If α and β are the zeros of x^2-x+k, and 3α+2β=20, find k.

To prove that 5^7 + 5^6 is divisible by 6, we can factor out the common term 5^6 and apply the divisibility rule for 6. Here's the solution:

1. Factor out the common term 5^6:
5^7 + 5^6 = 5^6(5 + 1) = 5^6 * 6

2. Apply the divisibility rule for 6:
A number is divisible by 6 if it is divisible by both 2 and 3. Since 6 is already a multiple of 6 (6*1), the expression 5^6 * 6 is divisible by 6.

Thus, we have proved that 5^7 + 5^6 is divisible by 6.

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Two  vectors in opposite directions that are orthogonal to u are v = 5i + 4j and w = -4i + 5j.

To find two vectors in opposite directions that are orthogonal to u, we need to use the cross product. The cross product of two vectors is a vector that is perpendicular to both of them. We can choose any two non-collinear vectors as long as they are orthogonal to each other and the given vector.

Let's find the cross product of u and a vector v. The cross product of two vectors a and b is given by:

a x b = |a| |b| sinθ n

where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and n is a unit vector perpendicular to both a and b in the direction given by the right-hand rule.

Since we want v to be orthogonal to u, we need to choose v such that u x v = 0. This means that the angle between u and v is either 0 or 180 degrees, and |v| is arbitrary.

Let v = 5i + 4j. Then, we have:

u x v = (1/4 i - 4/5j) x (5i + 4j)

= (-16/20)i - (5/20)j + (1/20)k

= (-4/5)i - (1/4)j + (1/20)k

Since u x v is not equal to zero, v is not orthogonal to u. To find another vector that is orthogonal to u, we can take the cross product of u and w, where w = -4i + 5j. Then, we have:

u x w = (1/4 i - 4/5j) x (-4i + 5j)

= (-5/20)i + (16/20)j + (1/20)k

= (-1/4)i + (4/5)j + (1/20)k

Since u x w is also not equal to zero, we need to adjust the signs of v and w to make them orthogonal to u. We can do this by taking the opposite of v and w. Therefore, two vectors in opposite directions that are orthogonal to u are v = 5i + 4j and w = -4i + 5j.

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|x-3|+|x+2|-|x-5| can be broken down into three separate cases based on the value of x:

(x-3) + (x+2) - (x-5) = 2x + 4

(x-3) + (x+2) - (5-x) = 2x - 6

-(x-3) - (x+2) - (5-x) = -3x - 4

We break down the expression into three separate cases based on the value of x. This is because the absolute value function creates "turning points" at which the behavior of the expression changes. We analyze the expression for each case and simplify it to obtain the final answer. The answer depends on the value of x, and we must consider the expression separately for each case.

If x ≥ 5, then the expression becomes:

(x-3) + (x+2) - (x-5) = 2x + 4

If -2 ≤ x < 3, then the expression becomes:

(x-3) + (x+2) - (5-x) = 2x - 6

If x < -2, then the expression becomes:

-(x-3) - (x+2) - (5-x) = -3x - 4

Therefore, the final answer depends on the value of x. If x is greater than or equal to 5, then the answer is 2x + 4. If x is between -2 and 3, then the answer is 2x - 6. And if x is less than -2, then the answer is -3x - 4.

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(a) To find a non-zero vector orthogonal to the plane through the points P, Q, and R, we need to take the cross product of two vectors in the plane. One way to do this is to subtract one point from another to get a vector, and then take the cross product of the two resulting vectors. For example, we could subtract point Q from point P to get the vector PQ, and subtract point R from point P to get the vector PR. Then, taking the cross product of PQ and PR will give us a vector orthogonal to the plane:PQ = <-3, 0, -9>PR = <-3, -6, 0>PQ x PR = <54, 27, 18>Therefore, the vector <54, 27, 18> is orthogonal to the plane through the points P, Q, and R.(b) To find the area of the parallelogram with sides PQ and PR, we need to find the length of the projection of PQ onto PR, and then multiply by the length of PR. The projection of PQ onto PR is given by:proj_PR(PQ) = (PQ · u) uwhere u is a unit vector in the direction of PR, and · denotes the dot product. Since PR = <-3, -6, 0>, we can take u = <-1/sqrt(10), -3/sqrt(10), 0>, which is a unit vector in the direction of PR. Then:proj_PR(PQ) = (PQ · u) u = (-3/sqrt(10)) <-1/sqrt(10), -3/sqrt(10), 0> = <9/10, 27/10, 0>The length of this vector is sqrt((9/10)^2 + (27/10)^2 + 0^2) = 3sqrt(10), so the area of the parallelogram is:A = |PQ| |proj_PR(PQ)| = sqrt((-3)^2 + 0^2 + (-9)^2) * 3sqrt(10) = 27sqrt(10)

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A. Orthogonal vector = (0, -27, 18)

B. The area of the parallelogram with sides PQ and PR is 1053 square units.

(a) To find a non-zero vector orthogonal to the plane through points P, Q, and R, we need to compute the cross product of vectors PQ and PR.

Vector PQ = Q - P = (-3, 0, -9)
Vector PR = R - P = (-3, -6, 0)

Cross product PQ x PR = (i, j, k) × ((-3, 0, -9), (-3, -6, 0))
= i(0 * 0 - (-9) * (-6)) - j(-3 * 0 - (-3) * (-9)) + k(-3 * -6 - 0 * (-3))
= i(0) - j(27) + k(18)
Orthogonal vector = (0, -27, 18)

(b) To find the area of the parallelogram with sides PQ and PR, we can use the magnitude of the cross product of PQ and PR.

Magnitude of PQ x PR = ||(0, -27, 18)||
= √(0^2 + (-27)^2 + 18^2)
= √(729 + 324)
= √1053

The area of the parallelogram with sides PQ and PR is 1053 square units.

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To find the sale price of each item, we need to use some basic algebra. Let's call the sale price of each item "x".

First, we need to find the total cost of the table rental for all six friends. Since each friend needs to pay $7.20, the total cost of the table rental is 6 * $7.20 = $43.20.

Next, we need to find the total revenue from selling the items. Each friend sells 3 items, so the total number of items sold is 6 * 3 = 18. The total revenue is the number of items sold multiplied by the sale price, so the total revenue is 18x.

We know that the total profit is $25.20, which is the total revenue minus the total cost of the table rental. So we can set up the equation:

18x - $43.20 = $25.20

Simplifying this equation, we get:

18x = $68.40

Dividing both sides by 18, we get:

x = $3.80

Therefore, the sale price of each item is $3.80.

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Answer:

8.19 units.

Step-by-step explanation:

I didn't use desmos, i completed the question with all steps and working and didn't require desmos. Hope this Helps. Question was solved using trignometric ratios.

Answer:

8 38/48

Step-by-step explanation:

The closest value to the total cost of Francine's calls is $191.07

To find the total cost of Francine's calls, we need to know the total number of minutes she spent on the phone. We can calculate this by adding up the lengths of all 11 calls:

Total minutes = 11 + 14 + 12 + 20 + 25 + 16 + 11 + 22 + 18 + 19 + 15 = 193

The total cost of Francine's calls can then be found by multiplying the total number of minutes by the cost per minute:

Total cost = 193 * 0.99

= 191.07

Therefore, the closest value to the total cost of Francine's calls is $191.07.

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To show that Harold is wrong about the cost of chocolate biscuits being $2 each, we can use the given information:

You bought 3 packets of plain biscuits and 5 packets of chocolate biscuits, and the total cost was $34.

Let's use the variables P for the cost of plain biscuits and C for the cost of chocolate biscuits.

We can write the equation:

3P + 5C = $34

Harold claims that the cost of chocolate biscuits is $2 each. So, let's substitute C = $2 into the equation:

3P + 5($2) = $34

Now, we can solve for P:
3P + $10 = $34
3P = $24
P = $8

So, the cost of plain biscuits is $8 each. This means you bought 3 packets of plain biscuits for $24 and 5 packets of chocolate biscuits for $10, which adds up to the total cost of $34.

Since the cost of plain biscuits came out to be a whole number, Harold's claim that chocolate biscuits cost $2 each is not necessarily wrong.

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The decay factor corresponding to each percent decrease is 0.87. The correct option is c. The customer would save a total of $17.35, or 17.35% off the original price.

To determine the decay factor corresponding to each percent decrease, we need to use the formula:

decay factor = (100% - percent decrease)/100%

For the first discount of 13%, the decay factor would be:

decay factor = (100% - 13%)/100% = 0.87

For the additional neighborhood discount of 5%, the decay factor would be:

decay factor = (100% - 5%)/100% = 0.95

So the correct answer is option C, with a decay factor of 0.87 for the 13% discount and 0.95 for the neighborhood discount.

These decay factors represent how much of the original price is left after the discount has been applied. For example, if the original price was $100, after the 13% discount, the price would be:

price after first discount = $100 x 0.87 = $87

Then, after the additional 5% discount, the final price would be:

final price = $87 x 0.95 = $82.65

So the customer would save a total of $17.35, or 17.35% off the original price.

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To answer this question, determine the quantity asked for:

Answers are:
Yes - How many hours a week do people exercise?
No - How many hours are there in a day?
Yes - How many rainbows have students seen this month?

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if you meant simplification

[tex]4^3\sqrt{-88}\implies 64\sqrt{-88}\implies 64\sqrt{(-1)(2^2)(2)(11)}\implies 64(2)\sqrt{(-1)(2)(11)} \\\\\\ 128\sqrt{(2)(11)}\cdot \sqrt{-1}\implies 128\sqrt{22}~i[/tex]

Based on the information provided, it is not clear what "they" refer to. However, it seems that there are two distinct events or phenomena being described here.

The first involves the albatross, which is known to be a migratory bird that visits certain regions at certain times of the year. It is unclear what exactly is being "kept" by the albatross, but it could be some sort of information or knowledge that is unique to this bird.

The second event or phenomenon is described as happening every night and involving various activities such as shooting, swimming, fighting, and singing.

It is unclear what this refers to exactly, but it could be some sort of nightly ritual or tradition that takes place in a particular community or region.

The statement that "separately, they would reveal many to you" suggests that each of these phenomena has its own unique insights or knowledge to offer.

However, the statement that "together, they will reveal only one to you" implies that there is some sort of deeper connection or meaning between these two events that is not immediately apparent.

The question of "whose one is it?" suggests that there is some sort of ownership or responsibility associated with this revelation.

It is unclear who this refers to, but it could be interpreted as asking who is responsible for uncovering the deeper meaning or connection between these two phenomena.

In summary, the information provided is somewhat cryptic and open to interpretation. However, it seems that there are two distinct events or phenomena being described, and that there is some sort of deeper connection or meaning between them that is not immediately apparent.

The question of who is responsible for uncovering this connection remains unanswered.

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The population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 14 manatees per square mile

The population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 70 manatees per square mile. To calculate this, we first need to find the area of the half circle. Using the formula for the area of a circle, A=πr^2, where r is the radius, we can find the area of the full circle with a radius of 5 miles:

A = 3.14 x 5^2
A = 78.5 square miles

Since we only want to consider the area of the half circle, we divide this by 2:

A = 78.5 / 2
A = 39.25 square miles

Now we can calculate the population density by dividing the number of manatees by the area:

Density = 550 / 39.25
Density ≈ 14

Therefore, the population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 14 manatees per square mile. Rounded to the nearest whole number, this is 14.

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The relation is not an equivalence relation. (a) {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}:

Reflexive: (2,2), (3,3) are present, but (1,1) and (4,4) are not present. Hence, not reflexive.

Symmetric: (2,3) is present, but (3,2) is also present. Hence, not symmetric.

Transitive: No counterexample to transitivity exists. Hence, it is transitive.

Therefore, the relation is not an equivalence relation.

(b) {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}:

Reflexive: (1,1), (2,2), (3,3), (4,4) are present. Hence, reflexive.

Symmetric: (1,2) is present, but (2,1) is also present. Hence, symmetric.

Transitive: No counterexample to transitivity exists. Hence, it is transitive.

Therefore, the relation is an equivalence relation.

(c) {(2,4),(4,2)}:

Reflexive: (2,2) and (4,4) are not present. Hence, not reflexive.

Symmetric: (2,4) is present, but (4,2) is not present. Hence, not symmetric.

Transitive: No counterexample to transitivity exists. Hence, it is transitive.

Therefore, the relation is not an equivalence relation.

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The value of x to the nearest tenth is 1.4

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Sin(θ) = opp/hyp

cos(θ)= adj/hyp

tan( θ) = opp/adj

x is opposite side to angle F and 32 is adjascent to angle F.

therefore;

tan F = opp/adj

tan23 = x/3.2

x = tan23 × 3.2

x = 1.4

therefore the value of x to the nearest tenth is 1.4

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The expression that is equivalent to the one Regina wrote is y + 27/4

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

y + 9 x 3/4

This means that

Expression = y + 9 x 3/4

When expanded, we have

Expression = y + 27/4

Using the above as a guide, we have the following:

The expression that is equivalent to the one Regina wrote is y + 27/4

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The amount of carrying charges for a $10,000 college loan with a 5% down payment, a 10% APR, and a 36-month repayment period is approximately $1,571.44.

To calculate the amount of carrying charges for a $10,000 college loan with a 5% down payment, a 10% annual percentage rate (APR), and a 36-month repayment period, follow these steps:
1. Determine the down payment: 5% of $10,000 = $500.
2. Subtract the down payment from the loan amount: $10,000 - $500 = $9,500. This is the principal loan amount.
3. Calculate the monthly interest rate: 10% APR / 12 months = 0.833% or 0.00833 as a decimal.

4. Calculate the monthly payment using the loan payment formula: P = r * PV / (1 - (1 + r)⁻ⁿ), where P is the monthly payment, r is the monthly interest rate, PV is the present value (principal loan amount), and n is the number of monthly payments. P = 0.00833 * $9,500 / (1 - (1 + 0.00833)⁻³⁶) = $307.54.
5. Determine the total amount paid over the loan term: Monthly payment * Number of monthly payments = $307.54 * 36 = $11,071.44.
6. Calculate the carrying charges: Total amount paid - Principal loan amount = $11,071.44 - $9,500 = $1,571.44.

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The zeros of the function y = (x − 4)(x² − 12x + 36) are 4 and 6.

To find the zeros of the function y = (x - 4)(x² - 12x + 36), we need to set y to zero and solve for x.

0 = (x - 4)(x² - 12x + 36)

Now, solve for each factor separately:

1) x - 4 = 0
x = 4

2) x² - 12x + 36 = 0
This is a quadratic equation, and we can factor it as (x - 6)(x - 6).
So, x - 6 = 0
x = 6

The zeros of the function are x = 4 and x = 6. The zeros of a function are the values of its variables that meet the equation and result in the function's value being equal to 0.

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Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads, reflecting the 0.4 probability of landing on heads and the 0.6 probability of landing on tails.

Kyra's "trick" coin has a probability of 0.4 for landing on heads, meaning that it is more likely to land on tails. If Kyra flips the coin many times, she will likely generate a frequency table that shows a higher frequency of tails than heads.

A possible frequency table might look like this:

| Outcome | Frequency |
|---------|-----------|
| Heads   | 40        |
| Tails   | 60        |

In this table, the coin landed on heads 40 times and tails 60 times, representing the probability of 0.4 for heads and 0.6 for tails (since the total probability must equal 1). The more times Kyra flips the coin, the closer the relative frequencies will get to the actual probabilities. It's important to note that this is just an example, and the exact frequency counts may vary in practice due to randomness.

To summarize, Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads.

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Answer:

f(-2) = 9

Step-by-step explanation:

f(x) = -3x + 3                  Solve for f(-2).

f(-2) = -3(-2) + 3

f(-2) = 6 + 3

f(-2) = 9

A population of 600 organisms is expected to increase by 50.

We can use the equation I = k * n to determine whether the anticipated growth (I) of an organism population is directly related to the size of the current population (n).

Where k is a proportionality constant that connects the anticipated growth to the current population.

We can use the details provided in the problem to determine the value of k. We can write: 20 = k * 240 to represent an increase of 20 in a sample of 240 organisms.

We can calculate k as follows: k = 20 / 240 = 1 / 12.

Now that we know the value of k, we can use it to calculate the predicted growth for a population of 600:

I = (1 / 12) * 600 = 50

Therefore, a population of 600 organisms is expected to increase by 50.

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80 shoppers in the dairy section at random is a biased sample (d).

To estimate the number of people who buy milk in a grocery store, we need an unbiased sample that represents the entire population. The options provided are 70 shoppers at random as they leave the store, 8 shoppers as they enter the store, 60 women with children at random, and 80 shoppers in the dairy section at random.

The most unbiased sample is selecting 80 shoppers in the dairy section at random because the dairy section is where people are most likely to buy milk, and selecting a random sample of shoppers in that section ensures that we are not introducing any biases.

The other options may introduce biases since they are not necessarily representative of the entire population of shoppers in the store. Therefore, selecting 80 shoppers in the dairy section at random provides the most unbiased estimate of the number of people who buy milk in the grocery store. So d is correct option.

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He will have $566 in this account if he keeps it for 5 years.

To determine how much money he will have in this account if he keeps it for 5 years, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the final amount

P = the principal, $500

r = the interest rate, 2.5% = 0.025

n = the number of times the interest is compounded per year, in this case quarterly (n = 4)

t = the time period in years, 5

Substituting the values :

A = 500(1 + 0.025/4)^(4 * 5)

A = 500(1 + 0.00625)²⁰

A = 500(1 .00625)²⁰

A = $566

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Answer:

E

Step-by-step explanation:

This figure contains a square and a quarter of a circle

The perimeter is the sum of all side lengths

Let's find the circumference of the quarter of a circle:

[tex]0.25c = 2\pi \times r[/tex]

The diameter is equal to a square's side length:

d = 8

hence, r = 0,5 × 8 = 4

[tex]0.25c = 2 \times 3.14 \times 4 = 25.12[/tex]

Now, we can find the perimeter:

P = 25,12 + 8 + 8 = 41,12

The resulting graph of g(x) will resemble the graph of f(x), but it will be stretched vertically and pushed leftward by three units.

Each element of a set (referred to as the domain) is mapped by a rule known as a function to a particular element of this other set (called the range). A function is, in other words, a connection between two subsets in which every member of the domain has a unique relationship to every element of the range. One popular approach to write a function is via function notation, which entails writing the function name surrounded by the incoming signal in parentheses, as in the following example: f (x). For instance, the formula f(x) = 2x + 1 takes the input x and produces the result 2x + 1.

given,

The fundamental quadratic function f(x) = x² serves as the parent function for g(x) = (x + 3)².

Since the argument of the function (x + 3) is x shifted left by 3 units, the graph of f(x) is horizontally displaced to the left by 3 units.

Given that the coefficient of the squared term is 1, the graph of the resulting function is shifted up by a factor of 1.

As a result, the transformation can be represented as a 3 unit horizontal shift and a 1 unit vertical stretch.

The resulting graph of g(x) will resemble the graph of f(x), but it will be stretched vertically and pushed leftward by three units.

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.m The correct equation to determine the amount of material used, m, to construct the fully enclosed tent is:

                         c. =2(8∙9)+2(7.5∙8)+(12∙9∙6)

To determine the amount of material used to construct the fully enclosed tent, we need to consider the surface area of the tent. The tent is fully enclosed, so we need to calculate the area of all the sides.

Option a. =(8∙9)+2(7.5∙8) calculates the area of the top and two sides of the tent. This does not include the front and back of the tent, so it is not the correct equation.

Option b. =(8∙9)+2(7.5∙8)+2(12∙9∙6) calculates the area of the top, two sides, front and back of the tent, but it also includes an extra term of 2(12∙9∙6) which is not necessary for a fully enclosed tent. This option overestimates the amount of material used.

Option c. =2(8∙9)+2(7.5∙8)+(12∙9∙6) calculates the area of the top, bottom, and all four sides of the tent. This is the correct equation to determine the amount of material used in a fully enclosed tent.

Option d. =3(8∙9)+2(12∙9∙6) overestimates the amount of material used because it includes an extra term of 3(8∙9) which is not necessary for a fully enclosed tent.

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Answer:

52.5

Step-by-step explanation:

15×7=105÷2

=52.5 ans it means fifteen times seven divided by two

ccabo de 25 años será la edad del padre tres veces mayor que la edad del hijo.El número es 36.Hay 8 hombres, 16 mujeres y 72 niños.Hay 27 cerdos y 8 pavos

Sea x el número de años transcurridos. La edad del padre será 35 + x y la edad del hijo será 5 + x. La ecuación que representa la situación es: 35 + x = 3(5 + x). Resolviendo la ecuación, tenemos:

35 + x = 15 + 3x

2x = 20

x = 10

Por lo tanto, después de 10 años, la edad del padre será tres veces mayor que la edad del hijo.

Sea x el número desconocido. La ecuación que representa la situación es: 2x - (1/2)x = 54. Resolviendo la ecuación, tenemos:

(4/2)x - (1/2)x = 54

(3/2)x = 54

x = 36

Por lo tanto, el número desconocido es 36.

Sea x el número de hombres. Según la información dada, el número de mujeres es el doble, es decir, 2x, y el número de niños es el triple, es decir, 3(x + 2x) = 9x. La ecuación que representa la situación es: x + 2x + 9x = 96. Resolviendo la ecuación, tenemos:

12x = 96

x = 8

Por lo tanto, hay 8 hombres, 16 mujeres y 72 niños en la reunión.

Sea x el número de cerdos y y el número de pavos. Según la información dada, tenemos las siguientes ecuaciones: x + y = 35 (por el total de cabezas) y 4x + 2y = 116 (por el total de patas). Resolviendo este sistema de ecuaciones, obtenemos:

x + y = 35

4x + 2y = 116

Multiplicamos la primera ecuación por 2:

2x + 2y = 70

Restamos la segunda ecuación de la primera:

2x + 2y - (4x + 2y) = 70 - 116

-2x = -46

x = 23

Sustituyendo el valor de x en la primera ecuación, tenemos:

23 + y = 35

y = 12

Por lo tanto, hay 23 cerdos y 12 pavos en la granja.

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It will take approximately 5.28 years for the depth of the lake to reach 26.7 meters.

The situation represents decay because the depth of the lake decreases over time.

The rate of growth or decay, r, is equal to -0.02 (negative because it represents a decrease of 2% per year).

So the depth of the lake each year is 0.98 times the depth in the previous year (100% - 2% = 98%).

To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:

26.7 = 30 *[tex](0.98)^n[/tex]

Solving for n, we get:

[tex](0.98)^n = \frac{26.7 }{ 30}[/tex]

n =[tex]log\frac{(\frac{26.7}{30}) }{ log(0.98)}[/tex]

Using a calculator, we find n ≈ 5.28 years.

Therefore, it will take approximately 5.28 years for the depth of the lake to reach 26.7 meters.

Learn more about Lake depth decay.

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