Find the value of ƒ (6).
y = f(x)

Answer:

A scale factor of a famous statue uses a scale factor of 210:1. If the height of the drawing is 1.2 feet, what is the actual height of the statue?

The actual height of the statue is 252 feet (1.2 x 210).

Answer:

252

Step-by-step explanation:

The weight's height at t=1.5 seconds is roughly 8.07 feet.

a. The equation for simple harmonic motion can be used to find a model that describes the motion of the weight:

Acos(wt + phi) + D = y(t)

A represents amplitude, w represents angular frequency, phi represents phase shift, and D represents vertical shift.

The amplitude A is equal to half the difference between these two heights because we know the weight oscillates between a minimum height of 4 feet and a maximum height of 10 feet:

A = (10 - 4)/2 = 3

The weight completes a full cycle in three seconds, hence the weight's angular frequency is:

T = 2pi/3 and w = 2pi

where T, or the period, denotes the length of time for one complete cycle.

We can set phi to 0 because the weight begins at the lowest height of 4 feet. Last but not least, the vertical shift D is just the average of the highest and lowest heights:

D = (10 + 4)/2 = 7

Combining everything, we obtain:

Y(t) equals 3cos((2pi/3)*t) + 7

b. By simply entering t=1.5 into our model, we can determine the height of the weight at t=1.5 seconds:

3cos((2pi/3)*1.5)*y(1.5) + 7 + 8.07

Hence, the weight's height at t=1.5 seconds is roughly 8.07 feet.

c. We can plot y(t) for t ranging from 0 to 6 seconds in order to graph the model for two periods (two full cycles). The graph is shown here:

code

      |

  10  |             *  

      |           *   *

      |         *       *

      |        *         *

  7   |       *           *

      |     *               *

      |    *                 *

      |   *                   *

  4   | *                       *

      +-----------------------------

        0    1    2    3    4    5  

The graph demonstrates that the weight oscillates sinusoidally with a period of 3 seconds between 4 and 10 feet. In contrast, the minimum height is reached at t=0 seconds and the maximum height is obtained at t=1.5 seconds and t=4.5 seconds.

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Answer:

See Screenshot

Step-by-step explanation:

Answer:

y- intercept = 7

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

here m = - 2 , then

y = - 2x + c ← is the partial equation

to find c substitute (4, - 1 ) into the partial equation

- 1 = - 2(4) + c = - 8 + c ( add 8 to both sides )

7 = c

that is the y- intercept = 7

The length of the center of the circle is approximately 391.9 cm

Let's call the point where the two chords intersect point P. To find the length of the center of the circle to point P, we need to find the distance between point O (the center of the circle) and point P.

To do this, we first need to find the distance from point O to each chord. We can use the Pythagorean theorem to do this. Let's call the midpoint of the first chord point A, and the midpoint of the second chord point B. Then:

OA = √((360 cm)^2 - (280 cm)^2) = 160√2 cm

OB = √((360 cm)^2 - (320 cm)^2) = 40√17 cm

Next, we need to find the distance from each midpoint (A and B) to point P. We can use the formula for the length of a chord that passes through the center of a circle:

AP = BP = √(2r^2 - d^2)

where r is the radius of the circle and d is the length of the chord. Then:

AP = BP = √(2(360 cm)^2 - (560 cm)^2) = 320√2= 320√(2) cm

Now we can use the Pythagorean theorem again to find the distance from point O to point P:

OP = sqrt(OA^2 - AP^2) = √(160√2 cm)^2 - (320√(2) cm)^2) ≈ 391.9 cm

Therefore, the length of the center of the circle to the crossing point of the chords is approximately 391.9 cm.

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Answer: 280cm

Step-by-step explanation:

Use some graph paper and draw the question scaling it down by 0.01.

5.6cm 6.4cm and the radius 3.6cm.

Then multiply the answer by 100.

This is one way to get the answer, but I am sure there are more and one way to get to the 280cm.

Good luck, and share your ideas.

The tens digit is 8, and the units digit is 9 (since y = x + 1). Therefore, the original number is 89.

In mathematics, an equation is a statement that asserts the equality of two expressions. An equation contains one or more variables, which are unknown values that can vary or change. The variables are usually represented by letters such as x, y, or z. The expressions on either side of the equals sign can be constants, variables, or functions of variables.

The simplest form of an equation is a linear equation, which can be represented as ax + b = c, where a, b, and c are constants and x is a variable. To solve a linear equation, we need to find the value of the variable that makes the equation true. This is done by performing arithmetic operations on both sides of the equals sign to isolate the variable.

Equations can also be more complex, involving multiple variables, powers, and functions. In such cases, algebraic techniques such as factoring, substitution, and solving systems of equations may be required to find the solution.

Equations are used to describe relationships between variables in many fields of study, including physics, chemistry, economics, and engineering. They are also used in everyday life, such as in calculating the cost of a purchase after applying a discount, or determining the amount of time it takes to travel a certain distance

Let's represent the tens and units digits with x and y, respectively. According to the problem statement, we have two equations:

y = x + 1 (the units digit is one more than the tens digit)

10x + y + 10y + x = 187 (the sum of the original number and the reversed number is 187)

Simplifying the second equation, we get:

11x + 11y = 187

x + y = 17 (dividing both sides by 11)

Substituting y = x + 1 into the second equation, we get:

x + (x + 1) = 17

2x + 1 = 17

2x = 16

x = 8

So the tens digit is 8, and the units digit is 9 (since y = x + 1). Therefore, the original number is 89.

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Therefore , the solution of the given problem of probability comes out to be  probability of  not experiencing either symptom is 0.05.

Calculating the likelihood that a statement is true or that an event will take place is the focus of probability theory, a branch of mathematics. Any number range between 0 and 1, where 1 is frequently used to describe certainty and 0 is commonly used to denote possibility, can be used to represent chance. A probability diagram shows the likelihood that a particular event will occur.

Here,

To determine if the two symptoms are independent, we need to check if the probability of one symptom changes depending on whether the other symptom is present or not.

P(pain | swelling) = 40% / 75% = 0.533

P(pain | no swelling) = 20% / 25% = 0.8

Since the conditional probabilities are not equal, the two symptoms are dependent.

(c) The probability that the victim will experience at least one of these symptoms is the probability of experiencing pain, swelling, or both. This can be calculated as:

P(pain or swelling) = P(pain) + P(swelling) - P(pain and swelling)

= 60% + 75% - 40%

= 95%

So the probability of experiencing at least one of these symptoms is 95%.

(d) The probability that a victim will have intense pain given that they have swelling is the conditional probability:

P(pain | swelling) = 40% / 75%

= 0.533

So the probability of having intense pain given that there is swelling is 0.533.

(e) The probability that a victim will experience pain but not swelling can be calculated as:

P(pain and no swelling) = P(pain) - P(pain and swelling)

= 60% - 40%

= 20%

So the probability of having intense pain without swelling is 20%.

(f) The probability that a victim will not experience either symptom is the complement of the probability of experiencing at least one symptom:

P(neither pain nor swelling) = 1 - P(pain or swelling)

= 1 - 0.95

= 0.05

So the probability of not experiencing either symptom is 0.05.

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Applying the secant-tangent theorem, the value of x is calculated as: x = 5.

The secant-tangent theorem states that if there is a point outside of a circle and we draw a tangent line and a secant line from that point to the circle, then the length of the tangent line squared is equal to the length of the secant line multiplied by the length of the part of the secant line that is outside the circle.

Therefore, applying the secant-tangent theorem:

4 * (x + 4) = 6²

4x + 16 = 36

4x = 36 - 16

4x = 20

4x/4 = 20/4

x = 5

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We can write tan(− 0.9) as -

tan(− 0.9) = -1.26.

Tangent function is defined as the ratio of perpendicular and base. Mathematically, we can write -

tan(Ф) = p/b

tan(Ф) = perpendicular/base

Given is to find the value of tan(− 0.9).

We know -

1 radian = 57.2958 degrees

0.9 radian = 57.3 x 0.9 = 51.57 degrees

0.9 radian = 51.57 degrees                           ... { 1 }

So, we can write -

tan(− 0.9) = tan( - 51.57°) = - 1.26

Therefore, we can write tan(− 0.9) as -1.26.

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The solution to the system of equations is x = 34.5, y = 19.

None of the operations that can be performed on these equations violate the rules of the elimination method, so they are all allowed.

To solve the system of equations using the elimination method, you can multiply equation 2 by 2 to obtain:

-2x + 4y = 26 (equation 2, multiplied by 2)

Then, you can add this equation to equation 1 to eliminate the x variable:

(2x - 3y) + (-2x + 4y) = 12 + 26

Simplifying and solving for y, you get:

y = 19

Substituting this value of y into either equation, you can solve for x:

2x - 3(19) = 12

2x - 57 = 12

2x = 69

x = 34.5

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Por lo tanto, el precio original del abrigo era de 72€

Para resolver este problema, primero debemos determinar el precio original del abrigo antes de las rebajas.

Sea x el precio original del abrigo.

Después de aplicar la primera rebaja del 40%, el precio del abrigo sería:

x - 0.4x = 0.6x

Luego, tras la subida del 30%, elprecio se incrementaría al 130% del precio anterior:

1.3(0.6x) = 0.78x

Sabemos que este precio final es de 56,16€, por lo que podemos resolver la ecuación:

0.78x = 56.16

x = 72

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Answer:

(1,-1)

Step-by-step explanation:

Answer:

Option B: [tex](1, -1)[/tex]

Step-by-step explanation:

The two equations are linear simultaneous equations, which can be solved by either the substitution, elimination or graphical method.

Substitution method:

Formulate an expression for x in terms of y by isolating x and making it the subject of the equation:

   [tex]x + 2y = -1[/tex]

                  ∴ [tex]x = -2y -1[/tex] —— (equation i)

                     [tex]2x - 3y = 5[/tex] ——- (equation ii)

Substitute (equation i) in (equation ii) to solve for y:

                                      [tex]2(-2y -1) - 3y = 5[/tex]

Expand the brackets or parenthesis using the Distributive Law:

                                       [tex]-4y -2 -3y = 5[/tex]

Bring all the like terms together. Isolate the y terms together on one side of the equation and move all the other terms to the other side:

[tex]-4y -3y = 5 + 2[/tex]

= [tex]-7y = 7[/tex]

= [tex]y = -\frac{7}{7}[/tex]

∴ y = [tex]-1[/tex]

Substitute this calculated value of y into any of the equations to solve for x:

      [tex]x = -2(-1) - 1[/tex]

         [tex]= 2 - 1[/tex]

   ∴ x = [tex]1[/tex]

Together these x and y values form an ordered pair. In addition, this ordered pair is a solution to both the equations and also considered the point of intersection of the two equations

∴ Option B: [tex](1, -1)[/tex]

Answer:

x = -7.8

Step-by-step explanation:

Solve for x:

[tex]x-(-3.9)=-3.9[/tex]

Simplify

[tex]x+3.9=-3.9[/tex]

Subtract 3.9 on both sides

[tex]x=-7.8[/tex]

Note that the above is solved using a Truth Table. The truth table is attached accordingly. Note that the argument is invalid

A truth table is a table that displays all possible combinations of truth values for the propositional variables in a logical expression and the resulting truth value of the expression.

It is an important tool in logic as it enables us to determine the validity of arguments, identify tautologies and contradictions, and simplify complex expressions. Truth tables provide a systematic and rigorous method of evaluating logical expressions and reasoning about their truth values.

With respect to the table, since there is at least one row where all premises are true but the conclusion is false (the fifth row in this case), the argument is invalid

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The value of x is given as follows:

x = 3.

Applying the secant-tangent theorem, we have that the following equation holds true:

4x² = 6².

Hence we can solve the equation for x to obtain the value of x, as follows:

4x² = 36

x² = 36/4

x² = 9

x = sqrt(9)

x = 3.

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The possible ratios of dogs to cats that could be made is

12 : 48 , 6 : 24 , 3 : 12 , 1 : 4 and soon.

What is ratio in math?

When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. A percentage is an equation in which two ratios are made equal to one another.

                             As an illustration, you could express the ratio as 1: 3 if there is 1 guy and 3 girls (for every one boy there are 3 girls)

Given ,

If there are 12 dogs and 48 cats at a pet daycare,

So, we have to fill out all of the possible ratios of dogs to cats that could be made.

ratio could be  = 12 : 48

12 : 48  = 1 : 4

6 : 24 = 1 : 4

3 : 12 = 1 : 4

1 : 4 = 1 : 4

Therefore, The possible ratios of dogs to cats that could be made is

12 : 48 , 6 : 24 , 3 : 12 , 1 : 4 and soon.

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Answer:

To determine how much the tourist will receive while leaving Nepal, we need to know the total cost of the 5 Nepali caps including VAT, and then subtract any taxes or fees that the tourist may have paid when purchasing the caps.

Given that the tourist bought 5 Nepali caps at a cost of Rs 452 per piece including VAT, we can calculate the total cost of the caps as:

Total cost of 5 Nepali caps = 5 x Rs 452 = Rs 2260

Since the cost already includes VAT, we do not need to add any additional taxes or fees.

Therefore, if the tourist does not make any additional purchases or incur any other expenses while in Nepal, they will receive back Rs 2260 when leaving Nepal.

Answer:

b = 82

Step-by-step explanation:

Given

a =46 degrees

c = 52 degrees

Since XY is a straight line, it has an angle of 180 degrees

So, Angle a + angle b + angle c = 180 degrees

46 + angle b + 52 = 180

angle b + 98 = 180

angle b = 180 - 98

angle b = 82 degrees

Therefore angle b is equal to 82 degrees

Answer:

180 in a straight line

Step-by-step explanation:

there is 180⁰ in straight line.

46 + 52 is 98

180 - 98 is 82

Answer:

Step-by-step explanation:

We can start by simplifying the denominator, which is cot(a) + tan(a):

cot(a) + tan(a) = cos(a)/sin(a) + sin(a)/cos(a)

To combine these terms, we can find a common denominator of sin(a)cos(a):

cos^2(a)/(sin(a)cos(a)) + sin^2(a)/(sin(a)cos(a))

= (cos^2(a) + sin^2(a))/(sin(a)cos(a))

= 1/(sin(a)cos(a))

Now we can substitute this expression into the original equation:

sec(a)/(cot(a) + tan(a))

= sec(a) / (1/(sin(a)cos(a)))

= sec(a) * (sin(a)cos(a))

= (1/cos(a)) * (sin(a)/cos(a))

= sin(a)/cos^2(a)

= csc(a)sec(a)

Therefore, the simplified form of sec(a)/(cot(a) + tan(a)) is csc(a)sec(a).

Answer:

257.14 calories (about 260)

Step-by-step explanation:

To solve this problem, we can use proportions.

If a 28 g serving of almonds has 160 calories, then:

28 g / 160 calories = 45 g / x calories

where x is the number of calories in a 45 g serving of almonds.

We can cross-multiply and solve for x:

28 g * x = 160 calories * 45 g

x = (160 calories * 45 g) / 28 g

x = 257.14 calories

Therefore, a 45 g serving of almonds has 257.14 calories (rounded to two decimal places).

Answer:

18

Step-by-step explanation:

Simplifying, the exponents expression, we have 63y/xu⁴

Exponents are the powers to which a particular number is raised.

Given the expression 3y⁻⁴ . 3y⁵x⁸u⁻⁵.7x⁻⁹. To simplify the expression, we use the product rule of exponents which states that for a given base x, xᵃ xᵇ = xᵃ⁺ᵇ.

So, applying this rule to the expression and collecting common bases, we have that

3y⁻⁴ . 3y⁵x⁸u⁻⁵u.7x⁻⁹ = 3y⁻⁴ × 3y⁵ × x⁸ × 7x⁻⁹ × u⁻⁵ × u

= 3 × 3 × y⁻⁴ × y⁵ × 7 × x⁸ × x⁻⁹ × u⁻⁵ × u

Applying the product rule, we have that

= 3 × 3 × y⁻⁴ ⁺ ⁵ × 7 × x⁸ ⁺(⁻⁹) × u⁻⁵ ⁺ ¹

= 3 × 3 × y⁻⁴ ⁺ ⁵ × 7 × x⁸ ⁻ ⁹ × u⁻⁵ ⁺ ¹

= 3 × 3 × y × 7 × x⁻¹ × u⁻⁴

= 3 × 3 × 7 × y × x⁻¹ × u⁻⁴

Using the reciprocal rule that x⁻ⁿ = 1/xⁿ, we have that

= 3 × 3 × 7 × y × x⁻¹ × u⁻⁴

= 63 × y × 1/x × 1/u⁴

= 63y/xu⁴

So, simplifying, we have 63y/xu⁴

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Answer:

A. 100%

Step-by-step explanation:

Business owners will recieve 100% of their profits, because they paid for them. The recommended profits are 50% because they need to pay employees as well.

a. The period of y = 2tan(4x) on the interval [0,pi] is pi/2.

Justification: The period of a tangent function is pi/b where b is the coefficient of x. Therefore, the period of 2tan(4x) is pi/4, and the function completes two periods on the interval [0,pi], giving a total period of pi/2.

b. The location of vertical asymptotes for y = 2tan(4x) on the interval [0,pi] is x = (2n + 1)pi/8 where n is an integer.

Justification: The vertical asymptotes of a tangent function occur where the cosine of the angle in the argument is equal to zero. In this case, the cosine is equal to zero at (2n + 1)pi/8, giving the location of the vertical asymptotes.

c. The domain of y = 2tan(4x) on the interval [0,pi] is [0, pi/4) U (pi/4, pi/2) U (pi/2, 3pi/4) U (3pi/4, pi].

Justification: The tangent function has vertical asymptotes at odd multiples of pi/2, so the function is undefined at these points, and thus excluded from the domain.

d. The range of y = 2tan(4x) on the interval [0,pi] is (-∞, ∞).

Justification: The range of the tangent function is the set of all real numbers.

e. The x-intercepts of y = 2tan(4x) on the interval [0,pi] are x = k(pi/8), where k is an integer.

Justification: The x-intercepts of a tangent function occur where the function is equal to zero, which happens at integer multiples of pi.

f. The y-intercept of y = 2tan(4x) on the interval [0,pi] is (0,0).

Justification: The y-intercept of a function occurs where x=0, so the y-intercept of this function is (0, 0), as 2tan(4*0) = 0.

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Answer:

66.67

Step-by-step explanation:

8 people stayed home, 8/12=2/3=66.67(approx)

Answer:

Step-by-step explanation:

3f(x) stretches the graph of the function by a factor of 3 since it is being multiplied to f(x) and 3 is greater than one.

f(x+3) shifts the graph of the function 3 units to the left since 3 equals h in the equation: f(x-h), so you switch the sign and it becomes negative thus moving left.

f(3x) compresses the graph by a factor of 1/3 since it is inside the parenthesis.

f(x) + 3 shifts the graph of the function up by 3 units since it is being added to f(x).

The complete sentence with inequality is,

⇒ 1 - a > 0

A relation by which we can compare two or more mathematical expression is called an inequality.

Given that;

The number line is shown in figure.

By figure, the value of a is in between - 1 and - 2.

Hence, It is negative number.

Thus, We can assume,

⇒ a = - x

Hence, We get;

⇒ 1 - a

⇒ 1 - (- x)

⇒ 1 + x > 0

Thus, The complete sentence with inequality is,

⇒ 1 - a > 0

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Answer:

B) She should take the itemized deductions because they would be larger than the standard deduction.

Step-by-step explanation:

Given Kristina will file a single so her standard deduction for 2022 is $12,950.

For itemized deduction, she can include her mortgage interest of $6,320.00.

For her medical expenses, it has to be more than 7.5% of her gross income of $32,500 => 7.5% of $32,500 = $2437.5.

so her deduction for medical expenses is $19,500 - $2437.50 = $17,062.5

so her total itemized deduction is $6,320.00 + $17,062.50 = $23,382.50

Her itemized deduction is larger than standard deduction for single filing. So itemized deduction is the best option for Kristina.

Answer:

a. The sequence x_n = n^-2 converges quadratically to zero as n approaches ∞.

b. The sequence x_n = 2^-n converges linearly to zero as n approaches ∞.

c. The sequence x_n = 2^-2n converges quadratically to zero as n approaches ∞.

d. The sequence x_n = 2^-a_n with a_0 = a_1 = 1 and a_n+1 = a_n+a_n-1 for n≥2 converges superlinearly to zero as n approaches ∞.

Step-by-step explanation:

a. The sequence {x_n} = {n^(-2)} converges quadratically to zero as n approaches ∞.

To see this, we can calculate the limit of the ratio between consecutive terms:

lim_{n -> ∞} (x_{n+1} / x_n) = lim_{n -> ∞} ((n+1)^(-2) / n^(-2)) = lim_{n -> ∞} ((n / (n+1))^2) = 1

Since the limit is equal to 1, we know that the convergence rate of {x_n} is at least quadratic. This can also be seen from the fact that the error in the nth term is O(n^(-3)), which is smaller than the error in a linearly converging sequence (which is O(n^(-1))).

b. The sequence {x_n} = {2^(-n)} converges linearly to zero as n approaches ∞.

To see this, we can calculate the limit of the ratio between consecutive terms:

lim_{n -> ∞} (x_{n+1} / x_n) = lim_{n -> ∞} (2^(-(n+1)) / 2^(-n)) = lim_{n -> ∞} (1/2) = 1/2

Since the limit is strictly less than 1, we know that the convergence rate of {x_n} is linear.

c. The sequence {x_n} = {2^(-2n)} converges quadratically to zero as n approaches ∞.

To see this, we can calculate the limit of the ratio between consecutive terms:

lim_{n -> ∞} (x_{n+1} / x_n) = lim_{n -> ∞} (2^(-(2(n+1))) / 2^(-2n)) = lim_{n -> ∞} (1/4) = 1/4

Since the limit is equal to 1/4, we know that the convergence rate of {x_n} is at least quadratic. This can also be seen from the fact that the error in the nth term is O(n^(-2)), which is smaller than the error in a linearly converging sequence (which is O(n^(-1))).

d. The sequence {x_n} = {2^(-a_n)}, where a_0 = a_1 = 1 and a_{n+1} = a_n + a_{n-1} for n >= 2, converges superlinearly to zero as n approaches ∞.

To see this, we can use the fact that a_n grows exponentially with n. We can prove by induction that a_n >= phi^n, where phi is the golden ratio. Since the terms of {x_n} are decreasing and bounded below by 0, they must converge to a limit L. Taking the limit of the ratio of consecutive terms, we get:

lim_{n -> ∞} (x_{n+1} / x_n) = lim_{n -> ∞} (2^(-a_{n+1}) / 2^(-a_n)) = lim_{n -> ∞} (2^(-a_n - a_{n-1})) = 0

This shows that the convergence rate of {x_n} is superlinear.

So, we can express matrix B as a linear combination of matrices A1 and A2 as shown above.

An arrangement of integers into rows and columns is called a rectangular matrix. Every single number in a matrix is referred to as a "matrix element" or "entry." The example matrix, designated as A, has two rows and three columns.

To express matrix B as a linear combination of matrices A1, A2, and A3, we need to find coefficients x, y, and z such that:

B = xA1 + yA2 + z*A3

Substituting the matrices B, A1, A2, and A3 into this equation, we get the following system of linear equations:

x + z = 1

x - y + z = 1

-3x + y + z = 0

x + y + z = 1

Solving this system of equations, we get:

x = 1/2

y = 1/2

z = 0

Therefore, we have:

B = (1/2)*A1 + (1/2)A2 + 0A3

Substituting the values of A1, A2, and A3, we get:

B = (1/2)[1 0] + (1/2)[0 1] + 0*[1 -1]

= [1/2 1/2]

So, we can express matrix B as a linear combination of matrices A1 and A2 as shown above.

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We can write B as a linear cοmbinatiοn οf A1, A2, and A3 as:

B = 2 A2 + A3

In linear algebra, a linear cοmbinatiοn οf matrices is a sum οf scalar multiples οf thοse matrices. That is, given matrices A1, A2, ..., An and scalars c1, c2, ..., cn, their linear cοmbinatiοn is defined as:

c1A1 + c2A2 + ... + cnAn

Tο write B as a linear cοmbinatiοn οf A1, A2, and A3, we need tο sοlve the system οf linear equatiοns:

x1 A1 + x2 A2 + x3 A3 = B

where x1, x2, and x3 are the cοefficients οf the linear cοmbinatiοn.

Rewriting the matrix equatiοn, we get:

x1 + x3 = 1

x1 - x3 = 1

-3x1 + x2 + x3 = -3

x2 + x3 = 1

Sοlving this system οf equatiοns, we get:

x1 = 0

x2 = 2

x3 = 1

Therefοre, we can write B as a linear cοmbinatiοn οf A1, A2, and A3 as:

B = 2 A2 + A3

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