Find the properties for the ellipse with the equation x^2/169 + y^2/144 = 1

latus rectum =

a. 288/13
b. 328/12
c. 288/12

Answer:

  36 units

Step-by-step explanation:

You want to know the perimeter of the right triangle with leg lengths 9 units and 12 units.

The perimeter of the triangle is the sum of the lengths of its sides. We can count squares to find the lengths of the horizontal and vertical legs. The length of the hypotenuse can be found using the Pythagorean theorem:

  c² = a² +b²

  c² = 9² +12² = 81 +144 = 225

  c = √225 = 15 . . . . length of the hypotenuse

Then the perimeter is ...

  P = a +b +c = 9 +12 +15 = 36 . . . units

The perimeter of the triangle is 36 units.

__

Additional comment

The leg lengths have the ratio 9:12 = 3:4, telling you this is a 3:4:5 right triangle. This means you know the side lengths are 3:4:5 = 9:12:15, and their sum is 9+12+15 = 36.

It is handy to memorize a few of the Pythagorean triples that often show up in algebra, trig, and geometry problems: {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}.

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The radius of the circle to the nearest 10th is 5.4

The formula for calculating the length of an arc of a circle is:

length of arc = radius x angle in radians

l = rθ

where

r = radius of the circle

l = length of arc

θ = angle in radians

From the question, we are given:

length of arc = 11.9

angle in radians (θ) = 2.2radians

The we can plug in the values

11.9 = r x 2.2

make r the subject of the formula

r = 11.9/2.2

r = 5.41 (to 2 decimal places)

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The 90 % confidence interval is ( 19.92 , 20.18 )

A confidence interval is defined as the range of values that we observe in our sample and for which we expect to find the value that accurately reflects the population.

The relationship between a value and the mean of a set of values is expressed numerically by a Z-score. The Z-score is computed using the standard deviations from the mean. A Z-score of zero indicates that the data point's score and the mean score are identical.

The Z-score is calculated using the formula:

z = (x - μ)/σ

where z: standard score

x: observed value

μ: mean of the sample

σ: standard deviation of the sample

Given data ,

Let the number of samples n = 15

Let the mean of the sample be μ = 20.05

Let the standard deviation σ = 0.3

Now , the z score of 90 % confidence interval is z = 1.645

The 90% confidence interval is calculated by the equation

A = μ ± z ( σ / √n )

Substituting the values in the equation , we get

P = μ + z ( σ / √n )

Q = μ - z ( σ / √n )

Now , the value of P is

P = 20.05 + 1.645 ( 0.3/√15 )

P = 20.05 + ( 0.4935 / 3.873 )

P = 20.05 + 0.12742

P = 20.18

Now , the value of Q is

Q = μ - z ( σ / √n )

Q = 20.05 - 1.645 ( 0.3/√15 )

Q = 20.05 - ( 0.4935 / 3.873 )

Q = 20.05 - 0.12742

Q = 19.92

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If someone asked me how it could be possible for an annual interest rate to be higher than 100%, I would explain that it is actually quite common in certain situations, particularly in the case of loans with very short terms or loans with high fees.

For example, let's say you borrowed $100 from a lender and agreed to pay back $110 in one week. The lender is essentially charging you 10% interest for the one-week loan period, but if you annualize that rate, it comes out to over 520%. This is because the lender is charging you a very high interest rate for a very short period of time.

Another example would be if you took out a payday loan, which typically have very high fees attached to them. For instance, you might borrow $500 and have to pay back $575 in two weeks. The interest rate on this loan might be calculated as the $75 fee divided by the $500 borrowed, which comes out to 15%. However, if you annualize that rate, it comes out to over 390%.

In both of these examples, the interest rate is very high because the loan term is very short and/or the fees are very high. It's important to note that borrowing at such high interest rates can be extremely costly and can lead to a cycle of debt, so it's generally recommended to avoid loans with high interest rates whenever possible.

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Answer:

y=60

Step-by-step explanation:

The model that represents Ms. Griffin's situation is 0.8 divided by 4, which equals 0.2 liters of hot tea in each teacup.

To find the amount of tea in each teacup, you need to divide the total amount of tea (0.8 liters) by the number of teacups (4). The model for this is 0.8 ÷ 4. Follow these steps:

1. Divide 0.8 by 4:
0.8 ÷ 4 = 0.2

2. Interpret the result:
Each teacup will have 0.2 liters of hot tea.

So, the model that represents Ms. Griffin's situation is 0.8 divided by 4, which equals 0.2 liters of hot tea in each teacup.

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If Her pool is 10 m by 5 m and is surrounded by a 1. 5 m border of paving stones. Therefore, Piscine will need a total of 104 square meters of paving stones.

a) The area of the pool plus the surrounding border of paving stones can be calculated as follows:

Total area = (length + 2 x border width) x (width + 2 x border width)

Total area = (10 + 2 x 1.5) x (5 + 2 x 1.5)

Total area = 13 x 8

Total area = 104 square meters

Therefore, Piscine will need a total of 104 square meters of paving stones.

b) We need to convert the dimensions of each paving stone to meters:

25 cm = 0.25 m

40 cm = 0.4 m

The area of each paving stone is:

0.25 m x 0.4 m = 0.1 square metres

The number of paving stones required can be calculated by dividing the total area by the area of each paving stone:

Number of paving stones = Total area / Area of each paving stone

Number of paving stones = 104 / 0.1

Number of paving stones = 1040

In theory, Piscine will need 1040 paving stones.

c) To determine whether the answer in part b) will be enough, we need to see if we can fit the paving stones into the available space without any gaps. We can arrange the paving stones in rows, with each row containing 10 stones (since the width of the pool is 5 m and the width of each paving stone is 0.4 m).

There will be 26 rows of paving stones around the pool (since the length of the pool plus the two borders is 13 m and the length of each paving stone is 0.25 m). Therefore, the total number of paving stones required is: 26 rows x 10 stones per row = 260 stones

Since the number of paving stones required is less than the number calculated in part b), Piscine will have enough paving stones to complete the border around her pool without any waste.

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The equation of the quadratic function in the stretch part is f(x) = x² + 4x - 11

From the question, we have the following parameters that can be used in our computation:

Zeros: -2 ± √15

This means that

Zeros: -2 - √15 and -2 + √15

The equation of the function is calculated as

f(x) = product of (x - zeros)

So, we have

f(x) = (x - (-2 -√15)) * (x - (-2 + √15))

When expanded, we have

f(x) = (x + 2 + √15)) * (x + 2 - √15))

Evaluate the products

f(x) = x² + 4x - 11

Hence, the function is f(x) = x² + 4x - 11

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The Paige walks at a speed of approximately 0.04167 miles per minute to get to the park.

To calculate Paige's speed, we used the formula:

Speed = Distance / Time

Given that Paige walks to the park 2/3 mile away, we substitute Distance with 2/3 mile and Time with 16 minutes. We get:

Speed = 2/3 mile / 16 minutes

Simplifying the expression by converting minutes to hours, we get:

Speed = 2/3 mile / (16/60) hours

Simplifying further by multiplying both the numerator and denominator by 60, we get:

Speed = [tex](2/3) * (60/1)[/tex] mile/hour / (16/1) minutes

Speed = 0.04167 mile/minute (rounded to 5 decimal places)

"Paige walks to the park 2/3 mile away it takes her 16 minutes to get there how many miles per minutes" is that Paige walks at a speed of approximately 0.04167 miles per minute.

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Answer:

Step-by-step explanation:

f(x) = ax² + bx + c      a= -8

f(x) = -8x² + bx + c     that a controls the direction and the stretch.

So the function will be stretched by 8.  The negative represents the direction because, it's negative, it will be facing down.  

Not sure how your class describes it but it could be facing down, concaved down, or expands downward.  

Answer:

935 Rocks and shells

Step-by-step explanation:

To find the total number of rocks and seashells collected by Max and Lilah, we can use the addition operation. Let S be the number of seashells and R be the number of rocks. Then, the math sequence for this problem is:

S + R = Total

Substituting the given values, we get: 290 + 645 = Total

Simplifying the right-hand side, we get:

Therefore, Max and Lilah collected a total of 935 rocks and seashells in all.

To find the total number of rocks and seashells collected by Lilah and Max, we simply need to add the number of seashells and rocks they each collected. Let S represent the number of seashells and R represent the number of rocks. Then, the equation is:

S + R = 290 + 645

Simplifying this expression, we get:

S + R = 935

935 rocks and seashells.

Answer:

Step-by-step explanation:

We can use trigonometry to solve this problem. Let's call the distance from the base of the plateau to our position "x". We can then use the tangent function to find x:

tan(20°) = opposite / adjacent

In this case, the opposite side is the height of the plateau (50 m) and the adjacent side is x. So we can write:

tan(20°) = 50 / x

To solve for x, we can rearrange this equation:

x = 50 / tan(20°)

Using a calculator, we get:

x = 143.45 meters (rounded to two decimal places)

Therefore, if the angle of elevation to the top of the plateau is 20 degrees, and the plateau is 50 meters high, we are approximately 143.45 meters away from the base of the plateau.

Answer:

The distance is 137.3739 feet.

Step-by-step explanation:

I hope this answer is right.

The ratios of successive numbers in the Fibonacci sequence eventually get closer to a. 1.61

In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1 (0, 1, 1, 2, 3, 5, 8, 13, ...). When you take the ratio of successive numbers in the sequence (e.g., 5/3 or 8/5), it converges to approximately 1.618, also known as the Golden Ratio or Phi.

The closest option in your list is 1.61, which is option (a).

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Answer:

sin (y) = oppoaite / hypotenus

sin (y) = oppoaite / hypotenus sin (y) = opp/hyp

And for geting cos

cos(x) = adjecent / hypotenus

cos(x) = adjecent / hypotenus cos(x) = adj/hyp

Answer:

Total wood = 4.5 Left over wood = 0.7 let 1 piece of wood = x

hence we have 5x+0.7=4.5

5x=4.5 - 0.7

5x = 3.8 x= 3.8/5 = 0.76 ft

Step-by-step explanation:

Answer:

A: 5x + 0.7 = 4.5

B: You know the equation is correct because it takes into account that Warren cut five pieces of wood that are equal in length from a 4.5 feet long piece of wood. The total length of the five pieces of wood should equal the length of the original piece of wood, minus the leftover wood he has. In other words, 5x represents the total length of the five pieces of wood cut, and 0.7 represents the amount of wood left over after the cutting.

C: 0.76 feet

Step-by-step explanation:

Part A:

Let x be the length of each of the five pieces of wood cut from the 4.5 feet long piece of wood. The equation that could be used to determine the length of each piece is:

5x + 0.7 = 4.5

Part B:

You know the equation is correct because it takes into account that Warren cut five pieces of wood that are equal in length from a 4.5 feet long piece of wood. The total length of the five pieces of wood should equal the length of the original piece of wood, minus the leftover wood he has. In other words, 5x represents the total length of the five pieces of wood cut, and 0.7 represents the amount of wood left over after the cutting.

Part C:

5x + 0.7 = 4.5

5x = 4.5 - 0.7

5x = 3.8

x = 3.8/5

x = 0.76

Therefore, each of the five pieces of wood that Warren cut is 0.76 feet long.

The limit of sec(y sec²y - tan²y - 1) as y approaches 1 is undefined. The function is not continuous at the point being approached.

Explanation: To evaluate the limit, we can first simplify the expression inside the secant function using the trigonometric identity:sec²θ - tan²θ = 1/cos²θ - sin²θ/cos²θ = (1 - sin²θ)/cos²θ = cos²θ / cos²θ = 1Substituting this expression back into the limit, we get:lim sec(y sec²y - tan²y - 1)y→1= lim sec(y - 1)y→1As y approaches 1, the argument of the secant function approaches 0, which means that the secant function approaches infinity. Therefore, the limit is undefined.Since the limit is not defined, the function is not continuous at the point being approached. A function is continuous at a point if and only if the limit of the function as x approaches that point exists and is equal to the value of the function at that point. In this case, since the limit does not exist, the function is not continuous at y = 1.

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To find the volume of the cylinder, we need to use the formula:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

If we have a cylinder with radius r and height h, and another cylinder with the same radius r but double the height (2h), the volume of the second cylinder is:

V' = πr^2(2h) = 2πr^2h

So, to answer the questions:

The volume of the cylinder is V = π(5 cm)^2(8 cm) = 100π cubic cm, which is approximately 314.16 cubic cm rounded to two decimal places.

The volume of the cylinder with the same radius and double the height is V' = 2π(5 cm)^2(8 cm) = 200π cubic cm, which is approximately 628.32 cubic cm rounded to two decimal place

The solution to the system of equations shown above is the ordered pairs [-2, -9] and [3, -4].

In order to graph the solution to the given system of equations on a coordinate plane, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection;

y = -x² + 2x - 1   ......equation 1.

2x - 2y = 14 ......equation 2.

Based on the graph shown in the image attached above, we can logically deduce that the solution to this system of equations is the point of intersection of the lines on the graph representing each of them, which is given by the ordered pairs (-2, -9) and (3, -4).

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The angle in each quadrant with a common reference angle with 306° are

To find angles in each quadrant with a common reference angle with 306°, we first need to determine the reference angle for 306°.

Reference angle is the acute angle between the terminal side of an angle and the x-axis. We can find the reference angle for any angle θ by subtracting the nearest multiple of 180° from θ and taking the absolute value of the result. In this case:

|306° - 180°| = 126°

So, the reference angle for 306° is 126°.

Now, we can find an angle in each quadrant with a common reference angle of 126°:

1st quadrant: The angle with a reference angle of 126° in the 1st quadrant is simply 126°.

2nd quadrant: To find the angle with a reference angle of 126° in the 2nd quadrant, we need to subtract the reference angle from 180° (since all angles in the 2nd quadrant are between 90° and 180°).

180° - 126° = 54°

So, an angle with a reference angle of 126° in the 2nd quadrant is 54°.

3rd quadrant: To find the angle with a reference angle of 126° in the 3rd quadrant, we need to subtract the reference angle from 180° and then add 180° (since all angles in the 3rd quadrant are between 180° and 270°).

180° + 126° = 306°

So, an angle with a reference angle of 126° in the 3rd quadrant is 306°.

4th quadrant: To find the angle with a reference angle of 126° in the 4th quadrant, we need to subtract the reference angle from 360° (since all angles in the 4th quadrant are between 270° and 360°).

360° - 126° = 234°

So, an angle with a reference angle of 126° in the 4th quadrant is 234°.

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If each bag of hot chips requires around 500 ml of soda to drink with, then 46,000 ml of soda would be needed to buy 20 bags of 4 bags of hot chips and 3 bottles of soda.

To determine how much soda is needed if you buy 20 bags of 4 bags of hot chips and 3 bottles of soda, we need to know the volume of each bottle of soda.

Assuming each bottle of soda contains 2 liters of soda, the total volume of soda needed can be calculated as follows:

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It will take 6 years for the enrollment at both schools to be equal.

To solve this problem, we need to set up an equation to represent the enrollment at both schools over time.

From the problem it can be deduced that

For Central Middle School:

Nichols Middle School

set these two equations equal to each other and solve for x:

800 + 75x = 890 + 60x

Simplifying the equation

15x = 90

Dividing both sides by 15, we get:

x = 6

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The future value of Payton's investment will be $1,048.29 when he cashes in the bond on the maturity date 15 years from now.

To calculate the future value of Payton's 15-year treasury bond, we can use the formula for compound interest:

FV = PV * (1 + r/n)^(n*t)

where FV is the future value, PV is the present value (or face amount), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time period in years.

In this case, the present value is $700, the interest rate is 2.5% or 0.025, the interest is compounded quarterly, so n = 4, and the time period is 15 years.

Plugging in the values, we get:

FV = $700 * (1 + 0.025/4)^(4*15)

FV = $700 * (1 + 0.00625)^60

FV = $700 * 1.49756

FV = $1,048.29

Therefore, the future value of Payton's investment will be $1,048.29 when he cashes in the bond on the maturity date 15 years from now.

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Answer:23.4

Step-by-step explanation: 18 divided by 130

The amount of the discount is $23.40.

The problem asks us to find the amount of the discount given a percentage discount on a known price. To do this, we use the formula for calculating a percentage of a number.

To calculate the discount amount, we need to find 18% of the original price, which is $130 per week.

We can start by calculating 18% of $130:

Discount = 0.18 * $130 = $23.40

Therefore, the amount of the discount is $23.40.

The directional derivative of the function f(x,y) in the direction of the vector v at the point (0,1/3) is:

Duf(0,1/3) = ∇f(0,1/3) · u = [0, 3e/2] · [-5/13, 12/13] = (3e/2)(12/13) ≈ 1.38

To find the directional derivative of the function f(x, y) = 3e^x sin(y) at the point (0, 1/3) in the direction of the vector v = (-5, 12), we first need to calculate the gradient of the function, which is a vector containing the partial derivatives with respect to x and y.

The partial derivative with respect to x:
∂f/∂x = 3eˣ sin(y)

At point (0, 1/3), ∂f/∂x = 3e⁰ sin(1/3) = 3 sin(1/3)

The partial derivative with respect to y:
∂f/∂y = 3eˣ cos(y)

we first need to find the gradient of f at that point:
∇f = [∂f/∂x, ∂f/∂y] = [3ex sin(y), 3ex cos(y)]

Evaluated at (0,1/3), we get:
∇f(0,1/3) = [0, 3e/2

At point (0, 1/3), ∂f/∂y = 3e⁰ cos(1/3) = 3 cos(1/3)
So the gradient vector is ∇f = (3 sin(1/3), 3 cos(1/3)).

Next, we need to normalize the direction vector v:
|v| = √((-5)² + (12)²) = 13

Normalized vector v: (-5/13, 12/13)

Finally, we calculate the directional derivative (D_vf) as the dot product of the gradient vector and the normalized direction vector:
D(vf)= ∇f • (-5/13, 12/13) = (3 sin(1/3) × (-5/13)) + (3 cos(1/3) × (12/13))
D(vf) = (-15/13) sin(1/3) + (36/13) cos(1/3)
That is the directional derivative of the function at the given point in the direction of the vector v

Duf(0,1/3) = ∇f(0,1/3) · u = [0, 3e/2] · [-5/13, 12/13] = (3e/2)(12/13) ≈ 1.38

Therefore, the directional derivative of f(x, y) in the direction of v at the point (0,1/3) is approximately 1.38.

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Answer:

x=85

y=27

;)

Step-by-step explanation:

x+y=112

y=x-58

add 58 to the other side

58+y=x

Subtract y

x-y=58

x+y=112

Now if we add these we get

2x=170

x=85

Then if we substitute 85 in x+y=112

85+y=112

112

-85

____

27

Check your Answer on

y=x-58

27=85-58

27=27

This is the Answer

Please DM me if I should reexplain THANK YOU!

Hope this helps!

The transformation of the triangle is reflection over the x-axis

Given data ,

Let the transformation be represented as A

Now , the triangle is given as XYZ

where the coordinates are X ( 6 , 1 ) , Y ( 2 , 4 ) and Z ( -2 , 0 )

Now , the coordinates of the transformed triangle is

X' ( 6 , -1 ) , Y' ( 3 , -4 ) and Z' ( -2 , 0 )

The reflection of point (x, y) across the x-axis is (x, -y)

Hence , the transformation is reflection over x-axis

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The recursive formula for an, the nth term of the sequence is a(n) = a(n - 1) * 2 where a(1) = -1

From the question, we have the following parameters that can be used in our computation:

-1, -3, -9, -3³ ...

The above definitions imply that we simply multiply 3 to the previous term to get the current term

Using the above as a guide,

So, we have the following representation

a(n) = a(n - 1) * 3

Where

a(1) = -1

Hence, the sequence is a(n) = a(n - 1) * 2 where a(1) = -1

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(a) The change in the cart's momentum is -200 kg m/s.

(b) The average force that the wall exerts on the cart is 1333.33 N.

(c) The impulse that acted on the cart is in the opposite direction to the cart's initial momentum.

(a) The change in momentum can be calculated as the final momentum minus the initial momentum. The initial momentum of the cart is zero since it was at rest, and the final momentum is calculated as (mass of cart) x (final velocity) = 100 kg x 2 m/s = 200 kg m/s. Therefore, the change in momentum is -200 kg m/s.

(b) The average force can be calculated using the impulse-momentum theorem, which states that the impulse acting on an object is equal to the change in its momentum. The impulse is calculated as (mass of cart) x (final velocity - initial velocity) = 100 kg x (2 m/s - 0 m/s) = 200 kg m/s.

The time taken for the cart to come to a stop is given as 0.15 s. Therefore, the average force exerted by the wall is 1333.33 N.

(c) The direction of the impulse is opposite to the initial momentum of the cart, which was in the direction of the hill. Since the cart was moving downhill, the impulse that acted on it was in the upward direction.

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The probability that a student took AP Chemistry given they did not get into their first-choice college is 0.0566.

The probability that a student took AP Chemistry given they did not get into their first-choice college is calculated using the formula below:

P(Not 1st | Chem) =0.1375

P(Chem) = 0.25

P(Not 1st) = P(Chem and Not 1st) + P(Phys and Not 1st) + P(ES and Not 1st) + P(Bio and Not 1st)

P(Not 1st)= 0.1375 + 0.1575 + 0.2400 + 0.0700

P(Not 1st)= 0.6050

Substituting the values in the formula above:

P(Chem | Not 1st) = 0.1375 * 0.25 / 0.6050

P(Chem | Not 1st) = 0.0566

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Mr. Penny will require a total of 20 square feet of wooden planks for all 18 students to construct their treasure boxes.

To determine the number of planks required, we need to calculate the total amount of wood needed for all 18 students' treasure boxes.

Each treasure box has two identical rectangular sides.

Each side is cut from a wooden plank that is 5/9 feet long and 1 foot wide.

Therefore, the area of each side is [tex](5/9) \times 1 = 5/9[/tex] square feet.

Since there are two identical sides for each treasure box, the total area of wood needed for one treasure box is [tex](5/9) \times 2 = 10/9[/tex] square feet.

To find the total wood needed for 18 students' treasure boxes, we multiply the area per treasure box by the number of treasure boxes:

Total wood needed [tex]= (10/9) \times 18 = 20[/tex] square feet.

So, Mr. Penny will require a total of 20 square feet of wooden planks for all 18 students to construct their treasure boxes.

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Question: What is the number of planks required for Mr. Penny's 18 students to each construct one treasure box if the rectangular sides of the treasure box will be cut from wooden planks that are 5/9 feet long and 1 foot wide?