Answer:
87.1 degrees
Step-by-step explanation:
First, you know the sum of the angles is 180 degrees. So, 5x+4x+10+10x-5=180.
Then, simplify to 19x+5=180. So, 19x=175. That means by dividing both sides by 19 x is approximately 9.21.
Finally, plug in x into angle 3.
10x9.21-5
=92.1-5
=87.1
So, the third angle is approximately 87.1.
y= -6x - 43
y= -8x - 59
what is the answer to x and y. Please include step by step.
Step-by-step explanation:
Don't worry we will take care of everything. It's easy. listen carefully
first of all write your questions carefully.
i think it should be like this-
y= -6x +43 ------ (i)
y= -8x -59---------(ii)
1. We have two unknowns x and y. and two equations .
2. CONCEPT:- there are several ways to solve it like.
Elimination methodmultiplication methodsubstitution method ( we will go with this)ACCORDING TO SUBSTITUTION METHOD:-
Simply, write the equation in term of single variable(unknown). * [ which is already done for us]
second step, substitute the expression of obtained variable in any one equation.
example: substitute the value of y from equation (i) in (ii)
we will obtained equation -
=> y= -8x -59 [ put value of y from (i) ]
=> -6x +43 = -8x -59
(solve and find x)
=> -6x +8x = -59 - 43
=> 2x = -102
=> x = -51
Third step in substitute method is, now substitute this obtained value of x in any equation (i) or (ii) to get unknown y.
y will be = 349
* your questions was wrong so i have made it by myself. if this does not match your question sorry from my side. buti have shown you the concept to solve this type of questions. You will be able to solve this by your self now by applying concepts.
(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and inflection points.
f(x)= x^4-2x^2+3
Part (c) is where I'm having issues on how to derive the second derivative f'(x)=4x(x-1)(x+1)
can you please show all steps.
The inflection points are at x = -[tex]\sqrt{(1/3)}[/tex] and x = sqrt(1/3), and f is concave up on (-[tex]\sqrt{1/3}[/tex]), [tex]\sqrt{(1/3)}[/tex]), and concave down elsewhere.
(a) To find the intervals on which f is increasing or decreasing, we need to find the first derivative of f and determine its sign.
[tex]f(x) = x^4 - 2x^2 + 3[/tex]
[tex]f'(x) = 4x^3 - 4x[/tex]
Setting f'(x) = 0, we get:
[tex]4x(x^2 - 1) = 0[/tex]
This equation has roots at x = 0, x = -1, and x = 1. We can use the first derivative test to determine the intervals of increasing and decreasing.
When x < -1, f'(x) is negative, so f is decreasing.
When -1 < x < 0, f'(x) is positive, so f is increasing.
When 0 < x < 1, f'(x) is negative, so f is decreasing.
When x > 1, f'(x) is positive, so f is increasing.
Therefore, f is decreasing on (-∞, -1) and (0, 1), and increasing on (-1, 0) and (1, ∞).
(b) To find the local maximum and minimum values of f, we need to look for critical points and evaluate the function at those points.
Critical points occur where f'(x) = 0 or is undefined. We have already found that f'(x) = 0 at x = 0, x = -1, and x = 1.
f(0) = 3, f(-1) = 6, f(1) = 2
Therefore, f has a local maximum at x = -1, with a value of 6, and local minimums at x = 0 and x = 1, with values of 3 and 2, respectively.
(c) To find the intervals of concavity and inflection points, we need to find the second derivative of f and determine its sign.
[tex]f(x) = x^4 - 2x^2 + 3[/tex]
[tex]f'(x) = 4x^3 - 4x[/tex]
[tex]f''(x) = 12x^2 - 4[/tex]
Setting f''(x) = 0, we get:
[tex]12x^2 - 4 = 0[/tex]
[tex]x^2 = 1/3[/tex]
x = ±sqrt(1/3)
We can use the second derivative test to determine the intervals of concavity.
When x < -sqrt(1/3), f''(x) is negative, so f is concave down.
When -sqrt(1/3) < x < sqrt(1/3), f''(x) is positive, so f is concave up.
When x > sqrt(1/3), f''(x) is negative, so f is concave down.
Therefore, the inflection points are at x = -sqrt(1/3) and x = sqrt(1/3), and f is concave up on (-sqrt(1/3), sqrt(1/3)), and concave down elsewhere.
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There is a light pole that casts a shadow that is 75 feet. At the same time, there is a stop sign that is 8 feet tall and casts a shadow that is 12 feet. How tall is the light pole? If necessary, round your answer to the nearest tenth.
Answer: Therefore, the height of the light pole is 50 feet.
Step-by-step explanation:
We can use proportions to solve the problem.
Let h be the height of the light pole.
Then, we have:
h / 75 = 8 / 12
To solve for h, we can cross-multiply:
12h = 75 * 8
h = (75 * 8) / 12
h = 50
(-4,2) and (3,-3) what’s the slope
Answer:
7 over -5
Step-by-step explanation:
Rashaad leans a 16-foot ladder against a wall so that it forms an angle of 66° with the ground. what's the horizontal distance between the base of the ladder and the wall?
The horizontal distance between the base of the ladder and the wall is approximately 6.58 feet
We can use trigonometric function to solve this problem. Let's call the horizontal distance we are looking for "x".
First, we can use the fact that the ladder forms an angle of 66° with the ground to find the vertical height it reaches. We know that the ladder is 16 feet long, and we can use the sine function to find the vertical height
sin(66°) = height/16
height = 16×sin(66°) = 15.12 feet (rounded to two decimal places)
Now, we can use the same angle and the cosine function to find the horizontal distance x
cos(66°) = x/16
x = 16×cos(66°) = 6.58 feet (rounded to two decimal places)
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At a parade 1/4 of the people have red hair, 1/6 of them have brown hair the rest of the people have black hair. What fraction of the people have black hair
A procession typically has one-fourth of the participants with red hair, one-sixth with brown hair, and the remaining participants with black hair. 7/12 of the population has black hair.
We are given that 1/4 of the people have red hair and 1/6 of them have brown hair.
Let's add these fractions together to find the fraction of people who have either red or brown hair:
[tex]\\\frac{1}{4} + \frac{1}{6}\\ =\frac{3+2}{12} \\ =\frac{5}{12}[/tex]
This means that 5/12 of the people have either red or brown hair.
Since the remaining people must have black hair, we can subtract this fraction from 1 to find the fraction of people who have black hair:
1 - 5/12
=12-5/12
= 7/12
Therefore, the fraction of people who have black hair is 7/12.
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simplify (1/2)³x(1/2)² and write in exponential form
Answer:
To simplify (1/2)³x(1/2)², we can multiply the coefficients and add the exponents with the same base:
(1/2)³x(1/2)² = (1/2)^(3+2) = (1/2)^5
Therefore, (1/2)³x(1/2)² simplified is equal to (1/2)^5, which can also be written as 1/32 in fraction form.
Step-by-step explanation:
In order to multiply we use the formula for indices' multiplication that is
[tex] {n}^{a} \times {n}^{b} = {n}^{a + b} [/tex]
Hence,
[tex]{( \frac{1}{2} )}^{3}( { \frac{1}{2}}^{2}) = {( \frac{1}{2} )}^{5} [/tex]
[tex] ( \frac{{1}^{5} }{{2}^{5} } ) = {2}^{ - 5} [/tex]
If 25 tapered pins can be machined from a steel rod 15 ft long, how many tapered pins can be made from a steel rod 9 ft long?
If 25 tapered pins are machined from a steel rod 15 ft long, then total 15 tapered pins can be made from a steel rod 9 ft long.
The unitary method is a method where you find the value of one unit and then find the value of the number of units you want. The formula for the units method is to find the value of a single unit and then multiply the value of the single unit by the number of units to get the desired value. We have specify that 25 tapered pins can be made up from a steel rod 15 ft long. Let 'x' represent the number of pins that can be made from 9 ft long steel rod. Now, using the unitary method,
number of pins machined from a 15 ft long steel rod = 25
number of pins machined from a 1 ft long steel rod = 25/15
So, number of pins machined from a 9 feet long steel rod, x = (25/15)×9
=> x = 3×5 = 15
Hence, required tapered pins made from a steel rod 9 ft long is equals to the 15 pins.
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Which of the following direct variations has a constant of variation that is equal to -3? Click on the graph until the correct graph appears.
The graph given in this problem represents a proportional relationship with a constant of variation that is equals to -3.
What is a proportional relationship?A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other. This means that if one quantity is multiplied by a certain factor, the other quantity will also be multiplied by the same factor.
The equation that defines the proportional relationship is given as follows:
y = kx.
In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.
For the graph given, when x increases by 1, y decays by 3, hence the constant is given as follows:
k = -3.
Then the equation is given as follows:
y = -3x.
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Zahra is 4/5 as tall as Zayden. Zayden is 3/5 as tall as Eric. Eric is how many times as tall as Zahra?
Answer:
1/5 is the right answer in my opinion but I'm not sure
HELP PLEASE!! ALSO IF U CAN PLEASE ANSWER MY RECENT QUESTIONS OF TODAY IF UR GENEROUS THANKS BAES
Answer:b
Step-by-step explanation:
just took the quiz
Explain what value or values of c make the equation x^2+12x+c=0
Answer:x^2+12x-28=01. 12/2=62. (x+6)^2=x^2+12x+363. If we subtract 64 from this expression it becomesx^2+12x+36-64=x^2+12x-28 and so equals 0We now have (x+6)^2-64=0This is our old friend a^2-b^2=(a+b)(a-b)(x+6)^2-8^2=(x+6+8)(x+6-8)=(x+14)(x-2)=0 and so x=-14 or x=2
Step-by-step explanation:
if z,z+3 are the roots of the equation x²-5x+k then; z= and k=
The roots of the equation x² - 5x + k are z = (5/a - 3)/2 and z+3 = (5/a + 3)/2, and the value of k is 9/4.
Define roots of the equationThe roots of an equation are the values of the variable that satisfy the equation and make it true. In the case of a quadratic equation of the form ax² + bx + c = 0, the roots are the values of x that satisfy the equation and make it equal to zero.
Given: z and z+3 are the roots of the equation x² - 5x + k
The sum of the roots z and z+3 is:
z + (z + 3) = 2z + 3
And the product of the roots z and z+3 is:
z (z + 3) = z² + 3z
By Vieta's formulas, we know that the sum of the roots of a quadratic equation ax² + bx + c = 0 is -b/a and the product of the roots is c/a. Therefore, we can equate these expressions to the sum and product of the roots of the given equation:
2z + 3 = 5/a (since b = -5 and a = 1)
z² + 3z = k/a (since c = k and a = 1)
Solving for z in the first equation, we get:
2z = 5/a - 3
z = (5/a - 3)/2
Substituting this value of z into the second equation, we get:
[(5/a - 3)/2]² + 3[(5/a - 3)/2] = k/a
Simplifying and solving for k, we get:
k = a[(5/a - 3)/2]² + 3a[(5/a - 3)/2]
= (5a - 9a + 18)/4
= 9/4
Therefore, the roots of the equation x² - 5x + k are z = (5/a - 3)/2 and z+3 = (5/a + 3)/2, and the value of k is 9/4.
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as the set designer for his school's fall play, kenji is making a treasure chest shaped like a rectangular prism. the chest needs to hold approximately 6 cubic feet, or 10,368 cubic inches, of sand that will be spilled out during the first act. also, since an actor will stand on the chest in the second act, the chest needs to be 24 inches tall. kenji decides that the chest will be twice as long as it is wide. to the nearest tenth of an inch, what is the width of the chest?
The width of the chest shaped like a rectangular prism, to the nearest tenth of an inch is 32.1.
Kenji has made a treasure chest, shaped like a rectangular prism that must hold around 10,368 cubic inches of sand. In the second act of the school play, an actor will stand on the chest.
Therefore, the chest must be 24 inches tall. The length of the chest is twice its width.
To determine the width of the chest, you should first determine the length of the chest.
Then, using the formula for the volume of a rectangular prism, you can solve for the width of the chest.
Length of the chest
The formula for the volume of a rectangular prism is V = lwh
Given that the chest needs to hold around 10,368 cubic inches of sand, we can write 10,368 = lwh
Since the chest is twice as long as it is wide, the length (l) of the chest is equal to 2w.
Substituting 2w for l, we have:
10,368 = (2w)w
hence 5,184 = wh²(5,184/h)
= w*h(5,184/h²) = w
Since the chest must also be 24 inches tall, we can write:
h = 24 feet
Substituting this value into the equation, we have:
(5,184/24²) = w
hence w = 32.1 inches
Therefore, the width of the chest to the nearest tenth of an inch is 32.1 inches.
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ricky has 23 word hours each week to dedicate to his classes.Homework takes 6.5 hours and each class (c) is 1.5 hours long.How many classes does ricky take?
Ricky has 23-word hours each week to dedicate to his classes. Homework takes 6.5 hours and each class (c) is 1.5 hours long. Thus, 05 classes Ricky takes every week.
There are 60 minutes in 1 hour. To convert from minutes to hours, divide the number of minutes by 60. For example, 120 minutes equals 2 hours because 120/60 = 2.
According to the Question:
Let us consider
a = hour she dedicates to his classes = 6.5 hours
and
b = hours spend on homework= 1.5 hours
Total hours in a week devoted to his classes = 23/7
Therefore, 23/7 ×1.5
= 5 classes.
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Can someone answer this question, please? Thanks!
Answer:
m>22 and m[tex]\leq[/tex]34
Step-by-step explanation:
Explain how to find the diameter of a circle if you know the circumference.
If the value of C, the circumference is known, then the diameter is:
D = C/pi
How to get the diameter of the circle if you know the circumference?For a circle whose radius is R, the formula for the circumference is:
C = 2*pi*R
Where pi = 3.14
Particularly, we know that the diameter of a circle is twice the radius, os:
D = 2*R
We can rewrite the circumference equation to get:
C = pi*(2*R) = pi*D
So, solving that equation for the diameter, we will get:
D = C/pi
That equation gives the diameter if we know the value of the circumference.
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40% of the people in a town are females. 10% of the females are left -handed. 20. 8% of all the people are left -handed. Work out the percentage of the people who are not female who are left -handed
Answer:
16.8%
Step-by-step explanation:
When fourty percentage of the people in a town are females. 10% of the females are left -handed. Then total 16.8% of peoples in the town are not female and are left-handed.
We have a percentage data of people in a town. Percentage of female peoples in the town = 40 %
Percentage of female left handed = 10 %
Percentage of total left handed = 20.8 %
Percentage is defined as a ratio of a number to 100 or it expressed as a fraction of 100. Let Total population of the town = X
So, female peoples in the town = 40% of X = (40/100) × X
= 0.4X
The peoples who are not female = X - 0.4X = 0.96X
Number of female left-handed = 10% of 0.4X = (10/100)× 0.4X
= 0.04X
Number of all left-handed in the town
= 20.8% of X = (208/100)× X
= 0.208X
Left-handed peoples who are not female
= 0.208X - 0.04X
= 0.168X
The percentage of people who are not female and who are left-handed
= (0.168X /X )×100
= 16.8%
Hence, required percentage is 16.8%.
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You are trying to hang a tire swing. To get the rope over a tree branch that is 15 feet high, you tie the rope to a weight and throw it over the branch. You release the weight at a height of 5.5 feet. What is the minimum upward velocity needed to reach the branch?
a) 7.5 feet per seconds
b) 19.7 feet per seconds
c) 20.5 feet per seconds
d) 24 feet per seconds
Since the minimum upward velοcity cannοt be negative, the answer is (A) 7.5 feet per secοnd. If we had taken the absοlute value οf the velοcity, the answer wοuld be (B) 19.7 feet per secοnd.
What is Velοcity?Velοcity is a physical quantity that describes the rate οf change οf an οbject's pοsitiοn with respect tο time. It is a vectοr quantity that has bοth magnitude and directiοn, with the magnitude representing the speed οf the οbject and the directiοn indicating its mοtiοn. In οther wοrds, velοcity tells us hοw fast an οbject is mοving and in which directiοn.
Tο find the minimum upward velοcity needed tο reach the branch, we can use the cοnservatiοn οf energy principle. At the height οf 5.5 feet, the weight has pοtential energy, which will be cοnverted tο kinetic energy as it falls. When the weight reaches the branch at a height οf 15 feet, all οf its initial pοtential energy will be cοnverted tο kinetic energy. Therefοre, we can equate the pοtential energy at the initial height with the kinetic energy at the final height:
[tex]mgh = (1/2)mv^2[/tex]
There m is the mass οf the weight (which cancels οut), g is the acceleratiοn due tο gravity (32 ft/s²), h is the initial height (5.5 ft), and v is the velοcity at the final height (15 ft).
Substituting the values, we get:
(32 ft/s²) * (15 ft - 5.5 ft) = (1/2) * v²
v² = 2 * (32 ft/s²) * (15 ft - 5.5 ft) = 800 ft²/s²
Taking the square rοοt οf bοth sides gives:
v = √(800) = 28.28 ft/s
Hοwever, we need the minimum upward velοcity, sο we must subtract the initial dοwnward velοcity (due tο gravity) οf 32 ft/s:
minimum upward velοcity = v - g = 28.28 ft/s - 32 ft/s = -3.72 ft/s
Since the minimum upward velοcity cannοt be negative, the answer is (A) 7.5 feet per secοnd, which is the magnitude οf the upward velοcity.
Nοte: If we had taken the absοlute value οf the velοcity, the answer wοuld be (B) 19.7 feet per secοnd. Hοwever, the minimum upward velοcity is the cοrrect answer in this cοntext.
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safety information about a ladder indicates that a ladder should not be set up at an angle less than 45 degrees or at an angle greater than 75 degrees. What range of height can a 20-foot ladder reach?
According to the question the height of the ladder is 34.641 feet, or approximately 19.98 feet.
What is height?Height is the measure of vertical distance, either how "tall" something or someone is, or the distance between the top and bottom of an object. It is measured in either centimeters, meters, or feet and inches, and can be used to describe the elevation of land or the altitude of an aircraft.
Let x be the height of the ladder.
For a 20-foot ladder at an angle of 45°, we have the following trigonometric equation:
tan(45°) = x/20
Solving for x we get:
x = 20tan(45°)
x = 20 × 1
x = 20
Therefore, the height of the ladder is 20 feet.
For a 20-foot ladder at an angle of 75°, we have the following trigonometric equation:
tan(75°) = x/20
Solving for x we get:
x = 20tan(75°)
x = 20 × 1.73205
x = 34.641
Therefore, the height of the ladder is 34.641 feet, or approximately 19.98 feet.
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The range of height can be determined by trigonometric ratios and the rage is 20ft to 74.6 ft(approximately).
What is Trigonometric Ratios?
Trigonometric Ratios are based on the value of the ratio of sides of a right-angled triangle where Hypotenuse is the longest side Perpendicular is opposite side to the angle and Base is the Adjacent side to the angle. The ratios of these three sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.
Given that a ladder should not be set up at an angle less than 45 degrees or at an angle greater than 75 degrees.
The ladder is 20 foot.
here the trigonometric function tangent will be used.
By the formula, tan Ф = Perpendicular/ adjacent side [ where Ф is the acute angle]
Let, the height of the ladder is x foot.
Then by formula of trigonometric ratio,
tan 45° = x/20
⇒ x/20 =1 [ since tan 45° =1 ]
⇒ x= 20
Again,
tan 75° = x/ 20
⇒ x= 20× tan 75°
⇒ x= 74.6 [ since tan 75° = 3.732]
Hence, the range of height is 20 foot to 74.6 foot ( approx ).
The rough diagram is attached below.
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3, 9, 15,.
Find the 45th term.
The 45th term of the sequence is 267.
The 45th term of a sequence can be calculated using the formula:
Tn = a + (n – 1)d
Where,
Tn = the nth term
a = the first term
n = the term position
d = the common difference
In this sequence, the first term is 3, the common difference is 6, and the term position is 45.
Therefore, the 45th term of the sequence is:
T45 = 3 + (45 – 1)6
T45 = 3 + (44)6
T45 = 3 + 264
T45 = 267
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John and Anvil share some money in the ratio 10: 6.
John gets $36 more than Anvil. How much do they get each?
Answer:
54 , 90
Step-by-step explanation:
Let John = 10x
Let Anvil = 6x
10x = 36 + 6x
10 - 6x = 36
4x = 36
x=9
Anvil = 6*9 = 54
John = 10*9 = 90
Four times the sum of 5 and some number is 20. What is the number?
Applying mathematical operations, "Four times the sum of 5 and some number is 20," the number is 0.
What are the mathematical operations?The mathematical operations include addition, subtraction, division, and multiplication.
Mathematical or algebraic operations use mathematical operands to manipulate values, variables, constants, and numbers to solve mathematical problems.
Is zero a number?Zero is considered to be a neutral number since it is neither positive nor positive.
Zero is important as a number placeholder.
4 (5 + x) = 20
20 + 4x = 20
4x = 20 - 20
4x = 0
x = 0
Thus, in the mathematical operations of multiplying the sum of 5 and some number by 4, the applicable number is zero.
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Add the polynomials functions, as indicated below. (Q+R)(x)
Answer:
x^3 + 5x - 23
Step-by-step explanation:
(Q+R)(x) is the same as Q(x) + R(x).
you just add them like so:
(x^3 - 9x^2 - 14) + (9x^2 + 5x - 9)
= x^3 + 5x - 23
Find the value of k such that the line through (6, 4) and (k,2) is parallel to the line y=2x .
the value of k such that the line through (6, 4) and (k, 2) is parallel to the line y = 2x is k = 5.
How to solve the question ?
To find the value of k such that the line through (6, 4) and (k, 2) is parallel to the line y = 2x, we need to find the slope of the line y = 2x, which is 2.
Since two lines are parallel if and only if they have the same slope, we know that the line through (6, 4) and (k, 2) must also have a slope of 2.
Using the slope formula:
slope = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) = (6, 4) and (x₂, y₂) = (k, 2), we get:
2 = (2 - 4) / (k - 6)
Multiplying both sides by k - 6 gives:
2(k - 6) = -2
Expanding and simplifying gives:
2k - 12 = -2
Adding 12 to both sides gives:
2k = 10
Dividing both sides by 2 gives:
k = 5
Therefore, the value of k such that the line through (6, 4) and (k, 2) is parallel to the line y = 2x is k = 5.
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expand 3(x+6)
please help me on the eqation
Answer:
3x + 18
Step-by-step explanation:
3 ( x + 6 )
= 3*x + 3*6
= 3x + 18
Answer:
3x + 18
Step-by-step explanation:
expand 3(x+6)
3(x + 6) =
3 * x + 3 * 6
3x + 18
pQRs is Rhombous '0' is the bisect of diagonal Qs and pR if p0=x or y-1 0Q=2x+7 and 0s =4y-5 what is value of x and y
Answer:
Step-by-step explanation:
The measures of the angles of a triangle are shown in the figure below. Solve for x.
36°
(6x+3)°
87°
Answer:
Step-by-step explanation:The sum of the measures of the angles in a triangle is always 180 degrees. So, we can set up an equation and solve for x:
36 + (6x+3) + 87 = 180
Combine like terms:
126 + 6x = 180
Subtract 126 from both sides:
6x = 54
Divide both sides by 6:
x = 9
Therefore, the value of x is 9.
Anna Has d dimes Brittany has 4 times as many dimes as Anna has write and expression for the total number of dimes Anna and Brittany have them simplify the expression
The expression represent total number dime both have is 5d.
What is expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. Unknown variables, integers, and arithmetic operators are the components of an algebraic expression. There are no symbols for equality or inequality in it.
Here Anna Has d dimes and Brittany has 4 times as many dimes as Anna.
Then,
Number of dime Anna has = d Number of dime Brittany has = 4d
Now the total number of dimes both have is:
Total dimes = d+4d
= 5d
Hence the expression represent total number dime both have is 5d.
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What is the average rate of change of [tex]f(x)=\frac{1}{2}x-7[/tex] from [tex]x=2[/tex] to [tex]x=10[/tex] ?
Answer:
1/2
Step-by-step explanation:
You want to know the average rate of change of f(x) = 1/2x -7 on the interval [2, 10].
Linear functionA linear function has the same rate of change everywhere. Its instantaneous rate of change, average rate of change, and slope are all the same constant.
The given function is written in slope-intercept form, so we can identify the slope as the coefficient of x: 1/2.
The average rate of change is 1/2.
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Additional comment
If you feel the need to "show work", you can evaluate ...
AROC = (f(10) -f(2))/(10 -2) = (-2 -(-6))/8 = 4/8 = 1/2