add parentheses to make true.

We can use trigonometry to solve this problem. Let the length of the rope be x. Then, we can draw a right triangle where the height of the tree is the opposite side, the length of the rope is the hypotenuse, and the angle of elevation is 60 degrees. The adjacent side can be found using the trigonometric function cosine:

cos 60° = adjacent / hypotenuse

1/2 = adjacent / x

Multiplying both sides by x, we get:

x/2 = adjacent

To find the length of the hypotenuse, we can use the Pythagorean theorem:

hypotenuse^2 = adjacent^2 + opposite^2

Substituting in the values we know, we get:

x^2 = (x/2)^2 + (9√5)^2

Simplifying the right side, we get:

x^2 = x^2/4 + 405

Multiplying both sides by 4, we get:

4x^2 = x^2 + 1620

Subtracting x^2 from both sides, we get:

3x^2 = 1620

Dividing both sides by 3, we get:

x^2 = 540

Taking the square root of both sides, we get:

x = √540 = 6√60

Therefore, the length of the rope that Rachel needs to hang the piñata is 6√60 feet. Rounded to the nearest tenth, this is approximately 74.1 feet.

Answer:

  43.4 ft

Step-by-step explanation:

You want to know the length of rope needed to hang a piñata from the top of a tree that is 9√5 feet tall, if the rope makes an angle of 60° with the ground.

The hypotenuse of the triangle can be found by considering the sine of the 60° angle. The sine of the angle is given by ...

  Sin = Opposite/Hypotenuse

Then the hypotenuse of the triangle can be found from ...

  sin(60°) = tree height / hypotenuse

  hypotenuse = tree height/sin(60°) = 9√5/sin(60°)

We presume the length of rope Rachel wants is the total of the lengths from the anchor point to the tree top and back to the ground. That will be ...

  9√5 + 9√5/sin(60°) = 9√5·(1 +1/sin(60°)) ≈ 43.4 . . . . feet

Rachel needs about 43.4 feet of rope to hang the piñata.

In order to obtain a 96.5% confidence interval of width of at most 0.16 for the probability of flipping a head in a coin flipping experiment, the coin needs to be flipped 533 times.

How to calculate the number of times we have to flip a coin?

We can use the formula:

n = ((Z_α/2) / E)²

Where, n is the number of times we need to flip a coin,

Z_α/2 is the Z-score at the α/2 level of significance for the normal distribution, and E is the desired margin of error or the maximum allowable deviation from the population proportion.

In this problem, the margin of error is 0.16, and the confidence level is 96.5%.

The α value is 1 - 0.965 = 0.035.

We need to find the Z-score corresponding to α/2 = 0.0175 using a standard normal distribution table.

Z_0.0175 = -1.96n = ((-1.96) / 0.16)²n ≈ 533

Therefore, the coin needs to be flipped 533 times to obtain a 96.5% confidence interval of width of at most 0.16 for the probability of flipping a head.

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Terry and Ricky traveled 35% of the total distance on the first day of their canoe trip.

Terry and Ricky went on a 60-mile canoe trip with their class, and on the first day, they traveled 21 miles. To find the percent of the total distance they traveled on the first day, follow these steps:
Divide the distance traveled on the first day (21 miles) by the total distance of the trip (60 miles).
Multiply the result by 100 to convert it into a percentage.
Step-by-step explanation:
Divide 21 miles by 60 miles:
  21 / 60 = 0.35
Convert the result into a percentage by multiplying by 100:
  0.35 x 100 = 35%

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Answer:

Step-by-step explanation:

two circles: 2(πr²)

1 rectangle:Lw

3.14(7)(7)

3.14(49)

2(153.86)=307.72

307.72(14)

4308.08

Answer:

the GCF is 5y and the factored form is 5y(2y^4 + 6y^2 - 3).

Step-by-step explanation:

The greatest common factor (GCF) of the terms 10y^5, 30y^3, and -15y is 5y.

To factor out the GCF, we can divide each term by 5y:

10y^5 / (5y) = 2y^4

30y^3 / (5y) = 6y^2

-15y / (5y) = -3

So we have:

10y^5 + 30y^3 - 15y = 5y(2y^4 + 6y^2 - 3)

Therefore, the GCF is 5y and the factored form is 5y(2y^4 + 6y^2 - 3).

Answer: GCF=5

Factored form= 5y(2y^4+6y^2-3

Step-by-step explanation:

We can use a proportion to estimate the number of students who could be expected to bring their lunch:

9 out of 15 students brought their lunch from home.

Let x be the number of students in the school who bring their lunch from home.

So we can set up the proportion:

9/15 = x/500

We can solve for x by cross-multiplying:

9 * 500 = 15x

x = (9 * 500) / 15

x = 300

Therefore, we can expect that 300 out of 500 students in the school bring their lunch from home.

Let x be pounds of hamburger and y be pounds of hot dogs.

The equation is: 8x + 6y ≤ 75.

An equation is a mathematical statement that uses the equal sign (=) to show that two expressions are equal. Equations can include variables, which are symbols that represent unknown or changing values. Solving an equation means finding the value of the variable that makes the equation true. Equations are commonly used in many areas of mathematics, science, and engineering, as well as in everyday life.

Let x be the pounds of hamburger purchased and y be the pounds of hot dogs purchased.

The cost of hamburger at $8 per pound would be 8x dollars.

The cost of hot dogs at $6 per pound would be 6y dollars.

The total cost of the purchases should not exceed $75, so we can write:

8x + 6y ≤ 75

This is the equation that represents Breanna's situation, where x represents the pounds of hamburger purchased and y represents the pounds of hot dogs purchased, and the total cost of the purchases should not exceed $75.

Therefore, Let x be pounds of hamburger and y be pounds of hot dogs.

The equation is: 8x + 6y ≤ 75.

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The evaluation of the function gives:

f(-4) = 1, f(1) = 2, f(2) = 6, f(5) = 29/3

A function is an expression that shows the relationship between the independent variable and the dependent variable.  A function is usually denoted by letters such as f, g, etc.

The evaluation is as follow:

For f(-4):

since x = -4. This means x ≤ -4. Thus, we use the function |2x + 7|. That is:

f(-4) = |2*(-4) + 7|

f(-4) = |-8 + 7|

f(-4) = | -1 |

f(-4) = 1

For f(1):

x = 1. Thus, -4< x ≤ 1. Use 1 + x²

f(1) = 1 + x²

f(1) = 1 + 1²

f(1) = 2

For f(2):

x = 2. Thus, 1< x < 3. Use the value 6

f(2) = 6

For f(5):

x = 5. Thus, x ≥ 3. Use (1/3)x + 8.

f(5) = (1/3)*5 + 8

f(5) = 29/3

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Evaluating the function at various values of x over different intervals,

f(-4) = 1, f(1) = 2, f(2) = 6, f(5) = 29/3

To evaluate the function f(x) at various values of x, we need to use the definition of the function for each interval of x.

For x ≤ -4:

f(x) = |2x + 7|

f(-4) = |2(-4) + 7| = |-1| = 1

For -4 < x ≤ 1:

f(x) = 1 + x²

f(1) = 1 + 1² = 2

For 1 < x < 3:

f(x) = 6

f(2) = 6

For x ≥ 3:

f(x) = (1/3)x + 8

f(5) lies in the interval x ≥ 3, so we use the definition of f(x) for this interval:

f(5) = (1/3)(5) + 8 = 29/3

Therefore,

f(-4) = 1

f(1) = 2

f(2) = 6

f(5) = 29/3

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In the scenario where a nutritionist selected a random sample of adults and asked them about their eating and exercise habits, the data show that people who eat organic fruits and vegetables are more likely to exercise regularly than those who do not eat organic fruits and vegetables, this scenario describes an observational study.

What is an observational study?Observational study, also known as non-experimental research, is a type of research design where researchers observe the behavior of a particular group of individuals without influencing it in any way. A researcher is not allowed to interfere with or manipulate any variables.

He just observes the environment and the behavior of the individuals under study. This is why observational studies are sometimes also referred to as naturalistic observations.

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The required solution of the linear programming problem for the given objective function and subject to constraints are,

Linear programming problem is Maximize 15x1 + 2x2

Subject to:

7x1 + x2 < 23

3x1 - x2 < 5

x1, x2 > 0

Objective function value for rounding up fraction 1/2 solution is 53

Objective function value for rounding up all fraction solution is 23.

Optimal objective function value 53 is lower than optimal value 95.5.

Optimal objective function value is always less than or equal to the LP's optimal objective function value as ILP problem is a more constrained version.

To solve the problem as an LP,

we can ignore the integer constraints

And solve the problem as a continuous linear program.

The problem can be written as,

Maximize 15x1 + 2x2

Subject to:

7x1 + x2 < 23

3x1 - x2 < 5

x1, x2 > 0

Rounding up fractions greater than or equal to 1/2,

The following feasible solution is,

x1 = 3, x2 = 4

The objective function value for this solution is 53.

However, this is not the optimal integer solution since both x1 and x2 are not integers.

Rounding down all fractions, we get the following feasible solution,

x1 = 1, x2 = 4

The objective function value for this solution is 23, which is less than the LP's optimal objective function value of 95.5.

This is not the optimal integer solution either.

Optimal objective function value for the ILP is lower than that for the optimal LP, solve the ILP problem.

In any one constraints

When x1 = 0 ⇒ x2 = 23

x2 = 0 ⇒ x1 = 3.3

Optimal value is ,

15(3.3) + 2(23)

= 49.5 + 46

= 95.5

Optimal objective function value is lower than optimal value.

The optimal objective function value for the ILP problem is always less than or equal to the corresponding LP's optimal objective function value .

Because the ILP problem is a more constrained version of the linear programming problem.

The ILP problem restricts the variables to be integers, which reduces the feasible region and makes the problem more difficult to solve.

The optimal objective function values for the LP and ILP problems are equal.

If the LP problem has an optimal solution that satisfies the integer constraints.

In general, the optimal objective function value of the MILP problem can be better or worse than that of the LP or ILP problem.

It depends on the specific problem instance.

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The above question is incomplete, the complete question is :

Given the following all-integer linear program:

Max 15x1 + 2x2

s. t.

7x1 + x2 < 23

3x1 - x2 < 5

x1, x2 > 0 and integer

a. solve the problem as an lp, ignoring the integer constraints.

b. what solution is obtained by rounding up fractions greater than or equal to 1/2? is this the optimal integer solution?

c. what solution is obtained by rounding down all fractions? is this the optimal integer solution? explain.

d. show that the optimal objective function value for the ilp (integer linear programming) is lower than that for the optimal lp.

e. why is the optimal objective function value for the ilp problem always less than or equal to the corresponding lp's optimal objective function value? when would they be equal? comment on the optimal objective function of the milp (mixed-integer linear programming) compared to the corresponding lp and ilp.

The tour guide should row towards the point on the coast closest to point Q, which is the point of perpendicular intersection between the coast and the line passing through points Q and the initial position of the tour guide.

To reach point Q in the least time, the tour guide should row towards the point on the coast that is closest to point Q. Let's call this point P.

We can use the Pythagorean theorem to find the distance from point P to point Q

distance PQ = sqrt((3 miles)^2 + (1 mile)^2) = sqrt(10) miles

Now, let's consider two scenarios

Scenario 1: The tour guide rows directly towards point Q.

The distance the tour guide needs to row is 2 miles + sqrt(10) miles. This will take the tour guide

time = (2 miles + sqrt(10) miles) / (3 miles/hour) = (2/3) + (1/3)*sqrt(10) hours

Scenario 2: The tour guide rows towards point P, then walks the rest of the way to point Q.

The distance the tour guide needs to row is 2 miles - x miles, where x is the distance from point P to the nearest point on the coast. By similar triangles, we know that

x / 3 miles = 1 mile / sqrt(10) miles

Solving for x, we get

x = 3/sqrt(10) miles

So the distance the tour guide needs to row is

2 miles - 3/sqrt(10) miles = 2 - (3/sqrt(10)) miles

The distance the tour guide needs to walk is

sqrt(10) miles - 1 mile = sqrt(10) - 1 miles

The time it takes the tour guide to row and walk is:

time = (2 - (3/sqrt(10))) / (3 miles/hour) + (sqrt(10) - 1) / (3 miles/hour)

Simplifying this expression, we get

time = (2/3) - (1/3)*sqrt(10) + (sqrt(10)/3) - (1/3) = (1/3) + (1/3)*sqrt(10) hours

Comparing the two scenarios, we see that scenario 2 takes less time, so the tour guide should row towards point P in order to reach point Q in the least time.

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Using the scatter plot, we can see that the scatter plot has a cluster, and the variables are related. As x increases, y increases.

The scatter plot is reportedly one of the statistical graphs' most useful discoveries. John Frederick W. Herschel, an English scientist, initially proposed the scatter plot in 1833. Herschel used it to look at the orbits of the double stars. He plotted the direction of the double star with relation to the measurement year. The scatter plot was used to interpret the key link between the two measurements. The scatter plot continues to rule in both business and science, despite the widespread use of bar charts and line plots. The relationship between the points on a scale can be easily understood by merely glancing at them.

Here in the graph shown,

Two clusters can be seen in the scatterplot, and their variables are related. Y increases, but the growth is gradual, as x increases in the bottom one. As x rises in the top cluster, y rises as well, but more quickly.

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The relationship between variables, and any patterns or trends in the scatterplot, researchers and analysts can gain a deeper understanding of the data they are working with.

A scatterplot is a graphical representation of a set of data points, where each point represents an observation or measurement of two variables. Scatterplots are commonly used to visualize the relationship between two variables. The scatterplot can reveal whether there is a correlation or relationship between the two variables.

When examining a scatterplot, there are a few key things to look for. One important aspect is the presence of a cluster. If there is a cluster of points in the scatterplot, it may indicate that there is a group of observations that share similar characteristics.

Additionally, it is important to determine whether the variables on the scatterplot are related or not. If the variables are not related, there will be no discernible pattern in the scatterplot. However, if the variables are related, there will be a pattern or trend visible in the scatterplot.

In cases where there is a positive correlation, as x increases, y increases as well. This is seen as a positive slope or trend line in the scatterplot. It is important to note that correlation does not necessarily imply causation, and further analysis may be required to understand the underlying factors influencing the relationship between the two variables.

Overall, scatterplots are a powerful tool for visualizing and analyzing data, and can provide valuable insights into the relationship between two variables. By understanding the presence of clusters,

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Answer: 1. -8, 2. 12, 3.-4 , 4.-7 , 5.3 , 6.15 , 7. 1, 8. 12, 9. -15, 10.318 ,

Step-by-step explanation:

A(n) asymptote is a line that a graphed curve gets closer to but never touches as the value of a variable gets extremely small or extremely large. Asymptotes can be horizontal, vertical, or slanted, and they are often used to describe the behavior of functions or graphs as they approach certain limits or values.

Answer:

A(n) asymptote is a line that a graphed curve gets closer to but never touches as the value of a variable gets extremely small or extremely large. Asymptotes can be horizontal, vertical, or slanted, and they are often used to describe the behavior of functions or graphs as they approach certain limits or values.

The equatiοn |x - 5| = 0 means the distance between x and 5 is zerο, which implies that x = 5. And since x is less than οr equal tο 5, the sοlutiοn set is x = {5}.

An absοlute value equatiοn is an equatiοn that cοntains an absοlute value expressiοn. An absοlute value expressiοn is the distance οf a number frοm zerο οn a number line, and it is represented by twο vertical bars enclοsing the number, like |x|.

The equatiοn in the fοrm οf |x - b| = c that has the sοlutiοn set x ≤ 5 is:

|x - 5| = 0

Tο see why, cοnsider that |x - 5| is the distance between x and 5 οn the number line.

Hence, the equatiοn |x - 5| = 0 means the distance between x and 5 is zerο, which implies that x = 5. And since x is less than οr equal tο 5, the sοlutiοn set is x = {5}.

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For x-intercept, [tex]y=0[/tex]

[tex]\implies b-x=0[/tex]

[tex]\implies b=6[/tex]

For vertical intercept [tex]y\rightarrow \pm \infty[/tex]

[tex]\implies cx+d=0[/tex]

[tex]\implies-3x+d=0[/tex]

[tex]\implies d=3c[/tex]

Now, [tex]y=\dfrac{6-x}{cx+3c}[/tex]

[tex]\implies x(xy+1)=6-3cy[/tex]

[tex]\implies x=\dfrac{6-3xy}{cy+1}[/tex]

At horizontal asymptote, [tex]x\rightarrow\pm\infty\implies cy+1=0[/tex]

[tex]\implies c(-1)+1=0[/tex]

[tex]\implies c=1[/tex]

[tex]\implies d=3[/tex].

For the given linear equation, the solution is (3,3) i.e. B.

What is a linear equation, exactly?

A linear equation is a mathematical equation in which the highest power of the variable(s) is one. In other words, it is an equation of the form:

y = mx + b

where y and x are variables, m is the slope or gradient of the line, and b is the y-intercept (the point at which the line intersects the y-axis). The slope represents the rate of change of the line, or how steep it is, and the y-intercept represents the value of y when x is zero.

Now,

We can solve this problem by plugging in the x and y values of each ordered pair and seeing if they make the equation true.

Let's start with option A, which is (2,4):

-3x+5y = 2x+3y (original equation)

-3(2) + 5(4) = 2(2) + 3(4) (substituting x=2 and y=4)

-6 + 20 = 4 + 12

14 = 16

The left side of the equation is not equal to the right side, so (2,4) is not a solution.

Now let's move on to option B, which is (3,3):

-3x+5y = 2x+3y (original equation)

-3(3) + 5(3) = 2(3) + 3(3) (substituting x=3 and y=3)

-9 + 15 = 6

6 = 6

The left side of the equation is equal to the right side, so (3,3) is a solution.

Therefore, the answer is option B, which is "Only (3,3)".

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Answer:

Step-by-step explanation:

If you substitute x = 7, then the reasoning is correct, but if you think that x = 7 should turn out at the end (answer), then this is not correct, because it turns out [tex]\frac{19}{3}[/tex]

The probability of getting P(sum > 9) when a cube is rolled twice is 6/36.

Mathematical study of random occurrences and their results is the subject of probability theory. It entails analysing and quantifying risk, chance, and uncertainty. For modelling and studying complex systems and events in disciplines like physics, engineering, economics, and finance, probability theory offers a framework. Concepts like random variables, probability distributions, expected values, variance, and covariance are all part of the theory. In many aspects of daily life, including forecasting weather patterns, data analysis for scientific study, and calculating the likelihood of certain outcomes in games of chance, probability theory is applied.

The total number of outcomes for a 6 sided cube rolled twice is 6 x 6 = 36.

For a sum greater than 9 the possibilities are:

(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) = 6 possibilities.

Thus,

P(sum > 9) = 6 / 36.

Hence, the probability of getting P(sum > 9) when a cube is rolled twice is 6/36.

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Answer:

-720

Step-by-step explanation:

Simplify the two negatives with positive:

120 * 6 * -1

Multiply 6 and -1 with -6:

120 * -6

= -720

Lilly needs 686 dark power crystals to raise 6174 zombies.

To solve this problem, we need to determine how many dark power crystals are needed to raise a specific number of zombies. We know that each crystal can raise 9 zombies. So, to determine the number of crystals needed to raise a given number of zombies, we simply divide that number by 9.

In this case, we want to know how many crystals Lilly needs to raise 6174 zombies. So we divide 6174 by 9

= 6174 / 9

Divide the numbers

= 686

This means that Lilly needs 686 dark power crystals to raise 6174 zombie

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The given question is incomplete, the complete question is:

Lilly is using dark power crystals to raise an army of zombies.each crystal can raise 9 zombies. how many crystal does Lily need to raise 6174 zombies​

An equation that shoes the total amount paid y if they played x number of games is y = 2x + 5

In mathematics, an equation is-a-statement that shows the equality between two-expressions. An equation typically has one or more unknowns that need to be solved for in order to satisfy the equality.

Here, y = total amount paid

x = number of games

Each game costs $2

So total cost of all games played = $2x

But to enter the game they need to rent shoes $5.

So, total amount paid = $5 + $2x

Thus, An equation that shoes the total amount paid y if they played x number of games is y = 2x + 5

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answer = (3x - 5) (3x + 3).

simplified = 3(x + 1) (3x - 5).

Answer:

72 i think

Step-by-step explanation:

i think your supposed to multiply 9 and 8 because count the sides on the shaded part ir

ts 8

a) Yes. (b)           G        Y        R

   G  [0.60     0.00     0.40]

   Y  [1.00     0.00     0.00]

   R  [0.30     0.20     0.50]

(c) P(G, G, G, G) = 0.216. P(G, G, G, Y)= 0.144. P(G, G, G, R) = 0.120. P(G, G, Y, R) = 0.000. P(G, Y, R, R) = 0.000. P(Y, R, R, R) = 0.000

(a) Yes, this situation represents a Markov chain because the probabilities of moving to the next state (traffic light) depend only on the current state (traffic light) and not on any previous states.

(b) The transition probability matrix is:

         G        Y        R

   G  [0.60     0.00     0.40]

   Y  [1.00     0.00     0.00]

   R  [0.30     0.20     0.50]

where G represents green, Y represents yellow, and R represents red. The transition diagram can be drawn with arrows between states labeled with the transition probabilities.

(c) There are two classes: {G, Y} and {R}, where {G, Y} is a transient class and {R} is an absorbing class.

(d) Starting with the first light being green, the probabilities after three more lights are:

P(G, G, G, G) = 0.60 * 0.60 * 0.60 = 0.216

P(G, G, G, Y) = 0.60 * 0.60 * 0.40 = 0.144

P(G, G, G, R) = 0.60 * 0.40 * 0.50 = 0.120

P(G, G, Y, R) = 0.60 * 0.00 * 0.50 = 0.000

P(G, Y, R, R) = 0.00 * 0.50 * 0.30 = 0.000

P(Y, R, R, R) = 0.00 * 0.50 * 0.50 = 0.000

The interpretation of these probabilities is, for example, that the probability of passing through four consecutive green lights is 0.216, and the probability of passing through three consecutive green lights followed by a yellow light is 0.144.

(e) Let p_G, p_Y, and p_R be the proportions of green, yellow, and red lights, respectively. Then we have the system of equations:

0.60 p_G + 0.00 p_Y + 0.40 p_R = p_G

1.00 p_Y + 0.00 p_Y + 0.00 p_R = p_Y

0.30 p_G + 0.20 p_Y + 0.50 p_R = p_R

p_G + p_Y + p_R = 1

Solving this system of equations, we get:

p_G = 0.44

p_Y = 0.14

p_R = 0.42

Therefore, approximately 44% of the street lights are green, 14% are yellow, and 42% are red.

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On solving the provided question we can say that Therefore, the percent decrease in video games sales in consoles is approximately 10.43%.

Percentage is a way of expressing a number as a fraction of 100. It is often denoted using the percent symbol (%). For example, if a student scores 80 out of 100 in a test, the percentage score is calculated as (80/100) x 100, which is equal to 80%. This means the student scored 80% of the total marks available in the test. Percentages are commonly used to express rates of change, proportions, and percentages of a whole, and they are used in many fields, such as finance, science, and statistics.

$1,150 million - $1,030 million = $120 million

$120 million ÷ $1,150 million = 0.1043

0.1043 x 100% = 10.43%

Therefore, the percent decrease in video games sales in consoles is approximately 10.43%.

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Answer: I'm not sure what birdhouse you're talking about, but I'll tell you the average amount of wood that would be needed to build a birdhouse.

Many cavity-nesting birds will add their own nest material, but the woodpeckers, waterfowl and owls prefer nest boxes with 2-3 inches of dry sawdust or woodchips in the bottom.

Answer:no

Step-by-step explanation:

for example Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 34