At x = 0 and x = 2, the curve defined by y² = x³ - 3x² has a horizontal tangent.
We have,
First, let's find the derivative of y with respect to x.
We can rewrite the equation as:
y² = x³ - 3x²
y = (x³ - 3x²)^(1/2)
Now, differentiating y with respect to x using the chain rule:
dy/dx = (1/2) * (x³ - 3x²)^(-1/2) * (3x² - 6x)
Next, we set the derivative equal to 0 to find the values of x where the curve has a horizontal tangent:
(1/2) * (x³ - 3x²)^(-1/2) * (3x² - 6x) = 0
We can disregard the constant factor of (1/2) since it doesn't affect the solution.
Thus, we have:
(x³ - 3x²)^(-1/2) * (3x² - 6x) = 0
To have a horizontal tangent, the slope of the curve should be 0, which means the derivative is equal to 0.
Now, we have two factors to consider:
(x³ - 3x²)^(-1/2) = 0, which means the expression inside the parentheses is equal to 0. However, this does not yield a real solution.
(3x² - 6x) = 0, which means the quadratic term equals 0.
Solving this quadratic equation:
3x² - 6x = 0
3x(x - 2) = 0
This equation has two solutions: x = 0 and x = 2.
Therefore,
At x = 0 and x = 2, the curve defined by y² = x³ - 3x² has a horizontal tangent.
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What is the summation of all the roots of the equation [tex]x^{2022}[/tex] = 1
Answer:
x = 1 or x = -1
Step-by-step explanation:
The only number square root and get 1 is 1 and -1. Because 2022 is an even number, so -1 will work in this case, and the final answer will be positive 1!
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What is 1/3 of 18? :)
if sin (2x) = a and 0 < x < pi/2 then sin x is equal to which of the following and why?
a) a
b) a/2
c) 2a
d) a cos x
e) a sex x
explain pleaseee
The solution to the equation are x=a
How to determine the value of sin (2x) = a ?The given equation is sin (2x) = a and 0 < x < pi/2
Sin(2x) = sin x, 0≤x≤2п ........................... (eq. 1)
Using the trigonometrical formula,
sin(2x) = sin xcosx
Let us subtract the formula from equation 1
2sinxcosx = sinx
Therefore, 2sinxcosx-sinx = 0
Let us remove common terms to get
sinx(2cox-1)=0
Now, sinx=0
x= 0, [tex]\pi[/tex], [tex]2\pi[/tex]
Also, 2cos x =1
This means that cos x = 1/2
By implication, x=cos⁻¹(1/2)
That is to say that x= [tex]\frac{\pi }{3} , \frac{5\pi }{3}[/tex]
In conclusion, the value of [tex]\pi[/tex] is a as a single integer
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can someone pls help me with this one?
Answer:
y = x + 2
Step-by-step explanation:
Use the slope-intercept form:
y=mx+b
M represents the slope of the line
B represents the y-intercept
simplify (1+√3)(1-√3)
Answer:
- 2
Step-by-step explanation:
the factors of a difference of square are
a² - b² = (a + b)(a - b)
(1 + [tex]\sqrt{3}[/tex] )(1 - [tex]\sqrt{3}[/tex] ) ← are the factors of a difference of squares , then
(1 + [tex]\sqrt{3}[/tex] )(1 - [tex]\sqrt{3}[/tex] )
= 1² - ([tex]\sqrt{3}[/tex] )²
= 1 - 3
= - 2
What is the vertex form of the equation? y = – x 2 + 12 x – 4 show steps of how you solved please
The vertex form of the equation is in the form of y = a(x-h)^2 + k.
Step 1: Use the formula below to solve for a.
a = -1/4 * (4 - 12) = -1
Step 2: Use the formula below to solve for h.
h = -b/(2a) = 12/(2(-1)) = -6
Step 3: Use the formula below to solve for k.
k = f(h) = -(-6)^2 + 12(-6) - 4 = 36
Therefore, the vertex form of the equation is y = -(x+6)^2 + 36.
In mathematics, what is the vertex?A vertex, or particular point, is a place where two or more lines or edges meet in a mathematical object. Angles, polygons, polyhedra, and graphs are where vertices are most frequently seen. Nodes and vertices in a graph are the same thing.
What is the vertex location?Recall that a parabola's General Form is y = ax2 + bx + c. The x-coordinate of the vertices, which is x = - b/2a, must first be discovered in order to find the vertex from this form. You will use this number to replace x in the parabola equation once you have determined the x-coordinate of the vertex.
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Solve pls Brainliest
Answer:
The ordered pair that satisfies the system of equations is (7, -1)
Step-by-step explanation:
⭐What is a system of equations?
a system of equations is 2 or more equations that intersect at a point (x,y)When we solve for an ordered pair that satisfies the system of equations, we are finding at what point do the equations intersect.
⭐How can we solve a system of equations?
eliminationsubstitutiongraphingFor this problem, I find elimination to be the easiest. I will demonstrate how to use elimination to solve this system of equations.
⭐What is the elimination method?
the elimination method consists of literally eliminating a variable from both equations to solve for another variableWe have the system: [tex]{ x+7y = 0\\\\2x - 8y = 22[/tex]. Let's eliminate the variable "x".
To do so, multiply the first equation by -2, then add both equations.
[tex]-2(x+7y = 0)\\-2x -14y = 0[/tex]
Now add this new equation with the second equation. This eliminates the x variable.
[tex]-2x - 14y = 0 \\+2x -8y = 22\\\\-22y = 22\\y = -1[/tex]
∴ y = -1
Next, we need to solve for the x variable. To do so, substitute the y variable into one of the equations we added and solve for x.
I will substitute y into the equation [tex]2x - 8y = 22[/tex], but you can also substitute it into [tex]-2x-14y = 0[/tex]
[tex]2x - 8y = 22[/tex]
[tex]2x-8(-1) = 22[/tex]
[tex]2x + 8 = 22[/tex]
[tex]2x = 14[/tex]
[tex]x = 7[/tex]
∴ The ordered pair that satisfies the system of equations is (7, -1)
⭐if this response helped you, please mark it the "brainliest"!⭐
if a gardener fences in the total rectangular area shown in the illustration instead of just the square area, he will need twice as much fencing to enclose the garden. how much fencing will he need?
The fencing needed to enclose the garden is equal to the perimeter of the rectangle and it will be 5x + 56.
The perimeter of the rectangle: The sum of all the sides of the rectangle will be known as the perimeter of the rectangle.
Let L be the length and W be the width of the rectangle.
Then the perimeter of the rectangle will be
The perimeter of the rectangle = Sum of all the sides
If the gardener fences in the total rectangular area shown in the illustration instead of just the square area, he will need twice as much fencing to enclose the garden.
From the diagram, the length is 28 + x. And the width is x. Then the perimeters of the rectangle will be
P = 28 + x + 28 + x + x + x + x
P = 56 + 5x
The fencing needed is equal to the perimeter and will be 5x + 56.
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the ratio of 6th and 7th graders are 11 to 9. if there was 108 7th graders how many 6th graders will there be
The number of the students in the sixth grade will be 132 students.
What are ratio and proportion?A ratio is a group of sequentially ordered numbers a and b expressed as a/b, where b is never equal to zero. When two objects are equal, a statement is said to be proportional.
The proportion of sixth and seventh graders is 11 to 9. In the event that there were 108 seventh graders.
The ratio of the number of students in the sixth and seventh will remain constant.
Let 'x' be the number of students in sixth grade. Then the equation is given as,
x / 108 = 11 / 9
Simplify the equation, then we have
x / 108 = 11 / 9
x = 108 × 11 / 9
x = 12 × 11
x = 132
The number of the students in the sixth grade will be 132 students.
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Jaheem invests $10,000 in two different money markets. The first earned 12.5% in one year, the second earned 8.25%. If he earned $925 in interest, how much did he invest in each account?
Answer:
Let X be the amount Jaheem invested in the first money market, and Y be the amount he invested in the second money market. We can set up the following system of equations to represent the given information:
X + Y = 10000 (1)
0.125X + 0.0825Y = 925 (2)
We can solve this system of equations using any method of your choice, such as elimination or substitution.
Using elimination, we can multiply equation (1) by 0.125 and equation (2) by 10,000 to eliminate Y:
1.25X + 1250Y = 1250000
125X + 82500Y = 9250000
Subtracting the first equation from the second gives us:
124X + 82350Y = 8037500
Y = (8037500 - 124X)/82350
Substituting this expression for Y into equation (1) and solving for X gives us:
X + (8037500 - 124X)/82350 = 10000
X = $7,500
Therefore, Jaheem invested $7,500 in the first money market and $2,500 in the second money market.
Step-by-step explanation:
SELF EXPLANATORY
what is another name for a chord that passes through the center of the circle? A) Diameter B) Circumference C) center D) radius
Answer:
A
Step-by-step explanation:
diameter
The another name for the chord that passes through the center of the circle is the diameter.
The correct option is (A).
What is a circle?A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point.
As per the given data:
We have to find out the alternate name for a chord that passes through the center of a circle out of the given options.
Diameter:
It's a chord that passes through the center and touches the ends of the circumference of the circle.
Hence, this is correct.
Circumference:
It's the length of the boundary of a circle, hence this cannot be a chord.
This is incorrect.
Center:
It's a point and a chord is a line, hence this cannot be true.
Radius:
A radius is not a chord, as it does not touch the endpoints on the circumference of the circle.
The another name for the chord that passes through the center of the circle is the diameter.
Hence, The another name for the chord that passes through the center of the circle is the diameter.
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The Quotient of a number b and four plus seven is two
The algebraic form of the statement is b/4 + 7 = 2 and the value of b is -20.
What is the numerical value of the b?Given the statement; the quotient of a number b and four plus seven is two.
First, we convert the statement into an algebraic equation.
Quotient of a number b and 4 ⇒ b/4Plus 7 ⇒ + 7 Is two ⇒ = 2Now, combine the terms and equality sign.
b/4 + 7 = 2
Next, solve for the numerical value of b.
b/4 + 7 = 2
Subtract 7 from both sides
b/4 + 7 - 7 = 2 - 7
Simplify the left-hand side
b/4 = 2 - 7
Subtract 7 from 2
b/4 = -5
Next, multiply both sides by 4
b/4 × 4 = -5 × 4
b = -5 × 4
b = -20
Therefore, the numerical value of b is -20.
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Liam deposits $3,500 in a saving account that pays 1.5% interest, compounded quarterly.
a.Find the balance after a year.
B. Compare the interest compounded annually.
Therefore the balance after a year comes out to be $3552.80 and the balance comes out to be after compounded annually is $3552.50
What is compound interest ?Compound interest is the interest on savings that is computed using both the original principal and the interest accrued over time.
It is believed that Italy in the 17th century was where the idea of "interest on interest" or compound interest first emerged. It will accelerate the growth of a sum more quickly than simple interest, which is only applied to the principal sum.
Money multiplies more quickly through compounding, and the higher the
Here,
The Liam deposits $3,500 in a saving account that pays 1.5% interest,
Thus rate=1.5% and principle amount =3500
Therefore it is compounded quartely ,
The balance after year=
C=P[tex]e^{rt}[/tex]
C=3500*[tex]e^{1.5*4}[/tex]
C=3552.80
Therefore the balance after a year comes out to be $3552.80
Therefore the balance comes out to be after compounded annually is
$3552.50
So there is only $0.30 difference between the amounts
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(PLEASE HELP)
Consider circle O with diameter GB. Line AD is tangent to circle O at point A and line DB is tangent to circle O at point B. The measure of ∠GBA is 38°. Chord GC bisects ∠G. Determine each measure
The angles GBD, GBD and G measures 90 , 90, 52 degree respectively.
GB is the diameter given.
AD is the tangent given.
The tangent makes 90 degree angle with diameter.
hence the angle GBD will be : 90°
Similarly, GBA will also be 90°
The angle GBA is given as 38°
Hence the angle G will be = 90 - 38 =52
All points in a plane that are at a specific distance from a specific point, the centre, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant. The radius of a circle is the separation between any point on the circle and its centre.
The radius must typically be a positive integer. A degenerate situation is a circle with radius equal to zero (one point). Except when otherwise specified, this article discusses circles in Euclidean geometry, namely the Euclidean plane.
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If the discriminant, b^2-4ac is ___, a quadratic will have two real roots, two points of intersection with the x-axis.
a) negative
b) positive
c) complex
d) zero
If the discriminant, b² - 4ac is positive
How to complete the statement?From the question, we have the following equation of discriminant
The discriminant (d) is calculated as
d = b² - 4ac
The solutions of a quadratic equation are dependent on the following conditions
If d = 0, the quadratic equation has 1 real solutionIf d < 0, the quadratic equation has imaginary solutionsIf d > 0, the quadratic equation has 2 different real solutionsThis means that "d > 0, the quadratic equation has 2 different real solutions" implies that the discriminant is positive
Hence, the complete statement is (b) positive
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please help me PLEASE
The value of x in the linear inequality 5/8x - 1 ≥ 9 is x ≥ 16
What is Linear InequalityLinear inequalities are inequalities that involve at least one linear algebraic expression, that is, a polynomial of degree 1 is compared with another algebraic expression of degree less than or equal to 1. There are several ways to represent various kinds of linear inequalities.
The inequality given is
5/8x - 1 ≥ 9
5/8x ≥ 9 + 1
5/8x ≥ 10
5x ≥ 10 * 8
5x ≥ 80
x ≥ 16
The value of x is greater than or equal to 16
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If y = x-4, which point corresponds to the location where y = -7? (Hint: Keep in mind that this question gives you the
value of y, not x.)
Answer:
(- 3, - 7 )
Step-by-step explanation:
given
y = x - 4 ← substitute y = - 7 into the equation and solve for x
- 7 = x - 4 ( add 4 to both sides )
- 3 = x
then point is (x, y ) = (- 3, - 7 )
Answer:
(-3, -7)
Step-by-step explanation:
y=-7
y=x-4
-7 = x - 4 ==> since y=-7 and y=x-4, -7 = x-4
-7 + 4 = x - 4 + 4 ==> solve for x by adding 4 on both sides
-3 = x
x = -3
x=-3, y=-7 ==> (-3, -7)
Find the area enclosed by the circle (x-2)^2 + y^2 = 4 and the line x = 3.
Step-by-step explanation:
(3-2)^2+y^2=4
(9-4)+y^2=2^2
the power of y and 2 should be cut now
5+y=2
5-2=y
3=y
y=3
:. the area of enclosed by a circle is 3 cm
Evaluate 5² + 2(-8 ➗ 2)
Answer:
17
Step-by-step explanation:
Using PEMDAS:
5² + 2(-8 ÷ 2)
5² + 2(-4)
25 - 8
17
which category, groceries or dining out, has a higher population mean annual credit card charge? dining out what is the point estimate of the difference between the population means? round to the nearest whole number.
groceries has a higher population mean annual credit card charge
What is population mean?
A population is the complete set group of individuals, whether that group comprises a nation or a group of people with a common characteristic. In statistics, a population is the pool of individuals from which a statistical sample is drawn for a study.
The population mean can be calculated by the sum of all values in the given data/population divided by a total number of values in the given data/population.
Given that Bank of America’s Consumer Spending Survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment (US Airways Attache, December 2003).
Sample size n =42
Mean difference d = 850
Std dev s = 1125
Point estimate of the difference is 0
95% confidence level would be
= 0A ± tcritical * stderrror
= A ± 2.021*173.59
= (-350.83 , 350.83)
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HELP MEEEEEEEEEEEEEEEE PLEASEEEEEEEEEEEE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer: The final balance is $9,013.73.
The total compound interest is $7,513.73.
Step-by-step explanation:
0.4 (2-q) = 0.2 (q+7)
Answers- A. -3 B. -1 C. 3 D. All real numbers
The table below shows the amount of money needed to buy certain amounts of fabric. If the price is the same for each yard, how much money would 12 yards of fabric cost?
Correct answer will get brainliest
The equation is y = (5/3)x. Then the cost of the 12 yards of fabric will be $20. Then the correct option is B.
What is the equation of a line passing through two points?Let the equation of the line pass through (x₁, y₁) and (x₂, y₂).
Then the equation of the line is given as,
[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]
The two points are (3, 5) and (6, 10).
Let 'x' be the number of yards of fabric and 'y' be the cost in dollars. Then the equation is given as,
(y - 5) = [(10 - 5) / (6 - 3)](x - 3)
y - 5= (5/3)x - 5
y = (5/3)x
The cost of the 12 yards of fabric is given as,
y = (5/3) × 12
y = 5 × 4
y = $20
The equation is y = (5/3)x. Then the cost of the 12 yards of fabric will be $20. Then the correct option is B.
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1) if f(x) = x² - 1 and g(x) = x + 1, then when x doesnt equal -1, f(x) / g(x) =
helpp
Answer:
f(x) / g(x) = x - 1 where x ≠ -1.
Step-by-step explanation:
f(x) / g(x) = x² - 1 / x + 1
= (x + 1)(x - 1) / (x + 1)
= x - 1.
what is the point-slope form of -4,7?
An equation for this line in the point-slope form will be y-7 = -2(x + 4).
What is the Point-slope form?
The equation of the straight line has its slope and given point. If we have a non-vertical line that passes through any point(x1, y1) and has a gradient m. then general point (x, y) must satisfy the equation. Point-slope is a specific form of linear equations in two variables:
y - y₁ = m(x - x₁)
Which is the required equation of a line in a point-slope form.
We need to find the equation of the line that passes through (-4, 7) and whose slope is -2.
Well, we will simply plug m = -2, x₁= -4, and y₁ = 7 into point-slope form.
y-y₁ = m(x-x₁)
Now substitute;
y-7 = -2(x - (-4))
y-7 = -2(x + 4)
Therefore, an equation for this line in the point-slope form will be y-7 = -2(x + 4).
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suppose that the trace of a matrix is , and the determinant is . find the eigenvalues of . smaller eigenvalue = , larger eigenvalue = .
The eigenvalues of A are 5 and 10.
What is eigenvalues ?
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that, when the linear transformation is applied to it, changes at most by a scalar factor. The factor by which the eigenvector is scaled is known as the associated eigenvalue, frequently denoted by lambda.
The eigenvalues of the 2 x 2 matrix are described by the following characteristic polynomial:
[tex]$$t^2-{tr}(A) \cdot t+{det}(A)=0$$[/tex]
The roots of the characteristic polynomial are deflned by the quadratic formula:
[tex]t=\frac{{tr}(A) \pm \sqrt{[{tr}(A)]^2-4-{det}(A)}}{2} \text { (2) }$$[/tex]
If we know that [tex]${tr}(A)=15$[/tex]and [tex]${det}(A)=50$[/tex], then the elgenvalues of A are:
[tex]$$\begin{aligned}& t=\frac{15 \pm \sqrt{15^2-4 \cdot(50)}}{2} \\& t_1=10 \vee t_2=5\end{aligned}$$[/tex]
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How can I solve this ? What is the angle whose tangent is 456/321?
The angle whose tangent is 456/321 is 54.85°
The tangent of an angle in trigonometry is the ratio of the lengths of the adjacent side to the opposing side. In order for the value of the cosine function to not be 0, it is the ratio of the sine and cosine functions of an acute angle.
The Tangent Formula is given as:
Tan A = Opposite Side/Adjacent side
In terms of sine and cosine, tangent may be represented as:
Tan A = Sin A / Cos A
We know that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side whereas the cosine of the angle is the ratio of the length of the adjacent side to the ratio of the hypotenuse side.
That is, Sin A = Opposite Side/ Hypotenuse Side
Cos A = Adjacent Side/ Hypotenuse Side
Therefore, tan A = Opposite Side/ Adjacent Side
[tex]tan^{-1} (1.42056)[/tex] =54.85°
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e throw a fair die three times. if we observe the first throw is a 4, what is the probability that there is at least two 4s in the three throws?
The probability that there is at least two 4s in the three throws is 0.05 or 5%
Probability
In statistics, probability deals with the possibility of happening of the events.
Given,
Here we know that the throw a fair die three times. if we observe the first throw is a 4.
And we need to find the probability that there is at least two 4s in the three throws.
According to the given question, we know that the dice is thrown 3 time.
Then the sample space of the given event is 216.
Here we know that the first throw has 4.
And we need to find the probability of getting two 4s in the 3 throws,
So, the possible number of events are.
=> {444, 441, 442, 443, 445, 446, 414, 424, 434, 454, 464}
So, the total possible number is 11.
Then the probability is,
=> 11/216
=> 0.05 or 5%
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(Factoring Algebraic Expressions MC)
Factor −3x2 + 12x.
The factor is -3x(x - 4) and value of x is 4.
Define Algebraic Expression
An expression that includes variables, constants, and algebraic operations is known as an algebraic expression (addition, subtraction, etc.)
Define Factorization
Factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Given expression,
-3x² + 12x
Take -3x common
⇒ -3x(x - 4)
Now, if you want to find the value of x then write it in standard form like this
-3x(x - 4) = 0
Now, find x
x - 4 = 0
x = 4
Hence, the factor is -3x(x - 4) and value of x is 4.
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the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1. before this change, how much taller was the maple tree than the cherry tree, in centimeters?
Before the change maple tree was 240 cm longer than cherry tree when the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1.
Define ratio.One quantity can be expressed as a fraction of another using a ratio, which is a number. Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects.
Given,
Current ratio = 5:2
Let y equal the height of the cherry tree, and x represent the height of a maple tree.
x/y = 5/2
New ratio = 3:1
x+20/y-20 = 3/1
Cross multiplying,
x + 20 = 3(y - 20)
x = 3(y - 20) -20
Equating values in x/y = 5/2
3(y-20) - 20/y = 5/2
5y = 2(3y - 60) -20
5y = 6y - 160
y = 160
Now, for value of x,
Equating values in x = 3(y - 20) -20
x = 3(160 - 20) - 20
x = 480 - 60 - 20
x = 400
For the height,
400 - 160
240
Before the change maple tree was 240 cm longer than cherry tree when the heights of a maple tree and a cherry tree currently have a ratio of 5:2. if the maple tree grew 20 centimeters, and 20 centimeters was cut off the top of the cherry tree, the ratio of their heights would be 3:1.
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