The solution of equation of trigonometric functions, tan²(x) + tan(x) − 20 = 0, is equals to the x = 1.33 radians , 1.37 radians (x = arc tan(4) = 1.33 , x = arc tan(-5)= 1.37).
We know that trigonometric functions are periodic functions, solutions of trigonometric equations are then infinite and periodic. In these equations, it is essential to know the reduction formulas of each quadrant, which allows each angle in the first quadrant to be related to its corresponding angle in the other three quadrants. We have an equation which contains triagmometric function,
tan²(x) + tan(x) − 20 = 0 --(1) and we have to solve it. First we change the variable as y = tan(x). Rewrite the equation (1), y² + y - 20 = 0 --(2) which is an quadratic equation. The quadratic formula helps to solve the equation (2).
=> y = ( -1 ± √1 - 4(-20))/2
=> y = ( -1 ± √81)/2
=> y = ( -1 ± 9)/2
=> y = ( -1 + 9)/2 or (-1 -9)/2
=> y = 4, -5
Now x = tan⁻¹(y), and so, x = tan⁻¹(4) = 1.33 radians, or tan⁻¹(-5) = 1.37 radians. Hence, required value is ( 1.33, 1.37).
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the set is a basis of the space of upper-triangular matrices. find the coordinates of with respect to this basis.
The set is a basis of the space of upper-triangular matrices. The coordinates of with respect to this basis is B⁻¹ × p
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients that includes only the operations of addition, subtraction, multiplication, and power of variables with a positive integer. Polynomials appear in many areas of mathematics and science. For example, they are used to create polynomial equations that encode a wide variety of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions that appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial circles and algebraic varieties, which are central concepts in algebra and algebraic geometry.
According to the Question:
Converting the polynomials into vectors by taking their coordinate vectors with respect to the standard basis of P³, {1, x, x²}.
Thus B = [-1, 0, -2], [-2, 3, -4], [-2, 9, -8].
And p is [-6, 21, -24].
⇒ [p(x)]B = B⁻¹ × p
Complete Question:
the set B = [tex]\left[\begin{array}{ccc}1&1&\\0&0\end{array}\right][/tex], [tex]\left[\begin{array}{ccc}0&1\\0&-1\end{array}\right][/tex], [tex]\left[\begin{array}{ccc}0&0&\\0&-2\end{array}\right][/tex] is a basis of the space of upper triangular 2 × 2 matrices . Find the coordinates of
M = [tex]\left[\begin{array}{ccc}-6&-3&\\0&-5&\end{array}\right][/tex] with the respect to this basis.
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A system of equations is shown. 2x-y= 15 y=9 What is the value of x in the solution to this system?
Answer:
x=12
Step-by-step explanation:
2x-y=15
y=9
2x-9=15
2x=24
x=12
For numbers 3, 4, and 5, find the value of the indicated length(s) in ⨀C. A and B are points of tangency. Simplify all radicals.
I just need help with these three problems!
As a result, AB = 52 and AC = BC = 5 as AC and BC are both the circle's radii since A and B are points of tangency .
what is circle ?A circle is a closed object made up of all points in a plane that are separated from the center by a predetermined distance, known as the radius. The diameter is the distance across the circle that passes through its center, while the circumference is the distance around the circle. The ratio of a circle's circumference to its diameter is always pi (), or roughly 3.14. Pi is also known as the proportionality constant. Circles are significant geometric forms that are used frequently in mathematics, science, and daily life.
given
AC and BC are both the circle's radii since A and B are points of tangency. Hence, AC = BC = 5.
We can apply the Pythagorean theorem to segment AB as follows:
[tex]AB^2 = 52 + 52 \\AB^2 = AC^2 + BC^2[/tex]
AB² = 50
AB = √50 = 5√2
As a result, AB = 52 and AC = BC = 5 as AC and BC are both the circle's radii since A and B are points of tangency .
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25 POINTS, PLS EXPLAIN! Which sum or difference is modeled by the algebra tiles? Select the correct answer. (-2x + 3) + (3x − 4) (-2x + 3) − (3x − 4) (2x − 3) + (-3x + 4) (2x − 3) − (-3x + 4)
Answer: Consider the expression (2x − 3) − (-3x + 4).
First, distribute the negative sign and rewrite the expression:
2x − 3 + 3x – 4.
Then, model the polynomial.
There are no zero pairs, so the difference is 5x − 7.
Step-by-step explanation:
PLATO
ANSWER THIS QUESTION FAST WILL GIVE BRAINLIEST
In each triangle, M, N, and P are the midpoints of the sides. Name a segment parallel to the one given.
Answer: MN || VT
Step-by-step explanation:
We can see that MN is vertical, and the only angle that is vertical, is VT. And we can see that they don't obstruct their lines (meaning that they don't intersect), and keep going for infinity. So, VT would be the only parallel line to the one given.
Hope this helps
In Exercise 5.1, we determined that the joint distribution of Y1, the number of contracts awarded to firm A, and Y2, the number of contracts awarded to firm B, is given by the entries in the following table.
The marginal probability function of Y1 was derived in Exercise 5.19 to be binomial with n = 2 and p = 1/3. Find
a E ( Y 1 ).
b V ( Y 1 ).
c E ( Y 1 − Y2).
Reference
Contracts for two construction jobs are randomly assigned to one or more of three firms, A, B, and C. Let Y1 denote the number of contracts assigned to firm A and Y2 the number of contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2 contracts. a Find the joint probability function for Y1 and Y2.
b Find F(1, 0).
The marginal probability function E(Y1) = 2/3, variance of a binomial distribution is V(Y1) = 4/9, and E(Y1-Y2) = 0, using the joint probability function for Y1 and Y2 and the fact that they have binomial distributions with n=2 and p=1/3.
The marginal probability function of Y1 is binomial with n=2 and p=1/3, as given in Exercise 5.19. Therefore, we have:
E(Y1) = np = 2(1/3) = 2/3.
The variance of a binomial distribution with parameters n and p is given by np(1-p), so we have:
V(Y1) = np(1-p) = 2(1/3)(2/3) = 4/9.
We can use the linearity of expectation to find E(Y1-Y2):
E(Y1 - Y2) = E(Y1) - E(Y2)
We know that E(Y1) = 2/3 from part (a), but we need to find E(Y2). Using the same reasoning as in part (a), we find that the marginal probability function of Y2 is also binomial with n=2 and p=1/3. Therefore, we have:
E(Y2) = np = 2(1/3) = 2/3.
Substituting these values into the expression for E(Y1-Y2), we get:
E(Y1 - Y2) = 2/3 - 2/3 = 0.
Therefore, E(Y1-Y2) is equal to 0.
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Label all of the angle measures on the transversal.
Remember you can find all of the angles since you know one of them is 45 degrees.
The transversal angle of 45 is angle6.
What are angles?Two lines intersect at a location, creating an angle.
An "angle" is the term used to describe the width of the "opening" between these two rays. The character is used to represent it.
Angles are frequently expressed in degrees and radians, a unit of circularity or rotation.
In geometry, an angle is created by joining two rays at their ends. These rays are referred to as the angle's sides or arms.
An angle has two primary components: the arms and the vertex. T
he two rays' shared vertex serves as their common terminal.
Hence, The transversal angle of 45 is angle6.
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I will mark you brainiest!
Determine the MOST PRECISE name for the quadrilateral below.
A) rhombus
B) parallelogram
C) square
D) trapezoid
E) kite
The answer is A, rhombus.
This year, the ratio of Alan's age to Bernice's age is 1:2. Four years ago, the total age of Alan and Bernice was 55 years. How old is Alan this year?
Answer:
21 years old
Step-by-step explanation:
Set Alan's age as x, Bernice's age as y
2x=y
x-4+y-4=55
x+y=63
3y=63
x=21
y=42
mathematicians divide the world of numbers into at least seven subsets. list seven subsets and define each
Mathematicians divide the world of numbers into seven subsets, namely natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers.
Natural numbers: These are the counting numbers, starting from 1 and continuing indefinitely. They can also be denoted as "N" or "ℕ".
Whole numbers: These are the natural numbers, as well as zero. They can also be denoted as "W" or "ℤ⁺⁰".
Integers: These are the whole numbers, as well as their negative counterparts. They can also be denoted as "Z" or "ℤ".
Rational numbers: These are numbers that can be expressed as a fraction, where the numerator and denominator are integers (and the denominator is not zero). They can also be denoted as "Q" or "ℚ".
Irrational numbers: These are numbers that cannot be expressed as a fraction of two integers. They are often represented by decimal expansions that neither terminate nor repeat. Examples include π and the square root of 2.
Real numbers: These are all the rational and irrational numbers, taken together. They can also be thought of as numbers that can be located on a number line. They are often denoted as "R" or "ℝ".
Complex numbers: These are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (defined as the square root of -1). Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers. They are often denoted as "C" or "ℂ".
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Find the value of f(4) for the function F(a) = 4 (a+3)-8
A boat is heading towards a lighthouse, whose beacon-light is 139 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 5 degrees.
What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest hundredth of a foot if necessary.
The bοat's hοrizοntal distance frοm the lighthοuse (and the shοre) is apprοximately 1592.53 feet.
What is trigοnοmetryTrigοnοmetry is οne οf the mοst impοrtant branches in mathematics that finds huge applicatiοn in diverse fields. The branch called “Trigοnοmetry” basically deals with the study οf the relatiοnship between the sides and angles οf the right-angle triangle.
Hence, it helps tο find the missing οr unknοwn angles οr sides οf a right triangle using the trigοnοmetric fοrmulas, functiοns οr trigοnοmetric identities. In trigοnοmetry, the angles can be either measured in degrees οr radians. Sοme οf the mοst cοmmοnly used trigοnοmetric angles fοr calculatiοns are 0°, 30°, 45°, 60° and 90°.
We can use trigοnοmetry tο sοlve fοr the hοrizοntal distance. Let x be the hοrizοntal distance frοm the bοat tο the lighthοuse.
Then, tan(5°) = οppοsite/adjacent = 139/x
Sοlving fοr x, we get:
x = 139/tan(5°) ≈ 1592.53 feet
Therefοre, the bοat's hοrizοntal distance frοm the lighthοuse (and the shοre) is apprοximately 1592.53 feet.
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Ciara has $4,300 in savings. If she deposits the money into a long-term savings account with 2.13% APY and monthly compounding, what will the accrued value of her account be in five years?
What is the answer? Need help!!!
What is this question asking? What does it mean by floor plan? A step-by-step explanation would be very much appreciated.
Answer:
this is when you want to draw a sketch of a building
it takes 6 painters 4 1/2 to paint these classroom. calculate how long 3 painters will take to complete the same job
a man allowed 10 discount on the fixed price of an piece of a land and offered 5 commision to the broker if he received 256500 what was the fixed price of land
The fixed price of the land is 296257.50.
What is discount?
Discount is the state of having a bond's price lower than its face value. The difference between the purchase price and the item's par value is the discount.
Discounts are different types of price reductions or deductions from a product's cost. It is frequently employed in consumer transactions when consumers receive discounts on a range of goods. The % discount rate is provided.
Here the discount on the fixed price = 10%
Commission for the broker = 5%
If total amount = 100%+10% = 110%
Then received amount = [tex]\frac{110}{100}\times256500[/tex]
=> Received amount = 282150
Now 5% commission then,
=> [tex]\frac{105}{100}\times 282150[/tex] = 296257.50.
Hence the fixed amount of the land before discount is 296257.50.
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pls help !! i will mark brainilest
Answer:
m = 2/3
Step-by-step explanation:
Answer:
[tex] \frac{2}{3} [/tex]
Step-by-step explanation:
slope is
[tex] m = \frac{rise}{run} = \frac{y 2 - y1}{x2 - x1} [/tex]
(0,0) & (3,2)
[tex] m = \frac{2 - 0}{3 - 0} = \frac{2}{3} [/tex]
Graph the system below and write its solution
Y=1/4x-3
-x+4y=-4
The graph of the system of equations is shown below:
The solution of the system is given by the point of intersection of the two lines which is (20/7, -7/2).
Can someone please help
Answer:
1.11
2.4
3.5
pls correct me if I'm wrong
Answer:
38. (b) 11
39. (c) 4
40. (c) 5
Step-by-step explanation:
38.)
[tex]\implies \: \sf \sqrt{3xx - 8} = 5 \\ \\ \implies \: \sf 3xx - 8 = {(5)}^{2} \\ \\ \implies \: \sf 3xx - 8 = 25 \\ \\ \implies \: \sf 3xx = 25 + 8 \\ \\ \implies \: \sf 3xx = 33 \\ \\ \implies \: \sf xx = \dfrac{33}{3} \\ \\ \implies \: \sf xx = 11\\ [/tex]
Hence, Required answer is option (b) 11.
39.)
[tex]\implies \: \sf \sqrt{4xx -7 } - 3 = 0 \\ \\ \implies \: \sf \sqrt{4xx - 7} = 3 \\ \\ \implies \: \sf 4xx - 7 = {(3)}^{2} \\ \\ \implies \: \sf 4xx - 7 = 9 \\ \\ \implies \: \sf 4xx = 9 + 7 \\ \\ \implies \: \sf 4xx = 16 \\ \\ \implies \: \sf xx = \dfrac{16}{4} \\ \\ \implies \: \sf xx = 4 \\ [/tex]
Hence, Required answer is option (c) 4.
40.)
[tex]\implies \: \sf \sqrt{6xx + 6} - 6 = 0 \\ \\ \implies \: \sf \sqrt{6xx + 6} = 6 \\ \\ \implies \: \sf 6xx + 6 = {(6)}^{2} \\ \\ \implies \: \sf 6xx + 6 = 36 \\ \\ \implies \: \sf 6xx = 36 - 6 \\ \\ \implies \: \sf 6xx = 30 \\ \\ \implies \: \sf xx = \dfrac{30}{6} \\ \\ \implies \: \sf xx = 5 \\ [/tex]
Hence, Required answer is option (c) 5.
determine if the transformation is one to one and/or onto. justify your answers. give an explanation for each of these properties.
To determine whether a transformation is one-to-one or onto, one must analyze its behavior and properties, such as passing the horizontal line test for one-to-one or checking if the range equals the codomain for onto.
In mathematical terms, a transformation refers to a function that maps elements from one set, called the domain, to another set, called the range. A transformation is said to be one-to-one if no two distinct elements in the domain are mapped to the same element in the range. This means that each element in the range is associated with a unique element in the domain.
On the other hand, a transformation is onto if every element in the range is mapped to by at least one element in the domain. In other words, for each element in the range, there exists at least one element in the domain that maps to it.
To determine whether a transformation is one-to-one or onto, one can analyze its properties and behavior. For example, a transformation is one-to-one if and only if it passes the horizontal line test. This means that no two points in the domain map to the same point on a horizontal line. To determine if a transformation is onto, one can check if the range of the transformation equals the codomain.
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The given question is incomplete, the complete question is:
How to determine the transformation is one to one and/or onto?
find the following answers
Answer:
hope it helps
Step-by-step explanation:
based on the given condition formulate
The number of the words formed with letters of the word 'PROBLEM' with neither start with O nor end with E.
The number of words formed with letters of the word 'PROBLEM' which neither start with O nor end with E is 3600.
The word PROBLEM has 7 letters, which can be arranged in 7! (factorial) ways, i.e. [tex]7*6*5*4*3*2*1 = 5040.[/tex]
Now to calculate the number of words formed with letters of the word 'PROBLEM' which neither start with O nor end with E, we can first calculate the number of words which start with O or end with E.
For words starting with O, there are 6 letters left and these can be arranged in 6! ways, i.e.[tex]6*5*4*3*2*1 = 720.[/tex]
For words ending with E, there are again 6 letters left and these can be arranged in 6! ways, i.e. [tex]6*5*4*3*2*1 = 720[/tex].
Now the total number of words which start with O or end with E will be the sum of two, i.e. 720 + 720 = 1440.
Therefore, the number of words formed with letters of the word 'PROBLEM' which neither start with O nor end with E will be the total number of words minus the total number of words which start with O or end with E, i.e. 5040 - 1440 = 3600.
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Angela is riding on a circular Ferris wheel that has a 59-foot radius. After boarding the Ferris wheel, she traveled a distance of 44.3 feet along the arc before the Ferris wheel stopped for the next rider.
a) Make a drawing of the situation and illustrate relevant quantities.
b) The angle that Angela swept out along the arc had a measure of how many radians?
c) The angle that Angela swept out along the arc had a measure of how many degrees?
a) Drawing of the situation is shown in below figure.
b) 0.75 radians
c) 42.97 degrees
Define the term conversion?Conversion is the process of changing a value from one unit or system of measurement to another.
a) The situation of the drawing: The center of the Ferris wheel is labeled "O", and its radius is 59 feet. Angela boards the Ferris wheel at point A and travels a distance of 44.3 feet along the arc to point B. The angle that she sweeps out along the arc is labeled θ.
(Drawing of the situation is shown in below figure)
b) The length of an arc of a circle by the formula: s = rθ
Given, the radius of the circle is 59 feet and the length of the arc that Angela travels is 44.3 feet. So,
θ = s / r
θ = 44.3 / 59
θ ≈ 0.75 radians
Therefore, the angle that Angela sweeps out along the arc has a measure of approximately 0.75 radians.
c) To convert radians to degrees, we use the formula:
θ (in degrees) = θ (in radians) × 180 / π
θ (in degrees) = 0.75 × 180 / π
θ (in degrees) ≈ 42.97 degrees
Therefore, the angle that Angela sweeps out along the arc has a measure of approximately 42.97 degrees.
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a) Drawing of the situation is shown in below figure.
b) 0.75 radians
c) 42.97 degrees
Conversion is the process of changing a value from one unit or system of measurement to another.
a) The situation of the drawing: The center of the Ferris wheel is labeled "O", and its radius is 59 feet. Angela boards the Ferris wheel at point A and travels a distance of 44.3 feet along the arc to point B. The angle that she sweeps out along the arc is labeled θ.
(Drawing of the situation is shown in below figure)
b) The length of an arc of a circle by the formula: s = rθ
Given, the radius of the circle is 59 feet and the length of the arc that Angela travels is 44.3 feet. So,
θ = s / r
θ = 44.3 / 59
θ ≈ 0.75 radians
Therefore, the angle that Angela sweeps out along the arc has a measure of approximately 0.75 radians.
c) To convert radians to degrees, we use the formula:
θ (in degrees) = θ (in radians) × 180 / π
θ (in degrees) = 0.75 × 180 / π
θ (in degrees) ≈ 42.97 degrees
Therefore, the angle that Angela sweeps out along the arc has a measure of approximately 42.97 degrees.
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what is 8 x 1 ????????????
Answer:8
Step-by-step explanation:8x1=8
Question 25 (2 points)
Suppose the Math Department has 17 full-time faculty members. If 3 are selected to
attend a conference in Las Vegas, in how many different ways can you selected the 3
individuals?
3
17
680
4080
Answer:
680 ways
Step-by-step explanation:
C(17, 3) gives 17! / (14! 3!), or (17*16*15)/6 = 680 ways to select the 3 individuals.
Hope this helped!
begin by finding the area under the curve from to , . this area can be written as the definite integral
The area under the curve y = 1/ (x^2 + 6x -16) from x = t to x = 6 is 1/10( ln(4) - 1/10 ln(t+8))
To find the area under the curve y = 1/ (x^2 + 6x -16) from x = t to x = 6, where t > 2, we need to evaluate the definite integral:
∫[t,6] 1/ (x^2 + 6x -16) dx
To solve this integral, we can use partial fraction decomposition. First, we factor the denominator:
x^2 + 6x -16 = (x+8)(x-2)
Then, we can write:
1/ (x^2 + 6x -16) = A/(x+8) + B/(x-2)
Multiplying both sides by (x+8)(x-2), we get:
1 = A(x-2) + B(x+8)
Setting x = -8, we get:
1 = A(-10)
So, A = -1/10.
Setting x = 2, we get:
1 = B(10)
So, B = 1/10.
Therefore, we can write:
1/ (x^2 + 6x -16) = -1/10(x+8) + 1/10(x-2)
Substituting this into the integral, we get:
∫[t,6] 1/ (x^2 + 6x -16) dx = ∫[t,6] (-1/10(x+8) + 1/10(x-2)) dx
Integrating, we get:
= [-1/10 ln|x+8| + 1/10 ln|x-2|] from t to 6
= 1/10 ln|6-2| - 1/10 ln|t+8|
= 1/10 ln(4) - 1/10 ln(t+8)
Therefore, the area is: 1/10( ln(4) - 1/10 ln(t+8))
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_____The given question is incomplete, the complete question is given below:
begin by finding the area under the curve from to y = 1/ (x^2 + 6x -16) from x = t to x = 6, t>2 this area can be written as the definite integral
The number 0 is an element of the set of natural numbers.
OA. True
B. False
SUBI
it is false. 0 is not a natural number. it is a whole number
Use the given acceleration function to find the velocity vector v(t), and position vector r(t). Then find the position at time t = 4.
a(t) = eti − 6k
v(0) = 2i + 9j + k, r(0) = 0
The velocity vector and position vector and position vector at t=4 is r(4) = (e₄+2)i+36j - 44k.
a(t) = eti - 6k
Since we know that v(t) = ∫ a(t) dt
= ∫ (eti - 6k) dt
= ∫6ti-6tk+c
where c is the arbitrary vector valued constant
since it is given that
v(0) = 2i + 9j + k
therefore from above
v(0) = e * 0i - 6(0) * k + c
2i + 9j + k =i+c
C= i +9j+k
therefore,
v(t) = eti - 6tk + i + 9j + k
= (et + 1) * i + 9j + (- 6t + 1) * k
Since we know that velocity vector can be found by integration of acceleration vector.
Since, v(t) = (et + 1) * i + 9j + (- 6t + 1) * k
and we know that
R(t) = ∫ v(t)dt
= ∫ of [(a + 1)i + 9j+(-6t + 1)k]dt =(a+t)i+9tj+(-3ta+t)k+C
where C is an arbitrary vector constant.
Now,
Since it given that r(0)=0 therefore
r(0) =(e0+1)+9(0)j)+(-3(0)2+0)x+C
0=2i+ C
C= -2i
therefore
r(t)= (et+t)i+9tj+(-3t+t)k-2i
r(t)=(a+t-2)1+9tj+(-3t+t)k
Since we know that position vector can be found by integration of velocity vector
r(4) = (e4+4-2)i+9(4)j + (-3(4)+4)k
r(4) = (e4+2)1+36j-44k
Now we have found the velocity vector and position vector and position vector at t=4 which are as follows:
v(t) =(et+1)i+9j+(-6t+1)k
r(t) =(et+t-2)i+9tj+(-3t2+t)k
r(4) = (e₄+2)i+36j - 44k
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if cot0=3/4 and the terminal point determined by 0 is in quadrant 3, then
If cotθ = 3/4 then cosθ = -3/5 is the right option according to the rules of trigonometry.
What is Trigonometry?Trigonometry is a branch of mathematics that deals with the study of relationships between the sides and angles of triangles. It is primarily concerned with the study of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent, and their applications in various fields such as engineering, physics, and navigation.
What are angles of triangle?A triangle is a three-sided polygon, and its angles are the angles formed by the intersection of its sides. The sum of the angles in a triangle is always 180 degrees.
First, we know that cot(0) = adjacent / opposite = 3/4.
In quadrant 3, the adjacent side is negative and the opposite side is positive, so we can draw a right triangle in quadrant 3 with adjacent side -3 and opposite side 4.
The hypotenuse can be found using the Pythagorean theorem.
h² = adjacent²+ opposite²
h² = (-3)^2 + 4^2
h²= 9 + 16
h² = 25
h = 5
So we have a right triangle in quadrant 3 with adjacent side -3, opposite side 4, and hypotenuse 5.
Using the definitions of the trigonometric functions, we can find the values of the other functions:
sin(0) = opposite / hypotenuse = 4/5
cos(0) = adjacent / hypotenuse = -3/5
tan(0) = opposite / adjacent = -4/3
csc(0) = hypotenuse / opposite = 5/4
sec(0) = hypotenuse / adjacent = -5/3
cot(0) = adjacent / opposite = 3/4 (given)
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