homework.
b. The more hours spent on homework, the lower the grade in school.
c. The higher the grade level in school, the less hours spent on homework.
d. No relationship.
Answers
The requried relation is, the higher the grade level in school, the more hours spent on homework. Option A is correct.
From the given data, we can see that the hours spent on homework per day increase as the grade level in school increases. This suggests that there is a positive correlation between grade level and homework hours.
In general, higher grade levels in school involve more advanced coursework and greater academic expectations, which can require more time and effort outside of class to complete homework and study. Therefore, it is reasonable to expect that students in higher grade levels will spend more hours on homework per day compared to students in lower grade levels.
Thus, the requried relation is, the higher the grade level in school, the more hours spent on homework. Option A is correct.
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The graph below shows the cost of a taxi in New York City. What is the rate of change of the graph? A. $2.00 per mile B. $2.50 per mile C. $4.00 per mile D. $1.00 per mile
Answers
Answer:
c
=
0.6
d
+
1.2
Explanation:
The
3
mile cab ride includes a basic charge as well as the cost for the distance covered.
For a
6
mile trip, the cost is
$
4.80
which means that :
The cost of
3
miles is
$
4.80
−
$
3.00
=
$
1.80
Therefore the cost for
1
mile is
$
1.80
÷
3
=
$
0.60
The basic fee for hiring the cab is
$
3.00
−
$
1.80
=
$
1.20
So the total cost,
c
for a trip of
d
miles will be:
c
=
0.6
d
+
1.2
Check:
If
d
=
3
c
=
0.6
(
3
)
+
1.2
=
3
If
d
=
6
c
=
0.6
(
6
)
+
1.2
=
4.8
Use
m
=
c
2
−
c
1
d
2
−
d
1
with the two given points to find the slope
m
.
Plug this slope
m
(along with one of the points) into
c
0
=
m
d
0
+
b
to find the intercept
b
.
Explanation:
Our goal is an equation of the form
c
=
m
d
+
b
.
The given data says that when our input (distance) is 3 miles, our output (cost) is $3.00. This gives us the data point
(
d
1
,
c
1
)
=
(
3
,
3
)
.
Similarly, since a 6-mile cab ride costs $4.80, this gives us the data point
(
d
2
,
c
2
)
=
(
6
,
4.8
)
.
If our model is linear (which we are told it is), then these two points must be on the line. And two points is all we need to define a line, so these two points can help us write the equation for this linear model.
Step 1: Use
m
=
c
2
−
c
1
d
2
−
d
1
to find the rate
m
that the cab charges per mile.
For any two points, the slope of the line between them is the ratio of "how far the points are from each other vertically" to "how far they are from each other horizontally". In other words, rise over run.
In this question, since cost is modeled as a function of distance, cost will be plotted along the vertical axis (rise), and distance will be along the horizontal axis (run).
m
=
c
2
−
c
1
d
2
−
d
1
=
4.8
−
3
6
−
3
m
=
c
2
−
c
1
d
2
−
d
1
=
1.8
3
=
0.6
So for each extra mile we travel, the cost will go up by $0.60.
Step 2: Use
c
0
=
m
d
0
+
b
to find the base cost
b
for each cab ride.
Knowing that the cost per mile is $0.60, we can use this with one of the given data points to find the base charge
b
. I'll pick the point
(
3
,
3
)
, but it would also work to use
(
6
,
4.8
)
.
c
0
=
m
(
d
0
)
+
b
3
=
0.6
(
3
)
+
b
3
=
1.8
+
b
1.2
=
b
So each cab ride starts with a base charge of $1.20.
Step 3: Place the known values for
m
and
b
into the general equation
c
=
m
d
+
b
.
We now have enough information to write a formula for the cost
c
as a function of distance
d
:
c
=
m
d
+
b
becomes
c
=
0.6
d
+
1.2
or
c
=
$
0.60
d
+
$
1.20
In other words, the cost
c
of a cab ride is the cost per mile ($0.60) times the number of miles
d
, plus the base charge of $1.20.
4) how many strings of length 12 over the alphabet {a, b, c} have at exactly 4 occurrences of a or 4 occurrences of b or 4 occurrences of c? how will your solution change if the length of the string is 17?
Answers
We can sum up the results of all three cases to get the total number of strings with exactly 4 occurrences of a or 4 occurrences of b or 4 occurrences of c. Total number of strings = C(12, 4) * 2^8 + C(12, 4) * 2^8 + C(12, 4) * 2^8.
To find the number of strings of length 12 over the alphabet {a, b, c} that have exactly 4 occurrences of a, 4 occurrences of b, or 4 occurrences of c, we can use combinatorics.
First, let's consider the case of a string length of 12.
We have three possibilities: exactly 4 occurrences of a, exactly 4 occurrences of b, or exactly 4 occurrences of c. We will calculate each case separately and then sum up the results.
Case 1: Exactly 4 occurrences of a.
We have 12 positions in the string and we need to choose 4 of them to place the a's. The remaining 8 positions can be filled with b's or c's, giving us 2 options for each position. Therefore, the number of strings with exactly 4 occurrences of a is C(12, 4) * 2^8.
Case 2: Exactly 4 occurrences of b.
Similar to Case 1, we have 12 positions and we need to choose 4 of them to place the b's. The remaining 8 positions can be filled with a's or c's, giving us 2 options for each position. So, the number of strings with exactly 4 occurrences of b is C(12, 4) * 2^8.
Case 3: Exactly 4 occurrences of c.
Again, we have 12 positions and we need to choose 4 of them to place the c's. The remaining 8 positions can be filled with a's or b's, giving us 2 options for each position. Hence, the number of strings with exactly 4 occurrences of c is C(12, 4) * 2^8.
If the length of the string is 17, the approach will remain the same. We will calculate the number of strings with exactly 4 occurrences of a, 4 occurrences of b, or 4 occurrences of c using the formula mentioned above. The only difference will be in the length of the string and the number of positions to be chosen for each case. We will have C(17, 4) instead of C(12, 4), and the remaining positions will be filled with the remaining characters from the alphabet.
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how many independent variables can you test in a single experiment?
Answers
The number of independent variables that can be tested in a single experiment depends on various factors, including the research design, resources, and the complexity of the study. In general, it is best to limit the number of independent variables to maintain control over the experiment and facilitate clear interpretation of results.
In simpler experiments, researchers often focus on a single independent variable to study its effect on the dependent variable while controlling for other factors. This allows for a clear cause-and-effect relationship to be established.
However, in more complex experiments, researchers may introduce multiple independent variables to examine their combined effects or interactions. This can provide insights into how different factors influence the outcome simultaneously.
The exact number of independent variables that can be tested in a single experiment varies, but it is generally recommended to keep the number manageable and avoid excessive complexity. This ensures that the experiment remains feasible, interpretable, and effectively addresses the research question or hypothesis.
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A baseball team plays the same opponent six times in a season. The set {0,1,2,3,4,5,6} describes the possible number of wins for the six games.
Set A contains the number of wins when exactly four games are won.
Set B contains the number of wins when at least four games are won.
Which of these statements are true? Choose all that are correct.
The union of set A and B is an empty set
The complement of the union of set A and B is {0,1,2,3} set A
The complement of set B is {0,1,2,3}
The intersection of set A and set B is an empty set
The complement of set B is {1,2,3}
Answers
- The complement of set B is {0,1,2,3}
- The complement of set B is {1,2,3}
A group of students take a test. The group consists of 8 girls and 12 boys. The mean mark for the girls is 18. The mean mark for the boys is 16. 5. Calculate the mean mark for the whole group.
Answers
First, let's calculate the total number of marks for each group:
Girls: 8 students * 18 marks/student = 144 marksBoys: 12 students * 16 marks/student = 192 marksThen, we can calculate the mean mark for the whole group by adding up the total number of marks for each group and dividing by the total number of students:
(144 marks + 192 marks) / (8 students + 12 students) = 336 marks / 20 students = 16.8 marks/student
Therefore, the mean mark for the whole group is 16.8.
The mathematical mean is a measure of central tendency that is calculated by adding up all the values in a data set and dividing by the number of values in the set. It is also commonly known as the "average."
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Answer:
34.5
Step-by-step explanation:
(a) 4x - 3y = 6,
7x + 3y = 27.
(d)
12x + 19y= 22,
18x + 23y = 44.
(g) 53x +41y = 147,
41x + 53y =135.
g
Answers
Then we substitute x=+3/4+3/2 to the second equation (7x+3y=27)and we do the calculations to find y
7(3/4+3/2)+3y=27
21/4y+21/2+3y=27
21y+42+12y=108
33y=66
Y=2
We substitute y (2) to the first equation 4x-3y=6 to find x
4x -3(2)=6
4x=12
X=3
Help please select the Awnser choices right awnsers only AWNSER BOTH QUESTIONS PLEASE
Answers
Answer:
Step-by-step explanation:
1. B
2. True
Answer: the awnser is D
Step-by-step explanation:
the awnser would be d becuase the others dont make sense at all enegry cannot be destroyed at all but it can change form hope this helps
hey can someone help me pls *ANSWER ASAP*
Answers
This is a translation of 2 units to the left and 5 units up, so the correct option is the first one.,
Which is the translation applied?Remember that a vertical translation of N units is:
g(x) = f(x) + N
if N < 0 the translation is down.
if N > 0 the translation is up.
And a horizontal translation of N units is:
g(x) = f(x + N)
If N > 0 the translation is to the right.
if N < 0 the translation is to the left.
Here we have the transformation:
f(x) = x²
g(x) = (x + 2)² + 5
So this is a translation of 2 units to the left and 5 units up.
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Mauricio can string 8 guitars in 2 hours how many can he string in 15 ?
Answers
Answer:
He strings 52 guitars in 15 hours
Which of the following results in the difference of two squares?
a. (7x - 3y)^2
b. (4x - 9y)(4x - 9y)
c. 5x^2(4x - 3)
d. (2x + 8y)(2x - 8y)
Answers
Answer:d
Step-by-step explanation:
Jada saw a music video that her favorite artist recently posted online. The equation v (t) = 320 ⋅ 4t represents the number of views v of the video, t days since Jada saw the video. What does v (2) equal? (only enter number)
Answers
Answer:
2,560
Step-by-step explanation:
The equation v(2) is equal to 2560.
What is a function?A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
v(t) = 320 x 4t ________(1)
v(t) represents the number of views v of the video, t days since Jada saw the video.
We see that,
v(t) is a function where t is the input and v(t) is the output.
Now,
Substitute t = 2 in (1)
v(t) = 320 x 4 x 2
= 320 x 8
= 2560
Thus,
The v(2) is equal to 2560.
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PLS HELP ASAP!!! 70 POINT QUESTION NO SCAM
WHAT IS THE MISSING NUMBER TO THE SUBTRACTION
Answers
Answer: 0.985
Step-by-step explanation: The answer shown is accurate.
Answer:
1-0.985
Step-by-step explanation:
0.985+0.015=1. How I figured it out is 1-x=0.015, it must be 0.985. Hope it helps!
Help with this please. I’d appreciate if you showed the steps and the answer. Thanks!
Answers
Answer:
Diagonal=11.3m
Area of square (A) =1/2*d^2
=1/2*(11.3)^2
=63.845m^2
We know,
Area of square A= l^2
63.845=l^2
L=
\( \sqrt{63.845 } \)
L=7.99m
Now,
Perimeter of square =4l
=4*7.99
=31.96m
Helppppp!!!!!! Quick :(((
Answers
Which one of these represents a linear function?
Answers
Out of all of these, it would make most sense for this option to be the linear function
This is the only equation that makes a line
Order these from least to greatest: .25, 3/8, 7/12, 5/16, .5
Answers
To make it easier change the fraction into decimals!
3/8 = .375
5/16 =.3125
7/12 = .583
.5 = 0.5
.25 = 0.25
So, according to my calculations I believe it goes like this: 0.25, 0.375, 0.3125, 0.5, .583
Really I am not sure if this is correct but I hope this helps you understand you can order it whatever way you want but make sure you understand the explanation :) have a great day!!
Set 2 Maths has 30 pupils in it.
Set 1 Maths has 80% of the number of pupils in Set 2
How many pupils does Set 1 Maths have?
Answers
What is the quotient for this division problem? x+33x2−2x+5 A −3x−7+26x+3 B −3x+11+38x+3 C 3x+7+26x+3 D 3x−11+38x+3
Answers
The quotient of given polynomials division is x+2. So, the correct answer is option A.
What is a quotient?A quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division, or as a fraction or a ratio.
Given that, (x²+3x+2)÷(x+ 1)
x²+3x+2 can be written as
x²+1x+2x+2 (Splitting of middle term)
= x(x+1)+2(x+1)
= (x+1)(x+2)
Now, (x+1)(x+2)÷(x+1)
= [(x+1)(x+2)]/(x+1)
= (x+2)
Therefore, option A is the correct answer.
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What is the quotient of the following division problem (x²+3x+2)÷(x+ 1).
a. x+ 2
b. x +1
c. x²+3x +2
d. 0
What does the expression n+3 represent??
Answers
Answer:
(?) +3
Explanation:
it would be whatever N equals then add 3
3. A leaking tap drips water at 0,5 ml/sec. Convert this rate to l/h.
Answers
Answer: 1.8 L/h
Step-by-step explanation:
To convert the rate of water dripping from a tap from millilitres per second (ml/sec) to litres per hour (L/h), we need to use conversion factors.
Step 1:
First, let's convert the rate from millilitres per second to litres per second.
There are 1000 millilitres in a litre, so we can divide the rate in millilitres per second by 1000 to get the rate in litres per second:
\(\LARGE \boxed{\textsf{0.5 ml/sec $\div$ 1000 = 0.0005 L/sec}}\)
Step 2:
We can convert the rate from litres per second to litres per hour. There are 3600 seconds in an hour, so we can multiply the rate in litres per second by 3600 to get the rate in litres per hour:
\(\LARGE \boxed{\textsf{0.0005 L/sec $\times$ 3600 = 1.8 L/h}}\)
Therefore, the rate of water dripping from the tap is 1.8 L/h.
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Given the following equation, select all statements that are true.
y=−7x+2
Question 3 options:
positive slope
negative slope
positive y-intercept
negative y-intercept
positive x-intercept
negative x-intercept
Answers
y=-7x+2
m=-7, so that's a negative slope.
b=2, so the y-intercept is (0,2). That's a postiive y-intercept.
Solve 0=-7x+2 to find the x-intercept.
7x=2
x=2/7
The x-intercept is (2/7,0), so that's a positive x-intercept.
Lina has 12 coins in her purse. D of them are dimes and the rest are quarters.
How many quarters does Lina have?
How much money does Lina have in quarters?
Answers
Answer:
she has 12 - D quarters.
She has 24 (12-D) cents in quarters
Step-by-step explanation:
You know that the number of coins, and you know the number of dimes - even if the amount is a variable - you can find the amount of quarters with an equation.
If you knew that Lina had 6 dimes instead of D, you would say 12 - 6 and get the number of quarters. You can do the same thing here.
and If you know Lina has 6 quarters, you could find the amount of money in cents by multiplying 6 by 25. You can do that here too, but with variables and an equation.
Answer:
she has 12 - D quarters.
Step-by-step explanation:
An object is dropped from 40 feet below the tip of the pinnacle atop a 716-ft tall building. The height h of the object after t seconds is given by the equation h=−16t2+676. Find how many seconds pass before the object reaches the ground.
Answers
Answer:
Below
Step-by-step explanation:
When it reaches the ground h = 0
h = 0 = -16t^2+676
16t^2 = 676
t^2 = 42.25 t = 6.5 s
What is 19.254 rounded to the nearest tenth and the nearest hundredth? 100 POINTS
a 19.2 and 19.25
b 19.2 and 19.26
c 19.3 and 19.25
d 19.3 and 19.26
Answers
Answer:
A: 19.3 and 19.25
Step-by-step explanation:
When rounding decimals if the number in the decimal place to the right is higher than 5 you round up. If not keep keep the same.
The rounding off 19.254 to the nearest tenth and the nearest hundredth will be 19.3 and 19.25 respectively.
What is rounding off ?
Rounding off means a number is made into simpler form by keeping its value fixed but closer to the next number.
To round off the number to the nearest tenth we need to look at the hundredths place digit. If the digit is 1, 2, 3, 4 then we don't have to do anything, if the digit is 5, 6, 7, 8, 9 we must round up.
here, The number 19.254 at tenth place is 5 so round up the hundredths place digit, hence the rounded off to the nearest tenth will be 19.3.
and also,
To round off the number to the nearest hundredth we need to look at the thousandth place digit. If the digit is 1, 2, 3, 4 then we don't have to do anything, if the digit is 5, 6, 7, 8, 9 we must round up.
here, The number 19.254 at hundredth place is 4 so round up the thousandth place digit, hence the rounded off to the nearest hundredth will be 19.25.
Therefore, the rounding off 19.254 to the nearest tenth and the nearest hundredth will be 19.3 and 19.25 respectively.
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10
10
-10
What is the slope of the line shown on the graph?
A)
-1
B)
0
1
D
undefined
Answers
Answer:
A
Step-by-step explanation:
Calculate the slope m using the slope formula
m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)
with (x₁, y₁ ) = (0, 0) and (x₂, y₂ ) = (- 5, 5) ← 2 points on the line
m = \(\frac{5-0}{-5-0}\) = \(\frac{5}{-5}\) = - 1 → A
If Albert owns x model airplanes and Maxim owns x + 13 model airplanes, then Albert and Maxim own___ model airplanes together. Albert owns ___ less model airplanes than Maxim.
Answers
Answer:
2x + 13
13
Step-by-step explanation:
Albert and Maxim own
x + (x + 13) = 2x + 13 model airplanes together
Albert owns
(x + 13) - x = 13 less model airplanes than Maxim.
Answer:
They own 2x+13 model airplanes together
Albert owns 13 less model airplanes than Maxim
i need all the help. thank you:))
