Using the circumference, the area of the hub cap is obtained as approximately 437.21 square centimeters.
What is circumference?
The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area.
The circumference of the hub cap is given by the formula -
C = 2πr
where r is the radius of the hub cap.
We can solve for the radius by dividing both sides by 2π -
r = C / 2π
r = 74.04 / 2(3.14)
r ≈ 11.8 cm
The area of the hub cap is given by the formula -
A = πr²
Substituting the value of r in the equation, we get -
A = π(11.8)²
A = 3.14 × 139.24
A ≈ 437.21 cm²
Therefore, the area of the hub cap is 437.21 square centimeters.
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Which value of x is the solution to this equation? 5x2 = 30x – 45
Answer:
x = 11/6 or 1 5/6 or 1.83333...
Step-by-step explanation:
The solution to the equation 5x^2 = 30x – 45 is x = 3.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
We have the Equation: 5x² = 30x - 45.
We can start by simplifying the equation by subtracting 30x and adding 45 from both sides to get:
5x² - 30x + 45 = 0
We can further simplify by dividing both sides by 5 to get:
x² - 6x + 9 = 0
This can be factored as (x - 3)(x - 3) = 0, which means that the only solution is x = 3.
Therefore, the solution to the equation 5x^2 = 30x – 45 is x = 3.
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An arc of length 21 meters subtends a central angle of 287° in a
circle.
a) Find the circumference of the circle & area.
The circumference and area of the circle with the arc of length 21 meters and a central angle of 287° are 4.35 meters and 1.16 square meters.
An arc of length 21 meters subtends a central angle of 287° in a circle. The circumference of the circle and area are required to be determined.
Here is the solution:
Let us suppose that the radius of the circle is r. Thus, we can write:
r = (L/θ) r = (21/287)
Let us substitute the values of r in the formulae of the circumference of the circle and area of the circle.
Circumference of the circle:
C = 2πr
C = 2π (21/287)
C ≈ 4.35 meters
Area of the circle:
A = πr²
A = π (21/287)²
A ≈ 1.16 square meters
Therefore, the circumference of the circle is 4.35 meters and the area of the circle is 1.16 square meters.
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What is the value x + y? (Round your answer to the nearest tenth)
Answer:
x + y = 110
Step-by-step explanation:
[tex]x=\sqrt{(65^{2} - 56^{2} } \\x=33\\\\\\y=\sqrt{(85^2-36^2} \\y=77\\\\x+y=33+77\\x+y=110[/tex]
jack bought 200 canned drinks to sell at his food and. the profit, y dollars, from the sale of the canned drinks as a function of the number of cans sold, x, is represented by the equation y=1.25x-100. How many cans must jack secc to recoup the cost of buying 200 canned drinks?
Answer:
To recoup the cost of buying 200 canned drinks, Jack needs to make a profit of zero, meaning that the revenue from selling the drinks needs to equal the cost of buying them.
Let's start by setting the profit equation equal to zero and solving for x:
0 = 1.25x - 100
1.25x = 100
x = 80
Therefore, Jack needs to sell 80 cans of drinks to recoup the cost of buying 200 canned drinks.
Step-by-step explanation:
How do you rewrite an equation into slope intercept form
Answer: y=mx+b
Step-by-step explanation:
To change the equation into slope-intercept form, we write it in the form y=mx+b .
Create a stretch of 3 on function f(x) = 2x to make g(x) Group of answer choices
According to the given information, the function g(x) that represents a horizontal stretch of 3 on f(x) = 2x is [tex]\rm g(x) = \frac{2}{3}x$[/tex]. Thus, option A is correct.
What is functiοn?A functiοn is a relatiοn between a set οf inputs and a set οf pοssible οutputs, with the prοperty that each input is related tο exactly οne οutput.
Tο create a hοrizοntal stretch οf 3 οn the functiοn f(x) = 2x, we can replace x in the functiοn with x/3, which will cause the οutput tο take three times the input value. This gives us the functiοn:
[tex]$$ g(x) = f\left(\frac{x}{3}\right) = 2\left(\frac{x}{3}\right) = \frac{2}{3}x $$[/tex]
Therefore, the function g(x) that represents a horizontal stretch of 3 on f(x) = 2x is [tex]g(x) = \frac{2}{3}x$.[/tex]
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Complete question:
Create a stretch of 3 on function f(x) = 2x to make g(x)
[Group of answer choices]
a. [tex]\rm g(x) = \tfrac{2}{3}x$[/tex]
b. [tex]\rm g(x) = \tfrac{3}{2}x$[/tex]
c. [tex]\rm g(x) = {2}x^3$[/tex]
d. [tex]$\rm g(x) = \sqrt[3]{{2}x} $[/tex]
In the diagram, E is the midpoint of DF. What is the value of x?
Answer:
[tex]\large\boxed{\mathtt{x = 10}}[/tex]
Step-by-step explanation:
[tex]\textsf{We are asked for the value of x.}[/tex]
[tex]\textsf{Because E is a Midpoint,} \ \overline{DE} \cong \ \overline{EF}.[/tex]
[tex]\textsf{Let's set them equal to each other.}[/tex]
[tex]\mathtt{45=3(x+5)}[/tex]
[tex]\large\underline{\textsf{Begin Solving:}}[/tex]
[tex]\mathtt{45= ( 3 \times x ) +(3 \times 5)}[/tex]
[tex]\mathtt{45= 3x+15}[/tex]
[tex]\large\underline{\textsf{Subtract 15 from Both Sides:}}[/tex]
[tex]\mathtt{3x=30}[/tex]
[tex]\large\underline{\textsf{Divide by 3:}}[/tex]
[tex]\large\boxed{\mathtt{x = 10}}[/tex]
Answer:
x = 10
Step-by-step explanation:
To find:-
The value of x.Answer:-
We are here given that, E is the midpoint of DF. The value of DE is 45 and that of EF is 3(x+5) . We are interested in finding out the value of "x" .
According to Euclid's first axiom , Things which are half of the same thing are equal to one another. So here;
[tex]\sf:\implies DF = DF \\[/tex]
[tex]\sf:\implies \dfrac{DF}{2}=\dfrac{DF}{2} \\[/tex]
[tex]\sf:\implies DE = EF \\[/tex]
On substituting the respective values, we have;
[tex]\sf:\implies 45 = 3(x+5) \\[/tex]
[tex]\sf:\implies 45 = 3x + 15 \\[/tex]
[tex]\sf:\implies 3x = 45-15\\[/tex]
[tex]\sf:\implies 3x = 30 \\[/tex]
[tex]\sf:\implies x =\dfrac{30}{3}\\[/tex]
[tex]\sf:\implies \red{x = 10}\\[/tex]
Hence the value of x is 10 .
What will be one-fifth of a complete angle in degrees
Answer:
72 degrees
Step-by-step explanation:
If a complete angle is 360 degrees, one-fifth of a complete angle would be:
360 ÷ 5 = 72Therefore one-fifth of a complete angle is 72 degrees.
Which expression is equivalent to (6−2 • 35)−2? HELP NOW
In response to the stated question, we can state that here Hence, the linear equation (6 2 • 35) 2 equals 1/4096
What is a linear equation?A linear equation is one that satisfies the algebraic formula y=mx+b. m is the y-intercept, while B is the slope. The foregoing sentence is sometimes referred to as a "linear equation with two variables" because y and x are variables. Bivariate linear equations are two-variable linear equations. Examples of linear equations include 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. An equation is said to be linear if its formula is y=mx+b, where m denotes the slope and b the y-intercept. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept.
We must do the following operations in the correct order to simplify the formula (62 • 35)2:
To start, we must multiply 2 by 35, which results in the number 70.
The result is -64 when we remove 70 from 6.
The final step is to raise the unfavorable integer -64 to the power of -2.
Hence, we can write:
[tex](6-2 * 35)-2 = (-64)^{-2}[/tex]
To make this statement easier to understand, keep in mind that taking the reciprocal of a number raised to a positive power is the same as raising it to a negative power. With those words:
[tex]a^-n = 1/a^n\\(-64)^-2 = 1/(-64)^2\\1/(-64)^2 = 1/4096[/tex]
Hence, the expression (6 2 • 35) 2 equals 1/4096.
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If the radius of Sphere B is 5 times larger than the radius of Sphere A, how many times larger is the volume of Sphere B compared to that of Sphere A?
The volume of Sphere A is V(A) = (4/3)πr³ and the volume of Sphere B is V(B) = 125(4/3)πr³. So the volume of Sphere B is 125/1 = 125 times larger than that of Sphere A.
what is sphere?
A sphere is a three-dimensional geometric shape that is perfectly round, like a ball. It is defined as the set of all points in space that are equidistant from a given point, called the center of the sphere.
The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius.
Let the radius of Sphere A be denoted by "r", then the radius of Sphere B is 5r.
The volume of Sphere A is V(A) = (4/3)πr³.
The volume of Sphere B is V(B) = (4/3)π(5r)³ = (4/3)π125r³ = 5³(4/3)πr³ = 125(4/3)πr³.
Therefore, the volume of Sphere B is 125/1 = 125 times larger than that of Sphere A.
So, the volume of Sphere B is 125 times larger than that of Sphere A.
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HELPPP! Cari knows that it is a 45 mile drive from her house to the airport. She also knows that it is a 45 mile drive from her house to her grandparents house in the woods. How many miles is it directly from the airport to her grandparents house in the woods? Show your work.
Answer:
We can use the Pythagorean theorem to solve this problem. Let's assume that Cari's house is at point A, the airport is at point B, and her grandparents' house is at point C. We know that the distance from A to B is 45 miles and the distance from A to C is 45 miles. We want to find the distance from B to C, which we can call x.
We can form a right triangle with sides AB, AC, and BC. The distance we want to find, x, is the length of the hypotenuse of this triangle (side BC).
Using the Pythagorean theorem, we know that:
AB^2 + AC^2 = BC^2
Substituting in the values we know:
45^2 + 45^2 = x^2
Simplifying:
2025 + 2025 = x^2
4050 = x^2
Taking the square root of both sides:
x = sqrt(4050)
x is approximately equal to 63.64 miles (rounded to two decimal places).
Therefore, the direct distance from the airport to Cari's grandparents' house is approximately 63.64 miles.
Can someone please please help! Thank you so much! The picture is attached and I've written out the question! thank you!
The cuboid below is made of nickel and has a mass of 534 g.
Calculate its density, in g/cm³.
If your answer is a decimal, give it to 1 d.p.
Answer:
8.9 g/cm^3
Step-by-step explanation:
Volume = 5 x 2 x 6 = 60 cm^2
density = m/V = 534/60 = 8.9 g/cm^3
20 poäng One positive number is 5 more than another. The sum of their squares is 53. What is the larger number?
One positive number is 5 more than another. The sum of their squares is 53. Then the larger number is 8.
Lets take x be the smaller number, then x + 5 = the larger number. Thus, we have the equation:
x2 + (x + 5)2 = 53
By Simplifying the equation, we get: 2x2 + 10x + 25 = 53
By Solving this quadratic equation, we get x = 3 and the larger number is 8.
Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).
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Find the x- and y-intercepts of the parabola y=−10x2−16x−5
The x-intercepts are approximately (-1.08, 0) and (-0.42, 0), and the y-intercept is (0, -5).
To find the x-intercepts of a parabola, we set y = 0 and solve for x. Similarly, to find the y-intercept, we set x = 0 and solve for y.
Setting y = 0 in the given equation, we get,
0 = −10x^2 − 16x − 5
We can solve for x by using the quadratic formula:
x = [-(-16) ± sqrt((-16)^2 - 4(-10)(-5))] / (2(-10))
Simplifying, we get,
x = [-(-16) ± sqrt(256 - 200)] / (-20)
x = [-(-16) ± sqrt(56)] / (-20)
x = [8 ± 2sqrt(14)] / (-10)
So the x-intercepts are:
x = [8 + 2sqrt(14)] / (-10) ≈ -1.08
x = [8 - 2sqrt(14)] / (-10) ≈ -0.42
To find the y-intercept, we set x = 0 in the given equation:
y = −10(0)^2 − 16(0) − 5
y = -5
So the y-intercept is -5.
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An angle measures 8° less than the measure of its complementary angle. What is the
measure of each angle?
and
X+y=9??? Can u give me a correct meaning of variable/s.
Answer:
In the equation X + y = 9, X and y are variables. Variables are used to represent unknown quantities or quantities that can vary. In this case, X and y represent two unknown numbers whose sum is equal to 9. We can use algebraic techniques to solve for either X or y, or both, depending on the situation. The values of X and y could represent anything, such as the number of apples and oranges in a basket, the length and width of a rectangle, or the ages of two people.
A man of mass 60kg runs with a constant velocity. He has kinetic energy of 750j. Calculate his velocity
Answer:
the velocity of the man is 5 m/s.
Step-by-step explanation:
The kinetic energy of a moving object is given by the equation:
K.E. = 0.5 * m * v^2
where
K.E. is the kinetic energy
m is the mass of the object
v is the velocity of the object
In this problem, we are given that the man has a mass of 60 kg and a kinetic energy of 750 J.
So,
750 J = 0.5 * 60 kg * v^2
Solving for v:
v^2 = (2 * 750 J) / 60 kg
v^2 = 25 J/kg
v = sqrt(25 J/kg)
v = 5 m/s
Therefore, the velocity of the man is 5 m/s.
Find the geometric mean of 8 and 32. The geometric mean is _
Answer:
20
Step-by-step explanation:
you add them and divide by how ever many you added
Find the image of (3, -6) reflected across the y-axis.
The image of the point after the reflection is (-3, -6).
How to determine the image of the pointTo reflect a point across the y-axis, we simply negate its x-coordinate while keeping the y-coordinate the same.
Thus, the image of point (3, -6) reflected across the y-axis would be (-3, -6).
We can visualize this by drawing a line passing through the point (3, -6) and the y-axis.
The reflection of the point will be at the same distance from the y-axis but on the opposite side.
So, the reflection of the point (3, -6) across the y-axis is (-3, -6).
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A cylinder has a radius of 5 yards. Its volume is 1,177. 5 cubic yards. What is the height of the cylinder?
The height of the cylinder is 27 yards.
To find the height of a cylinder, we need to use the formula V = πr^2h, where V is the volume, π is the constant pi (3.14), r is the radius, and h is the height.
In this problem, we know the radius is 5 yards, and the volume is 1,177.5 cubic yards. We can rearrange the formula to calculate the height as h = V / (πr^2). Substituting the given values into the equation, we get: h = 1177.5 / (3.14 * 25) = 27 yards. Therefore, the height of the cylinder is 27 yards.
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Can someone show me an example on how to solve a y=mx+b equation?
Answer:
hey, good afternoon!
Step-by-step explanation:
here is an example: the cost of a pizza is 7.50 and you can add a topping for 0.65 cents. find the cost of the pizza with 6 toppings
y=mx+b
the x stands for the toppings and y is the cost. the rate is 0.65 cents and the orginal amount of money is b. so the equation would look like this:
y= .65x+7.50. x=6
y = 65(6)+7.50
=3.9+ 7.50= $11.50(answer)
the m is the rate of change in this question is 0.65 cents
b is initial/ fixed/ value once again in this qeustion it 7.50$
hope this helps. have a great day!
There were 5 choirs at the choir concert on Friday night. Each choir sang 5 songs.
Which shows how many songs the choirs sang in all?
A.
5 - 5
B.
5 × 5
C.
5 + 5
D.
5 × 6
I need help with this for math
The answer of the given question based on finding the new coordinate of c is (1, 5). and to determine if the image of Triangle ABC is similar or congruent to the original triangle is AB / A'B' = BC / B'C' = CA / C'A' = √(2).
What is Congruent triangles?Congruent triangles are triangles that have the same shape and size. More specifically, if two triangles have exactly the same size and shape, then they are congruent triangles. This means that all the corresponding sides and angles of two triangles are equal. Congruent triangles can be thought of as being identical to each other, but they may be positioned and oriented differently in space.
To rotate a point 180° degrees counterclockwise about the origin, we can simply multiply its coordinates by the matrix:
[ -1 0 ]
[ 0 -1 ]
So, the coordinates of the rotated point C would be:
[ -1 0 ] [ 3 ]
[ 0 -1 ]*[ 8 ] =[ -3 ]
[ 1 ]
we translate this point 4 units to the right and 4 units up by adding 4 to the x-coordinate and 4 to the y-coordinate:
[ -3 + 4 ]
[ 1 + 4 ] = [ 1, 5 ]
Therefore, the new coordinates of point C are (1, 5).
To determine if the image of triangle ABC is similar or congruent to the original triangle, we can check if the ratios of the corresponding sides are the same. Let's calculate the lengths of the sides of the original triangle:
AB = √((8-3)² + (3-3)²) = 5
BC = √((3-8)² + (8-3)²) = 5*√(2)
CA = √((3-3)² + (3-8)²) = 5
Now, let's calculate the lengths of the sides of the image triangle, using the new coordinates:
A'B' = √((8-1)² + (3-5)²) = 5*√(2)
B'C' = √((1-1)² + (5-8)²) = 3
C'A' = √((1-8)² + (5-3)²) = 5*√(2)
We can see that the ratios of the corresponding sides are the same:
AB / A'B' = BC / B'C' = CA / C'A' = √(2)
Therefore, the image of triangle ABC is similar to the original triangle.
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answers for these please
The values of the variables are x = 2, z = 6.92, x = 10.5 and y = 12.6
How to find the values of the variablesFigure 12
In this figure, we make use of the following theorem of intersecting secants
AB * AC = AD * AE
By substitution, we have
(x + 4) * (x + 4 + x - 2) = 4 * 9
Evaluate the sum
(x + 4) * (2x + 2) = 36
So, we have
(x + 4) * (x + 1) = 18
Expand
x^2 + 5x + 4 = 18
Evaluate the like terms
x^2 + 5x - 14 = 0
Factorize
(x + 7)(x - 2) = 0
Solve for x
x = -7 or x = 2
x cannot be negative
So, we have
x = 2
Figure 13
In this figure, we make use of the following theorem of intersecting secant-tangent
XU * WU = UV^2
So, we have
8 * 6 = z^2
Evaluate
z^2 = 48
Take the square root
z = 6.92
Figure 14
Using the same theorem used in (13), we have
2 * (2 + x) = 5^2
So, we have
4 + 2x = 25
Evaluate
2x = 21
Divide
x = 10.5
Figure 15
Using the same theorem used above
y^2 = 8 * 20
So, we have
y^2 = 160
Take the square root
y = 12.6
Hence, the value of y is 12.6
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what is the exact value of tangent of 11 times pi over 12 question mark negative quantity of 2 minus radical 3 end quantity negative quantity of 2 plus radical 3 end quantity 2 minus radical 3
The exact value of tangent of 11 times pi over 12 is [tex]-2+\sqrt{3}[/tex].
We have to find the exact value of [tex]tan\frac{11\pi}{12}[/tex]
[tex]tan\frac{11\pi}{12}=tan\frac{6\pi+5\pi}{12}[/tex]
[tex]tan\frac{11\pi}{12}=tan(\frac{6\pi}{12}+\frac{5\pi}{12})[/tex]
[tex]=tan(\frac{\pi}{2}+\frac{5\pi}{12})[/tex]
We have [tex]tan(\frac{\pi}{2}+x)=-cotx[/tex]
[tex]tan\frac{11\pi}{12}=-cot(\frac{5\pi}{12})[/tex]
[tex]tan\frac{11\pi}{12}=-cot(\frac{2\pi+3\pi}{12})[/tex]
[tex]tan\frac{11\pi}{12}=-cot(\frac{\pi}{6}+\frac{\pi}{4})[/tex]
We have identity [tex]cot(x+y)=\frac{cotxcoty-1}{coty+cotx}[/tex]
[tex]tan(\frac{\pi}{2}+x)= -\frac{cot\frac{\pi}{6}cot\frac{\pi}{4}-1}{cot\frac{\pi}{4}+ cot\frac{\pi}{6}}[/tex]
[tex]=\frac{-\sqrt{3}(1) - 1}{1+\sqrt{3}}[/tex]
Rationalize the denominator and simplify, we get:
[tex]=-2+\sqrt{3}[/tex]
Therefore, the exact value of [tex]tan\frac{11\pi}{12}[/tex] is [tex]-2+\sqrt{3}[/tex].
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Complete question is:
Find the exact value of [tex]tan\frac{11\pi}{12}[/tex] ?
express the sum of the angles of this triangle in two different ways
Answer:
X=60
Step-by-step explanation:
Firstly you must know that a triangle has 180° angle in it's all side.. so in this case you can say x+½x+(1,5)x= 180°
here x equal to 60°
this is a simple way to expres how can find the x angle
And to second way about how to find the x we have to know the lineer algebra. so lineer algebra is a strong way and useful to not only geometric shapes and for all mathematic calculate..
3. Consider the quadratic equation x2 + 2x - 35 = 0. Solve by factoring and using the zero-product property. What are solutions to quadratic equations called? Show your work.
To solve the equation, factor [tex]\bold{x^2+2x-35}[/tex] use the formula [tex]\bold{x^2 +(a + b) x + ab = (x + a)(x + b)}[/tex]. To find a and b, set up a system to be solved.
[tex]a + b = 2[/tex]
[tex]ab = -35[/tex]
Since ab is negative, a and b have opposite signs. Since [tex]a+b[/tex] is positive, the positive number has a greater absolute value than the negative. Show all pairs of integers whose product is −35.
[tex]\bold{-1.35}[/tex][tex]\bold{-5.7}[/tex]Calculate the sum of each pair.
[tex]\bold{-1 + 35 = 34}[/tex][tex]\bold{-5 + 7 = 2}[/tex]The solution is the pair that gives sum 2.
[tex]\bold{a = -5}[/tex][tex]\bold{b = 7}[/tex]Rewrite the factored expression [tex](x + a)(x + b)[/tex] with the values obtained.
[tex]\bold{(x -5)(x + 7)}[/tex]To find solutions to equations, solve [tex]x-5=0[/tex] and [tex]x+7=0[/tex].
[tex]\bold{x = 5}[/tex][tex]\bold{x = -7}[/tex]What are the solutions of quadratic equations called?The "solutions" of a Quadratic Equation are the values where the equation equals zero. They are also called "roots", or even "zeros".
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Find each Measure. 15 points
Answer:
M<E = 63
M<F = 63
M<G = 117
Step-by-step explanation:
M<F:
= 180-117
=63
M<E:
=180-117
=63
M<G:
=180-63
=117
Problem #2 (15 Points)
2. A report by a tourism company says that 28% of United States citizens plan to travel
internationally annually. The tourism company selects a random sample of 40 people and
calculates the sample proportion (p) of people planning to travel internationally.
a. Tell whether n-p≥10 and n-(1 p)≥10 are true or false.
F +
+Show your work here:
-
b. Describe the shape of the sampling distribution of the sample proportion of people
who expect to travel internationally the next year. Write an explanation for your
reasoning.
Show your work here:
Answer:
Step-by-step explanation:
its b
Solve from [0, 2pi)
cos x/2 = (sqrt(2))/2
Thus, cos x/2 = (sqrt(2))/2 has the solution for x = 4kx ± π/2 in the intervals [0, 2pi).
Explain about the cosine function?The ratio of the hypotenuse to the side adjacent to the acute angle together in right triangle is the cosine function, which is a trigonometric function.
This cosine function for only a specified range of angles is represented graphically by the cos graph. A trigonometric graph's horizontal axis is the angle, which is typically denoted by the symbol, and the y-axis is really the sine function of just that angle. The maximum and minimum values are 1 and 1.
There are amplitudes for both the sine and cosine functions. This is due to the scope limitations of each function. Both functions typically have an amplitude of 1, though this can vary depending on the way the function is constructed.
Given function:
cos x/2 = (sqrt(2))/2
OR,
cos x/2 = √2/2
cos x/2 = cos (2kx ± π/4) , k ∈ R (real number)
x/2 = 2kx ± π/4
x = 4kx ± π/2
Thus, cos x/2 = (sqrt(2))/2 has the solution for x = 4kx ± π/2 in the intervals [0, 2pi).
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