The amount of chlorine in the tank as a function of time is:
C(t) = 0.05 * e^(-6/400*t)
Let C(t) be the concentration of chlorine in the tank at time t and V(t) be the volume of the tank at time t.
At t=0, we know that:
C(0) = 0.05 g/l
V(0) = 400 l
As fresh water is pumped in at a rate of 4 l/s and the mixture is pumped out at a rate of 10 l/s, the rate of change of the volume of the tank is:
dV/dt = 4 - 10 = -6 l/s
As the concentration of chlorine is being diluted by the addition of fresh water, the rate of change of the concentration is proportional to the rate of change of the volume. So we can write:
dC/dt = - k * C(t) * (dV/dt)
where k is a proportionality constant.
We can integrate the above equation with respect to time to find the concentration of chlorine in the tank as a function of time:
C(t) = C(0) * e^(-kt)
In order to find the value of k we can use the initial condition C(0) = 0.05 g/l and V(0) = 400l
0.05 = C(0) * e^(-k*0)
0.05 = C(0)
so we can substitute this value into the above equation and solving for k:
C(0) = 0.05 = 0.05 * e^(-k*0)
0.05 = 0.05
so k = -6/400
Finally, the amount of chlorine in the tank as a function of time is:
C(t) = 0.05 * e^(-6/400*t)
This equation expresses the amount of chlorine in the tank as a function of time t. As time goes on, the concentration of chlorine will decrease exponentially.
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Nick and Lauren hiked 3. 75 kilometers to a lake 45° east of south from their campsite. Then they hiked 30° west of north to the nature center 6. 5 kilometers from the lake. How many kilometers are they away from their campsite? Round to the nearest tenth
Answer:
Step-by-step explanation: 23.4
A chemical company make two brand of antifreeze. The firt brand i 60% pure antifreeze, and the econd brand i 85% pure antifreeze. In order to obtain 80 gallon of a mixture that contain 70% pure antifreeze, how many gallon of each brand of antifreeze mut be ued?
To obtain 80 number of gallons of a mixture that contains 70% pure antifreeze, you must use 56 gallons of the 60% pure antifreeze and 24 gallons of the 85% pure antifreeze.
1. Calculate the total amount of antifreeze needed to make 80 gallons of the 70% mixture.
80 gallons x 0.7 = 56 gallons of antifreeze
2. Calculate the amount of 60% pure antifreeze needed.
56 gallons x 0.6 = 33.6 gallons of 60% pure antifreeze
3. Calculate the amount of 85% pure antifreeze needed.
56 gallons x 0.85 = 47.6 gallons of 85% pure antifreeze
4. Subtract the amount of 60% antifreeze from the total amount of antifreeze needed to determine the amount of 85% antifreeze needed.
56 gallons - 33.6 gallons = 22.4 gallons of 85% pure antifreeze
5. Round the amounts to the nearest whole number.
33.6 gallons of 60% pure antifreeze and 24 number of gallons of 85% pure antifreeze.
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32 to 28 decrease percentage
Answer:
Step-by-step explanation:
To calculate a percentage decrease, first calculate the difference (decrease) between the two numbers you are comparing. Next, divide the decrease by the original number and multiply the answer by 100.
In this case, our difference is 32-28=4
4 divided by 32= 4/32, and multiplying by 100 gives us 400/32=12.5
Therefore, the percent decrease is 12.5%
4x+40=y
HELPPP PLEASE
Here we are given a right angled triangle in which there are two unknown angles x and y . we need to find out the values of x and y. Firstly, as we know that the angle sum property of a triangle is 180° . So that,
[tex]\longrightarrow x + y +90^o = 180^o\\[/tex]
[tex]\longrightarrow x + y = 180^o -90^o = 90^o \dots \\[/tex]
[tex]\longrightarrow y = 90^o - x \dots (i) \\[/tex]
And we also have,
[tex]\longrightarrow 4x + 40 = y \\[/tex]
from equation (i) , we have ,
[tex]\longrightarrow 4x + 40 = 90 - x \\[/tex]
[tex]\longrightarrow 4x + x = 90 +40 \\[/tex]
[tex]\longrightarrow 5x = 130\\[/tex]
[tex]\longrightarrow x =\dfrac{130}{5}\\[/tex]
[tex]\longrightarrow \underline{\underline{x = 26^o}} \\[/tex]
substitute this value in equation (i) ,
[tex]\longrightarrow y = 90 - 26 \\[/tex]
[tex]\longrightarrow \underline{\underline{y = 64^o}} \\[/tex]
And we are done!
Maria orders 8 pizzas for 24 people coming to her party.
a) How many people can be fed
b) How many pizzas would she need if 33 people come to the party.
Using the arithmetic operation of division -
a) 3 people can be fed by 1 pizza.
b) Maria would need 11 pizzas for 33 people.
What is arithmetic operation?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
The number of pizzas ordered = 8
The number of people are = 24.
Number of people that can be fed if 1 pizza was there =
Number of people = Number of people that came/ Number of pizza
Use the arithmetic operation of division -
Number of people = 24/8
Number of people = 3
Therefore, 3 people can be fed.
Now, the number of people = 33
Number of pizzas required for 33 people can be obtained as -
Use the arithmetic operation of division -
Number of pizzas = Number of people/ Number of people fed by 1 pizza
Number of pizzas = 33/3
Number of pizzas = 11
Therefore, Maria will require 11 pizzas.
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Joseph has 93332 apples. she gets 1849032 more how many does he have?
Answer:
1,942,364
Step-by-step explanation:
1 8 4 9 0 3 2
+ 9 3 3 3 2
------------------------
1 9 4 2 3 6 4
Joseph now has 1,942,364 apples.
No. of apples Joseph had in the beginning = 93332
No. of apples he gets more = 1849032
Total no. of apples he has now = 93332 + 1849032
= 1,942,364
speedy boat rental charge 15$ deposit fee plus 2$ for each hour of use to rent a paddle boat Write an equation to find how many hours,h,
The number of hours that the rented the boat, if the total cost was $33 would be represented by the equation: 33 = 2h + 15
How to write the equation ?To write the equation on the number of hours that the boat would have been rented for to get a total cost of $ 33, use the formula :
Total cost = Deposit fee + ( Cost per hour x Number of hours boat was used)
Total cost = $ 33
Cost per hour = $ 2
Number of hours boat was in use = h
The equation therefore is:
33 = 2 h + 15
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The full question is:
Speedy Boat Rental charges a $15 deposit fee plus $2 for each hour of use to rent a paddleboat. Write an equation to find out for how many hours, h, they rented the Boat if the total cost was $33.
The graphs of lines k1 and k2 are shown on the grid. Which system of equations is best represented by this graph?
F. 3x-y=2
4x+9y=36
G. 3x-y=6
4x+9y=4
H. x-3y=-18
9x+4y=9
J. x+y=10
9x+4y=13
The equations which represents the given graph is 3x-y=2 and
4x+9y=36 .
What is Linear equation ?
Linear equation can be defined as the equation in which the highest degree is one
Given ,
The graphs of lines k1 and k2 are shown on the grid.
From the given graph ,
The Two lines are intersecting at a point x = 49/31 and y=85/31
So, lets put the values of and y in the given linear equations
so we get,
3x-y=2
4x+9y=36
put values x = 49/31 and y=85/31
so ,
3*(49/31) - 85/31 = 2
= 62/31
= RHS
which is equals to RHS
4*(49/31)+9*(85/31) = 36
=36* 31 / 31
= RHS
Which equals to RHS
Hence, The equations which represents the given graph is 3x-y=2
4x+9y=36 .
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The sum of $4 500 is divided among Anesha. Sian and Joanne. Sian received half, Anesha received $1050 and Joanne received the remainder. Calculate:
(a) Sian's share
(b) Joanne's share
(c) the ratio in which the $4500 was divided among the three persons
(d) the percentage of the total amount that Anesha received.
Answer:
a: 2250. b: 1200. c: 5:2.3:2.7 d: 23.3%
Step-by-step explanation:
Sian gets 1/2 so 4500/2 = 2250
Anesha = 1050
Joanna = 2250-1050 = 1200
Answer the following questions:
1. For a triangle oriented similar to the triangle in Example 3, find the value of h if a = 6 and b = 24.
2. Leo is inspecting a 1 cm ∶ 1.5 m scale floor plan of his new house. The dimensions of his bedroom in the scaled plan are 2 cm by 3 cm. What is the area of Leo’s actual bedroom in square meters?
3. You jogged 2.4 miles in 30 minutes. At that rate, how long will it take you to jog 3.2 miles?
4. An airplane flies 100 miles in half an hour. How far can it fly in 1 1⁄4 hours at the same speed?
5. The currency in Tajikistan is the Somoni. The exchange rate is approximately 0.17 Somoni = 1 Philippine peso. How much money in Somoni would you get with ₱500?
6. A map scale designates 1 inch = 25 km. The distance between the two towns on the map is 3 inches. If a liter of gasoline can last you for 50 kilometers, how many liters of gasoline would you need to drive from the first town to the second?
Answer: Using the formula for similar triangles, you can find the value of h by multiplying b by the ratio of corresponding sides. If a = 6 and b = 24, then the ratio of corresponding sides is 6/24 = 1/4. Therefore, h = b * (1/4) = 24 * (1/4) = 6
To find the area of Leo's actual bedroom in square meters, you can use the scale factor of 1 cm : 1.5 m and multiply the dimensions of the scaled plan by this factor. If the dimensions of the scaled plan are 2 cm by 3 cm, then the actual dimensions of the bedroom are 2 cm x 1.5 m/cm x 3 cm x 1.5 m/cm = 3 m x 4.5 m = 13.5 square meters
To find how long it will take to jog 3.2 miles at the same rate, you can use the formula distance = rate x time. If you jogged 2.4 miles in 30 minutes, then you can find the rate by dividing distance by time, which is 2.4 miles / 30 minutes = 0.08 miles/minute. Therefore, time = distance / rate = 3.2 miles / 0.08 miles/minute = 40 minutes
To find how far an airplane can fly in 1 1/4 hours, you can use the formula distance = rate x time. If the airplane flies 100 miles in half an hour, then the rate is 100 miles / 0.5 hours = 200 miles/hour. Therefore, distance = rate x time = 200 miles/hour x 1.25 hours = 250 miles
To find how much money in Somoni you would get with ₱500, you can use the exchange rate of 0.17 Somoni = 1 Philippine peso. Therefore, 0.17 Somoni x ₱500 = ₱85 Somoni
To find how many liters of gasoline you would need to drive from the first town to the second, you can use the map scale and
Step-by-step explanation:
Brynn has $1,200 in her savings account and plans to save an additional $350 each month in order to purchase a car. Write an equation to represent the total amount she has saved each month
An equation to represent the total amount she has saved each month is 350x + 1200.
We have to write an equation to represent the total amount she has saved each month.
Brynn has $1,200 in her savings account and plans to save an additional $350 each month in order to purchase a car.
To write an expression we have to read the statement very carefully and also write the expression after reading the statement.
Let she save the money in x months.
If she save $350 each month, then the total saving in x months = 350x
She has $1200 in saving account. Now the expression is
= 350x + 1200
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Use point-slope form to write the equation
of a line that passes through the point
(11, 18) with slope 3.
By using the point-slope form, the equation of a line that passes through the point (11, 18) with slope 3 is equal to y - 18 = 3(x - 11).
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁)
Where:
m represents the slope.x and y represents the data points.From the information provided above, we have the following slope and data points on the line:
Points = (11, 18).Slope, m = 3At data point (11, 18), a linear equation of this line can be calculated in point-slope form as follows:
y - y₁ = m(x - x₁)
y - 18 = 3(x - 11)
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GEOMETRY 9.3.5 POINT ON A CIRCLE JOURNAL ASSIGNMENT
FULL DOCUMENT PLS
Suppose that this grid represents the customer’s yard. The bottom left corner is the origin point, (0, 0), and the x- and y- axes represent the two fences. The rose bush is at point (16, 18).
the solution of the given problem of coordinates comes out to be the coordinate P = (7, 0.4).
How do coordinates work?A coordinate system is a technique used in geometry to precisely locate points or other topological objects on a continuum, such as Euclidean space, using one or more numbers or coordinates. A point or item on an a double plane is located using a pair of integers known as coordinates. The x and y positions are two integers that describe a point's location on a 2D plane.
David want to find the coordinate are a useful tool for segmenting lines into smaller units.
P's coordinate is P = (7, -0.4)
The criteria are as follows:
A = (8,0)
=> B = (3,-2) (3,-2)
m: n=1:4
Using P = mx2+nx1 my2+ny
m+n
m+n, line ratio is computed.
The result is:
P 1x3+4x8 /1x-2+4x0 = 1+4 / 7
=> P = 35, 2
=>P = (7, -0.4) (7, -0.4)
1+4
P therefore has the coordinate P = (7, 0.4).
Therefore , the solution of the given problem of coordinates comes out to be the coordinate P = (7, 0.4).
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Part B
Here’s a different way to partition the same function. Write a description of the partitioned function using known function types. Include function transformations in your description
With a negative leading coefficient, quadratic polynomials, and the horizontal offset for a scaled 1/x function.
Section 1's wiggles might be attributed to a variety of distinct purposes. A fourth-degree polynomial is maybe the simplest. It would have to have a negative leading coefficient in order to have downward-trending end behavior.
The curve in section 2 appears to represent either an exponential function or a scaled and translated version of 1/x. We prefer a version of 1/x because the latter is anticipated to approach the horizontal asymptote more quickly than is depicted above.
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DE is a midsegment of the triangle
If BC=6x-2 and DE=2x+2 find the length of BC
If BC=6x-2 and DE=2x+2, then the length of BC is 16.
DE is a midsegment of the triangle.
As we know that;
A midsegment is a line segment that joins two of the midpoints of a triangle's sides.
The midsegment linking the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is half that of the third side, according to the midsegment theorem.
So according to Midpoint Theorem
DE = BC/2
Now putting the value
2x+2 = (6x-2)/2
Multiply by 2 on both side, we get
4x + 4 = 6x-2
Add 2 on both side, we get
6x = 4x + 6
Subtract 4x on both side, we get
2x = 6
Divide by 2 on both side, we get
x = 3
Now putting the value of x in BC=6x-2
BC = 6(3) - 2
BC = 18 - 2
BC = 16
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(ii) Hence, given that 6log 3+7-3 log, a = 0, find the possible values of a.
Answer: Given that 6log3 + 7 - 3loga = 0, we can start by isolating the variable a. To do this, we'll add 3loga to both sides:
6log3 + 7 = 3loga
Next, we'll divide both sides by 3:
2log3 + (7/3) = loga
Now we have loga on one side of the equation and a constant on the other. To find the possible values of a, we'll need to use the properties of logarithms.
We know that logb(x) = y is equivalent to b^y = x, so we can rewrite the equation as:
a = 3^(2log3 + (7/3))
To find the possible values of a, we'll need to evaluate the exponent on the right side of the equation. Since we don't know the value of log3, we can't find the exact value of a. However, we can say that the possible values of a are all the values that can be expressed as 3 raised to the power of (2log3 + (7/3))
Step-by-step explanation:
Question 1 of 10
Which algebraic expression represents this word description?
The quotient of nine and the sum of a number and one
O A. 2+1
B. z1
C. +1
OD. +1
An algebraic expression which represents this word description "The quotient of nine and the sum of a number and one" is: D. 9 over x+1.
What is a quotient?In Mathematics, a quotient can be defined as a mathematical expression that is simply used to represent the division of a number by another number.
In order to translate this word problem into algebraic equation, we would assign a variable to the unknown number and then translate the word problem into algebraic equation as follows:
Let the variable n represent the unknown number.
Translating the word problem into an algebraic equation, we have the following;
Quotient = 9/(x + 1)
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Complete Question:
Which algebraic expression represents this word description?
The quotient of nine and the sum of a number and one
A. x+1 over 9 B. 9/x +1 C. x/9 +1 D. 9 over x+1
question 1 a number increases by 10%, and then decreases by 10%. will the result be greater than, less than, or equal to the original number
The number is decreased by 1%.
Now, According to the question:
Let the number be 100
Number is increased by 10% = 100+10% of 100
=100+10
=110
Now, the resultant number is reduced by 10% =110-10% of 110
=110-11
=99
The number obtained is 99
Decrease in value = 100 – 99 = 1
Percentage decrease = 1100 × 100=1%
Hence, the number is decreased by 1%.
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A batch of bread requires 3 cups of flour. How many batches can you make with 2 cups of flour?
Answer:
Step-by-step explanation:
To find out how many batches of bread you can make with 2 cups of flour, you need to divide the amount of flour you have by the amount of flour each batch requires.We know that a batch of bread requires 3 cups of flour. To find out how many batches you can make with 2 cups of flour, you can divide 2 by 3:2 / 3 = 0.67Therefore, you can make 0.67 batches of bread with 2 cups of flour. However, since you can't bake a fraction of a batch, you can't make any batches of bread with 2 cups of flour, as it is not enough. You will need at least 3 cups of flour to make a batch of bread.
At the end of an advertising campaign, weekly sales amounted to $25,000 and then decreased by 10% each week after the end of the campaign. A. Write the equation of the exponential function that models the weekly sales. B. Find the sales 6 weeks after the end of the advertising campaign
The final amount after a certain rate of decrease over a period of time can be calculated by Xₙ = X₀ (1-r)ⁿ. From calculations the sales after 6 weeks are 13286.025.
If the percent increase or decrease, period of time and initial amount are given, the final amount can be calculated using the equation Xₙ = X₀ (1±r)ⁿ. Here the percentage decrease is given. So the equation is
Xₙ = X₀ (1-r)ⁿ
Xₙ is the final amount
X₀ is the initial amount
r is the rate
n is the period of time
Here percent is given as 10%. So the rate, r = 10/100 = 0.10
Xₙ = X₀ (1-r)ⁿ
= 25000× (1-0.10)⁶
= 13286.025
So the sales after 6 weeks are 13286.025
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determine if the three side lengths form a right triangle. 15, 8, 7
yup
Step-by-step explanation: the triplets of 15,8,7 make a right triangle
u can also add 8 squared and 7 squared and if it equals to 15 squared then it’s right
A university class has had 18 adult education students enroll so far, as well as 18 other students. Considering this data, how many of the next 14 students to enroll should you expect to be adult education students?
The number of next 14 students that will be expected to be adults education students would be = 7 adult education students.
Who is an adult education student?An adult education student is the student that is undertaken by individuals that are mature men and women who take course for continuing education.
The first enrollment consist of the following;
Adult education student = 18
other = 18
Total for first enrollment = 18+18 = 36
If first 36 enrolled students = 18 adult students
second 14 enrolled students = X adults
Make X the subject of formula;
X adults = 14× 18/36
= 252/36
= 7
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Question 1 (2 points) ✓ Saved
Emilio has a bank account that pays 4% interest each year. He has $200 in his
account. How much interest will he earn after 1 year?
$10
$6
$4
$8
2.10
7) Graph the systems of linear equations on your calculator
gasoline is sold by the liter in many countries. how many liters are required to fill a 12.0-gal gas tank?
The Volume or liters required to fill a 12.0 - gal gas tank is 45.42 liters.
Gallon to liter conversion factors are:
[tex]1 \ Gallon = 3.785 \ Liter[/tex]
Assuming that the gallon measurement for the gas tank is 12.0 gal.
We now need to calculate the gas tank's liter capacity.
As, 1 gallon = 3.785 liter
[tex]So \ 12\ gallon = 12*3.785[/tex]
[tex]=45.42\ gallons.[/tex]
Therefore, 45.42 liters of liquid are needed to fill the gas tank.
Meaning of gasoline:
Petroleum is the source of gasoline, which is a mixture of flammable, volatile liquid hydrocarbons used as a fuel for internal combustion engines and as a solvent for oils and fats.
It is a by-product of the petroleum industry and is used as fuel for cars because it burns with a high degree of energy and mixes easily with air in a carburetor.
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How many ounces are in ten and half pounds?
Answer:
10 1/2 lbs = 168 oz
Answer:
168
Step-by-step explanation:
i think
the twenty-third term in an arithmetic sequence is $\frac23$ and the fifty-third term in the sequence is $\frac32$. what is the thirty-fifth term?
The thirty-fifth term in an arithmetic sequence is 1.
The formula for the nth term of an arithmetic sequence is:
[tex]t_{n}[/tex] = a + (n - 1)d
The twenty-third term in an arithmetic sequence is [tex]$\frac{2}{3}[/tex].
a + 22d = [tex]$\frac{2}{3}[/tex] ...... (1)
The fifty-third term in an arithmetic sequence is [tex]$\frac{3}{2}[/tex].
a + 52d = [tex]$\frac{3}{2}[/tex]........(2)
Subtracting both the equations.
Subtract equation (1) from equation (2):
a + 52d - (a + 22d) = [tex]$\frac{3}{2} - \frac{2}{3}[/tex]
a + 52d - a - 22d = [tex]$\frac{9-4}{6}[/tex]
30d = [tex]$\frac{5}{6}[/tex]
d = [tex]$\frac{1}{36}[/tex]
Substitute the value of d in equation (1).
We get:
a + 22[tex]$(\frac{1}{36})[/tex] = [tex]$\frac{2}{3}[/tex]
a + [tex]$\frac{11}{18}[/tex] = [tex]$\frac{2}{3}[/tex]
a = [tex]$\frac{2}{3}[/tex] - [tex]$\frac{11}{18}[/tex]
a = [tex]$\frac{12}{18} - \frac{11}{18}[/tex]
a = [tex]$\frac{1}{18}[/tex]
To find the 35th term substitute the values of a and d in the nth term equation.
The thirty-fifth term in an arithmetic sequence is:
= [tex]$\frac{1}{18}[/tex] + (35 - 1) [tex]$\frac{1}{36}[/tex]
= [tex]$\frac{1}{18}[/tex] + [tex]$\frac{34}{36}[/tex]
= [tex]$\frac{1}{18}[/tex] + [tex]$\frac{17}{18}[/tex]
= [tex]$\frac{18}{18}[/tex]
= 1
Therefore the thirty-fifth term is 1.
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What is the distance between PQ? Explain. Best explanation gets brainliest
The length of the line segment is given by the distance equation
D = 7.2 units
What is the distance of a line between 2 points?The distance of a line between 2 points is always positive and given by the formula
Let the first point be A ( x₁ , y₁ ) and the second point be B ( x₂ , y₂ )
The distance between A and B is D , and the distance D is
Distance D = √ ( x₂ - x₁ )² + ( y₂ - y₁ )²
Given data ,
Let the distance of the line segment between two points be D
Now , the equation will be
Let the first point be represented as P ( 1 , 6 )
Let the second point be represented as Q ( 7 , 2 )
Now , distance between P and Q is D , and the distance D is
Distance D = √ ( x₂ - x₁ )² + ( y₂ - y₁ )²
D = √ ( 1 - 7 )² + ( 6 - 2 )²
On simplifying the equation , we get
D = √ ( -6 )² + ( 4 )²
D = √ ( 36 + 16 )
D = √ 52
D = 7.2 units
Hence , the distance is 7.2 units
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Given that mLKN=145°, find the mLKM and mMKn
Answer: ∠LKM=56° and ∠MKN=89°
Step-by-step explanation:
Notice that ∠LKM and ∠MKN make up ∠LKN. That means ∠LKM and ∠MKN combines equals 145°.
2x+10+4x-3=145 [combing like terms]
6x+7=145 [subtract both sides by 7]
6x=138 [divide both sides by 6]
x=23
We are almost done. We still need to plug it in to find the angle values.
∠LKM
2x+10 [plug in x=23]
2(23)+10 [multiply]
46+10 [add]
56
∠LKM=56°
------------------------------------------------------------------------------------------------------------
∠MKN
4x-3 [plug in x=23]
4(23)-3 [multiply]
92-3 [subtract]
89
∠MKN=89°
Therefore, ∠LKM=56° and ∠MKN=89°.
The highschool student council has decided to have a dance-a-thon and needs to hire a disk jockey for the event. Council members have investigated 3 companies. Gabe's Mobile Music charges $60 per hour. Derek's DJ Service charges $100 plus $40 per hour. Solid Sounds charges $175 plus $30 per hour. Which company should they choose? Why?
Answer:
Solid Sounds
Step-by-step explanation:
As a test, let's say the dance-a-thon lasted 10 hours. Plug in 10 hours to each:
Gabe's: $60/hr x 10 hr = $600
Derek's: $100 + ($40/hr x 10 hr) = $500
Solid Sounds: $175 + ($30/hr x 10 hr) = $475
This shows that Solid Sounds is the cheapest (no matter how many hours the dance lasts)