Answer:
Option A.
Step-by-step explanation:
The given equation of ellipse is
[tex]4x^2+9y^2=36[/tex]
Divide both sides by 36.
[tex]\dfrac{4x^2}{36}+\dfrac{9y^2}{36}=1[/tex]
[tex]\dfrac{x^2}{9}+\dfrac{y^2}{4}=1[/tex]
[tex]\dfrac{x^2}{3^2}+\dfrac{y^2}{2^2}=1[/tex] ...(1)
The standard form of an ellipse is
[tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex] ...(2)
where, (h,k) is center, (h±a,k) are vertices and (h±c,k) are foci.
On comparing (1) and (2), we get
[tex]h=0,k=0,a=3,b=2[/tex]
Now,
Center [tex]=(h,k)=(0,0)[/tex]
Vertices [tex]=(h\pm a,k)=(0\pm 3,0)=(3,0),(-3,0)[/tex]
We know that
[tex]c=\sqrt{a^2-b^2}=\sqrt{3^2-2^2}=\sqrt{5}[/tex]
Foci [tex]=(h\pm c,k)=(0\pm \sqrt{5},0)=(\sqrt{5},0),(-\sqrt{5},0)[/tex]
Therefore, the correct option is A.
What is the constant of variation, k, of the line y=kx through (3,18) and (5,30)? 3 6
Answer:
6
Step-by-step explanation:
The constant of variation is the slope
k = (y2-y1)/(x2-x1)
= (30-18)/(5-3)
=12/2
= 6
The value of constant of variation, k, is,
⇒ k = 6
What is Equation of line?
The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Here, the constant of variation, k, of the line y = kx through (3,18) and (5,30)
Since, The constant of variation is the slope,
Hence, We get;
k = (y₂ - y₁)/(x₂ - x₁)
= (30 - 18)/(5 - 3)
= 12/2
= 6
Thus, the value of constant of variation, k, is,
⇒ k = 6
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What number must be added to the expression below to complete the square? x2-5x
Answer:
6.25
Step-by-step explanation:
(x-a)^2=x^2-2ax+a^2
2a=5
a=2.5
2.5 ^ 2 = 6.25
What is the area of the house (including the drawing room, TV room, balcony, hallway, kitchen, and bedroom)?
Answer:
A
Step-by-step explanation:
Sharon tried to solve an equation step by step.
9
9
15
5
=−3(e−2)
=−3e+6
=3e
=e
Step 1
Step 2
Step 3
Find Sharon's mistake.
Choose 1 answer:
Choose 1 answer:
Answer:
step 2
Step-by-step explanation:
9 = -3(e - 2)
9 = -3e + 6
9-6 = -3e
3 = -3e
divide both sides by -3
-1 = e
Suppose you finance a home in the amount of $425,000 at 4.3% compounded monthly for 30 years. Calculate your payment.
Answer: $1,540,194.30 .
Step-by-step explanation:
The formula to calculate the accumulated amount earned on principal (P) at rate of interest (r)[ in decimal] compounded monthly after t years :
[tex]A=P(1+\dfrac{r}{12})^{12t}[/tex]
Given: P= $425,000
r= 4.3% = 0.043
t= 30 years
[tex]A=425000(1+\dfrac{0.043}{12})^{12(30)}\\\\=425000(1.003583)^{360}\\\\=425000\times3.62398657958\approx1540194.30[/tex]
Hence, the payment would be $1,540,194.30 .
Suppose that the credit remaining on a phone card (in dollars) is a linear function of the total calling time (in minutes). When graphed, the function gives a line with a slope of -0.12. See the figure below. There is $28.96 in credit remaining on the card after minutes of calls. How much credit was there after 21 minutes of calls?
Answer:
Credit remaining after 21 minutes = $30.4
Step-by-step explanation:
Credit remaining on a phone card is a linear function of the total calling time.
When graphed, let the linear function representing the line is,
y = mx + b
Where 'm' = slope of the line
b = y-intercept
From the graph,
Slope of the line = -0.12
y = -0.12x + b
If this line passes through a point (33, 28.96),
28.96 = -0.12(33) + b
b = 28.96 + 3.96
b = 32.92
Therefore, the linear function is,
f(x) = -0.12x + 32.92
where x = calling time
Credit left in the card after 21 minutes,
f(21) = -0.12(21) + 32.92
= -2.52 + 32.92
= $30.4
20 applicants are interviewed for a job. The interviewer creates an ordered list (first to last) of the best 6 applicants. How many ways are
Answer:
38,760 different ways.Step-by-step explanation:
The question is incomplete. Here is the complete question.
20 applicants are interviewed for a job. The interviewer creates an ordered list (first to last) of the best 6 applicants. How many ways are there for the interviewer to create the list?
Since the question deals with selection, we will apply the combination rule. Generally, if r objects are to be selected from n pool of objects, this can be done in nCr number of ways.
nCr = n!/(n-r)!r!
According to the question, if the the interviewer is to select 6 applicants from a pool of 20 applicants that are interviewed for a job, this can be done in 20C6 number of ways.
20C6 = 20!/(20-6)!6!
20C6 = 20!/14!6!
20C6 = 20*19*18*17*16*15*14!/14!*6*5*4*3*2
20C6 = 20*19*18*1716*15/6*5*4*3*2
20C6 = 27,907,200/720
20C6 = 38,760 different ways.
Hence, the interviewer can create the list in 38,760 different ways.
In how many ways can a subcommittee of 6 students be chosen from a committee which consists of 10 senior members and 12 junior members if the team must consist of 4 senior members and 2 junior members?
Answer:
The number of ways is 13860 ways
Step-by-step explanation:
Given
Senior Members = 10
Junior Members = 12
Required
Number of ways of selecting 6 students students
The question lay emphasis on the keyword selection; this implies combination
From the question, we understand that
4 students are to be selected from senior members while 2 from junior members;
The number of ways is calculated as thus;
Ways = Ways of Selecting Senior Members * Ways of Selecting Junior Members
[tex]Ways = ^{10}C_4 * ^{12}C2[/tex]
[tex]Ways = \frac{10!}{(10-4)!4!)} * \frac{12!}{(12-2)!2!)}[/tex]
[tex]Ways = \frac{10!}{(6)!4!)} * \frac{12!}{(10)!2!)}[/tex]
[tex]Ways = \frac{10 * 9 * 8 * 7 *6!}{(6! * 4*3*2*1)} * \frac{12*11*10!}{(10!*2*1)}[/tex]
[tex]Ways = \frac{10 * 9 * 8 * 7}{4*3*2*1} * \frac{12*11}{2*1}[/tex]
[tex]Ways = \frac{5040}{24} * \frac{132}{2}[/tex]
[tex]Ways = 210 * 66[/tex]
[tex]Ways = 13860[/tex]
Hence, the number of ways is 13860 ways
Q-The general solution of inequality cos 2 x≤- sin x is
Answer:
x∈[2nπ−5π/6, 2nπ−π/6]∪{(4n+1)π/2}, n ϵ I
Step-by-step explanation:
1−2sin2 x≤−sin x ⇒ (2sin x+1)(sin x−1)≥0
sin x≤−1/2 or sin x≥1
−5π/6+2nπ≤x≤−π/6+2nπ or , n ϵ I x=(4n+1)π/2, n ϵ I⇒ -5π6+2nπ≤x≤-π6+2nπ or , n ϵ I x=4n+1π2, n ϵ I (as sin x = 1 is valid only)
In general⇒ In general x∈[2nπ−5π/6, 2nπ−π/6]∪{(4n+1)π/2}, n ϵ I
someone please help it's Multiple Coordinate Transformations
Answer:
Step-by-step explanation:
Coordinates of the vertices of the given rectangle are,
A(-4, 5), B(-2, 5), C(-2,1) and D(-4, 1).
If the given rectangle ABCD is translated by 1 unit right and 4 units down,
Rule for the translation will be,
(x, y) → [(x + 1), (y - 4)]
Therefore, coordinates of A, B, C and D after translation will be,
A(-4, 5) → A'(-3, 1)
B(-2, 5) → B'(-1, 1)
C(-2, 1) → C'(-1, -3)
D(-4, 1) → D'(-3, -3)
Then A', B', C' and D' are reflected about the y-axis.
Rule for the reflection about y-axis will be,
(x, y) → (-x, y)
New coordinates after the reflection will be,
A'(-3, 1) → A"(3, 1)
B'(-1, 1) → B"(1, 1)
C'(-1, -3) → C"(1, -3)
D'(-3, -3) → D''(3, -3)
Finally these points are rotated 270° clockwise about the origin,
Rule to be followed,
(x, y) → (-y, x)
Therefore, new coordinates will be,
A"(3, 1) → A"'(-1, 3)
B"(1, 1) → B'"(-1, 1)
C"(1, -3) → C"'(3, 1)
D"(3, -3) → D"'(3, 3)
Solve the polynomial x² - 7x + 12 = 0 Show each step used to find the solutions.
Answer:
x=4 and x=3
Step-by-step explanation:
The first step to find all factors of 12
1,2,3,4,6,12
and their negative form
-1,-2,-3,-4,-6,-12
we must add two of these numbers to get -7
and the two numbers that add up to -7 are -3 and -4
we can simply plot them into the polynomial (x-3)(x-4)
since x-3=0
we add three to both sides to get x=3
and since x-4=0
add four to both sides
we get x=4
and now check out work which
we can use First out in last (foil) method
(x-3)(x-4)
first
x^2
out
-4x
in
-3x
last
12
now we simplify to get x^2-7x+12
so it x=4 x=3 works
Answer:
x² - 7x + 12 = 0
doing middle term factorization
x²-4x-3x+12=0
taking common from each two term
x(x-4)-3(x-4)=0
taking common
(x-4)(x-3)=0
either
x-4=0
:. x=4
or
x-3=0
x=3
:.x=4,3
if 2x-7 is 5 more than x+4, what is the value of 3x+5
Answer:
53
Step-by-step explanation:
Let's start with the given relation:
2x -7 = (x+4) +5
x = 16 . . . . . . . . . add 7-x
3x +5 = 3(16) +5 = 53 . . . . . multiply by 3 and add 5
The value of 3x+5 is 53.
Draw the image of ABC under dilation whose center is P and scale factor is 4.
Answer:
Step-by-step explanation:
Let the point P is origin (0, 0), segment AB lies on the x-axis and segment PC on the y-axis,
Coordinates of A, B and C will be (1, 0), (2, 0) and (0, 2).
When triangle ABC is dilated by a scale factor 4 about the origin,
Rule for dilation;
(x, y) → (4x, 4y)
New coordinates of the points A, B and C will be,
A(1, 0) → A'(4, 0)
B(-2, 0) → B'(-8, 0)
C(0, 2) → C'(0, 8)
By plotting points A', B' and C' we can get the dilated image A'B'C'.
Dilation involves changing the size of a shape.
Assume point P is the center of origin, then we have the following coordinates of ABC
[tex]A = (1,0)[/tex]
[tex]B = (-2,0)[/tex]
[tex]C = (0,2)[/tex]
The scale factor (k) is given as:
[tex]k= 4[/tex]
So, the dilation rule is represented as:
[tex](x,y) \to (4 \times (x,y)[/tex]
Using the above dilation rule, the following coordinates represent the image of triangle ABC
[tex]A' = (4,0)[/tex]
[tex]B' = (-8,0)[/tex]
[tex]C = (0,8)[/tex]
See attachment for the image of the dilation.
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Sketch a graph to envision the following scenario. Place time on the x-axis. A rock climber is 25 feet above sea level. After 8 seconds he is at sea level. (He did not fall). What is the slope of the line depicted (no units)
Answer:
Please refer to the attached graph.
Slope = -3.125
Step-by-step explanation:
Given
Time is placed on x axis.
Initially, height of rock climber is 25 feet above sea level.
[tex]t_1 =0\ sec[/tex]
[tex]h_1[/tex] = 25 feet
After 8 seconds, he is at sea level.
[tex]t_2 =8\ sec[/tex]
[tex]h_2[/tex] = 0 feet
To find:
Graph of the given points and Slope of the line depicted.
Solution:
Kindly refer to the attached graph.
Here, we have been given two points to be plotted on the xy coordinate plane.
1st point is (0, 25): Time is 0 and height above sea level is 25 ft
2nd point is (8, 0): Time is 8 seconds and height above sea level is 0 ft (i.e. he comes to sea level)
First of all, mark these points and join them with a straight line.
Please refer to attached graph.
Slope of a line is given as:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
Here,
[tex]y_2 = 0\\y_1 = 25\\x_2 = 8\\x_1 = 0[/tex]
Using the formula:
[tex]m=\dfrac{0-25}{8-0}\\\Rightarrow m=-\dfrac{25}{8} = \bold{-3.125}[/tex]
The histogram shows that nine students had grades of 80 or higher.
The histogram shows there were 22 students in the class.
The histogram shows there were 25 students in the class.
The histogram is symmetrical.
The histogram has a peak.
The histogram shows the data is evenly distributed.
The histogram shows a gap in the data
Answer:
bde
Step-by-step explanation:
Answer:
B: The histogram shows there were 22 students in the class.
D: The histogram is symmetrical.
E:The histogram has a peak.
F: The histogram shows the data is evenly distributed.
Step-by-step explanation:
edg 2020
Find the principal invested if $495 interest was earned in 3 years at an interest rate of 6%.
Answer: $2750
Step-by-step explanation:
Formula to calculate interest : I = Prt , where P = Principal amount , r = rate of interest ( in decimal) , t= time.
Given: I= $495
t= 3 years
r= 6% = 0.06
Then, according to the above formula:
[tex]495 = P (0.06\times3)\\\\\Rightarrow\ P=\dfrac{495}{0.18}\\\\\Rightarrow\ P=2750[/tex]
Hence, the principal invested = $2750
A spinner has 10 equally sized sections, 5 of which are gray and 5 of which are blue. The spinner is spun twice. What is the probability that the first spin lands on gray and the second spin lands on blue? Write your answer as a fraction in the simplest form.
Answer:
[tex]P(Gray\ and\ Blue) = \frac{1}{4}[/tex]
Step-by-step explanation:
Given
[tex]Sections = 10[/tex]
[tex]n(Gray) = 5[/tex]
[tex]n(Blue) = 5[/tex]
Required
Determine P(Gray and Blue)
Using probability formula;
[tex]P(Gray\ and\ Blue) = P(Gray) * P(Blue)[/tex]
Calculating P(Gray)
[tex]P(Gray) = \frac{n(Gray)}{Sections}[/tex]
[tex]P(Gray) = \frac{5}{10}[/tex]
[tex]P(Gray) = \frac{1}{2}[/tex]
Calculating P(Gray)
[tex]P(Blue) = \frac{n(Blue)}{Sections}[/tex]
[tex]P(Blue) = \frac{5}{10}[/tex]
[tex]P(Blue) = \frac{1}{2}[/tex]
Substitute these values on the given formula
[tex]P(Gray\ and\ Blue) = P(Gray) * P(Blue)[/tex]
[tex]P(Gray\ and\ Blue) = \frac{1}{2} * \frac{1}{2}[/tex]
[tex]P(Gray\ and\ Blue) = \frac{1}{4}[/tex]
Find an equation for the line tangent to the curve at the point defined by the given value of d²y/dx².
At this point. x = 2 cos t, y = 2 sin t, t=π/4
Answer:
Step-by-step explanation:
Given:
x = 2cost,
t = (1/2)arccosx
y = 2sint
dy/dx = dy/dt . dt/dx
dy/dt = 2cost
dt/dx = -1/√(1 - x²)
dy/dx = -2cost/√(1 - x²)
Differentiate again to obtain d²y/dx²
d²y/dx² = 2sint/√(1 - x²) - 2xcost/(1 - x²)^(-3/2)
At t = π/4, we have
(√2)/√(1 - x²) - (√2)x(1 - x²)^(3/2)
If f is a function that f(f(x)) = 2x² + 1, which is the value of f(f(f(f(3)))? Please help!
[tex]f(f(3))=2\cdot3^2+1=19\\f(f(f(f(3))))=2\cdot19^2+1=723[/tex]
What is the solution to this equation? 2x + 4 = 16
Answer:
x=6
Step-by-step explanation:
2x+4=16
2x+4-4=16-6
2x=12
x=6
Proof:
2x+4=16
2(6)+4=16
12+4=16
16=16
Hope this helps ;) ❤❤❤
Answer:
find out what x is and it is 6 it is 6 because 2 times 6 is 12 and 12 plus 4 is 16
Step-by-step explanation:
2000 people attended a baseball game. 1300 of the people attending supported the home team, while 700 supported the visiting team. What percentage of people attending supported the home team?
Answer:
Percentage of home team supporters =65%
Percentage of visiting team supporters =35%
Step-by-step explanation:
Total attendees=2,000 people
Home team supporters=1,300
Visiting team supporters=700
What percentage of people attending supported the home team?
Percentage of people attending who supported the home team = home team supporters / total attendees × 100
=1,300/2,000 × 100
=0.65 × 100
=65%
Visiting team supporters = visiting team supporters / total attendees
× 100
=700/2000 × 100
=0.35 × 100
=35%
Alternatively,
Visiting team supporters = percentage of total attendees - percentage of home team supporters
=100% - 65%
=35%
The average person lives for about 78 years. Does the average person live for at least 1,000,000 days? (Hint: There are 367 days in each year.)
what i
Answer:
[tex]\large \boxed{\sf No}[/tex]
Step-by-step explanation:
There are 365 days in 1 year.
The average person lives for about 78 years.
Multiply 78 by 365 to find the value in days.
[tex]78 \times 365= 28470[/tex]
The average person lives for about 28470 days.
The probability of the stock market going up in a single day is 51%. What is the probability that the market will go up 4 consecutive days?
Answer:
[tex]Probability = 0.068[/tex]
Step-by-step explanation:
Let P(S) represent the probability that stock market will go up in a day
Given
[tex]P(S) = 51\%[/tex]
Required
Determine the probability that stock will go up 4 consecutive days
Start by converting P(S) from percentage to decimal
[tex]P(S) = 51\%[/tex]
[tex]P(S) = \frac{51}{100}[/tex]
[tex]P(S) = 0.51[/tex]
The above expression represents the probability in a day;
For 4 days; we have:
[tex]Probability = P(S) * P(S) * P(S) * P(S)[/tex]
[tex]Probability = (P(S))^4[/tex]
Substitute 0.51 for P(S)
[tex]Probability = (0.51)^4[/tex]
[tex]Probability = 0.06765201[/tex]
[tex]Probability = 0.068[/tex] -- Approximated
Hence, the probability that stock will rise for 4 consecutive days is 0.068
The table shows the height, in meters, of an object that is dropped as time passes until the object hits the ground. A 2-row table with 10 columns. The first row is labeled time (seconds), x with entries 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.6. The second row is labeled height (meters), h with entries 100, 98.8, 95.1, 89.0, 80.4, 69.4, 55.9, 40.0, 21.6, 0. A line of best fit for the data is represented by h = –21.962x + 114.655. Which statement compares the line of best fit with the actual data given by the table? According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground. According to the line of best fit, the object was dropped from a lower height. The line of best fit correctly predicts that the object reaches a height of 40 meters after 3.5 seconds. The line of best fit predicts a height of 4 meters greater than the actual height for any time given in the table.
Answer: A. According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground.
The statement first "According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground" is correct.
What is the line of best fit?A mathematical notion called the line of the best fit connects points spread throughout a graph. It's a type of linear regression that uses scatter data to figure out the best way to define the dots' relationship.
We have a line of best fit:
h = –21.962x + 114.655
As per the data given and line of best fit, we can say the object would have impacted the ground 0.6 seconds later than it did according to the line of best fit.
Thus, the statement first "According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground" is correct.
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Salema's score on a test was 80%. If the test was worth a total of 60 points, how many points did Salema earn?
Answer:
48
Step-by-step explanation:
Do 60*.80
60 represent the total points the test was worth
.80 represents the % number
The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Required percentage value = a
total value = b
Percentage = a/b x 100
100% = 60
80% = 48
The points Salema earned are 48 points.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/100 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
10% = 10/100 = 1/10
We have,
Salema's score = 80%
Total score in the test = 60 points
Salema's score.
= 80% of 60 points
= 80/100 x 60
= 48
Thus,
The points Salema earned are 48 points.
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In a random sample of 64 people, 48 are classified as 'successful.' Determine the sample proportion of 'successful' people.
Answer:
The sample proportion of 'successful' people is [tex]\frac{3}{4}[/tex].
Step-by-step explanation:
The sample consist of 64 people and 48 of them are 'successful'. Hence, the proportion of 'successful' people is:
[tex]p = \frac{n}{N}[/tex]
Where:
[tex]N[/tex] - People that forms the sample, dimensionless.
[tex]n[/tex] - People classified as 'successful', dimensionless.
Given that [tex]n = 48[/tex] and [tex]N = 64[/tex], the sample proportion of 'successful' people is:
[tex]p = \frac{48}{64}[/tex]
[tex]p = \frac{3}{4}[/tex]
The sample proportion of 'successful' people is [tex]\frac{3}{4}[/tex].
Scott start his banking account with 150 and is spending $7 per day on lunch . How would one describe the graph of this model?
Answer:
So this is giving us the slope the slope is y=-7x+150
Step-by-step explanation:
It is giving us the Y intercept which is $150 because thats how much he starts out with
It is giving us the slope -7 dollars because he is spending that everyday
Which of the following is equivalent to –2i(6 – 7i)?
Answer:
[tex]\boxed{\sf \bf \ \ -2i(6-7i)=-14-12i \ \ }[/tex]
Step-by-step explanation:
Hello, please consider the following.
[tex]-2i(6-7i)=-12i+14i^2=-14-12i[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
Answer:
A
Step-by-step explanation:
Answer = A
In designing an experiment involving a treatment applied to 4 test subjects, researchers plan to use a simple random sample of 4 subjects selected from a pool of 31 available subjects. (Recall that with a simple random sample, all samples of the same size have the same chance of being selected.) Answer the question below.
What is the probability of each simple random sample in thiscase?
Answer:
The probability is [tex]p(n ) = 3.18*10^{-5}[/tex]
Step-by-step explanation:
From the question we are told that
The population size is N = 31
The sample size n = 4
Generally the number of way by which the n can be selected from N is mathematically represented as
[tex]\left N} \atop {}} \right. C_n = \frac{N! }{(N-n)!n!}[/tex]
=> [tex]\left 31} \atop {}} \right. C_4 = \frac{31! }{(31-4)!4!}[/tex]
=> [tex]\left 31} \atop {}} \right. C_4 = \frac{31 * 30 * 29 * 28* 27! }{27! * 4*3 * 2 * 1 }[/tex]
=> [tex]\left 31} \atop {}} \right. C_4 = \frac{ 755160 }{ 24}[/tex]
=> [tex]\left 31} \atop {}} \right. C_4 = 31465[/tex]
The number of ways of selecting a particular sample size is is [tex]k = 1[/tex]
Therefore the probability of each simple random sample in this case is mathematically evaluated as
[tex]p(n ) = \frac{1}{31465}[/tex]
[tex]p(n ) = 3.18*10^{-5}[/tex]
suppose a chemical engineer randomly selects 3 catalysts for testing from a group of 10 catalysts, 6 of which have low acidity & 4 have high acidity. What is the probability that exactly2 lower acidic catalysts are selected?
Step-by-step explanation:
Total catalysts = 10
Probability of 2 lower acidic catalysts = 2/10 = 1/5
