A tree that is 8 tall is growing at a rate of 1 foot each year

A tree that is 10 feet tall is growing at a rate of 1/2 foot each year

How many years will it take the two trees to reach the same height?
Justify your response using mathematics.

Answer:

Step-by-step explanation:

To maximize the volume of the box, we need to find the height that will maximize the volume of the box.

Let's start by finding an expression for the volume of the box. The box has dimensions of 14-2x by 10-2x by x, where x is the height of the box. The volume of the box is:

V(x) = (14-2x)(10-2x)(x)

Expanding this expression, we get:

V(x) = 4x^3 - 48x^2 + 140x

To find the value of x that maximizes this expression, we can take the derivative of V(x) with respect to x and set it equal to zero:

V'(x) = 12x^2 - 96x + 140 = 0

We can solve this quadratic equation using the quadratic formula:

x = [96 ± sqrt(96^2 - 4(12)(140))]/(2(12)) = [96 ± 16sqrt(6)]/24

We can simplify this to:

x = 4 ± sqrt(6)/3

Since the dimensions of the box must be positive, we can discard the negative solution:

x = 4 + sqrt(6)/3

So the height of the box that will give a maximum volume is approximately 5.61 inches (rounded to two decimal places).

Therefore, the width of the rectangle is 10.

So, the answer is (O) 10.

To find the width of a rectangle, you will need to have at least one of the following pieces of information:

Length and area: If you know the length and area of the rectangle, you can use the formula for area (A = length x width) and solve for the width (W = A/length).

Perimeter and length: If you know the perimeter and length of the rectangle, you can use the formula for perimeter (P = 2 x length + 2 x width) and solve for the width (W = (P - 2 x length)/2).

Diagonal and length: If you know the diagonal and length of the rectangle, you can use the Pythagorean theorem ([tex]a^2 + b^2 = c^2[/tex]) to solve for the width. [tex](W = \sqrt(c^2 - length^2)),[/tex] where c is the diagonal and a and b are the length and width (in some order).

Ratio of length to width: If you know the ratio of length to width, you can use this ratio to solve for the width. For example, if the ratio of length to width is 3:2, and the length is 15, then the width would be 10 (because 15/3 = 5 and 5 x 2 = 10).

Measurements: If you have physical access to the rectangle, you can measure the width directly using a ruler or tape measure.

Let's assume the length of the rectangle as "L" and the width as "W".

From the given information, we know that:

The area of the rectangle is 40, so we have the equation:

[tex]L x W = 40[/tex]

The width is 2 less than 3 times the length, so we have the equation:

[tex]W = 3L - 2[/tex]

Now we can substitute the second equation into the first equation and get:

[tex]L x (3L - 2) = 40[/tex]

Expanding the left-hand side, we have:

3L^2 - 2L - 40 = 0

Dividing both sides by 1, we get:

[tex]3L^2 - 2L - 40 = 0[/tex]

Using the quadratic formula, we get:

[tex]L = 4 or L = -10/3[/tex]

Since the length of a rectangle cannot be negative, we take L = 4.

Then, we can use the equation W = 3L - 2 to find the width:

[tex]W = 3 x 4 - 2 = 10[/tex]

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Answer:

V = 4,423.4

SA = 1,508.0

Step-by-step explanation:

r = d/2
V = πr²h

(3.14)(8)²(22) = 4,423.362456

SA = 2πr(h + r)

2(3.14)(8)(22 + 8) = 1,507.964474

Answer:

Rounding to the nearest hundred thousand, the volume of the cone is approximately 19,400,000 cubic meters.

For part A the length of the perpendicular is approximately 156.18. Length of GH, the area of the rectangle is approximately 12336.22 square units.

FOR Part B the length of the perpendicular is approximately 182.48.

the area of the triangle is approximately 8954.96 square units.

How to solve the question?

To find the perpendicular of a right triangle, we need to use the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Let the length of the perpendicular be h, and the length of the other leg of the triangle be b. Then, we can set up the equation:

175²= h²+ 79²

Simplifying, we get:

30,625 = h²+ 6,241

Subtracting 6,241 from both sides, we get:

24,384 = h²

Taking the square root of both sides, we get:

h ≈ 156.18

Therefore, the length of the perpendicular is approximately 156.18.

The area of a rectangle is given by the formula:

Area = length × width

Substituting the given values, we get:

Area = 156.18 × 79

Area = 12336.22 square units

Therefore, the area of the rectangle is approximately 12336.22 square units.

To find the perpendicular, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, we know the length of the hypotenuse and one of the other sides, so we can solve for the length of the perpendicular.

Using the Pythagorean theorem:

h²= b²+ p²

where h is the length of the hypotenuse, b is the length of the base, and p is the length of the perpendicular.

Substituting the given values:

207²= 98²+ p²

Simplifying:

42849 = 9604 + p²

Subtracting 9604 from both sides:

33245 = p²

Taking the square root of both sides (since p is a length, we only take the positive square root):

p = sqrt(33245)

p ≈ 182.48

Therefore, the length of the perpendicular is approximately 182.48.

To find the area of the triangle, we can use the formula:

Area = (1/2) x base x height

Substituting the given values:

Area = (1/2) x 98 x 182.48

Simplifying:

Area = 0.5 x 98 x 182.48

Area ≈ 8954.96

Therefore, the area of the triangle is approximately 8954.96 square units.

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The probability that at most 50 people out of 2400 who have not taken drugs will test positive on a drug test that is accurate 98% of the time is approximately 0.109

To calculate the probability, we need to first determine the parameters of the binomial distribution. Let's define success as testing negative on the drug test, since that is what we want to happen. Therefore, the probability of success is 0.98, and the probability of failure (testing positive) is 0.02.

Next, we need to determine the number of trials (n) and the number of successes (x) we are interested in. In this case, n = 2400 (the number of people taking the test) and x = 0, 1, 2, ..., 50 (the number of people who test positive).

Using the binomial distribution formula, we can calculate the probability of getting at most 50 people testing positive as follows:

P(X ≤ 50) = Σ(i=0 to 50) [(n choose i) * p^i * (1-p)^(n-i)]

where (n choose i) = n! / (i! * (n-i)!) is the binomial coefficient.

Plugging in the values, we get:

P(X ≤ 50) = Σ(i=0 to 50) [(2400 choose i) * 0.98^i * 0.02^(2400-i)]

Using a computer program or calculator, we can evaluate this sum to get P(X ≤ 50) ≈ 0.109.

Therefore, the probability that at most 50 people out of 2400 who have not taken drugs will test positive on a drug test that is accurate 98% of the time is approximately 0.109.

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Answer:12

Step-by-step explanation:

have a good day

Answer:

it is given that diagram shows part of the Wheel of Theodorus. graphics for exercise 20 page 428 geometry part (a). It is required to identify which ...

Step-by-step explanation:

J would be the correct answer choice.

To put it simply, the number itself will always stay the same. You will always end up with either a negative or positive version of the number.

It will take both machines approximately 1.71 hours or 1 hour and 42 minutes to produce widgets together.

To solve this problem, we can use the formula:

1 / time taken by machine 1 + 1 / time taken by machine 2 = 1 / time taken by both machines

Let's denote the time taken by the old machine as x hours and the time taken by the new machine as y hours.

From the problem statement, we know that the old machine can produce widgets in 3 hours, so we have:

1 / x = 1 / 3

Solving for x, we get:

x = 3

Similarly, we know that the new machine can produce widgets in 4 hours, so we have:

1 / y = 1 / 4

Solving for y, we get:

y = 4

Now, we can plug in the values of x and y into the formula and solve for the time taken by both machines:

1 / 3 + 1 / 4 = 1 / t

Multiplying both sides by 12t, we get:

4t + 3t = 12

7t = 12

t = 12 / 7

Therefore, it will take both machines approximately 1.71 hours or 1 hour and 42 minutes to produce widgets together.

In conclusion, we used the formula for the combined work rate of two machines to calculate the time taken by both machines to produce widgets. We first found the individual work rates of the old and new machines and then substituted those values into the formula to solve for the time taken by both machines working together.

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Hi! I can help you with that.

Each package weighs 30 oz. So, if Paul buys 4 packages, he will be buying:

30 oz x 4 = 120 oz

To convert this into pounds, we need to divide the number of ounces by 16 (as there are 16 ounces in 1 pound). So,

120 oz ÷ 16 = 7.5 pounds

Therefore, Paul will be buying 7.5 pounds of hot dogs for his camping trip.

Answer:

3 more girls than boys.

Step-by-step explanation:

Let's change 4:5 to 4x(boys) and 5x(girls).

Since there are 27 students, the equation would be:

4x+5x=27

Then, you simplify the equation:

9x=27

x=3

We are not finished yet!

Since there is 4x boys and 5x girls,

do 4 times 3 which gets you 12 boys

and 5 times 3 which gets you 15 girls.

15 girls minus 12 boys is 3 more girls than boys!

Hope this helps :)

As per the figure provided the exact value of CE will be equivalent to 12.

According to the figure given in the question, Angles ABE and DBC are vertical angles and thus have the same measure. Since the given segment AE is parallel to a segment of CD, angles A and D are of the same distance by the alternate interior angle theorem. As a result, according to the angle-angle theorem, triangles ABE and DBC are equivalent, with vertex A corresponding to vertex B and vertex E to vertex D, respectively.

Hence, AB ÷ DB = EB ÷ CB

10 ÷ 5 = 8 ÷ CB

Since, CB=4 and CE= CB+BE

CE = 4 + 8

CE=12.

Therefore CE is equal to 12.

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The rate of change in the area of the circle is 22 cm²/s when the radius of the circle is increasing at a rate of 0.5 cm/s.

It is half of the diameter of the circle or sphere. The radius is commonly denoted by the letter "r".

The area of a circle is given by the formula A = πr², where A is the area and r is the radius.

If the radius is increasing at a rate of 0.5 cm, then the rate of change of the radius with respect to time is dr/dt = 0.5 cm/s.

Using the chain rule of differentiation, we can find the rate of change of the area with respect to time:

dA/dt = dA/dr * dr/dt

By varying the area formula, we can determine dA/dr:

dA/dr = 2πr

Plugging in the values of r and dr/dt, we get:

dA/dt = (2πr)(0.5) = πr

At the initial radius of 7cm, the rate of change in the area is:

dA/dt = π(7) = 22/7 * 7 = 22 cm²/s

Therefore, the rate of change in the area of the circle is 22 cm²/s when the radius of the circle is increasing at a rate of 0.5 cm/s.

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The total probability of winning is 0.75, which is not equal to 1, indicating that the game is not fair.

To decide in case this is a fair game, we ought to calculate the likelihood of each player winning.

Let H speak to the result of flipping a head and Tail speak to the result of flipping a tail. At that point, the conceivable results of three coin flips are:

HHH, HHT, HTH, THH, TTH, THT, HTT, TTT

The likelihood of getting each result is 1/8, accepting the coin flips are reasonable and autonomous.

Xenia wins on the off chance that there are 2 heads and 1 tail. Three conceivable results fulfill this condition:

HHT, HTH, and THH. Hence, the likelihood of Xenia winning is 3/8.

Yolanda wins in case there are 2 tails and 1 head. There are moreover three conceivable outcomes that fulfill this condition:

TTH, THT, and HTT. Subsequently, the likelihood of Yolanda winning is additionally 3/8.

Zoe wins if there are 3 heads or 3 tails. Two conceivable results fulfill this condition:

HHH and TTT. In this manner, the likelihood of Zoe winning is 2/8 = 1/4.

To check in case the diversion is reasonable, we include the probabilities of each player winning:

3/8 + 3/8 + 1/4 = 0.75

The full probability of winning is 0.75, which isn't break even with 1, indicating that the game is not fair.

One way to form it a fair diversion would be to alter the payouts or the rules of the amusement to guarantee that the full likelihood of winning includes up to 1.

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Erica covered a total distance of 4,500 meters in 60 minutes (1 hour).

Distance traveled is the total length of the path covered by an object in motion over a given period of time. It is a scalar quantity.

Erica ran at a rate of 3.5 meters per second for 1/2 hour, which is 30/2 = 15 minutes.

During this time, she covered a distance of:

distance = rate x time

distance = 3.5 m/s x 900 s

distance = 3,150 meters

Next, Erica walked at a rate of 1.5 meters per second for 1/2 hour, which is also 15 minutes.

During this time, she covered a distance of:

distance = rate x time

distance = 1.5 m/s x 900 s

distance = 1,350 meters

Adding the distances Erica ran and walked, we get:

total distance = 3,150 meters + 1,350 meters

total distance = 4,500 meters

Therefore, Erica covered a total distance of 4,500 meters in 60 minutes (1 hour).

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A test estimate of 16,900 is required to appraise the extent of the populace having a particular genetic marker.

To decide the test estimate required to assess the extent of the populace with a characterized edge of mistake and certainty level, we will utilize the equation:

[tex]n = (z^2 * p * (1 - p)) / (E^2)[/tex]

Or:

n is the sample size

z is the z-score related to the level of confidence

p is the estimated ratio of the population to the genetic marker

1 - p is the estimated proportion of the population with no genetic markers

E is the desired margin of error (in decimal)

By substituting the values ​​given in the formula, we get:

[tex]n = (z^2 * p * (1 - p)) / (E^2)[/tex]

[tex]n = (2.33^2 * 0.5 * 0.5) / (0,01^2)[/tex]

[tex]n = 1.69 * 10^4[/tex]

therefore, a test estimate of 16,900 is required to appraise the extent of the populace having a particular hereditary marker with an edge of mistake of 1% and a certainty level of 98. %, expecting 50% of the populace has the marker.

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Answer:

m∠ 1 = 165°

m∠ 2 = 15°

Step-by-step explanation:

We Know

∠1 + ∠2 = 180°

Let's solve

5x + (x - 18) = 180

5 + x - 18 = 180

6x - 18 = 180

6x = 198

x = 33

Now we put 33 in for x and solve for ∠1 and ∠2 !

m∠ 1 = 5x°

m∠ 1 = 5(33)

m∠ 1 = 165°

m∠ 2 = (x - 18)

m∠ 2 = 33 - 18

m∠ 2 = 15°

Answer:

m∠1 = 165°

m∠2 = 15°

Step-by-step explanation:

Note that, when the angle measurements are combined, the total measurement is 180°, based on the definition of a straight line.

It is given that m∠1 = 5x°, and m∠2 = (x - 18)°. Set the two angle measurements equal to the total measurement:

[tex]5x + (x - 18) = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](5x + x) - 18 = 180\\(6x) - 18 = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents

Multiplications

Divisions

Additions

Subtractions

~

First, add 18 to both sides of the equation:

[tex]6x - 18 = 180\\6x - 18 (+18) = 180 (+18)\\6x = 180 + 18\\6x = 198\\[/tex]

Next, divide 6 from both sides of the equation:

[tex]6x = 198\\\frac{6x}{6} = \frac{198}{6}\\ x = \frac{198}{6} = 33[/tex]

x = 33. Next, plug in 33 for x for both measurements:

[tex]m\angle1 = 5x\\m\angle1 = 5 * (33)\\m\angle1 = 165\\\\m\angle2 = x - 18\\m\angle2 = (33) - 18\\m\angle2 = 15[/tex]

Check. Both, when combined, should make 180°

165 + 15 = 180

180 = 180 (True)

~

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The solution for differential equation is the negative square root, since y(π) = -5. Thus, the final solution is;  y = 3 - √(9 - 6 tan(x))

A differential equation is a mathematical equation that relates a function or a set of functions with their derivatives.

Given differential equation;  dy/dx = sec²(x) (2+y)²

separate the variables and integrate both sides:

∫ 1/(2+y)² dy = ∫ sec²(x) dx

Using the substitution u = 2+y, du/dy = 1, we can rewrite the left-hand side as:

∫ 1/u² du = -1/u + C₁

Similarly, we can integrate the right-hand side using the identity ∫ sec²(x) dx = tan(x) + C₂, Substituting these expressions back into the original equation, we get:

-1/(2+y) + C₁ = tan(x) + C₂

To determine the values of C₁ and C₂, we use the initial condition y(π) = -5, which implies x = π. Substituting these values, we get:

-1/(2-5) + C₁ = tan(π) + C₂

-1/(-3) + C₁ = 0 + C₂

C₁ = C₂ + 1/3

putting the value of C₁ and C₂ into the previous expression, So,

-1/(2+y) + C₁ = tan(x) + C₁ - 1/3

-1/(2+y) = tan(x) - 1/3

Multiplying both sides by (2+y)², we get:

-(2+y) = (2+y)² tan(x) - (2+y)²/3

Simplifying and solving for y, we get:

y² - 6 - 6 tan(x) = 0

Solve it for y by using the quadratic formula,

y = 3 ± √(9 - 6 tan(x))

Therefore, the solution to the differential equation dy/dx = sec²(x) (2+y)² with the initial condition y(π) = -5 is:    y = 3 ± √(9 - 6 tan(x))

We choose the negative square root, since y(π) = -5. Thus, the final solution is:     y = 3 - √(9 - 6 tan(x))

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The event where two or more things happen at the same time is called a "compound event."  the correct answer is B.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

A compound event refers to a situation where two or more events occur at the same time. In other words, it is an event that consists of multiple outcomes happening simultaneously.

For example, if you flip a coin and roll a dice at the same time, the resulting event would be a compound event because it involves the outcomes of both actions occurring simultaneously. It is important to distinguish between compound events and independent or dependent events, as they have different probabilities and methods of calculation. Compound events are often used in probability theory and statistics to analyze and predict the likelihood of multiple outcomes occurring at the same time.

The event where two or more things happen at the same time is called a "compound event." Therefore, the correct answer is B.

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No, the square piece of plywood cannot enter through the door because its diagonal is longer than the height of the door.

To see if the plywood would enter through the door, we can use the Pythagorean theorem to find the length of the diagonal of the plywood:

diagonal^2 = side1^2 + side2^2

So, we have

diagonal^2 = 2.2^2 + 2.2^2

diagonal^2 = 4.84 + 4.84

diagonal^2 = 9.68

Take the square root

diagonal ≈ 3.11

Therefore, the diagonal of the plywood is approximately 3.11 meters.

Since the height of the door is only 3 meters, the plywood is too big to fit through the door. So we can conclude that the plywood cannot enter through the door.

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The area of the composite shape is 155 yd². And the right option is A-155 yd².

A composite shape is any shape that is made up of two or more geometric shapes.

The area of the composite shape can be calculated using the formula below

Note: The composite shape in the question, is made up of a tripezium and a parallelogram.

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

Hence, the area is 155 yd².

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The proper way to move on coordinate surface from point A's location to point b is to move 6 units to the right and 1 unit to the up. The correct option is a.

What does the word "coordinate" mean in mathematics?

Coordinates are a pair of numbers (also known as Cartesian coordinates), or occasionally a letter and an integer, that identify a particular point on a grid,also known as a coordinate system. The [tex]x[/tex] -axis (horizontal) and [tex]y[/tex] -axis are the two vectors on a coordinate plane, which has four quadrants. (vertical).

On the coordinate surface, we can use the following methods to move from Point A to Point B:

Point A, which is two units to the right of the Centre and nine units up, is a good place to start.

Point C, which is situated at Move five floors to the toward the right and five units up to get there. (5, 5).

Move three units to your right and three units up from Point C to Point B, which is situate

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Answer:

Step-by-step explanation:

Rahul correctly solved the problem, because it changed the sign in places when there was a "-" in front of the bracket.

( x² + 3x - 5) - (4x - 5) = x² + 3x - 5 - 4x + 5 =  x²-x

-(a+b) = -a-b

-(a-b+c-a+c)= -a+b-c+a-c

Answer:

x = 73°

Step-by-step explanation:

I added a photo of my notes

An inscribed angle is equal to half the arc on which it rests

So, according to the photo I've added (knowing that a full circle forms 360 degrees) we can write an equation:

2x + 214° = 360°

2x = 360° - 214°

2x = 146° / : 2

x = 73°

I not sure if this is the right answer, though...

The equation that represents the proportional relationship is [tex]y=\frac{1}{4}x[/tex].

What is proportion?

Proportion is a mathematical concept that describes the equality of two ratios. In other words, it is a statement that two ratios or fractions are equal.

For example, if we have two fractions [tex]\frac{a}{b}[/tex] and [tex]\frac{c}{d}[/tex], we can say that they are in proportion if:

=> [tex]\frac{a}{b}=\frac{c}{d}[/tex]

Now the ratio between x and y = 4:1

Here x = 28 then y = [tex]\frac{1}{4}\times28 = 7[/tex]

Then, x = 12 then [tex]y=\frac{1}{4}\times12 = 3[/tex]

We can see that the ratio between X and Y is always 4:1, which means that Y is one-fourth of X. We can write this as:

=>[tex]y=\frac{1}{4}x[/tex]

Therefore, the equation that represents the proportional relationship is:[tex]y=\frac{1}{4}x[/tex].

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The measure of angle LEP can be found using the angle addition postulate, which states that the measure of an angle formed by two adjacent angles is equal to the sum of their measures. Given that angle, LEW is 51 degrees and angle PEW is 27 degrees, we can find the measure of angle LEP by adding these two angles. Thus, the angle LEP is 78 degrees (51 + 27).

         L

        / \

       /   \

      /     \

     E-------P

        W

In mathematics, an angle is a geometric figure formed by two rays sharing a common endpoint, known as the vertex of the angle. There are two main ways to label angles: by giving the angle a name, usually a lowercase letter or a Greek letter, or by using the three letters that define the angle, with the middle letter being where the angle is located. Angles are typically measured in degrees using a protractor and are defined by their measure, which is not dependent upon the lengths of the sides of the angle. Coterminal angles share terminal sides but differ in size by an integer multiple of a turn.

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Hence, the response is (iv) 85 as the exterior angle is equal to the sum of the two opposite interior angles.

The apex of the angle is the intersection of two rays that together form the geometric shape known as an angle. Angles, which are commonly measured in degrees or radians, are employed in a variety of mathematical and scientific contexts, including engineering, physics, geometry, and trigonometry.

given

We can utilise the above details to construct an equation because the exterior angle is equal to the sum of the two opposite interior angles:

x + 35 = 120

35 is subtracted from both sides to yield:

x = 85

Hence, the response is (iv) 85 as the exterior angle is equal to the sum of the two opposite interior angles.

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The complete question is :-  

Use the Exterior Angle Theorem to solve for x.

Exterior angle = 120°

60

120

O 155

85

35°

the two ordered pairs that can be added to the given set and still have it represent a function are (3,4) and (-3,-2).

In order for a set of ordered pairs to represent a function, each input (x-coordinate) must be paired with exactly one output (y-coordinate).

Looking at the given set of ordered pairs, we can see that this condition is met:

The input -2 is paired with the output -1.

The input -1 is paired with the output 0.

The input 0 is paired with the output 1.

The input 1 is paired with the output 2.

The input 2 is paired with the output 3.

To add two more ordered pairs to this set and still have it represent a function, we need to make sure that the new pairs each have a unique x-coordinate that is not already in the given set.

For example, we could add the pairs (3,4) and (-3,-2):

The input 3 is paired with the output 4, which is not already in the original set.

The input -3 is paired with the output -2, which is not already in the original set.

Therefore, the two ordered pairs that can be added to the given set and still have it represent a function are (3,4) and (-3,-2).

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