Show that the sequence defined by
a1= 1
an+1 = 3-(1/an)
is increasing and an<3 for all n. Deduce that{an} is convergent and find its limit

Answer:

B 16

Step-by-step explanation:

It will take 8 days to eat one carton so 2 cartons will be 16 days.

The values of h is v3 in span fv1; v2g and h is fv1; v2; v3g is 4.

If we have given a vector v of initial point A and terminal point B

v = ai + bj

then the components form as;

AB = xi + yj

Here, xi and yj are the components of the vector.

We are given that;

v1 and v2 are not scalar multiples to each other

So, they are not linearly dependent

Now,

1(10h -42) -3(-14 +3h) +2 (18-20) =0

10h -9h-4 =0

h = 4

Therefore, by vectors the answer will be 4 so that the vector v3 is in {v1,v2}.

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The evaluation of equations by resolving into quadratic equations and factoring are presented as follows;

A quadratic equation is an equation which can be expressed as; f(x) = a·x² + b·x + c, where a, b, and c are real numbers and a ≠ 0.

1. √(30 - x) = x

x = √(30 - x)

x² = 30 - x

x² + x - 30 = 0

Factoring the above quadratic equation, we get;

(x + 6)·(x - 5) = 0

2. x = √(42 - x)

x² = 42 - x

x² + x - 42 = 0

(x + 7)·(x - 6) = 0

3. √(12 - r) = r

Squaring both sides of the equation, we get;

(12 - r) = r²

r² = 12 - r

r² + r - 12 = 0

(r + 4)·(r - 3) = 0

4. m = √(56 - m)

m² = 56 - m

m² + m - 56 = 0

(m + 8)·(m - 7) = 0

5. r = √(-1 - 2·r)

r² = -1 - 2·r

r² + 2·r + 1 = 0

(r + 1)² = 0

6. √(4·n + 8) = n + 3

4·n + 8 = (n + 3)² = n² + 6·n + 9

4·n + 8 = n² + 6·n + 9

n² + 6·n + 9 - (4·n + 8) = 0

n² + 6·n + 9 - 4·n - 8 = 0

n² + 2·n + 1 = 0

(n + 1)² = 0

7. -n + √(6·n + 19) = 2

-n + √(6·n + 19) = 2

√(6·n + 19) = n + 2

(6·n + 19) = (n + 2)² = n² + 4·n + 4

6·n + 19 = n² + 4·n + 4

n² + 4·n + 4 = 6·n + 19

n² + 4·n + 4 - (6·n + 19) = 0

n² - 2·n - 15 = 0

(x - 5)·(x + 3) = 0

8. 4 + √(-3·m + 10) = m

√(-3·m + 10) = m - 4

-3·m + 10 = (m - 4)² = m² - 8·m + 16

-3·m + 10 = m² - 8·m + 16

m² - 8·m + 16 = -3·m + 10

m² - 8·m + 16 + 3·m - 10 = 0

m² - 5·m + 6 = 0

(m - 3)·(m - 2) = 0

9. x - 5 = √(x + 1)

Squaring both sides, we get;

(x - 5)² = (√(x + 1))²

(x - 5)² = x + 1

x² - 10·x + 25 = x + 1

x² - 10·x + 25 - (x + 1) = 0

x² - 10·x + 25 - x - 1 = 0

x² - 11·x + 24 = 0

(x - 8)·(x - 3) = 0

10. n - 7 = √(3·n - 21)

(n -7)² = (3·n - 21)

n² - 14·n + 49 = 3·n - 21

n² - 14·n + 49 - (3·n - 21) = 0

n² - 14·n + 49 - 3·n + 21 = 0

n² - 17·n + 70 = 0

(n - 10)·(n - 7) = 0

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Loren could receive a monthly payout of approximately $4,646.81 from the annuity to last for 30 years at a 10% annual interest rate.

What is annuity?

A contract between you and an insurance company known as an annuity entails you making a lump sum payment or a series of payments in exchange for regular payments starting either right away or in the future.

To calculate the monthly payout that Loren can receive from an annuity for 30 years, use the formula for the present value of an annuity -

PV = PMT x [(1 - (1 + r)^(-n)) / r]

Where PV is the present value of the annuity (the amount of money Loren has at retirement), PMT is the monthly payout, r is the monthly interest rate (10%/12 = 0.00833), and n is the number of monthly payments (30 years x 12 months/year = 360 months).

Substituting the given values into the formula -

$500,000 = PMT x [(1 - (1 + 0.00833)^(-360)) / 0.00833]

Solving for PMT -

PMT = $500,000 / [(1 - (1 + 0.00833)^(-360)) / 0.00833]

PMT = $4,646.81

Therefore, The value is obtained as $4,646.81.

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Answer:

To simplify the expression −8(1+x)+7x, we can use the distributive property of multiplication over addition, which states that a(b+c) = ab + ac. Applying this property, we get:

−8(1+x) + 7x = −81 − 8x + 7x (distributing the -8)

= -8x - 8 + 7x (simplifying the multiplication)

= -x - 8 (combining like terms)

Therefore, the simplified form of the expression −8(1+x)+7x is −x − 8.

to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below.

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{70})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{100}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{100}-\stackrel{y1}{70}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{2}}} \implies \cfrac{ 30 }{ 2 } \implies 15[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{70}=\stackrel{m}{ 15}(x-\stackrel{x_1}{2}) \\\\\\ y-70=15x-30\implies {\Large \begin{array}{llll} y=15x+40 \end{array}}[/tex]

Answer:

See below

Step-by-step explanation:

It's basically the perimeter of a circle, or it's total distance around it.

Answer:

2.7 cm

Step-by-step explanation:

We have the relationship
tan 56 = 4/z

Therefore z = 4/ (tan 56) = 4/1.48256 = 2.698
= 2.7 cm correct to one decimal place

Answer:

C

Step-by-step explanation:

x + y + 10 = 0 ( subtract x + 10 from both sides )

y = - x - 10 → (1)

x² + y - 2 = 0 → (2)

substitute y = - x - 10 into (2)

x² - x - 10 - 2 = 0

x² - x - 12 = 0 ← in standard form

(x - 4)(x + 3) = 0 ← in factored form

equate each factor to zero and solve for x

x + 3 = 0 ⇒ x = - 3

x - 4 = 0 ⇒ x = 4

The fraction of these 36 chemical elements that are named for women or places is 27/36 or by simplifying it, 3/4.

A fraction is defined as a mathematical concept that can be used to express a portion of a whole or a number that can be written as the ratio of two integers.

Here,
To find the fraction of the 36 chemical elements that are named for women or places, we need to add the number of elements named for women (2) and the number of elements named for places (25), and then divide by the total number of elements (36),

Fraction = (2 + 25) / 36 = 27 / 36

Therefore, the fraction of these 36 chemical elements that are named for women or places is 27/36 or simplify it, 3/4.

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The height of the balloon is 1.3 miles above the ground.

Height is the measure of vertical distance, either how "tall" something or someone is, or how "high" the point is. For example, a person's height is typically measured using a stadiometer and is recorded in centimeters or feet and inches. The height of a mountain is usually measured in meters or feet. In aviation, height is measured from the surface of the earth, either above ground level (AGL) or above mean sea level (AMSL).

The answer to this question can be found by using the trigonometric formula of Tangent. To calculate the height of the balloon, we must find the length of the hypotenuse in the triangle formed by the two towns and the balloonist.

Using the formula for tangent, we can calculate the length of the hypotenuse. The formula is:

tan (angle) = opposite side/adjacent side

In this case, the opposite sides are the two towns, and the adjacent side is the height of the balloon.

For the first town, the formula is:

tan 36° = 1.6/x

Solving for x, we can calculate the length of the hypotenuse for the first town to be 2.8 miles.

For the second town, the formula is:

tan 32° = 1.6/x

Solving for x, we can calculate the length of the hypotenuse for the second town to be 2.9 miles.

Since the hypotenuse is the total distance between the two towns and the balloonist, the height of the balloon can be calculated by subtracting the total length of the two towns (1.6 miles) from the hypotenuse of the second town (2.9 miles).

The height of the balloon is 1.3 miles above the ground.

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Answer:

What is height?

Height:

When a rectangle is drawn with horizontal and vertical sides, the word height makes it clear which dimension is meant; height labels how high (how tall) the rectangle is. That makes it easy to indicate the other dimension—how wide the rectangle is from side to side—by using the word width. And if the side-to-side measurement is greater than the height, calling it the length of the rectangle is also acceptable, as it creates no confusion.

The answer to this question can be found by using the trigonometric formula of Tangent. To calculate the height of the balloon, we must find the length of the hypotenuse in the triangle formed by the two towns and the balloonist.

Using the formula for tangent, we can calculate the length of the hypotenuse. The formula is:

tan (angle) = opposite side/adjacent side

In this case, the opposite sides are the two towns, and the adjacent side is the height of the balloon.

For the first town, the formula is:

tan 36° = 1.6/x

Solving for x, we can calculate the length of the hypotenuse for the first town to be 2.8 miles.

For the second town, the formula is:

tan 32° = 1.6/x

Solving for x, we can calculate the length of the hypotenuse for the second town to be 2.9 miles.

Since the hypotenuse is the total distance between the two towns and the balloonist, the height of the balloon can be calculated by subtracting the total length of the two towns (1.6 miles) from the hypotenuse of the second town (2.9 miles).

The height of the balloon is 1.3 miles above the ground.

Answer:

------------------------------

As per question we have:

Find the required segment length using segment addition rule:

We can conclude that -

A test statistic measures the degree of agreement between a sample of data and the null hypothesis.

Given is that you are testing the claim that the mean GPA of night students is different from the mean GPA of day students

T scores (or T statistics) are used to test the difference between a

sample mean and another sample mean or some theoretical value.

So, we can write the T score as -

T{score} = 3.06 - 2.83

T{score} = 0.23

DF = 0.82 - 0.45

DF = 0.37

Level of significance = 10%

We can write the p - value as -

p - value = 0.89

Also -

The result is not significant at p < 0.10.

Therefore, we can conclude that -

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The least number of bars of soap that must be made on any given day for the factory to make a profit that day is 369 bars.

Two expressions in an equation are combined using the equal symbol ("="). The "left-hand side" and "right-hand side," respectively, of the equation refer to the two expressions that are on either side of the equals sign. Usually, we consider an equation's right side to be zero.

We have the equation ,

y = 0.84 x + 428

Let n be the least number of bars required to make a profit.

then,

2n = 0.84n + 428

dividing both sides by 2

n = 0.42n + 214

0.58n = 214

n = 368.96

n ≈ 369 bars

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There are 16 sweets that are fizzy and orange and the fraction is 4/15

From the question, we have the following parameters that can be used in our computation:

Fizzy = 2/5

Orange = 4/6 of Fizzy

So, we have

Orange = 2/5 * 4/6

Orange = 4/15

Using (a) as a guide, we have

Fizzy and orange​ = 4/15 * 60

Fizzy and orange​ = 16

Hence, the fizzy and orange​ is 16

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Answer:

60

Step-by-step explanation:

a tringle always adds ups to 180 add them then subtract.

Answer:

              The x-intercept of Line A is (-3, 0), and the x-intercept of Line B is (-1, 0). Therefore, the x-intercept of Line A is less than the x-intercept of Line B.

Step-by-step explanation:

              The x-intercept of a graph or function is where the line hits the x-axis on a coordinate plane. In other words, the y-value of the x-intercept is 0. Since we are looking for the x-intercept of Line A, we can use the given equation to solve for it by inserting the value y = 0 (as mentioned above, the x-intercept of a function will have the y-value as 0). By solving the equation, you'll get that [tex]-2=\frac{1}{3} (x-3)[/tex] ⇒ [tex]-6=x-3[/tex] ⇒ [tex]x = -3[/tex]. Therefore, the x-intercept of Line A is (-3, 0).

              The x-intercept on a graph is easier to find, as it is where the line touches the x-axis. So the x-intercept of Line B would be (-1, 0). Now that we have both x-intercepts, we can compare them and come up with the final answer that is required. As -3 is less than -1, we can conclude that the x-intercept of Line A is (-3, 0), and the x-intercept of Line B is (-1, 0). Therefore the x-intercept of Line A is less than the x-intercept of Line B.

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The original number that satisfies the condition the square of a non-zero number is equal to 68% of the original number is 1.68.

Variables and constants are used to create algebraic expressions using a variety of techniques.

It is a mathematical expression made up of terms and includes variables, constants, and algebraic operations like addition, subtraction, etc. When operations like addition, subtraction, multiplication, division, etc. are performed on any variable, the resulting equations are called algebraic expressions.

An algebraic expression (or variable expression) is the name given to a grouping of terms that have undergone operations including addition, subtraction, multiplication, and division.

Let us suppose a non zero number = x.

Given that, the square of a non-zero number is equal to 68% of the original number.

This can be represented algebraically as follows:

x² = x(1 + 68/100)

x² = x(1 + 0.68)

x² = x(1.68)

x = 1.68

Hence, the original number that satisfies the condition the square of a non-zero number is equal to 68% of the original number is 1.68.

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Answer:

Step-by-step explanation:

Since the probability of rain on any given day is 0.5, and the weather on each day is independent of the others, we can calculate the probability of it raining all three days by multiplying the probabilities of rain on each individual day:

P(rain on day 1) x P(rain on day 2) x P(rain on day 3) = 0.5 x 0.5 x 0.5 = 0.125

Therefore, the probability of it raining all three days of your trip to Seattle in October is 0.125, or 12.5%.

Answer:

while True:

   user_input = input("Please enter a sequence of comma-separated numbers: ")

   if user_input.replace(",", "").isdigit():

       break

   print("Invalid input! Please try again.")

num_list = user_input.split(",")

num_list = [int(num) for num in num_list]

total_sum = sum(num_list)

print("The sum of the numbers is:", total_sum)

Step-by-step explanation:

This program uses a while loop to repeatedly ask the user for input until they provide a valid sequence of comma-separated numbers (all-digit input). The .isdigit() function is used to check if the input is all-digit. If the input is valid, the program splits the input string at each ',' using the .split() function, converts each item in the resulting list into an integer using list comprehension, then adds all numbers together using the sum() function. Finally, the program outputs the total sum.

The key features of this rational function include the following:

Domain: (-∞, 2) ∪ (2, ∞)

Range: (-∞, 1) ∪ (1, ∞)

Y-intercept: 1.

X-intercept: 2.

Vertical asymptote: x = -2.

Horizontal asymptote​: y = -1.

In Mathematics, a rational function simply refers to a type of function  which is expressed as a fraction. This ultimately implies that, a rational function is composed of two (2) main parts and these include the following:

Additionally, the vertical extent of any graph of a rational function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

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The true statements about the populations of the town based on the linear equations in the table are;

A linear equation is an equation of the form; y = m·x + c, where the slope of the graph of the equation is m and the y-intercept of the graph is c.

The analysis of the equations in the table are as follows;

Equation for Pinehill is; P = 800 - 20t

The slope of the above equation is; -20 and the y-intercept is 800. The negative sign for the slope indicates that the population is decreasing;

Therefore, the initial population of 800 in Pinehill is decreasing at a rate of 20 per unit time, t

Population for Rye; P = 500 + 15t

The population of Rye is increasing at a rate of 15 per unit time

Population for Smithfield; P = 10t + 950

The population of Smithfield is increasing at a rate of 10 per unit time

Population for Troy; P = -50·t + 600

The -50 indicates that population is decreasing at a rate of 50 per unit time

The true statements are therefore;

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The answer comes to $43,592.00. 28 can be wiped from the account after 20 years, leaving nothing behind.

Think about taking out a $1,000 loan with a seven-year, 10% interest rate. The first year's interest would cost you $100. You would pay $1,100 in interest the next year, which is the sum of the initial amount plus interest. Because of this, your salary for the upcoming year will be $110 (1,100 times 0.10).

Calculate the Yearly Amount (A) given the Prepaid Value (P), at interest rate I over n periods using the equation (A/P, I n).

The equation is:

[tex]A=P \frac{i(1+i)^n}{(1+i)^n-1}[/tex]

Where:

P is the Present value = $500,000

A is the monthly Amount

i is the interest rate = 0.06

20 years are the number of periods (n).

Substituting values:

[tex]A=\$ 500,000 \frac{0.06(1.06)^{20}}{(1.06)^{20}-1}=\$ 43,592.28[/tex]

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The number of hamburgers sold in the winter is 57 million.

What is percentage ?

Percentage is a way of expressing a fraction or ratio as a portion of 100. It is denoted by the symbol "%".

For example, if there are 10 green balls in a bag of 20 balls, we can say that the percentage of green balls is:

(10/20) x 100% = 50%

This means that 50% of the balls in the bag are green.

Percentages are often used to represent proportions or amounts in many contexts, such as finance, statistics, and everyday life. They can be used to express changes, comparisons, and relationships between different quantities.

In addition, percentages are commonly used in calculations involving tax, interest rates, discounts, and other types of financial transactions. It is important to have a good understanding of percentages in order to manage personal finances, make informed business decisions, and interpret data accurately.

Let's suppose that the total number of hamburgers sold by the fast food chain in a year is represented by the height of the bars in the bar graph. We know that 25% of the hamburgers are sold in the fall, which means that the bar representing the fall sales must be 25% of the total height of the bars.

Since there are four seasons in a year, each season must represent 25% / 4 = 6.25% of the total height of the bars. Therefore, the height of each bar is 6.25% of the total number of hamburgers sold.

Let's denote the number of hamburgers sold in the winter by x. Then, the height of the winter bar is also x/100 of the total height of the bars.

We know that the winter bar is covered by a smudge, but we can still use the information we have to set up an equation. Since the heights of the bars must add up to the total height, we can write:

25% + 6.25% + 6.25% + 6.25% + x/100 = 100%

Simplifying the left-hand side, we get:

43% + x/100 = 100%

Subtracting 43% from both sides, we get:

x/100 = 57%

Multiplying both sides by 100, we get:

x = 57 million hamburgers.

Therefore, the number of hamburgers sold in the winter is 57 million.

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Using Chebyshev's Theorem, considering a standard deviation of 40, we have that at least 75% of the students scored between 360 and 520.

When we have no information about the population distribution, Chebyshev's Theorem is used. It states that:

At least 75% of the measures are within 2 standard deviations of the mean.

At least 89% of the measures are within 3 standard deviations of the mean.

An in general terms, the percentage of measures within k standard deviations of the mean is given by.

here, we have,

In this problem, considering a standard deviation of 40, we have that:

440 - 2 x 40 = 360.

440 + 2 x 30 = 520.

Within 2 standard deviations of the mean, no information about the distribution, hence, at least 75% of the students scored between 360 and 520.

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The solution of the system of equations is

A system of equations are a pair of equations.

Given the system of equations

4/x - 2/y = 12 (1) and

5/x + 1/y = 8 (2)

We proceed to solve as follows.

Since we have

4/x - 2/y = 12 (1) and

5/x + 1/y = 8 (2)

Multiplying equation (1) by 1 and equation (2) by 2, we have that

4/x - 2/y = 12 (1) × 1 and

5/x + 1/y = 8 (2) × 2

4/x - 2/y = 12 (4)  and

10/x + 2/y = 16 (5)

Now, adding equations (4) and (5), we have that

4/x - 2/y = 12 (4)  

+

10/x + 2/y = 16 (5)

14/x = 28

14 = 28x

x = 14/28

x = 1/2

Substituting x = 1/2 into equation (1), we have that

4/x - 2/y = 12 (1)

4/(1/2) - 2/y = 12 (1)

4 × 2 - 2y = 12

8 - 2y = 12

-2y = 12 - 8

-2y = 4

y = 4/-2

y = -2

So,

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The marginal revenue at 8 units is given as follows:

MR(8) = 288 dollars per unit.

The revenue function is defined as follows:

R(q) = -7q² + 400q.

The marginal revenue function is the derivative of the revenue function, hence it is obtained as follows:

MR(q) = -14q + 400.

At 8 units, the marginal revenue is found with the numeric value of MR(q) at q = 8, hence:

MR(8) = -14(8) + 400 = 288 dollars per unit.

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The tables that show a valid probability density function, are tables 1 to 3

From the question, we have the following parameters that can be used in our computation:

Table of values

For a table to show a valid probability density function, the following must be true

Sum of P(X = x) = 1

Using the above as a guide, we have the following:

Table 1: Sum = 0.37  + 0.06 + 0.01 + 0.56 = 1

Table 2: Sum = 3/8 + 3/8 + 1/4 = 1

Table 3: Sum = 1/8 + 1/8 + 1/8 + 3/8 + 1/4 = 1

Table 4: Sum = 0.03 + 0.01 + 0.61 + 0.31 + 0.21 = 1.17

Table 5: Sum = 1/10 + 3/10 + 4/5 + 1/5 = 1.4

Hence, the tables are tables 1 to 3

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The individual events are dependent. The probability of the combined event is of:

0.0137 = 1.37%.

The mass probability formula is presented as follows:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

As the students are chosen without replacement, the events are dependent.

The parameter values for this problem are given as follows:

N = 72 students, k = 18 seniors, n = sample size of 3.

The probability of three seniors is P(X = 3), hence:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 3) = h(3,72,3,18) = \frac{C_{18,3}C_{54,0}}{C_{72,3}} = 0.0137[/tex]

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Answer:

Step-by-step explanation:

(a) To find the minimum possible mass of the four barrels, we want to arrange them in a way that minimizes the total mass. One way to do this is to put the two lightest barrels together and the two heaviest barrels together. This way, the total mass is minimized because the lighter barrels balance out the heavier ones.

So, the minimum possible mass of the four barrels is:

2(30 kg) + 2(30 kg) = 120 kg

(b) To find the maximum possible mass of the four barrels, we want to arrange them in a way that maximizes the total mass. One way to do this is to put the heaviest barrel with the lightest barrel and the other two barrels together. This way, the total mass is maximized because the heaviest and lightest barrels are paired together, making the total mass greater than if they were all paired together.

So, the maximum possible mass of the four barrels is:

30 kg + 90 kg + 30 kg + 30 kg = 180 kg