50 Points! Multiple choice geometry question. Photo attached. Thank you!

Answer:

We don't need to worry about the displaystyle- {3} −3 anyway, because we dcided in the first step that displaystyle {x}ge- {2} x ≥ −2. So the domain for this case is displaystyle {x}ge- {2}, {x}ne {3} x≥ −2,x≠ 3, which we can write as displaystyle {left [- {2}, {3}right)}cup {left ({3},inftyright)} [−2,3)∪(3,∞).

Step-by-step explanation:

Answer:

KL = 6

Step-by-step explanation:

We see that the length of IL includes IJ, JK, and Kl and is 26.

Since IL = 26 and IJ + JK + KL = IL, we can subtract the sum of the lengths of IJ and Jk from IL to find KL:

IL = IJ + JK + KL

26  = 9 + 11 + KL

26 = 20 + KL

6 = KL

Thus, the length of KL is 6.

We can confirm this fact by plugging in 6 for KL and checking that we get 26 on both sides of the equation when simplifying:

IL = IJ + JK + KL

26 = 9 + 11 + 6

26 = 20 + 6

26 = 26

Thus, our answer is correct.

The measure of the arc DGF which substends the angle DEF at the circumference of the circle is equal to 190°

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference.

Given that the arc DGF = (62x + 4)°

62x + 4 = 2(30x + 5)

62x + 4 = 60x + 10

62x - 60x = 10 - 4 {collect like terms}

2x = 6

x = 6/2

x = 3

arc DGF = 62(3) + 4 = 190°

Therefore, the measure of the arc DGF which substends the angle DEF at the circumference of the circle is equal to 190°

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The linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].

In a linear equation, the y-intercept is where the line crosses the y-axis.

It is represented by the constant term in the equation.

So, to determine which linear function has the greatest y-intercept, we need to look at the constant term of each equation.

Let's consider each equation: [tex]y = 6x + 1[/tex]

The constant term in this equation is 1.

So, the y-intercept is 1.

On a coordinate plane, a line goes through points (0, 2) and (5, 0).

To find the equation of this line, we can use the point-slope form:

[tex]y - y1 = m(x - x1)[/tex]

where m is the slope and (x1, y1) is a point on the line.

Using the points (0, 2) and (5, 0), we get:

[tex]m = \frac{(0 - 2)}{(5 - 0)} =-\frac{2}{5}[/tex]

So, the equation of the line is:

[tex]y - 2 = (\frac{-2}{5} )(x - 0)[/tex]

[tex]y = (\frac{-2}{5} )x + 2[/tex]

The constant term in this equation is 2.

So, the y-intercept is 2.

On a coordinate plane, a line goes through points (1, 2) and (0, -3).

To find the equation of this line, we can use the point-slope form:

[tex]y - y1 = m(x - x1)[/tex]

where m is the slope and (x1, y1) is a point on the line.

Using the points (1, 2) and (0, -3), we get:

[tex]m = \frac{ (-3 - 2) }{(0 - 1)} = -5[/tex]

So, the equation of the line is:

[tex]y - 2 = (-5)(x - 1)y = -5x + 7[/tex]

The constant term in this equation is 7.

So, the y-intercept is 7.

[tex]y = 3x + 4[/tex]

The constant term in this equation is 4.

So, the y-intercept is 4.

Therefore, we can see that the linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].

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Answer:

Step-by-step explanation:

To determine the remaining sides of triangle THE given that sin(E) = 4 and TE = 4, we can use the sine ratio.

The sine ratio is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

In this case, sin(E) = 4/TE, which means the side opposite angle E is 4 and the hypotenuse TE is 4.

Using the Pythagorean theorem, we can find the length of the remaining side TH:

TH^2 = TE^2 - HE^2

TH^2 = 4^2 - 4^2

TH^2 = 16 - 16

TH^2 = 0

TH = 0

Therefore, the length of side TH is 0.

Answer:

180 meters

Step-by-step explanation:

To find the perimeter of the actual swimming pool, you need to first find the length and width of the actual swimming pool by multiplying the length and width of the pictured swimming pool by the scale factor of 1200 cm.

Now that we know that the perimeter of the actual pool is 18000 centimeters, we need to convert that to meters! Keep in mind that:

Now we can divide 18000 by 100:

18000 cm ÷ 100 = 180 m

Therefore, the perimeter of the actual swimming pool is 180 m.

Answer:

To write the polynomial function m(x) = x + x^2 - 1 in standard form, we rearrange the terms in descending order of degree:

m(x) = x^2 + x - 1

Polynomial name: Quadratic polynomial

Degree: 2 (the highest exponent is 2)

Leading coefficient: 1 (the coefficient of the highest-degree term)

Constant term: -1 (the term without any variable)

`Both Meg and Cassandra are incorrect in their assessments (option a).

Meg and Cassandra have both misunderstood the logical relationships between the statements. Let's analyze each statement and compare their claims:

Statement 1: "If she is stuck in traffic, then she is late."

Statement 2: "If she is late, then she is stuck in traffic."

Statement 3: "If she is not late, then she is not stuck in traffic."

Meg's claims: Meg states that Statement 3 is the inverse of Statement 2 and the contrapositive of Statement 1. However, this is incorrect. The inverse of Statement 2 would be "If she is not stuck in traffic, then she is not late," and the contrapositive of Statement 1 would be "If she is not late, then she is not stuck in traffic." So Meg's analysis is incorrect.

Cassandra's claims: Cassandra states that Statement 2 is the converse of Statement 1 and the inverse of Statement 3. However, this is also incorrect. The converse of Statement 1 would be "If she is late, then she is stuck in traffic," and the inverse of Statement 3 would be "If she is late, then she is stuck in traffic." So Cassandra's analysis is incorrect as well.

Therefore, both Meg and Cassandra are wrong in their assessments. The correct logical relationships are as follows:

- The contrapositive of Statement 1 is Statement 3.

- The converse of Statement 1 is Statement 2.

Hence, the correct answer is that both Meg and Cassandra are incorrect.

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Answer:

$3606.44

Step-by-step explanation:

The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.

To do this, we need to use the formula for simple interest:

[tex]\boxed{I = \frac{P \times R \times T}{100}}[/tex],

where:

I = interest earned

P = principal invested

R = annual interest rate

T = time

By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:

[tex]6180 = \frac{P \times 6.12 \times 28}{100}[/tex]

⇒ [tex]6180 \times 100 = P \times 171.36[/tex]     [Multiplying both sides by 100]

⇒ [tex]P = \frac{6180 \times 100}{171.36}[/tex]    [Dividing both sides of the equation by 171.36]

⇒ [tex]P = \bf 3606.44[/tex]

Therefore, James needs to invest $3606.44.

The correct statement about the mean is:

The mean equals the sum of all observations divided by the number of observations.

The mean is a commonly used measure of central tendency. It is calculated by summing up all the observations and then dividing the sum by the total number of observations. It provides an average value that represents the typical value of the data set.

To calculate the mean, it is not necessary to order the observations from least to most. The order of the observations does not affect the mean calculation.

The mean is not necessarily a whole number. It can be a decimal or a fraction, depending on the data set and the values of the observations. The mean represents the balance point of the data set and can take on any real number value.

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The angular velocity of the object is 31.25 radians/second.

Angular velocity is defined as the change in angular displacement per unit of time. In this case, the object rotates a total of 25 radians in 0.8 seconds. Therefore, the angular velocity can be calculated by dividing the total angular displacement by the time taken.

Angular velocity (ω) = Total angular displacement / Time taken

Given that the object rotates 25 radians and the time taken is 0.8 seconds, we can substitute these values into the formula:

ω = 25 radians / 0.8 seconds

Simplifying the equation gives:

ω = 31.25 radians/second

So, the angular velocity of the object is 31.25 radians/second.

Angular velocity measures how fast an object is rotating and is typically expressed in radians per second. It represents the rate at which the object's angular position changes with respect to time.

In this case, the object completes a rotation of 25 radians in 0.8 seconds, resulting in an angular velocity of 31.25 radians per second. This means that the object rotates at a rate of 31.25 radians for every second of time.

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Note the translated question is:

An object that is rotated moves 25 radians in 0.8 seconds. what is the angular velocity of said object?

Step-by-step explanation:

a)  23.6 (1.08)^x        for 2016    x = 26   ('x' is the number of years past 1990)

        23.6 (1.08)^(26) = 174.6  billion

b)  109 = 23.6 ( 1.08)^x

    4.6186 = 1.08^x

          x =   log 4.6186 / log1.08   = 19.88 yrs  

                                         means  1990 + 19.88 yrs = year 2010

Answer:

99

Step-by-step explanation:

since the height is 9 and the base is 11 we use the formula BH=V

substitute 9x11 and get 99

Answer:

See below

Step-by-step explanation:

The perimeter of a rectangle is [tex]P=2L+2W[/tex] where L is the length and W is the width.

Answer : Focus is (0,3)

To find the focus of the parabola defined by the equation (y - 3)² = -8(x - 2), we can compare it with the standard form of a parabolic equation: (y - k)² = 4a(x - h).

In the given equation, we have:

(y - 3)² = -8(x - 2)

Comparing it with the standard form, we can determine the values of h, k, and a:

h = 2

k = 3

4a = -8

Solving for a, we get:

4a = -8

a = -8/4

a = -2

Therefore, the vertex of the parabola is (h, k) = (2, 3), and the value of 'a' is -2.

The focus of the parabola can be found using the formula:

F = (h + a, k)

Substituting the values, we get:

F = (2 + (-2), 3)

F = (0, 3)

Therefore, the focus of the parabola defined by the equation (y - 3)² = -8(x - 2) is at the point (0, 3).

Answer:

Focus = (0, 3)

Step-by-step explanation:

The focus is a fixed point located inside the curve of the parabola.

To find the focus of the given parabola, we first need to find the vertex (h, k) and the focal length "p".

The standard equation for a sideways parabola is:

[tex]\boxed{(y-k)^2=4p(x-h)}[/tex]

where:

If p > 0, the parabola opens to the right, and if p < 0, the parabola opens to the left.

Given equation:

[tex](y-3)^2=-8(x-2)[/tex]

Compare the given equation to the standard equation to determine the values of h, k and p:

The formula for the focus is (h+p, k).

Substituting the values of h, p and k into the formula, we get:

[tex]\begin{aligned}\textsf{Focus}&=(h+p,k)\\&=(2-2,3)\\&=(0,3)\end{aligned}[/tex]

Therefore, the focus of the parabola is (0, 3).

Answer: y = 18.32 + 0.0932x

Step-by-step explanation:

Given that the base charge per month is $18.32 and the charge per kilowatt-hour is 9.32 cents, which is assigned to the variable x, we can write an equation to find the total monthly charge based on kilowatt-hours.

total = monthly charge + kilowatt-hour x the amount of kilowatts
y = 18.32 + 0.0932x

here we see that y is the total cost per month, our base monthly charge is $18.32, our kilowatt hour charge is 9.32 cents, which we write in terms of dollar amounts for the sake of the equation (just divide by 100), and our variable x represents the number of kilowatts.

The rectangles that are similar are Rectangle 3 and Rectangle 4.

To determine which rectangles are similar, we need to compare their corresponding side lengths.

Rectangle 1:

Length: 3 cm

Height: 5 cm

Rectangle 2:

Length: 2.5 cm

Height: 5.5 cm

Rectangle 3:

Length: 2.5 cm

Height: 2 cm

Rectangle 4:

Length: 5 cm

Height: 4 cm

To determine similarity, we need to compare the ratios of the corresponding side lengths of the rectangles.

Comparing Rectangle 1 with Rectangle 2:

Length ratio: 3 cm / 2.5 cm = 1.2

Height ratio: 5 cm / 5.5 cm ≈ 0.91

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 2 are not similar.

Comparing Rectangle 1 with Rectangle 3:

Length ratio: 3 cm / 2.5 cm = 1.2

Height ratio: 5 cm / 2 cm = 2.5

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 3 are not similar.

Comparing Rectangle 1 with Rectangle 4:

Length ratio: 3 cm / 5 cm = 0.6

Height ratio: 5 cm / 4 cm = 1.25

The length ratio and height ratio are not equal, so Rectangle 1 and Rectangle 4 are not similar.

Comparing Rectangle 2 with Rectangle 3:

Length ratio: 2.5 cm / 2.5 cm = 1

Height ratio: 5.5 cm / 2 cm = 2.75

The length ratio and height ratio are not equal, so Rectangle 2 and Rectangle 3 are not similar.

Comparing Rectangle 2 with Rectangle 4:

Length ratio: 2.5 cm / 5 cm = 0.5

Height ratio: 5.5 cm / 4 cm = 1.375

The length ratio and height ratio are not equal, so Rectangle 2 and Rectangle 4 are not similar.

Comparing Rectangle 3 with Rectangle 4:

Length ratio: 2.5 cm / 5 cm = 0.5

Height ratio: 2 cm / 4 cm = 0.5

The length ratio and height ratio are equal, so Rectangle 3 and Rectangle 4 are similar.

Therefore, the rectangles that are similar are Rectangle 3 and Rectangle 4.

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Answer:

Let's denote the height of the triangle as "h" inches.

According to the given information, the base of the triangle is 3 inches more than two times the height. Therefore, the base can be expressed as (2h + 3) inches.

The formula to calculate the area of a triangle is:

Area = (1/2) * base * height

Substituting the given values, we have:

7 = (1/2) * (2h + 3) * h

To simplify the equation, let's remove the fraction by multiplying both sides by 2:

14 = (2h + 3) * h

Expanding the right side of the equation:

14 = 2h^2 + 3h

Rearranging the equation to bring all terms to one side:

2h^2 + 3h - 14 = 0

Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:

h = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, the values are:

a = 2

b = 3

c = -14

Substituting these values into the quadratic formula:

h = (-3 ± √(3^2 - 4 * 2 * -14)) / (2 * 2)

Simplifying:

h = (-3 ± √(9 + 112)) / 4

h = (-3 ± √121) / 4

Taking the square root:

h = (-3 ± 11) / 4

This gives us two possible solutions for the height: h = 2 or h = -14/4 = -3.5.

Since a negative height doesn't make sense in this context, we discard the negative solution.

Therefore, the height of the triangle is h = 2 inches.

To find the base, we substitute this value back into the expression for the base:

base = 2h + 3

base = 2(2) + 3

base = 4 + 3

base = 7 inches

Hence, the base of the triangle is 7 inches and the height is 2 inches.

Step-by-step explanation:

-The answer for the height is 5.5 units.

-The base of the triangle is aproximately 2.5454 units.

To answer this problem, you have to set an equation with the information you're given. If you do it correctly, it should look like this:

7=1/2(3+2h)

-Now, you have to solve for h:

7=1.5+h

7-1.5=h

5.5=h

-Now that you have the height, you plug it in into the triangle area formula to solve for the base:

7=1/2(b)5.5

7=2.75b

7/2.75=b

b≈2.5454

-To make sure that the corresponding values for the base and height are correct, we plug the values in and this time we are going to solve for a(AREA):

A(triangle)=1/2(2.5454)(5.5)

A=1/2(13.9997)

A=6.99985 square units

-We round the result to the nearest whole number and we get our 7, which is the given value they gave us.

Answer:

The consecutive interior angles are supplementary, so we have:

3x + 20 + 2x = 180

5x + 20 = 180

5x = 160, so x = 32

Answer:

f(x) = 4x² - 2x + 11

f'(x) = 8x - 2

m = f'(3) = 8(3) - 2 = 24 - 2 = 22

41 = 22(3) + b

41 = 66 + b

b = -25

y = 22x - 25

Answer:

∠ MHU = 54°

Step-by-step explanation:

the angle MHU between the tangent and the secant is half the measure of the intercepted arc HM , then

∠ MHU = [tex]\frac{1}{2}[/tex] × 108° = 54°

Answer:

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.

The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:

V = 2π ∫[a, b] x * f(x) dx

In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.

Substituting these values into the formula, we have:

V = 2π ∫[0, 5] x * (x - 4)^3 dx

To evaluate this integral, we can expand the cubic term and then integrate:

V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx

V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx

Integrating each term separately:

V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5

Now we can substitute the bounds of integration:

V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]

Simplifying:

V = 2π [(1/5 * 3125) - 0]

V = 2π * (625/5)

V = 2π * 125

V = 250π

Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.

Answer:

Step-by-step explanation:

a. To write an equation describing the total cost of RV rental, we can use the given information:

Let x be the number of days of rental.

Let y be the number of miles driven.

The total cost of RV rental can be calculated as follows:

Total cost = (125 * x) + (0.32 * y) + 2500

b. To compare the cost of the two options, we need to calculate the total cost for each option.

Option 1:

x = 11 days

y = 3500 miles

Total cost = (125 * 11) + (0.32 * 3500) + 2500

Option 2:

x = 14 days

y = 3000 miles

Total cost = (125 * 14) + (0.32 * 3000) + 2500

Compare the total costs of both options to determine which one is cheaper.

c. To compromise and keep the rental cost under $5,000, we can adjust either the number of miles driven or the number of days of rental.

For the first compromise (lessening the miles), let's assume the new number of miles driven is y1.

The domain for the compromise in miles would be 0 ≤ y1 ≤ 3500, as you cannot drive more miles than the original option or negative miles.

For the second compromise (lessening the days of rental), let's assume the new number of days of rental is x1.

The domain for the compromise in days would be 0 ≤ x1 ≤ 14, as you cannot have more days of rental than the original option or negative days.

The justification for these domains is that they restrict the values within the range of the original options while allowing for adjustments in the desired direction.

d. To find out how many miles or how many days need to be eliminated to stay under the $5,000 budget, we can set up and solve equations.

For the compromise in miles (y1):

(125 * x) + (0.32 * y1) + 2500 ≤ 5000

For the compromise in days (x1):

(125 * x1) + (0.32 * y) + 2500 ≤ 5000

Solve each equation by isolating the variable to determine the maximum allowed value for y1 or x1, respectively.

Finally, after solving the equations, compare the maximum allowed values for y1 and x1 and consider other factors like practicality, time constraints, and preferences to make a recommendation to your parents about which compromise is best.

In a right triangle, the length of side "a" is 12.

The Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, can be used to find the length of side "a" in a right triangle with sides of 13 and 5 units.

Let's assign "a" as the unknown side. According to the Pythagorean theorem, we have the equation: [tex]a^{2}[/tex] = [tex]13^{2}[/tex] - [tex]5^{2}[/tex].

Simplifying the equation, we get [tex]a^{2}[/tex] = 169 - 25, which becomes [tex]a^{2}[/tex] = 144.

To solve for "a," we take the square root of both sides: a = √144.

The square root of 144 is 12. Therefore, side "a" has a length of 12 units.

In summary, using the Pythagorean theorem, we determined that side "a" in the right triangle with side lengths 13 and 5 units has a length of 12 units.

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Answer:

A.

Step-by-step explanation:

In this case, we have to use tan ([tex]\frac{opposite}{adjacent}[/tex] because we are asked for the opposite side (x) given the adjacent side (20 m).

So tan(75)=[tex]\frac{x}{20}[/tex]

Solve for x

x = 20 * tan(75)

x = 74.641...

x = 74.64 m

Answer:

The height is 74.64 meters

Step-by-step explanation:

We have a ΔABC with ∠B = 75°, hypotenuse = AB

[tex]cos\; 75\textdegree = \frac{\sqrt{3} -1}{2\sqrt{2} }\\\\\frac{1}{cos\; 75\textdegree} = \frac{2\sqrt{2} }{\sqrt{3} -1}[/tex]

cos B = adjacent/hyppotenuse

⇒ hypotenuse (AB) = adjacent/cosB = 20/cosB

[tex]= 20 \frac{2\sqrt{2} }{\sqrt{3} -1}\\\\= \frac{40\sqrt{2} }{\sqrt{3} -1}\\\\= 77.27[/tex]

⇒ AB = 77.27

By pythagoras theorem,

AB² = AC² + BC²

⇒ AC² = AB² - BC²

= 77.27² - 20²

AC² = 5570.65

⇒ AC = √5570.65

AC = 74.64

The step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 0.5.

To compute the step length 'a' using the line search method, we can follow these steps:

1: Calculate the gradient at point x(k).

  - Given x(k) = (1, -1)

  - Compute the gradient ∇f(x₁,x₂) at x(k):

    ∇f(x₁,x₂) = (∂f/∂x₁, ∂f/∂x₂)

    ∂f/∂x₁ = 3x₁ + 1

    ∂f/∂x₂ = 2x₂ - 1

    Substituting x(k) = (1, -1):

    ∂f/∂x₁ = 3(1) + 1 = 4

    ∂f/∂x₂ = 2(-1) - 1 = -3

  - Gradient at x(k): ∇f(x(k)) = (4, -3)

2: Compute the dot product between the gradient and the descent direction.

  - Given PL = (1, 0)

  - Dot product: ∇f(x(k)) ⋅ PL = (4)(1) + (-3)(0) = 4

3: Compute the norm of the descent direction.

  - Norm of PL: ||PL|| = √(1² + 0²) = √1 = 1

4: Calculate the step length 'a'.

  - Step length formula: a = -∇f(x(k)) ⋅ PL / ||PL||²

    a = -4 / (1²) = -4 / 1 = -4

5: Take the absolute value of 'a' to ensure a positive step length.

  - Absolute value: |a| = |-4| = 4

6: Finalize the step length.

  - The step length 'a' is the positive value of |-4|, which is 4.

Therefore, the step length 'a' for the line search method at point x(k) = (1, -1) with the descent direction PL = (1, 0) is 4.

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After a long study, tree scientists conclude that a eucalyptus tree will grow at the rate of 3ft per year, where t is time in years. So, the tree will grow 5 feet in the first year.

We have to find the number of feet the tree will grow in the first year, given that 5(t + 1)³. The rate of growth of a tree is given as 3ft/year. Therefore, in the first year, the tree will grow 3 feet.

To find the number of feet the tree will grow in the first year, we substitute t = 0 in the given expression.

5(t + 1)³ = 5(0 + 1)³= 5(1)³= 5(1)= 5. Therefore, the tree will grow 5 feet in the first year.

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