3.25 lab: leap year a year in the modern gregorian calendar consists of 365 days. in reality, the earth takes longer to rotate around the sun. to account for the difference in time, every 4 years, a leap year takes place. a leap year is when a year has 366 days: an extra day, february 29th. the requirements for a given year to be a leap year are:

1) To determine if the triangle is acute, obtuse, or right, we need to check if the sum of the squares of the two shorter sides is greater than, equal to, or less than the square of the longest side.

Let's first arrange the side lengths in ascending order:

28 in. < 34 in. < 42 in.

Now we can apply the Pythagorean theorem:

28² + 34² = 196 + 1156 = 1352

42² = 1764

Since 1352 is less than 1764, the sum of the squares of the two shorter sides is less than the square of the longest side. Therefore, the triangle is obtuse.


2)

We can use the tangent function to find the length of BC.

tan(B) = opposite/adjacent

We know that the opposite side of angle B is BC, and the adjacent side is AC, which is the height of the triangle. Since angle B is 45 degrees, we have:

tan(45) = BC/16

Simplifying the left side using the fact that tan(45) = 1, we get:

1 = BC/16

Multiplying both sides by 16, we get:

BC = 16

Therefore, the length of BC is 16 ft.


3)

Since the hypotenuse of each triangular quilt piece is 18 inches, we can use the Pythagorean theorem to find the length of each side. Let's call one leg of the triangle x (since the two legs will be equal), and the hypotenuse is 18. Then:

x² + x² = 18²

Simplifying:

2x² = 324

Dividing by 2:

x² = 162

Taking the square root:

x = √(162) = sqrt(2 * 3⁴)

We can simplify this by breaking it down into factors:

x = 3 * √(2) * 3

x = 9√(2)

Therefore, the length of each side of the triangular quilt piece is 9√(2) inches or 12.73 inches


4)

We can use the inverse sine function to find the angle that has a sine of 9/16. Using a calculator, we get:

sin^(-1)(9/16) ≈ 35.54°

Therefore, the missing value is approximately 35.54°

5)
We can use the inverse cosine function to find the angle that has a cosine of 7/18. Using a calculator, we get:

cos^(-1)(7/18) ≈ 61.48°

Therefore, the missing value is approximately 61.48°


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The closest distance that a cinema can put seats to the screen is approximately 3.2 meters.

the trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.

We can use trigonometry to find the closest distance that a cinema can put seats to the screen.

Let "d" be the closest distance between the screen and the seats. Then, we can form a right triangle with the distance "d" as the adjacent side, the height of the screen (7.3 m) as the opposite side, and the height of the customer's eyes above the floor (1.2 m) plus the height of the seat above the floor (which we can assume is 0.5 m) as the hypotenuse.

Using the tangent function, we can write:

tan(31°) = (7.3)/(d + 1.7)

Solving for "d", we get:

d = (7.3)/(tan(31°)) - 1.7 ≈ 3.2 m

Therefore, the closest distance that a cinema can put seats to the screen is approximately 3.2 meters.

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Answer:

B, $305.11

Step-by-step explanation:

since it is in annual interest, 36 months = 3 years.

use the formula for simple interest, which is x = Prt

278 * 0.0325 * 3 = 27.105

that is the interest that you gained.

then, add the original amount that you had, which was 278.

27.105+278 = 305.105

this rounds to $305.11, which is answer choice B

Answer: x^2 - 9

Step-by-step explanation:

Answer:

681.25 ft2.

Step-by-step explanation:

Answer:

Step-by-step explanation: answer is option b

Answer:

  9.  (x +3)² +4x = x² +10x +9

  10.  (5 +3m)/15

  11.  x = (-1±i√14)/3

Step-by-step explanation:

You want help with practice test questions involving quadratics and rational numbers.

For this, it is perhaps easiest to show equivalence by multiplying out the second form of the quadratic:

  (x +3)² +4x

  = (x +3)(x +3) +4x

  = x(x +3) +3(x +3) +4x

  = x² +3x +3x +9 +4x

  = x² +(3 +3 +4)x +9

  = x² +10x +9 . . . . . . . . equivalent form

As you know, fractions can be added by adding their numerators when they have a common denominator. Here, the denominators of 3 and 5 have no common factors, so the least common denominator is their product, 3·5 = 15. Writing equivalent fractions with denominators of 15, we have ...

  [tex]\dfrac{1}{3}+\dfrac{m}{5}=\dfrac{5}{5}\cdot\dfrac{1}{3}+\dfrac{3}{3}\cdot\dfrac{m}{5}\\\\=\dfrac{5}{15}+\dfrac{3m}{15}=\boxed{\dfrac{5+3m}{15}}[/tex]

It can be worthwhile to remember that you can always add fractions with the formula ...

  [tex]\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}[/tex]

The solution to 3x² +2x +5 = 0 is found using the quadratic formula with ...

  [tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-2\pm\sqrt{2^2-4\cdot3\cdot5}}{2\cdot3}\\\\=\dfrac{-2\pm\sqrt{-56}}{6}=\dfrac{-2\pm2i\sqrt{14}}{6}\\\\\boxed{x=\dfrac{-1\pm i\sqrt{14}}{3}}[/tex]

__

Additional comment

It can be useful to remember ...

  (a ±b)² = a² ±2ab +b²

where the sign of the middle term matches the sign in the binomial. This is also used for "completing the square" where the b² term is the square of the middle term after it is divided by 2a.

Answer:

Step-by-step explanation:

Finally got it right. lol

Zia weighs 4 ounces more than Pia .First, we need to convert Pia's weight from mixed numbers to an improper fraction and then convert the result to ounces to make it comparable to Zia's weight.

Pia's weight = 1 3/4 pounds

= (4*1 + 3)/4 pounds

= 7/4 pounds

= (7/4) * 16 ounces

= 28 ounces

Now we can compare Zia's weight and Pia's weight.

Zia's weight = 32 ounces

Since 32 ounces is greater than 28 ounces, Zia weighs more than Pia.

To find out how much more Zia weighs than Pia, we can subtract Pia's weight from Zia's weight.

Zia's weight - Pia's weight = 32 ounces - 28 ounces = 4 ounces

Therefore, Zia weighs 4 ounces more than Pia.

In summary, we first converted Pia's weight from pounds to ounces by multiplying it by 16. Then we compared the weights of Pia and Zia in ounces and found that Zia weighs more than Pia. Finally, we found the difference between their weights by subtracting Pia's weight from Zia's weight, which showed that Zia weighs 4 ounces more than Pia.

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Answer:

t ≤ 9

Step-by-step explanation:

After answering the provided question, we can conclude that As a result,  equation each chair costs $1.50 to rent, and each table costs $6.25 to rent.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an analytical solution (=). For example, the argument "2x + 3 = 9" asserts that the quote "2x + 3" equals the value "9". The goal of equation solving is to determine the value or moral standards of the variable(s) that will allow the equation to be true. Equations can be simple or complex, regular or multidimensional, and include one or more factors. In the equation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are employed in numerous different areas of mathematics, such as algebra, calculus, and geometry.

8c + 4t = 37 (equation 1)

3c + 2t = 17 (equation 2)

[tex]3c = 17 - 2t\\c = (17 - 2t) / 3\\8c + 4t = 37\\8[(17 - 2t) / 3] + 4t = 37\\(136 - 16t) / 3 + 4t = 37\\[/tex]

[tex]136 - 16t + 12t = 111\\-4t = -25\\t = 6.25\\3c + 2t = 17\\3c + 2(6.25) = 17\\[/tex]

[tex]3c + 12.5 = 17\\3c = 4.5\\c = 1.5\\[/tex]

As a result, each chair costs $1.50 to rent, and each table costs $6.25 to rent.

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Using percentage,

18. 78/100 of earth's atmosphere is made up of nitrogen.

19. 0.21 part of earth's atmosphere is made up of oxygen.

20. 100% of earth's atmosphere is here represented by the graph.

21. 1/100 of earth's atmosphere is made up of other gases.

The denominator of a percentage (also known as a ratio or fraction) is always 100. Sam, for instance, would have received 30 out of a possible 100 points if he had received 30% on his maths test. In ratio form, it is expressed as 30:100 and in fraction form as 30/100. In this case, the percentage symbol "%" is read as "percent" or "percentage."

This percent symbol can always be changed to a fraction or decimal equivalent by substituting "divided by 100."

Here in the graph as we can see,

Nitrogen = 78%

Oxygen = 21%

Other gases = 1%

Total = 100%

Fraction of nitrogen gas means, 78% in fraction = 78/100

Now, decimal of oxygen gas means, 21% in decimal = 0.21

The total percentage represented by graph = 100%

Other gases make up for 1/100 fraction of the total earth's atmosphere.

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An equation that shows an example of the associative property of addition is: a. (- 4 + i) + 4i = -4 + (i + 4i)

Associativity, which applies only to addition and multiplication, states that any sum or product of 3 or more numbers is unaffected by how the numbers are grouped in parentheses. In other words, if the same numbers are grouped differently for addition and multiplication, their results will always be the same.

The associative property of the addition states that the addition of numbers cannot affect by the grouping of the number.

A+(B+C) = (A+B) +C

Here, in the above equation the value of A+(B+C)  is always equal to the value of (A+B)+C whether the grouping of number is changes or not.

That is:

The expression that complies with this property is given by:

(-4+i) + 4i = -4 + (i+ 4i)

Complete Question:

Which equation shows an example of the associative property of addition?

a.(–4 + i) + 4i = –4 + (i + 4i)

b.(–4 + i) + 4i = 4i + (–4i + i)

c.4i × (–4i + i) = (4i – 4i) + (4i × i)

d.(–4i + i) + 0 = (–4i + i)

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Answer:

I think the answer is 4 tell me if it wasn't

The percentage of students scoring 84 or more in the exam is 99.18%.

Given, Mean test score = 72,

Standard deviation = 5.

We are supposed to find the percentage of students scoring 84 or more in the exam.

To find the percentage of students scoring 84 or more in the exam, we will use the following steps:

First, we need to find the z-score associated with 84.

Let us assume that z is the z-score corresponding to the value 84, then;

z = (84 - 72) / 5 = 2.4

Now, we have the value of z, we can find the percentage of students scoring 84 or more in the exam using the normal distribution table.

The percentage is the area under the normal distribution curve to the right of the z-score.

To find the area using the normal distribution table, we need to look for the value 2.4 in the z-table. Since 2.4 is not exactly listed in the z-table, we will use the value for 2.4 closest to it.

Using the z-table, the value closest to 2.4 is 0.9918.

Therefore, the percentage of students scoring 84 or more in the exam is;

P(Z > 2.4) = 0.9918 × 100 = 99.18%.

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Using the surface area of the rectangular prism we know that the dimensions of the base are 2in x 2in.

A three-dimensional solid form with six faces, including rectangular bases, is called a rectangular prism.

A rectangular prism also refers to a cuboid.

A cuboid and a rectangular prism have the same cross-section.

The surface area of the rectangular prism: A=2(wl+hl+hw)

Insert values: A=2(wl+l+hw)

232=2(wl+28l+28w)

232 = 2wl + 56l + 56w

We know that w and l are equal as: w = l as the base is square:

So, we can write:

232 = 2wl + 56l + 56w

232 = 2w² + 56w + 56w

232 = 2w² + 112w

2w² + 112w - 232 = 0

2w² + w(116 - 4) -232

2w² + 116w - 4w - 232

2w(w+58) -4(w+58)

Then, we have:

w+58 = 0 ⇒ w = -58

2w-4 = 0 ⇒ w = 2

Since the base measurements cannot be negative.

Then, the dimensions of the base are: 2in x 2 in

Therefore, using the surface area of the rectangular prism we know that the dimensions of the base are 2in x 2in.

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Answer:

  173.82 cm²

Step-by-step explanation:

You want the area of a regular octagon, given its side length is 6 cm.

A regular n-gon can be divided into n congruent isosceles triangles, each with its a.pex at the center of the polygon. The a.pex angle will have a measure of 360°/n.

The central angle of one sector of a regular octagon is 360°/8 = 45°. That means the triangle interior angle at A or B will be ...

  ∠OAM = (180° -45°)/2

  ∠OAM = 67.5°

The apothem of a regular polygon is the distance from its center to the midpoint of one side. Here, it is the length of segment OM.

We know ∠OAM is 67.5°. The side OM of triangle OAM is related to the side AM by the tangent function:

  Tan = Opposite/Adjacent

  tan(67.5°) = OM/AM

Since M is the midpoint of the 6 cm length AB, the measure of AM = 3 cm. That lets us find OM:

  OM = AM·tan(67.5°) = (3 cm)tan(67.5°)

  OM ≈ 7.2426 cm

The area of the octagon will be 8 times the area of a sector triangle. This, ...

  A = 8(1/2sa) . . . . . . . . . . where s is the side length, and a is the apothem

  A = 4(6 cm)(7.2426 cm)

  A ≈ 173.82 cm²

__

Additional comment

Effectively, we have found a formula for the area of any regular n-gon using only the side length.

  a = s/2·tan(90° -180°/n)

  A = ns/2·a = (1/2)ns·(1/2)s·tan(90° -180°/n)

  A = ns²/(4·tan(180°/n))

For this octagon, the formula gives ...

  A = 8·(6 cm)²/(4·tan(22.5°)) ≈ 173.82 cm²

The formula A = (n/2)sa is often written A = 1/2Pa, where P is the perimeter (=ns).

You may notice that many regular polygon area problems give inconsistent values for the side length and apothem. Specifying one is sufficient for finding the area. Specifying numerical values for both is trouble.

Answer:

Step-by-step explanation:

Advik spends $24.50 at the home goods store on a bird bath and $2.79 a bag for multiple mulch bags. His total before tax for the purchase is $66.35. 15 sacks of mulch were bought by Advik.

Let's start by first finding the total cost of the bird bath and the mulch bags.

The cost of the bird bath is $24.50.

Let's use "x" to represent the number of bags of mulch that Advik purchased. Each bag costs $2.79, so the total cost of the mulch bags is 2.79x.

The total cost of the purchase before tax is $66.35. So we can write an equation:

24.50 + 2.79x = 66.35

To solve for x, we can first subtract 24.50 from both sides:

2.79x = 41.85

Then we can divide both sides by 2.79:

x = 15

So Advik purchased 15 bags of mulch.

The complete question is:-

At a home goods store, Advik buys a bird bath for $24.50 and several bags of mulch that cost $2.79 each. The total for his purchase is $66.35 before tax.

Find how many bags he purchased.

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The comparison of variance that is true for the given average per month is "May > July".

Variance is a metric for gauging how dispersed a set of data is. The average of the squared discrepancies between each data point and the data set mean is used to compute it. Data points with a high variation are dispersed throughout a wider range of values, whereas those with a low variance are grouped more closely together. In statistics, variability is frequently used to examine data distribution and contrast the variability of various data sets. It is frequently stated in terms of the original data's square units, such square metres or square dollars.

From the given table the average per month is:

April: $6 million - $2.5 million = $3.5 million below average

May: $6 million - $2 million = $4 million below average

June: $6 million + $3.1 million = $9.1 million above average

July: $6 million - $1.6 million = $4.4 million below average

August: $6 million + $2.67 million = $8.67 million above average

Thus, comparing the numbers on the number line we observe that the average in May is greater than July.

Hence, the comparison that is true is "May > July".

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Answer:

Step-by-step explanation:

please Thanks me

The cost of constructing the path at the rate of Rs. 25 per m2 would be Rs. 38,465.

To find the cost of constructing the path, we need to determine the area of the path and then multiply it by the cost per square meter.

The total diameter of the pond and the path is 21m + 3.5m + 3.5m = 28m.

So the radius of the pond is half of the diameter, which is 21m/2 = 10.5m.

The radius of the pond with the path is (21m+3.5m)/2 = 12.25m.

Area of the path = Area of the outer circle - Area of the pond

[tex]= π(12.25)^2 - π(10.5)^2[/tex]

[tex]= π[(12.25)^2 - (10.5)^2][/tex]

= 3.14 x (24.5 + 10.5) x (24.5 - 10.5)

= 3.14 x 35 x 14

= 1538.6 m2 (rounded to one decimal place)

Therefore, the cost of constructing the path at the rate of Rs. 25 per m2 would be:

Cost = Area x Rate per m2

= 1538.6 x 25

= Rs. 38,465.

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The rate at which the diameter decreases when the diameter is 11 cm is 4 cm/min.

To solve this question, we need to use the formula for the surface area of a sphere, A = 4πr2. Since the surface area of the snowball is decreasing at a rate of 2 cm2/min, we can set up a differential equation to calculate the rate of change of the radius with respect to time:  dA/dt = 2 cm2/min = 8π dr/dt.

We can rearrange this equation to find dr/dt, the rate at which the radius of the snowball is changing:  dr/dt = 2 cm2/(8πmin).

We then use the equation for the diameter of a circle, d = 2r, and substitute the rate of change of the radius to get the rate of change of the diameter: dd/dt = 2(dr/dt) = 2 cm2/(4πmin).

Finally, we can plug in the given diameter, 11 cm, to find the rate of decrease in diameter:  dd/dt = 11 cm/(4πmin) = 4 cm/min. Therefore, the rate at which the diameter decreases when the diameter is 11 cm is 4 cm/min.

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Answer: 5.44 x 10 to the power of 8

Step-by-step explanation:

The Great Sphinx in Egypt is 66 ft tall, based on measurements of its shadow and a 6 ft-tall person's shadow at sunrise.

Let's denote the height of the Sphinx as "h".

At sunrise, the angle of elevation from the tip of the Sphinx to the top of the sun is 90 degrees. This means that the triangles formed by the Sphinx, its shadow, and the sun are all similar.

Using this fact, we can set up a proportion between the height of the Sphinx and the length of its shadow:

h/220 = 6/x

where "x" is the length of the shadow of the 6 ft-tall person.

Solving for "x", we get:

x = (220*6)/h

We also know that the length of the shadow of the person is 20 ft, so we can set up another proportion:

h/220 = 6/20

Solving for "h", we get:

h = (220*6)/20 = 66 ft

Therefore, the height of the Great Sphinx is 66 ft.

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The complete question is:

The Great Sphinx in Egypt's shadow extends 220 feet below the summit of the structure at sunrise. An someone who is 6 feet tall and standing close to the Sphinx has a shadow that extends 20 feet from the top of their head. The Great Sphinx's height is how tall?

a) 70.0 ft

b) 66.0 ft

c) 733.3 ft

d) 54.5ft

Answer:22,888

Step-by-step explanation: replace x with 6 and use calculator :)

The axiom AABC and ADEF shown in the figure are congruent by SAS axiom.

The SAS (Side-Angle-Side) axiom, also known as the SAS postulate or SAS congruence criterion, states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

In other words, if you have two triangles with the same angle between two sides of equal length, then those triangles are congruent.

This can be written mathematically as:

If triangle ABC is congruent to triangle DEF by SAS axiom, then:

AB = DE

AC = DF

angle BAC = angle EDF

angle ABC = angle DEF

angle ACB = angle DFE

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Answer:

c

Step-by-step explanation:

The rule for this relation could be S = n² + 1.

What is relation rule?

A relation rule is a description of how two or more quantities or variables are related to each other. A relation rule can take various forms, depending on the nature of the relation and the variables involved.

For example, a relation rule can be expressed as an equation, a formula, a table, a graph, or a verbal description. In each case, the relation rule specifies the conditions under which the variables are related, and how they vary in relation to each other.

Based on the given table:

n: 1, 0, 2

S: 1, 0, 3

We can observe that when n=1, S=1; when n=0, S=0; and when n=2, S=3.

There are different possible rules that can describe this relation, but one possible rule is:

S = n² + 1

Using this rule, we can verify that it matches the given table:

Therefore, the rule for this relation could be S = n² + 1.

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(a) The rates of changes of the volume are:

When r = 9 inches, dV/dt = 972π cubic inches per minute

When r = 36 inches, dV/dt = 15,552π cubic inches per minute

(b)  The rate of change of the volume relates to the radius through a quadratic function not a constant function.

a) We know that the formula for the volume of a sphere is V = (4/3)πr³. We can take the derivative with respect to time t to find the rate of change of volume.

dV/dt = 4πr² (dr/dt)

When r = 9 inches, we have:

dV/dt = 4π(9²)(3) = 972π cubic inches per minute

When r = 36 inches, we have:

dV/dt = 4π(36²)(3) = 15,552π cubic inches per minute

b) The rate of change of the volume of the sphere is not constant even though dr/dt is constant because the rate of change of the volume relates to the radius through a quadratic function not a constant function.

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The difference between the greatest and least number of books read is 6.

A graph is a visual representation of data that displays the relationship between two or more variables. Graphs are used in many fields, including mathematics, science, economics, engineering, and social sciences, to help researchers and practitioners analyze, interpret, and communicate data effectively.

Here,

To graph the data, we can use a bar graph or a line graph, where the x-axis represents the weeks and the y-axis represents the number of books read. Each bar or point on the graph represents the number of books read in a particular week. Using the data from the table, we can create the following line graph:

To find the difference between the greatest number of books read and the least number of books read, we need to identify the highest and lowest data points on the graph. From the graph, we can see that the highest data point is 10, which corresponds to Week 1, and the lowest data point is 4, which corresponds to Week 3. Therefore, the difference between the greatest and least number of books read is:

10 - 4 = 6

So the difference is 6 books.

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Complete question:

The table shows how many books students in a class read each week for two months. Graph the data.

Week           1              2                      3

Books       10               7                     9

Read          4               5                     6

What is the difference between the greatest numb books read and the least number of books read?

Step-by-step explanation:

mean is all the numbers added together, which is 35 and divide by the amount of numbers so 35 divide 5 is 7. Mean= 7

Medial is the number in the middle so 4 is the medial. Because it is in the middle of all the numbers.

Hope this helps.

The 2 pounds of $5.61 per pound birdseed are used in the mixture.10 - 2 = 8 pounds of $8.91 per pound birdseed are used in the mixture.

Answer:2 pounds of $5.61 per pound birdseed are used in the mixture.8 pounds of $8.91 per pound birdseed are used in the mixture.

Given information:Ten pounds of mixed birdseed sells for $8.25 per pound. The mixture is obtained from two kinds of birdseed, with one variety priced at $5.61 per pound and the other at $8.91 per pound.To find:Pounds of each variety of birdseed used in the mixture.Solution:Let 'x' be the number of pounds of the first kind birdseed used in the mixture.

Then, (10 - x) will be the number of pounds of the second kind birdseed used in the mixture.The price of the first kind of birdseed per pound is $5.61.Price of x pounds of the first kind of birdseed = 5.61x dollars.The price of the second kind of birdseed per pound is $8.91.Price of (10 - x) pounds of the second kind of birdseed = 8.91(10 - x) dollars.

The total cost of the mixture = $8.25 per pound × 10 pounds= $82.50 dollars.According to the question,$$5.61x+8.91(10-x)=82.5$$Multiply the numbers inside the brackets by multiplying with the negative sign.

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