solve the equation
a) y''-2y'-3y= e^4x
b) y''+y'-2y=3x*e^x
c) y"-9y'+20y=(x^2)*(e^4x)
Answer:
a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:
r^2 - 2r - 3 = 0
Factoring, we get:
(r - 3)(r + 1) = 0
So the roots are r = 3 and r = -1.
The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:
y_h = c1e^3x + c2e^(-x)
To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = Ae^4x
Taking the first and second derivatives of y_p, we get:
y_p' = 4Ae^4x
y_p'' = 16Ae^4x
Substituting these into the original differential equation, we get:
16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x
Simplifying, we get:
5Ae^4x = e^4x
So:
A = 1/5
Therefore, the particular solution is:
y_p = (1/5)*e^4x
The general solution to the non-homogeneous equation is:
y = y_h + y_p
y = c1e^3x + c2e^(-x) + (1/5)*e^4x
b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:
r^2 + r - 2 = 0
Factoring, we get:
(r + 2)(r - 1) = 0
So the roots are r = -2 and r = 1.
The general solution to the homogeneous equation y'' + y' - 2y = 0 is:
y_h = c1e^(-2x) + c2e^x
To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = (Ax + B)e^x
Taking the first and second derivatives of y_p, we get:
y_p' = Ae^x + (Ax + B)e^x
y_p'' = 2Ae^x + (Ax + B)e^x
Substituting these into the original differential equation, we get:
2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x
Simplifying, we get:
3Ae^x = 3xe^x
So:
A = 1
Therefore, the particular solution is:
y_p = (x + B)e^x
Taking the derivative of y_p, we get:
y_p' = (x + 2 + B)e^x
Substituting back into the original differential equation, we get:
(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x
Simplifying, we get:
-xe^x - Be^x = 0
So:
B = -x
Therefore, the particular solution is:
y_p = xe^x
The general solution to the non-homogeneous equation is:
y = y_h + y_p
y = c1e^(-2x) + c2e^x + xe^x
c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:
r^2 - 9r + 20 = 0
Factoring, we get:
(r - 5)(r - 4) = 0
So the roots are r = 5 and r = 4.
The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:
y_h = c1e^4x + c2e^5x
To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:
y_p = (Ax^2 + Bx + C)e^4x
Taking the first and second derivatives of y_p, we get:
y_p' = (2Ax + B)e^4x + 4Axe^4x
y_p'' = 2Ae^4x +
Use the graph to answer the question.
graph of polygon ABCD with vertices at 1 comma 5, 3 comma 1, 7 comma 1, 5 comma 5 and a second polygon A prime B prime C prime D prime with vertices at 8 comma 5, 10 comma 1, 14 comma 1, 12 comma 5
Determine the translation used to create the image.
7 units to the right
7 units to the left
3 units to the right
3 units to the left
The translation used to create the image A'B'C'D', from the pre-image, ABCD is; 7 units to the right
What is a translation transformation?A translation transformation is one in which the location of the points on the pre-image is changes but the size, and orientation of the pre-image is preserved.
The coordinates of the vertices of the polygon ABCD are; (1, 5), (3, 1), (7, 1), (5, 5)
The coordinates of the vertices of the polygon A'B'C'D' are; (8, 5), (10, 1), (14, 1), (12, 5)
Whereby the vertices of the image and the preimage are;
A(1, 5), B(3, 1), C(7, 1), D(5, 5), and A'(8, 5), B'(10, 1), C'(14, 1), D'(12, 5), the difference in the x and y-values indicates;
A' - A = (8 - 1, 5 - 5) = (7, 0)
B' - B = (10 - 3, 1 - 1) = (7, 0)
C' - C = (14 - 7, 1 - 1) = (7, 0)
D' - D = (12 - 5, 5 - 5) = (7, 0)
Therefore, the transformation used to create the image A'B'C'D' from the pre-image, ABCD is a translation; 7 units to the right
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What is the equivalent to this
None of the given options A, B, C, or D is correct as they all provide different answers.
what is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.
The question asks for the equivalent of 6 × 2. This means that we need to find a number that is equal to the result of multiplying 6 and 2 together.
When we multiply 6 and 2, we get:
6 × 2 = 12
So, the equivalent of 6 × 2 is 12.
However, none of the answer options provided matches this answer.
Option A suggests that the equivalent of 6 × 2 is 2 × 1, which is equal to 2, not 12.
Option B suggests that the equivalent of 6 × 2 is 3 × 2, which is equal to 6, not 12.
Option C suggests that the equivalent of 6 × 2 is 9 × 3, which is equal to 27, not 12.
Option D suggests that the equivalent of 6 × 2 is 18 × 1/2, which is equal to 9, not 12.
Therefore, none of the options provided is correct.
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Which geometric term describes ∠ T A G ?
Answer:
angle
Step-by-step explanation:
since there is a < sign, that makes it an angle. I'm not sure if that is the whole problem, or if It is missing a picture. Hope this helps!
Answer: i know it is acute
Complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B).
The truth table for (A ⋁ B) ⋀ ~(A ⋀ B) is:
A B (A ⋁ B) ⋀ ~(A ⋀ B)
0 0 0
0 1 0
1 0 0
1 1 0
The truth table is what?A truth table is a table that displays all possible combinations of truth values (true or false) for one or more propositions or logical expressions, as well as the truth value of the resulting compound proposition or expression that is created by combining them using logical operators like AND, OR, NOT, IMPLIES, etc.
The columns of a truth table reflect the propositions or expressions themselves as well as the compound expressions created by applying logical operators to them. The rows of a truth table correspond to the various possible combinations of truth values for the propositions or expressions.
To complete the truth table for (A ⋁ B) ⋀ ~(A ⋀ B), we need to consider all possible combinations of truth values for A and B.
A B A ⋁ B A ⋀ B ~(A ⋀ B) (A ⋁ B) ⋀ ~(A ⋀ B)
0 0 0 0 1 0
0 1 1 0 1 0
1 0 1 0 1 0
1 1 1 1 0 0
So, the only case where the expression is true is when both A and B are true, and for all other cases it is false.
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… Approximate the area of the shaded region.
The approximated area of the shaded region is 92.54 square units
Approximating the area of the shaded region.From the question, we have the following parameters that can be used in our computation:
Two isosceles right trianglesCircleThe area of the shaded region in the figure is calculated as
Shaded region = Circle - Isosceles right triangle 1 - Isosceles right triangles 2
Using the given dimensions, we have
Shaded region = 3.14 * 6^2 - 1/2 * 5^2 - 1/2 * 4^2
Evaluate
Shaded region = 92.54
Hence, the shaded region is 92.54 square units
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What is the mean of the values in the stem-and-leaf plot?
Enter your answer in the box.
Answer:
mean = 24
Step-by-step explanation:
the mean is calculated as
mean = [tex]\frac{sum}{count}[/tex]
the sum of the data set is
sum = 12 + 13 + 15 + 28 + 28 + 30 + 42 = 168
there is a count of 7 in the data set , then
mean = [tex]\frac{168}{7}[/tex] = 24
Find the amount accumulated after
investing a principle P for t years and an
interest rate compounded twice a year.
P = $100 r = 3% t = 5
k=2
Hint: A = P(1 + E) kt
A = $[?]
Answer:
A= $116.05
Step-by-step explanation:
A=P(1+E)kt
A=100(1+0.03/2)^10
A= $116.05
The monthly cost (in dollars) of a long-distance phone plan is a linear function of the total calling time (in minutes). The monthly cost for 35 minutes of calls is $16.83 and the monthly cost for 52 minutes is $18.87. What is the monthly cost for 39 minutes of calls?
Answer: We can use the two given points to find the equation of the line and then plug in 39 for the calling time to find the corresponding monthly cost.
Let x be the calling time (in minutes) and y be the monthly cost (in dollars). Then we have the following two points:
(x1, y1) = (35, 16.83)
(x2, y2) = (52, 18.87)
The slope of the line passing through these two points is:
m = (y2 - y1) / (x2 - x1) = (18.87 - 16.83) / (52 - 35) = 0.27
Using point-slope form with the first point, we get:
y - y1 = m(x - x1)
y - 16.83 = 0.27(x - 35)
Simplifying, we get:
y = 0.27x + 7.74
Therefore, the monthly cost for 39 minutes of calls is:
y = 0.27(39) + 7.74 = $18.21
Step-by-step explanation:
See the photo below
This problem involves integration and algebraic manipulation, and belongs to the subject of calculus. The solutions are:
[tex]A) $\int_{0}^{2} (f(x) + g(x)) dx = -3$[/tex]
[tex]B) $\int_{0}^{3} (f(x) - g(x)) dx = -4$[/tex]
[tex]C) $\int_{2}^{3} (3f(x) + g(x)) dx = -32$[/tex]
This is a problem that asks us to find the values of some definite integrals using given values of other definite integrals. We are given three definite integrals, and we are asked to compute three other integrals involving the same functions, using the given values.
The problem involves some algebraic manipulation and the use of the linearity of the integral.
It also involves finding the constant "a" that makes a definite integral equal to zero. The integral involves two functions, "f(x)" and "g(x)," whose definite integrals over certain intervals are also given.
See the attached for the full solution.
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In a country club of 141 people, 61 play football, 65 play base ball and 72 play hockey hockey.
22. play all the games while 11 play none of the games. An equal number play only two games (How many play only two games (i) How many play only football?
The number of people who play only football is |A' ∩ B' ∩ C'| = 25. So, 25 people play only football.
Describe Statistics?Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It involves the use of various methods and techniques to draw conclusions from data and make informed decisions.
The main goal of statistics is to provide a systematic approach to understanding and interpreting data, which can be used in a wide variety of fields, including business, social sciences, engineering, medicine, and many others. Statistics is used to study and analyze various types of data, including numerical, categorical, and ordinal data, as well as time series and spatial data.
Let A, B, and C be the sets of people who play football, baseball, and hockey, respectively. We know that:
|A| = 61, |B| = 65, |C| = 72
We also know that:
|A ∩ B ∩ C| = 22, |A ∪ B ∪ C| = 141, |A' ∩ B' ∩ C'| = 11
where A', B', and C' denote the complements of A, B, and C, respectively.
We can use the principle of inclusion-exclusion to find the number of people who play only two games. This principle states that:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
Using the given values, we can substitute and simplify to get:
141 = 61 + 65 + 72 - |A ∩ B| - |A ∩ C| - |B ∩ C| + 22
|A ∩ B| + |A ∩ C| + |B ∩ C| = 75
We also know that an equal number of people play only two games, so let x be that number. Then:
|A ∩ B| = |A ∩ C| = |B ∩ C| = x
Substituting into the previous equation, we get:
3x = 75
x = 25
Therefore, 25 people play only two games. To find the number of people who play only football, we need to subtract the number of people who play baseball and hockey from the number of people who play only two games:
|A' ∩ B ∩ C| = x = 25
|A' ∩ B ∩ C'| = 11
|A' ∩ C ∩ B'| = x = 25
|A ∩ B' ∩ C'| = x = 25
|A' ∩ B| = 65 - (25 + 11) = 29
|A' ∩ C| = 72 - (25 + 11) = 36
|A' ∩ B' ∩ C| = 61 - (25 + 11) = 25
|A ∩ B' ∩ C| = 141 - (29 + 25 + 36 + 11) = 40
Therefore, the number of people who play only football is:
|A' ∩ B' ∩ C'| = 25
So, 25 people play only football.
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Given the expression 3x+2 evaluate the expression for the given values of x when x=(-2)
Answer:
...............................
if m(x)= sin²(x), then m'(x)=? A. cos²x+sin²x. B.sinx²-cos²x C. 2cos²x-sinx D. cos²-sin²x
how am i supposed to prove that theyre collinear
Answer:
They are collinear if they are on the same line
Camille opened a savings account and deposited $8,063.00 as principal. The account earns 14.69% interest, compounded quarterly. What is the balance after 10 years?
Use the formula A=P1+
r
n
nt, where A is the balance (final amount), P is the principal (starting amount), r is the interest rate expressed as a decimal, n is the number of times per year that the interest is compounded, and t is the time in years.
Round your answer to the nearest cent.
Save answer
Answer:$26,141.13.
Step-by-step explanation:
Using the formula A = P * (1 + r/n)^(n*t), where A is the balance, P is the principal, r is the interest rate, n is the number of times per year that the interest is compounded, and t is the time in years, we can calculate the balance in the savings account after 10 years:
A = 8,063.00 * (1 + 0.1469/4)^(4*10)
A ≈ 26,141.13
Therefore, the balance in the savings account after 10 years, rounded to the nearest cent, is $26,141.13.
Find the surface area. Round to the nearest hundredth
Answer:
122.30 cm²
Step-by-step explanation:
Divide the polyhedron into shapes:
+) 2 triangles with the same area.
The area of the triangle is
(4.3×11)÷2 = 23.65 cm².
And with two triangles of the same area we take the sum of both areas
23.65 + 23.65 = 47.3 cm²
+) 3 rectangles with different areas.
(3×6) + (3×8) + (3×11) = 75 cm²
So the surface area is the sum of areas of the triangles and rectangles
47.3 + 75 = 122.3 = 122.30 cm²
PLEASE HELP
Find the Area
2cm
___cm^2
Answer:
3.14 cm^2
Step-by-step explanation:
1. Find radius:
If diameter is 2, divide it by 2 to get radius = 1
2. Find formula:
A=πr^2
3. Plug in:
A = π(1)^2
4. Solve (multiply):
A = π(1)^2:
3.14159265359
Or
3.14 cm^2
Answer:
3.14 cm^2
Step-by-step explanation:
A=[tex]\pi[/tex]r^2
r=2
2/2=1
A=[tex]\pi[/tex](1)^2
=[tex]\pi[/tex]1
≈3.14x1
≈3.14cm^2
The circle graph below shows the number of animals in Mushu's farm. Sheep Donkeys Camels Goats Cows If there were 24 goats, how many cows are there in the farm?
By assuming that the farmer has goat and cows in the ratio 3:4, the number of cows in the farm will be 32 cows.
If there were 24 goats, how many cows are there in the farm?To find out how many cows are in the farm, we need to know the total number of animals in the farm. Assuming the ratio of goats to cows is 3:4, we can write this as: [tex]3x : 4x[/tex]
Where 3x represents the number of goats, and 4x represents the number of cows. If we know that there are 24 goats, we can set up an equation to solve for x:
3x = 24
Dividing both sides by 3, we get:
x = 8
Now that we know the value of x, we can find the number of cows:
= 4x
= 4(8)
= 32
Therefore, there are 32 cows in the farm.
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Help me pleas whith this
A stainless-steel patio heater is shaped like a square pyramid. The length of one side of the base is 10 feet. The slant height is 12 feet. What is the height of the heater? Round to the nearest tenth of a foot.
The height of the heater is approximately 6.6 feet.
What is Pythagoras theorem?According to Pythagoras's Theorem, the square of the hypotenuse side in a right-angled triangle is equal to the sum of the squares of the other two sides. Perpendicular, Base, and Hypotenuse are the names of this triangle's three sides.
We can use the Pythagorean theorem to find the height of the pyramid. Let's call the height "h". Then, the slant height is the hypotenuse of a right triangle with base and height both equal to 10 feet, so we have:
h² + 10² = 12²
Simplifying and solving for h, we get:
h² + 100 = 144
h² = 44
h ≈ 6.6 feet
Therefore, the height of the heater is approximately 6.6 feet.
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Suppose that the function h is defined as follows.
if
-2
-1
h(x)= 0
1
2
Graph the function h.
-3.5
if-2.5
if-1.5
if -0.5
if 0.5 ≤x≤1.5
+
X
Ś
The required graph of the function given; h (x) has been attached.
Define a graph?In mathematics, graph theory is the study of graphs, which are mathematical structures used to represent pairwise interactions between objects. In this definition, a network is made up of nodes or points called vertices that are connected by edges, also called links or lines. In contrast to directed graphs, which have edges that connect two vertices asymmetrically, undirected graphs have edges that connect two vertices symmetrically. Graphs are one of the primary areas of study in discrete mathematics.
Here as per the question the graph of the function, h (x) has been attached.
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solve for the unknown to find the unit rate
1/5 ?
----- = -----
1/20 1
Answer: 4
Step-by-step explanation:
To find the unit rate, we can cross-multiply the fractions.
Multiplying the numerator of the first fraction by the denominator of the second fraction, we get 1/5.
Multiplying the numerator of the second fraction by the denominator of the first fraction, we get 1/20.
Now we have the equation 1/5 = 1/20.
To solve for the unknown, we can cross-multiply again, which gives us 20 * 1/5 = 4.
Therefore, the unit rate is 4.
If you have a standard score of Z = 1, what percentage of the population has scores less than you?
Step-by-step explanation:
From a z-score table
z-score = 1 corresponds to .8413 or 84.13 percentile
meaning 84 .13 % have a lesser score than you
The diameter of a circle is 38 feet.what is the circles circumfrence. Use 3.14 for pi
Answer:
The circumference of the circle is 119.32 ft.
Step-by-step explanation:
The circumference of a circle can be solved through the formula:
C = πd
where d is the diameter
Given: d = 38 ft
π = 3.14
Solve:
C = πd
C = 3.14 (38 ft)
C = 119.32 ft
Triangle PQR is drawn with coordinates P(0, 2), Q(0, 5), R(1, 4). Determine the translation direction and number of units if R′(−7, 4).
8 units down
8 units up
8 units to the right
8 units to the left
It follows that the translation direction is 8 units to the left and 0 units up or down.
Describe translation?A translation is a geometric change in Euclidean geometry where each point in a figure, shape, or space is moved uniformly in one direction. A translation can either be thought of as moving the origin of the coordinate system or as adding a constant vector to each point
The new vertices of a triangle with vertex locations of (0,0), (1,0), and (0,1), for instance, would be (2,3, (3,3), and (2,4) if the triangle were translated 2 units to the right and 3 units up.
We can use the following procedures to get the translation direction and number of units for R′(7, 4):
1. Determine the difference between R and R′'s x-coordinates: −7 − 1 = 8
2. Determine the difference between R and R′'s y coordinates: 4 − 4 = 0
It follows that the translation direction is 8 units to the left and 0 units up or down.
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Need help please
The half-life of Palladium-100 is 4 days. After 16 days a sample of Palladium-100 has been reduced to a mass of 2 mg.
What was the initial mass (in mg) of the sample? --------------
What is the mass 7 weeks after the start?-------------
1. The initial mass of the sample was 32 mg. 2. The mass of the sample 7 weeks after the start is approximately 0.162 mg.
What is radioactive decay?An unstable atomic nucleus releases particles or electromagnetic radiation as it undergoes radioactive decay, changing into a different nucleus. Since this process is unpredictable and spontaneous, the decay's timing cannot be anticipated. The radioactive substance's half-life, or the amount of time it takes for half of its radioactive atoms to decay, is used to calculate the rate of decay. Radiometric dating, nuclear energy production, and medical imaging all employ radioactive decay, which can cause the emission of alpha particles, beta particles, or gamma rays. Understanding the behaviour of matter at the atomic and subatomic level requires knowledge of radioactive decay.
1. The radioactive decay is given by the formula:
[tex]N(t) = N_0 * (1/2)^{(t/T)}[/tex]
Now, for half-life of Palladium-100 is 4 days and t = 16 and N(t) = 2 we have:
[tex]2 = N_0 * (1/2)^{(16/4)}\\2 = N_0 * (1/2)^4\\2 = N_0* 1/16\\N_0 = 2 * 16\\N_0 = 32 mg[/tex]
2. Foe 7 weeks:
7 weeks = 7 * 7 days = 49 days.
[tex]N(49) = N_0 * (1/2)^{(49/4)}\\N(49) = 32 * (1/2)^{(49/4)}\\N(49) = 0.162 mg[/tex]
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8. The population P (t) of a bacteria culture is given by P (t) = -1500t² + 60,000t + 10,000, where is the time in hours after the culture is started. Determine the time(s) at which the population will be greater than 460,000 organisms.
The time(s) at which the population will be greater than 460,000 organisms is between 10 and 30 hours after the culture is started.
What is inequality?An inequality is a comparison between two numbers or expressions that are not equal to one another. Symbols like, >,,, or are used to denote it, indicating which value is more or smaller than the other or just different.
To find the time(s) at which the population will be greater than 460,000 organisms, we need to solve the inequality:
P(t) > 460,000
Substituting the given equation for P(t), we get:
-1500t² + 60,000t + 10,000 > 460,000
Simplifying this inequality, we get:
-1500t² + 60,000t - 450,000 > 0
Dividing both sides by -1500 and flipping the inequality sign, we get:
t² - 40t + 300 < 0
We can solve this inequality by factoring the quadratic equation:
(t - 10)(t - 30) < 0
The roots of this equation are t = 10 and t = 30. Plotting these values on a number line, we can see that the solution to the inequality is:
10 < t < 30
Therefore, the time(s) at which the population will be greater than 460,000 organisms is between 10 and 30 hours after the culture is started.
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How much money would an investment of 300 at rate of 12 percent compound monthly be after 4 years
Answer:
data given
principal 300
rate12%
time 4 years
Step-by-step explanation:
from
A=P[1+r/100]^n
where
A is amount
p is principal
r is rate
n is interest period
now,
A= 300[1+12/(100×12)]^(12×4)
A=300×1.01^48
A=483.67
: .would be 483.67
Help me I don’t understand
Answer: C, 125
Step-by-step explanation: That is the slope of the line, which remains constant. The slope represents the distance over the time, and distance divided by time equals the speed. This means that the speed remains constant throughout.
Jasmine asked her classmates to name all the types of trees they found while on a field trip at a local park.
1/7 reported finding a birch tree.
7/9 reported finding a pine tree.
1/4 reported finding a maple tree.
11/23 reported finding an oak tree.
Based on the results, which statements are true? (Pick all that apply)
A. Most students found a pine tree.
B. More students found a maple tree than a pine tree.
C. More students found a birch tree than an oak tree.
D. More students found a pine tree than a birch tree.
E. More students found a maple tree than an oak tree.
The statements that are correct concerning the outcome of events between Jasmine and her classmates include the following:
Most students found a pine tree.
More students found a pine tree than a birch tree. That is option A and D respectively.
How to calculate the number of students per tree?The quantity of students that found birch tree = 1/7 = 0.14
The quantity of students that found pine tree = 7/9 = 0.8
The quantity of students that found maple tree = 1/4 = 0.25
The quantity of students that found oak tree = 11/23 = 0.48
Therefore, the statement that are correct about the outcome of the event between Jasmine and her classmates is as follows:
Most students found a pine tree.
More students found a pine tree than a birch tree.
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