(a) log₂(x) is a logarithmic function, (b) ∜x is a root function, (c) 2x^3/(1 - x^2) is a rational function, (d) 1 - 1.1t + 2.54t^2 is a polynomial of degree 2, (e) 5^t is an exponential function and (f) sin θ cos^2 θ is a trigonometric function.
A power function has the form f(x) = x^n, where n is a constant.
A root function has the form f(x) = √x or f(x) = x^(1/n), where n is a constant.
A polynomial is an algebraic function that can be written in the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where a_i are constants and n is the degree of the polynomial.
A rational function is a function that can be written as the ratio of two polynomials, i.e., f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials and q(x) ≠ 0.
An algebraic function is a function that can be expressed as the solution to a polynomial equation, i.e., it can be written as a polynomial, a root, or a combination of polynomials and roots.
A trigonometric function is a function that involves the ratios of sides in a right triangle, such as sine, cosine, tangent, etc.
An exponential function has the form f(x) = a^x, where a is a positive constant greater than 1.
A logarithmic function has the form f(x) = logₐx, where a is a positive constant greater than 1.
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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 8x^3, y = 0, x = 1; about x = 2
Answer is 24pi/5
I just don't know how to get it.
Volume of the solid obtained by rotating the region bounded by the given curves about the specified line is [tex]\frac{24\pi }{5}[/tex]
Now, According to the question:
The shell method formula sums the volume of a cylindrical shell over the radius of the cylinder, and the volume of a cylinder is given by V=2πrh V = 2 π r h where r is the radius and h is the height.
Take a vertical cross section of the region. The width of the cross section is dx, and the height is [tex]8x^3.[/tex]
Rotate the cross section about the line x = 2. The height of the shell is [tex]8x^3.[/tex] the radius is 2 - x, and the thickness is dx.
Volume of shell = 2π(2-x)([tex]8x^3[/tex])dx = 2π([tex]16x^3 - 8x^4[/tex])dx
Volume of solid = [tex]2 \pi \int\limits^1_0 [16x3 - 8x4]dx[/tex]
[tex]= 2\pi (4x^4 - (8/5)x^5)^1_0[/tex]
= 24π/5
Hence, Volume of the solid obtained by rotating the region bounded by the given curves about the specified line is [tex]\frac{24\pi }{5}[/tex].
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With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is okay. The ABC Electronics Company has just manufactured 1500 write-rewrite CDs, and 160 are defective. If 6 of these CDs are randomly selected for testing, what is the probability that the entire batch will be accepted?
To find the probability that the entire batch will be accepted, we need to find the probability that all 6 CDs selected for testing are not defective. This is known as the probability of a "success" in the binomial distribution, which is (1 - population proportion of defective CDs) raised to the power of the number of CDs selected for testing.
The population proportion of defective CDs is 160/1500 = 0.107. So the probability of a CD being not defective is 1- 0.107 = 0.893. Therefore, the probability of all 6 CDs selected for testing are not defective is (0.893)^6 = 0.827 which is about 83%
Therefore, the probability that the entire batch of 1500 CDs will be accepted if 6 CDs are randomly selected for testing is 0.827 or about 83%.
It's worth noting that this method is based on some assumptions and the actual probability might be different.
lim (1 - 1/(n + 1)) ^ (n ^ 2)
Answer: Lim (1 - 1/(n + 1)) ^ (n ^ 2) = 1^infinity = 1
Step-by-step explanation: The expression given is the limit of the sequence (1 - 1/(n + 1)) ^ (n ^ 2) as n approaches infinity.
We can start by looking at the exponent first. The n^2 will grow faster than any polynomial function, making the entire expression go to zero as n approaches infinity.
Now let's look at the base (1 - 1/(n + 1))
As n increases, the value of the base becomes closer and closer to 1, since 1/(n+1) becomes smaller and smaller.
Therefore, the limit of the expression is:
Lim (1 - 1/(n + 1)) ^ (n ^ 2) = 1^infinity = 1
As n goes to infinity, the expression goes to 1
Please note that this is true for the limit, for a specific value of n, the expression will not be 1.
Answer:
The limit of (1 - 1/(n + 1)) ^ (n ^ 2) as n approaches infinity is 0.
As n becomes larger, the term 1/(n + 1) becomes smaller and closer to zero, which means that (1 - 1/(n + 1)) becomes closer and closer to 1. Therefore, the expression (1 - 1/(n + 1)) ^ (n ^ 2) becomes closer and closer to 1^(n^2) = 1. And any number powered by infinity (n^2) will be approaching zero, so the limit of the expression is 0.
a group of librarians is interested in the numbers of books and other media that patrons check out from their library. they examine the checkout records of 150 150150 randomly selected adult patrons.
For the given situation of number of visitors checkout out the library and 150 are randomly selected the population and the sample is represented by :
Option B. The population is all the given adult visitors of the library ; and the sample is the 150 selected visitors.
Total population is represented by all the adult visitors check out the library.
Number of adult visitors examine by the librarian = 150 randomly selected adult visitors.
Here 150 randomly selected adult visitors represents the sample of the whole population.
Therefore, the population and the sample for the given situation of visiting the library is given by :
Option B. The population is all the given adult visitors of the library ; and the sample is the 150 selected visitors.
The above question is incomplete, the complete question is :
A librarian is interested in the numbers of books that visitors check out from the library. She examines the checkout records of 150
randomly selected adult visitors.
Identify the population and sample for this situation.
A The population is all visitors of the library; the sample is the adult visitors
of the library.
B The population is all adult visitors of the library; the sample is the 150
visitors selected.
C The population is all visitors who check out at least 1 book from the
Library; the sample is the 150 visitors selected.
D The population is the 150 visitors selected; the sample is the 150 visitors
who check out at least 1 book from the library.
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Suppose f(3) = 2 and f(x) is continuous at z = 3. a-3 Check the option from the following that best represents this information O A. lim f(x) = 2 O B. lim f(x) = 2 OC. lim f(3) = 2 OD. Įim f(x) = 3 1- 3 o NN N 1=0 h= 0.
f(x) is continuous at z = 3. a-3 Check the option from the following that best represents this information O A. limit f(x) = 2 O B. lim f(x) = 2 OC.
What is a limit?
A limit is given by the value of function f(x) as x tends to a value.
For this problem, at x = 0, we have that to the left the function goes to positive infinity, while to the right it goes to negative infinity, hence:
1. lim f(x) = ∞
x->0-
2. lim f(x) = -∞
x->0+
At x = 2, the function goes to infinity to the left and to the right, hence:
3. lim f(x) = ∞
x->2
To the left of the graph, the function goes to negative infinity, while to the right it goes to 1, hence:
4. lim f(x) = 1
x-> ∞
5. lim f(x) = -∞
x-> -∞
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Which inequality is graphed below?
-6
-5
Ox≤-2
Ο x > - 2
Stry
x < -2
Ox>-2
-2
The inequality graphed on the number line is the one on the fourth option, we will get:
x > -2
Which inequality is graphed on the number line?On the number line we can see that we have an open circle at x = -2 (that means that x = -2 is not a solution to the inequality) and then a line that extends to the right side of that value.
Then the inequality is just the set of all numbers larger than -2, then we will get:
x > -2
That is the graphed inequality, and thus, the correct option is the last one.
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Mr. Kimball receives a $3000 annual salary
increase on the anniversary of his hiring if he receives
a satisfactory performance review. His starting salary
was $41,250. Write an equation to show k, Mr.
Kimball's salary after t years at this company if his
performance reviews are always satisfactory.
Answer:
OK so he gets 3000 a year
so 3000t is what we need. thats 3000 times the number of years.
His starting salary is 41250, so we would add that on.
K = 3000t + 41250.
Step-by-step explanation:
find the value of x to the nearest kilometer
x
21.5km
27km
a: 2km
b: 7km
c: 16km
d: 267km
Answer:
C 16 km
Step-by-step explanation:
Pythagoras
c² = a² + b²
c being the Hypotenuse (the side opposite of the 90° angle), a and b are the legs.
in our case that is
27² = 21.5² + x²
729 = 462.25 + x²
x² = 266.75
x = sqrt(266.75) = 16.33248297... km ≈ 16 km
99 = 96 = 9[?] Enter the missing exponent.
f(3)=
Slove f(×)=3
×=
The solution for f(3) in the function is -7772
How to determine the solution for f(x) in the function?From the question, we have the following equation that can be used in our computation:
f(x) = 4 − (x + 3)⁵
Also from the question, we have
f(3)
This means that the value of x is 3, and we calculate f(x) when x = 3
Substitute the known values in the above equation, so, we have the following representation
f(3) = 4 − (3 + 3)⁵
Evaluate
f(3) = −7772
Hence, the solution is −7772
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Complete question
Given f ( x ) = 4 − ( x + 3 )^5, find f(3).
solve for all values of x x^2-2x-48=0
by completing the square
Answer: To solve for all values of x in the equation x^2 - 2x - 48 = 0 by completing the square, we can follow these steps:
Bring the x term to one side of the equation by adding 2x to both sides: x^2 - 2x = 48
Divide the coefficient of the x^2 term (which is 1) by 2 and square it to find the value to add to both sides of the equation to complete the square: (1/2)^2 = 1/4. Add 1/4 to both sides of the equation: x^2 - 2x + 1/4 = 48 + 1/4
Take the square root of both sides of the equation: (x - 1)^2 = 192 + 1
Solve for x by taking the square root of both sides of the equation: x - 1 = ± √193
Add 1 to both sides of the equation to find the values of x: x = 1 ± √193
So the solutions of the equation are x = 1 + √193 and x = 1 - √193
It is important to notice that these are the solutions in radical form, so they are not exact values of x.
Step-by-step explanation:
Write an equation of the line that passes through the point (1,5) and is (a) parallel to the line y=3x-5 and (b) perpendicular to the line y=3x-5
Equation of parallel line = y = 3x + 2 and equation of perpendicular line 3y = -x + 16, passing through the point (1,5).
(a) To find an equation of the line that passes through the point (1,5) and is parallel to the line y = 3x - 5, we need to use the slope of the given line. The slope of a line is represented by m and it is equal to the coefficient of x. In this case, the slope of the line y = 3x - 5 is 3.
Therefore, the slope of the line that is parallel to y = 3x - 5 is also 3.
We can use the point-slope form of a line, which is:
y - y1 = m(x - x1)
Where (x1, y1) is a point on the line and m is the slope.
Substituting the given point (1, 5) and the slope of 3, we get:
y - 5 = 3(x - 1)
Simplifying the equation, we get:
y = 3x + 2
So, the equation of the line that passes through the point (1, 5) and is parallel to the line y = 3x - 5 is y = 3x + 2
(b) To find the slope of a line that is perpendicular to the line y = 3x - 5, we need to take the negative reciprocal of the slope. The negative reciprocal of 3 is -1/3. So, the slope of a line that is perpendicular to y = 3x - 5 is -1/3
We can use the point-slope form of a line, which is:
y - y1 = m(x - x1)
Where (x1, y1) is a point on the line and m is the slope.
Substituting the given point (1, 5) and the slope of -1/3, we get:
y - 5 = -1/3(x - 1)
Simplifying the equation, we get:
3(y - 5) = -x + 1
3y = -x + 16
So, the equation of the line that passes through the point (1, 5) and is perpendicular to the line y = 3x - 5 is 3y = -x + 16
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if mp = 6x - 5, QR = 3x + 1, and RN=6, what is QN?
QN is the answer to the equation that results from combining the slopes and y-intercepts of the equations MP and QR.
If mp = 6x - 5, QR = 3x + 1, and RN=6, what is QN?QN is the answer to the equation that results from combining the slopes and y-intercepts of the equations MP and QR. In this problem, the slope of MP is 6 and the y-intercept is -5.The slope of QR is 3 and the y-intercept is 1. To find QN, we need to combine these equations by substituting the known variables. We start by replacing the x in the MP equation with the QR equation. This gives us 6(3x + 1) – 5 = 6(3x + 1) - 5.We can then simplify this equation to 18x + 6 – 5 = 18x + 1. If we subtract 18x from both sides, we get 6 – 5 = 1. Finally, if we add 5 to both sides, we get 11 = 6. Therefore, our answer is QN = 11. To check our answer, we can plug our value of 11 into the QR equation. 3(11) + 1 = 34 + 1 = 35, which is equal to the RN value of 6.To learn more about system of linear equations refer to:
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The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Answer:Option B , C and E are correct
Step-by-step explanation:
y^2-5y=750
750-y(y-5)=0
(y+25)(y-30)=0
1. Jamar has two bowls of fruit from which he can choose a
snack when he gets home from school. One bowl has 1 yellow,
1 red, and 2 green apples. The other bowl has 5 each of
boysenberries, strawberries, blueberries, and blackberries.
What is the probability that he gets a yellow apple and a
boysenberry when he reaches into the bowls?
Answer:
One bowl has 1 yellow, 1 red, and 2 green apples. The other bowl has 5 each of boysenberries, strawberries, blueberries, and blackberries.
Step-by-step explanation:
Andrew is preheating his oven before using it to bake. The initial temperature of the oven is 65° and the temperature will increase at a rate of 10° per minute after being turned on. What is the temperature of the oven 15 minutes after being turned on? What is the temperature of the oven t minutes after being turned on?
Answer:
see belwo
Step-by-step explanation:
65
increase 10/min
in 15min it'll increase 150
so 65+150 =215
over t mins it'll be 65+10*t
Answer: 215°
t = minutes after being turned on
If the oven temperature increases by 10 per minute, and the initial temperature is 65, this can be represented as:
65 + 10t
After 15 minutes:
65 + 10(15)
65 + 150
215°
If x = temperature of the oven, the temperature of the oven t minutes after being turned on is:
65 + 10t = x
Eric works at the deil on the weekends to earn extra money he makes 10 per hour making sandwiches and 14 per hour delivering orders eric puts half of his total earning in a savings account for college he wants to know how much he saves each week help i need answers fast
Answer:
you do it step by step so it can be expressed
24. in problem 23, how many different paths are there from a to b that go through the point circled in the following lattice? b ross, sheldon. first course in probability, a (p. 17). pearson education. kindle edition.
Total 18 different paths are there from a to b that go through the point circled in the following lattice
In the given question, we have to find how many different paths are there from a to b that go through the point circled in the following lattice.
The diagram of the lattice is given below:
Therefore, we must first go from A to the encircled point.
We must travel 2 horizontally and 2 vertically to do that.
Now, to get to this point, we have ways = [tex]C_{2}^{4}[/tex] ways
Total ways to go from A to the encircled point = [tex]\frac{4!}{2!(4-2)!}[/tex]
Total ways to go from A to the encircled point = 6 ways
Additionally, we must travel 2 horizontal and 1 vertical units to get from the encircling point to B.
As a result, total ways to go from the encircling point to B = [tex]C^{3}_{1}[/tex]
Total ways to go from the encircling point to B = [tex]\frac{3!}{1!(3-1)!}[/tex]
Total ways to go from the encircling point to B = 3 ways
So, different paths are there from a to b = 6×3
Different paths are there from a to b = 18 ways
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Which of the following are equivalent to
five quarts?
A 1 1/2-gallons
B 18 cups
C 160 ounces
D 32 tablespoons
Answer:
Step-by-step explanation:
5 quart to gallon is 1.25
5 quart to cup is 20
5 Quart to oz is 160
5 Quart to tablespoon is 320
Answer: C
A middle school took all of its 6th grade students on a field trip to see a symphony at a theater that has 4100 seats. The students filled 2296 of the seats in the theater. What percentage of the seats in the theater were filled by the 6th graders on the trip?
Answer:
56%
Step-by-step explanation:
4100 seats.
filled 2296
% = 2296/4100 = 56%
Answer:
56
Step-by-step explanation:
it is Right
An insurance company determines that N, the number of claims received in a week, is a random variable with P[N = n] = 1/2n+1, where n > 0 . The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week.
The probability that exactly seven claims will be received during a given two-week period=1/64.
Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can range from 0 to 1, with 0 denoting an impossibility and 1 denoting a certainty. For pupils in Class 10, probability is a crucial subject because it teaches all the fundamental ideas of the subject. One is the probability of every event in a sample space.
An insurance company determines that N, the number of claims received in a week, is a random variable with P[N = n] = 1/2n+1, where n > 0 .
Let N1 and N2 represent the number of claims during weeks one and two, respectively,
then since N1 and N2 are independent.
[tex]Pr[N_1+N_2=7]=\sum_{n=0}^7Pr[N1=n]Pr[N2=7-n]\\\\=\sum_{n=0}^7\frac{1}{2^{n+1}}\frac{1}{2^{8-n}}\\\\=\sum_{n=0}^7\frac{1}{2^{9}}\\\\=\frac{8}{2^9}=\frac{1}{64}[/tex]
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Let the universal set be U = {1, 2, 3,..., 10}, and let A = {1, 4, 7, 10}, B = {1, 2, 3, 4, 5}, and C= {2, 4, 6, 8}. List the elements of (A-C) - (An B). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
the universal set be U = {1, 2, 3,..., 10}, and let A = {1, 4, 7, 10}, B = {1, 2, 3, 4, 5}, and C= {2, 4, 6, 8}. the elements of (A-C) - (An B) ={7,10}
Set is a collection of well-defined objects and it has distinct elements. The set formulas include the union, intersection, complement, and difference of sets. Venn diagrams are popularly used to visualize set formulas to arrive at their proof.
The universal set be U = {1, 2, 3,..., 10}, and let A = {1, 4, 7, 10}, B = {1, 2, 3, 4, 5}, and C= {2, 4, 6, 8}.
So, to calculate (A-C)-(A∩B)
FIRST A-C= {1, 4, 7, 10} - {2, 4, 6, 8}.
A - C = { 1,7,10}
A∩B={1,4,}
(A-C)-(A∩B)= { 1,7,10} -{1,4,} = { 7, 10}
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show that -0.75 ( -4f + 12 ) and ( 5f + 9 ) - ( 2f + 18 ) are equivalent expressions.
The Linear terms -0.75(-4f + 12) and (5f + 9) - (2f + 18) are actually an equivalent expressions.
What is equivalent expressions?Equivalent algebraic expressions are ones that, despite their apparent differences, actually have the same value. As a result of their similarity, they will produce the same outcomes regardless of the values we choose to use as their variables.
If -0.75 ( -4f + 12 ) and ( 5f + 9 ) - ( 2f + 18 ) are equivalent expressions then
-0.75(-4f + 12) = (5f + 9) - (2f + 18)
3f − 9 = (5f + 9) - (2f + 18)
3f − 9 = 5f + 9 -2f - 18
3f − 9 = 5f - 2f + 9 - 18
3f − 9 = 3f − 9
Thus, The Linear terms -0.75(-4f + 12) and (5f + 9) - (2f + 18) are actually an equivalent expressions.
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What number needs to be added to 4 and 9 so that the ratio of the first number to the second becomes 2 : 3?
Answer:
6
Step-by-step explanation:
4+6=10
9+6=15
the ratio of 10 and 15= 2:3
Determine what the key terms refer to in the following study A study was conducted in a local community to analyze which voters would be likely to vote and how on an (____) Variable (____) Sample (____) Population
(____) Statistic (____) Parameter (____) Data a. voted, didn't vote, voted
b. a group of voters in the community who voted, randomly selected
c. the number of eligible voters who actually plan to vote d. the number of eligible voters in the study who actually plan to vote e. the attendance of a single voter f all eligible voters in the comunity
To determine the key terms refer to in the following conducted study in a local community will be as follows:
a. voted, didn't vote, voted will be variable.
b. a group of voters in the community who voted, randomly selected will be sample.
c. the number of eligible voters who actually plan to vote will be population.
d. the number of eligible voters in the study who actually plan to vote statistic.
e. the attendance of a single voter will be data
f. all eligible voters in the comunity will be parameter.
Any quantity calculated from sample values and taken into consideration for statistical purposes is referred to as a statistic (singular) or sample statistic. A population parameter estimate, a sample description, or a hypothesis evaluation are examples of statistical purposes. Statistics are expressed as the average (or mean) of sample values.
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Need help with #20 please
The length of the side DE is 9.
What is Parallelogram?Parallelogram is a type of polygon with four sides, four angles and four vertices. It is a type of Quadrilateral.
Opposite sides are parallel, and the opposite sides and opposite angles are equal.
Given is a figure of a quadrilateral DEFG.
We have from the figure, two triangles, ΔDGF and ΔDEF.
Diagonal of the quadrilateral DF is a common side for both triangles.
Given that DG = EF
Also ∠FDG = ∠EFD
So we get the two triangles are congruent by SAS theorem.
So the given figure is parallelogram.
So Length of DE = Length of GF = 4x + 1.
We have EF = DG
6x + 4 = 4x + 8
2x = 4
x = 2
Length of DE = 4x + 1 = 9
Hence the length of the side DE of the parallelogram is 9.
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Q"
R
X
S
15
12
9
N
Given AQRS~AXYZ, what is the value of tan(Q)?
OOO
m/n M/T +/n +/m
²
Answer:
3/4
Step-by-step explanation:
since they are similar, they have the exact same angles.
angle Q = angle X
tan(angle) = sin(angle)/cos(angle)
that is the same ratio here as
9/12 = 3/4
3/5 ? 2/3
Which symbol makes the sentence true?
Responses
A <<
B ==
C >>
D ≥
Answer:
Choice A
Step-by-step explanation:
1/5 = 9/15
2/3 = 10/15
so 3/5 < 2/3
help me pls…. i really need it
Easy if writing/typing put P L S and an .
If speaking maybe instead say please
A tub has 50 small balls. Some balls are white, and summer orange. Without being able to see into the tub, each student in a class of 25 is allowed to pick a ball out of the tub at random. The color of the ball is recorded, and the ball is put back into the tub. At the end, seven orange balls and 18 white balls were picked. What is the best estimate you can give for the number of orange balls in the number of white balls in the tub? Describe how to calculate this best estimate, and explain why your method of calculation makes sense in a way that a seventh grader my understand. Is your best estimate necessarily accurate? Why or why not?
The best estimate we can give for the number of orange balls in the tub is 7, and the number of white balls in the tub is 43.
To calculate this, we take the number of orange balls picked (7) and divide it by the total number of balls picked (25) to get the proportion of orange balls in the tub. Then we multiply that proportion by the total number of balls in the tub (50) to get our estimate for the number of orange balls in the tub.
7 / 25 = 0.28
0.28 x 50 = 14
Similarly, we take the number of white balls picked (18) and divide it by the total number of balls picked (25) to get the proportion of white balls in the tub. Then we multiply that proportion by the total number of balls in the tub (50) to get our estimate for the number of white balls in the tub.
18 / 25 = 0.72
0.72 x 50 = 36
This method of calculation makes sense because it uses the data we have (the number of orange and white balls picked) to estimate the overall makeup of the tub. By using the proportion of orange and white balls picked, we can make an educated guess about how many of each color ball are in the tub without seeing it.
However, it's important to note that this best estimate is not necessarily accurate. The real number of orange and white balls in the tub could be different. The method used here is a probability method, it's best estimate based on the sample data. The sample data is based on a random selection of 25 balls, and the number of orange or white balls that would be picked could vary depending on the random draw. The more samples we take, the more accurate our estimate would be.