The three Pythagorean trigonometric identities, which I’m sure one can find in any Algebra-Trigonometry textbook, are as follows:
sin² θ + cos² θ = 1
tan² θ + 1 = sec² θ
1 + cot² θ = csc² θ
where angle θ is any angle in standard position in the xy-plane.
Consistent with the definition of an identity, the above identities are true for all values of the variable, in this case angle θ, for which the functions involved are defined.
The Pythagorean Identities are so named because they are ultimately derived from a utilization of the Pythagorean Theorem, i.e., c² = a² + b², where c is the length of the hypotenuse of a right triangle and a and b are the lengths of the other two sides.
This derivation can be easily seen when considering the special case of the unit circle (r = 1). For any angle θ in standard position in the xy-plane and whose terminal side intersects the unit circle at the point (x, y), that is a distance r = 1 from the origin, we can construct a right triangle with hypotenuse c = r, with height a = y and with base b = x so that:
c² = a² + b² becomes:
r² = y² + x² = 1²
y² + x² = 1
We also know from our study of the unit circle that x = r(cos θ) = (1)(cos θ) = cos θ and y = r(sin θ) = (1)(sin θ) = sin θ; therefore, substituting, we get:
(sin θ)² + (cos θ)² = 1
1.) sin² θ + cos² θ = 1 which is the first Pythagorean Identity.
Now, if we divide through equation 1.) by cos² θ, we get the second Pythagorean Identity as follows:
(sin² θ + cos² θ)/cos² θ = 1/cos² θ
(sin² θ/cos² θ) + (cos² θ/cos² θ) = 1/cos² θ
(sin θ/cos θ)² + 1 = (1/cos θ)²
(tan θ)² + 1 = (sec θ)²
2.) tan² θ + 1 = sec² θ
Now, if we divide through equation 1.) by sin² θ, we get the third Pythagorean Identity as follows:
(sin² θ + cos² θ)/sin² θ = 1/sin² θ
(sin² θ/sin² θ) + (cos² θ/sin² θ) = 1/sin² θ
1 + (cos θ/sin θ)² = (1/sin θ)²
1 + (cot θ)² = (csc θ)²
3.) 1 + cot² θ = csc² θ
Simplify:
(cos^2(a)-cot^2(a))/sin^2(a)-tan^2(a)
Simplify the expression:
[tex]\dfrac{cot^2(a)(1+csc(a))(1-csc(a))}{(1+sec(a))(1-sec(a))}[/tex]
You deposit $100 in a savings account. The account earns 8% simple interest per year.
Answer:
124 and 125.97
Step-by-step explanation:
y
The amount of paint needed to cover a wall is proportional to its area. The wall is rectangular and has an area of 6z^2 + 6z square meters. Factor this polynomial to find possible expressions for the length and width of the wall. (Assume the factors are polynomials.)
We can factor the polynomial 6z² + 6z as follows:
[tex]6z^2 + 6z = 6z(z + 1)[/tex]
We can use this factorization to express the area of the wall as the product of two factors:
[tex]6z^2 + 6z = 6z(z + 1) = length × width[/tex]
Therefore, the length of the wall is 6z, and the width of the wall is z + 1.
I will mark you brainiest!
What is the length of EF?
A) 2.4
B) 3.8
C) 0.5
D) 1.2
Answer:
the answer is 2.4 for ef
pls mark me brainliest
what is the number of real solutions
X^2+6x=5
Answer options
1. No real solutions
2. Cannot be determined
3. Two real solutions
4. One real solution
In the given situation we know that the equation X²+6x=5 has (3) two real solutions.
What do we mean by real solutions?In algebra, a real solution is just an answer to an equation that is a real number.
Discriminant b² - 4ac has zero value in the case of a single actual solution.
One solution, x = -1, exists for the equation x² + 2x + 1 = 0.
The Determinant Calculator on Cuemath can be used to determine the determinant of a quadratic equation.
Use the formula: b²-4ac
Insert values:
b²-4ac
6²-4(1)(5)
36-20
16
We know that:
If b²-4ac is > o (Two real solutions)
Therefore, in the given situation we know that the equation X²+6x=5 has (3) two real solutions.
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Find the length of AD in the figure.
A. 34 units
B. 122 units
C. 130 units
D. 26 units
The length of quadrilateral ABCD of AD, which is closest to option A, is [tex]4*sqrt(13)[/tex] (34 units).
Is a triangle 90 degrees or 180?A triangle will always have an angle total of 180 degrees. A quadrilateral can be divided in half from corner to corner to form a triangle because the angle total of a quadrilateral is equal to 360°. A triangle is effectively half of a quadrilateral, therefore it makes sense that its angle measurements are also half. 180° is the half of 360°.
The Pythagorean theorem can be used to calculate the duration of AD.
The distance formula can be used to get the length of AB first:
[tex]AB = sqrt((4-(-4))2 + (6-(-2))2 = sqrt(8-(-8-8) + 8-(8) = 8*sqrt (2)[/tex]
The distance formula can then be used to get the length of AC:
[tex]Sqrt((12-(-4)) = AC^2 + (6-(-2))^2)= sqrt(16^2 + 8^2) = 8*sqrt(5) (5)\\[/tex]
Now, we can calculate the length of AD using the Pythagorean theorem:
[tex]AD^2 = AB^2 + BD^2AD^2 = (8sqrt(2))^2 + (4sqrt(5))^2AD^2 = 128 + 80AD^2 = 208AD = sqrt(208)[/tex]
Simplifying the radical, we get:
AD = sqrt(1613) = 4sqrt(13)
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The length of quadrilateral ABCD of AD, which is closest to option A, is (34 units). Thus, option A is correct.
What is the sum of the angles in a triangle?The sum of the angles in a triangle is always 180 degrees. Because a quadrilateral's total angles are 360°, it can be divided in half from a corner to create a triangle.
The distance formula can be used to get the length of AB first: [tex]AB = sqrt((4 — (-4))2+ (6— (-2))2 = sqrt(8 — (-8— 8)+8— (8) = 8* sqrt(2)[/tex] The distance formula can then be used to get the length of AC:
[tex]Sqrt((12 — (-4)) = AC2 + (6 — (-2))2) = sqrt(162 + 82) = 8* sqrt(5)(5)[/tex]
Now, we can calculate the length of AD using the Pythagorean theorem:
[tex]AD2 = AB2 + BD2AD2 = (8sqrt(2))2 + (4sqrt(5))2AD2 = 128 + 80AD2 = 208AD = sqrt(208)[/tex]
Simplifying the radical, we get: [tex]AD = sqrt(1613) = 4sqrt(13)[/tex]
Therefore, It makes obvious that since a triangle is functionally half of a quadrilateral, its angle measurements are also half. Half of a 360° circle is 180°.
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JJ fills in the grid of numbers below so that the sum of the first three numbers is 100, the sum of the middle three numbers is 200, and the sum of the last three numbers is 300. What is the filled in grid?
10 Fill in the three boxes 130
Answer:
Can you rephrase the question as it doesn't seem possible to solve. There may be a value missing.
Homer's car weighs 4,000 pounds. How many tons does
Homer's car weigh?
Answer:2
Step-by-step explanation:
Answer:
2 Tons
Step-by-step explanation:
Homer’s car weighs 2 tons because there are 2,000 pounds in a ton and 4,000 divided by 2,000 equals 2
Evaluate the expression.
|-2| – |-1|
Answer:
1
Step-by-step explanation:
We must solve the absolute values first
An absolute value is the numbers distance from zero
So a simplified version of your problem would be
2-1
So your answer is 1
The absolute value of a number is its distance from 0, so |-2| is 2 and |-1| is 1. Substituting these back in, the expression 2 - 1 equals 1.
Explanation:To evaluate the expression |-2| – |-1|, you first need to understand what the absolute value symbol does. The absolute value of a number is its distance from 0 on the number line, regardless of direction. So, both |-2| and |-1| become positive numbers, as distance is always positive.
Therefore, |-2| equals 2 and |-1| equals 1. If we substitute these values back into the original expression, 2 - 1, the result is 1.
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given the following limit lim(x;y)!(0;0) infinty y infinity y , show that the function f (x; y) does not have a limit as (x; y) ! (0; 0).
The limit of f(x, y) as (x, y) approaches (0, 0) depends on the path taken, the limit does not exist, and we can conclude that the function f(x, y) do not have a limit as (x, y) → (0, 0).
To show that the function f(x, y) does not have a limit as (x, y) → (0, 0), we need to show that the limit does not exist, either because the limit is infinite or because the limit does not exist.
We are given that the limit of f(x, y) as (x, y) → (0, 0) when y → infinity is infinity. This means that as y approaches infinity, the function f(x, y) becomes arbitrarily large, regardless of the value of x. However, this does not imply that the limit of f(x, y) exists as (x, y) → (0, 0).
To see why, consider the sequence of points (x_n, y_n) = (1/n, n) as n approaches infinity. As y_n → infinity, we have
lim (x_n, y_n) → (0, 0) f(x_n, y_n) = infinity.
However, if we consider the sequence of points (x_n, y_n') = (1/n, n^2) instead, as n approaches infinity, we have
lim (x_n, y_n') → (0, 0) f(x_n, y_n') = 0.
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Decrease R450 in the ratio 9:8
The value of R500 decrease to ratio 9:8 is x = 400.
What is cross multiplication?By using the cross multiplication approach, the denominator of the first term is multiplied by the numerator of the second term, and vice versa. Using the mathematical rule of three, we may determine the answer based on proportions. The best illustration is cross multiplication, where we may write in a percentage to determine the values of unknown variables.
Given that, decrease R450 in the ratio 9:8.
Let 9 = 450
Then 8 will have the value = x.
That is,
9 = 450
8 = x
Using cross multiplication we have:
9x = 450(8)
x = 50(8)
x = 400
Hence, the value of R500 decrease to ratio 9:8 is x = 400.
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she earned $480 in one week. If she worked 40 hours, how much did she earn per hour?
Answer:
12 dollars
Step-by-step explanation:
480 / 40 = 12
12 dollars
can you find the following limits?
1=?
2=?
3=?
The first limit, [tex]\lim_{x\to \→-2^- } -3(x+2)/x²+4x+4[/tex] , evaluates to negative infinity, while the second limit, [tex]\lim_{ x\to \-2^+}-3(x+2)/x²+4x+4[/tex] , evaluates to positive infinity.
What is function?Function in maths is a relation between two sets of values. It is a type of mathematical equation in which each input value has a unique output value. In a function, each input value must correspond to only one output value. This means that for any input value, the output must be the same.
This indicates that the function has a vertical asymptote at x=-2.
In order to understand why this is the case, we can first rewrite the function as follows:
f(x) = -3(x+2)/(x+2)(x+2)
The denominator of the function is (x+2)(x+2), which has a double root at x=-2. This means that the denominator is equal to zero when x=-2. As a result, the function f(x) will have a vertical asymptote at x=-2, since the denominator will be equal to zero and the function will approach negative or positive infinity. This is why the two limits mentioned above both evaluate to either negative or positive infinity.
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The limit of the given functions are:
[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4 = positive infinity
[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4 = negative infinity
[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4 = Undefined
What is function?Function is a relation between two sets of values. In a function, each input value must correspond to only one output value. This means that for any input value, the output must be the same.
[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4
= -3(-2+2)/(-2)²+4(-2)+4
= -3/0 + 8 + 4
= +∞ (infinity)
[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4
= -3(-2+2)/(-2+2)²+4(-2+2)+4
= -3/4 + 0 + 4
= -∞
[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4
= -3(-2+2)/(-2+2)²+4(-2+2)+4
= -3/0 + 0 + 4
= Undefined
Therefore, the limit of the functions given are:
[tex]\lim_{x \to \--2^-}[/tex]-3(x+2)/ x²+4x+4 = +∞
[tex]\lim_{x \to \--2^+}[/tex] -3(x+2)/ x²+4x+4 = -∞
[tex]\lim_{x \to \--2[/tex] -3(x+2)/ x²+4x+4 = Undefined
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Chaz is a college student. He has a checking account balance of -$52.00. His roommate Will's
checking account balance is -$59.25. Chaz thinks that Will owes more to the bank than Chaz
does. Is Chaz correct? Explain your answer.
Answer: No, Chaz is not correct. Although their balances are both negative, we cannot compare them simply based on their numeric values. The magnitude of the balance does not indicate who owes more to the bank, as it depends on various factors such as account activity, fees, and interest rates. We would need to know more information about their accounts, such as the interest rates and any fees, in order to determine who owes more to the bank.
Excluding the bank fees Chaz would technically be correct.
No, Chaz is not correct.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
No, Chaz is not correct.
Although both Chaz and Will have negative checking account balances, we cannot determine who owes more to the bank based solely on the balance amount.
The balance amount only indicates how much money they owe to the bank, but it does not give any information about the amount they initially deposited or any other financial transactions they may have made.
To determine who owes more to the bank, we would need to know the initial deposit amount, the transaction history, and any fees or interest charges that have been applied to the accounts.
Without this additional information, we cannot accurately compare the two balances or determine who owes more to the bank.
Thus,
No, Chaz is not correct.
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How this app works? Why I can’t find any answers? I need pay for points when I ask questions? I subscribe this app and when I’m lucky to the answers almost the answers are wrong
State if the triangles in each pair are similar
Answer:
They are similar
Step-by-step explanation:
It's because 27/18 = 12/8
27/18 = 1.5
12/8 = 1.5
If 50% of a number is 170 and 80% of the same number is 272, find 30% of that number.
Answer:
30% of the number is 102.
Of the clients at Dillon's salon, 4 clients have blond hair and 12 clients have hair in other
colors.
What is the probability that a randomly selected client at Dillon's salon has blond hair?
Write your answer as a fraction or whole number.
P(blond) =
In response to the stated question, we may state that As a result, the probability of a randomly picked client at Dillon's salon having blond hair is one-quarter.
What is probability?Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.
Dillon's salon has 4 clients with blond hair and 12 clients with different hair colours, for a total of 4+12=16 clients.
The chance of picking a blond-haired customer is equal to the number of blond-haired clients divided by the total number of clients:
P(blond) = number of blond clients / total number of clients = 4 / 16 = 1/4
As a result, the likelihood of a randomly picked client at Dillon's salon having blond hair is one-quarter.
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P(blond) = 4/16 = 1/4 .is the probability that a randomly selected client at Dillon's salon has blond hair.
What is probability ?Probability is a measure of the likelihood that an event or experiment will occur. It is expressed as a number between 0 and 1, with 0 meaning that the event is impossible to occur and 1 meaning that the event is certain to occur. Probability can be calculated using a variety of methods depending on the type of problem. For example, the probability of a coin being heads can be calculated by dividing the number of heads by the total number of coin flips.
Probability theory is an important part of statistics and is used to make predictions about the likelihood of certain events occurring. It is also used to assess risk and make decisions in a wide range of fields, from economics and finance to medicine and engineering.
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If 20 is 20% of 20% of an integer, what is that integer?
F. 20
G. 50
H. 200
J. 500
K. 1000
Answer:
J. 500
Step-by-step explanation:
20 percent is 0.2 when multiplying.
0.2(0.2(x)) = 20
0.04(x) = 20
x = 500
Uniform Distribution: Suppose that the random variable X follows a uniform distribution that takes on values from -2 to 3.
(3 points) Draw the graph of this uniform density function. I do not need a title for this one – but do want to see the scaling.
(3 points) What is P (-1.25 < X < 1.3)?
A company produces ceramic floor tiles that are supposed to have a surface area of 16 square inches. Due to variability in the manufacturing process, the actual surface area has a normal distribution with a mean of 16 square inches and a standard deviation of 0.008 square inches.
(5 points) What percent of the tiles have an area that is between 15.975 and 16.01 square inches?
(5 points) What is the probability that a tile will have an area that is more than 16.025 square inches?
i. Is this unusual? (1 point) How do you know?
(7 points) If 10,000 tiles are produced, how many can be expected to have a surface area less than 15.985 square inches?
(7 points) How many tiles must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches?
1. Graph is attached below. 2. 0.51 3. the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%. 4. 0.08% 5. 304 tiles 6. 0.4938x tiles.
Describe uniform density function?A uniform density function is a type of probability distribution that assigns equal probability to all values within a specified range. It is also known as a rectangular distribution because the graph of the probability density function appears as a rectangle with a constant height over the interval of possible values.
The probability density function of a uniform distribution is defined by two parameters: a minimum value (a) and a maximum value (b). The function assigns a probability of 1/(b-a) to each value within the range [a, b], and a probability of 0 to any value outside this range.
1. The graph of the uniform density function that takes on values from -2 to 3 is mentioned below
2. To find P(-1.25 < X < 1.3), we need to find the area under the uniform density function between -1.25 and 1.3. Since the density function is uniform, the area is simply the rectangle with base (1.3 - (-1.25)) = 2.55 and height 1/5 (since the range of values is 5). Therefore,
P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51
3. To find the percent of tiles that have an area between 15.975 and 16.01 square inches, we need to standardize the values using the z-score formula and then find the area under the normal curve between those two values:
z1 = (15.975 - 16) / 0.008 = -3.125
z2 = (16.01 - 16) / 0.008 = 1.25
Using a standard normal table or calculator, we find the area between these two z-scores to be approximately 0.8278.
Therefore, the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.
4. To find the probability that a tile will have an area that is more than 16.025 square inches, we again need to standardize the value using the z-score formula and then find the area under the normal curve to the right of that value:
z = (16.025 - 16) / 0.008 = 3.125
Using a standard normal table or calculator, we find the area to the right of this z-score to be approximately 0.0008.
Therefore, the probability that a tile will have an area that is more than 16.025 square inches is 0.08%.
i. Yes, this is unusual because the probability is very low, indicating that it is highly unlikely for a tile to have an area greater than 16.025 square inches.
5. To find the number of tiles that can be expected to have a surface area less than 15.985 square inches, we first need to standardize the value using the z-score formula:
z = (15.985 - 16) / 0.008 = -1.875
Using a standard normal table or calculator, we find the probability to the left of this z-score to be approximately 0.0304.
Therefore, out of 10,000 tiles, we can expect approximately 0.0304 * 10,000 = 304 tiles to have a surface area less than 15.985 square inches.
6. To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, we need to find the z-scores for the lower and upper bounds of this range:
z1 = (15.98 - 16) / 0.008 = -2.5
z2 = (16.02 - 16) / 0.008 = 2.5
Using a standard normal table or calculator, we find the probability between these two z-scores to be approximately 0.4938.
Therefore, out of x tiles, we can expect approximately 0.4938x tiles to have a surface area between 15.
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1. The graph of the uniform density function that takes on values from -2 to 3 is attached below.
2. P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51
3. The percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.
4. The probability that a tile will have an area that is more than 16.025 square inches 0.08%.
5. Number of tiles that can be expected to have a surface area less than 15.985 square inches
6. To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, 0.4938x tiles.
Describe uniform density function?A uniform density function is a type of probability distribution that assigns equal probability to all values within a specified range. It is also known as a rectangular distribution because the graph of the probability density function appears as a rectangle with a constant height over the interval of possible values.
1. The graph of the uniform density function that takes on values from -2 to 3 is mentioned below:
2. To find P(-1.25 < X < 1.3), we need to find the area under the uniform density function between -1.25 and 1.3. Since the density function is uniform, the area is simply the rectangle with base (1.3 - (-1.25)) = 2.55 and height 1/5 (since the range of values is 5). Therefore,
P(-1.25 < X < 1.3) = (2.55 * 1/5) = 0.51
3. To find the percent of tiles that have an area between 15.975 and 16.01 square inches, we need to standardize the values using the z-score formula and then find the area under the normal curve between those two values:
z1 = (15.975 - 16) / 0.008 = -3.125
z2 = (16.01 - 16) / 0.008 = 1.25
Using a standard normal table or calculator, we find the area between these two z-scores to be approximately 0.8278.
Therefore, the percent of tiles with an area between 15.975 and 16.01 square inches is 82.78%.
4. To find the probability that a tile will have an area that is more than 16.025 square inches, we again need to standardize the value using the z-score formula and then find the area under the normal curve to the right of that value: z = (16.025 - 16) / 0.008 = 3.125
Using a standard normal table or calculator, we find the area to the right of this z-score to be approximately 0.0008.
Therefore, the probability that a tile will have an area that is more than 16.025 square inches is 0.08%. Yes, this is unusual because the probability is very low, indicating that it is highly unlikely for a tile to have an area greater than 16.025 square inches.
5. To find the number of tiles that can be expected to have a surface area less than 15.985 square inches, we first need to standardize the value using the z-score formula:
z = (15.985 - 16) / 0.008 = -1.875
Using a standard normal table or calculator, we find the probability to the left of this z-score to be approximately 0.0304.
Therefore, out of 10,000 tiles, we can expect approximately 0.0304 * 10,000 = 304 tiles to have a surface area less than 15.985 square inches.
6. To find the number of tiles that must be produced to ensure that the manufacturer has 70,000 tiles that are between 15.98 and 16.02 square inches, we need to find the z-scores for the lower and upper bounds of this range:
z1 = (15.98 - 16) / 0.008 = -2.5
z2 = (16.02 - 16) / 0.008 = 2.5
Using a standard normal table or calculator, we find the probability between these two z-scores to be approximately 0.4938.
Therefore, out of x tiles, we can expect approximately 0.4938x tiles to have a surface area between 15.
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Find the domains of the following functions:
f(x) = 2x + 7.
h(x) = (x-6)/(2x - 8)
The domain of the given function f(x) = 2x + 7 is -∞ < x < ∞.
What is a domain?The range of values that we are permitted to enter into our function is known as the domain of a function.
Consider the function y = f(x), which has the independent variable x and the dependent variable y.
A value for x is said to be in the domain of a function f if it successfully allows the production of a single value y using another value for x.
So, the given function is:
f(x) = 2x + 7
To find the domain:
There are no undefined places or domain restrictions in the function. As a result, -∞ < x < ∞ is the domain.
Therefore, the domain of the given function f(x) = 2x + 7 is -∞ < x < ∞.
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Correct question:
Find the domains of the following function: f(x) = 2x + 7
y varies inversely with x.
y = 15when x = 10
Find y when x = 5
Answer:
y=10
Step-by-step explanation:
Do X4 and 15+ X have the same value when X is 5
Select the description of the graph created by the equation 3x2 – 6x + 4y – 9 = 0. Parabola with a vertex at (1, 3) opening left. Parabola with a vertex at (–1, –3) opening left. Parabola with a vertex at (1, 3) opening downward. Parabola with a vertex at (–1, –3) opening downward.
A parabola with a vertex at (1,3) and an opening downhill is depicted by the equation.
Describe a curve.A parabola is an equation of a curve with a spot on it that is equally spaced from a fixed point and a fixed line.
In mathematics, a parabola is a roughly U-shaped, mirror-symmetrical plane circle. The same curves can be defined by a number of apparently unrelated mathematical descriptions, which all correspond to it. A point and a line can be used to depict a parabola.
Equation given: 3x² - 6x + 4y - 9 = 0. When the given equation's graph is plotted, it is discovered that the parabola that is created is opened downward and has a vertex at the spot. ( 1,3). The graph and the following response are attached.
The equation that depicts a parabola with a vertex at (1,3) opening downward is option C, making it the right choice.
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Answer:
Parabola with a vertex at (1, 3) opening downward.
Step-by-step explanation:
Please help ASAP!!!! The average of two numbers is 13.One number is 10.What is the other number?
Answer:16
Step-by-step explanation:
average is similar to mean
you get the Total value and divide it with the number of values added
please help fast!! brainliest!! Find the slope of a line perpendicular to the line whose equation is 4x − 6y = −24
Fully simplify your answer.
The slope of the sole sequence's perpendicular line is [tex]-\frac{3}{2}[/tex].
What is the perpendicular direction?As two lines meet at right angles, the word "perpendicular" refers to an angle. Every direction, including up and down, across, and side to side, can be faced by a pair of perpendicular lines.
Is a straight line considered to be perpendicular?A perpendicular is a straight line in mathematics that forms a correct angle (90 °) with another line. In other words, two lines are parallel to one another if they connect at a right angle.
[tex]y = mx + b[/tex], where [tex]m[/tex] is the slope:
[tex]4x - 6y = -24[/tex]
[tex]-6y = -4x - 24[/tex]
[tex]y = (4/6)x + 4[/tex]
[tex]y = (2/3)x + 4[/tex]
So the slope of the given line is [tex]2/3[/tex].
To find the slope of a line perpendicular to this line, we need to take the negative reciprocal of [tex]2/3[/tex]:
[tex]-1/(2/3) = -3/2[/tex]
Therefore, the slope of a line perpendicular to the given line is [tex]-\frac{3}{2}[/tex].
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X and Y are independent exponential RVs with parameters 1 and 2, respectively. What is the joint CDF of X and Y? OFxy (a,b) = -4e-26, a > 0,6> 0 OFxy (a, b) = 2e-e-2 , a > 0,6 > 0 O Fxy (a,b) = (1 - e-a) (1 - e-26), a > 0, 6 > 0
The joint CDF of X and Y, two independent exponential random variables with parameters 1 and 2, respectively, is Fxy(a,b) = (1 - e^(-a))(1 - e^(-2b)), where a > 0 and b > 0. The correct answer is C).
Since X and Y are independent, the joint CDF of X and Y is the product of their marginal CDFs:
FXY(a,b) = FX(a)FY(b)
where FX and FY are the CDFs of X and Y, respectively.
The CDF of an exponential distribution with parameter λ is given by:
FX(x) = 1 - e^(-λx)
Therefore, the marginal CDFs of X and Y are:
FX(a) = 1 - e^(-a), λ = 1
FY(b) = 1 - e^(-2b), λ = 2
Taking the product, we get:
FXY(a,b) = FX(a)FY(b) = (1 - e^(-a))(1 - e^(-2b))
Therefore, the answer is:
Fxy (a,b) = (1 - e^(-a))(1 - e^(-2b)), a > 0, b > 0.
The correct option is C).
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1. The population of California was 10,586,223 in 1950 and 23,668,562 in 1980. Assume the population grows exponentially. a) Find an exponential function that models the population, P, in terms of t, the years after 1950. b) Use your model, to predict the population of California in 2020.
An exponential function can model California's population growth from 1950 to 1980. The predicted population in 2020 is around 43.3 million (43,301,300), assuming exponential growth.
To find an exponential function that models the population, we can use the formula:
P(t) = P0 * e^(rt)
where P0 is the initial population, r is the growth rate, and t is the time in years. We can use the information given to find P0, r, and then plug in t = 0 to find the exponential function.
P0 = 10,586,223 (initial population in 1950)
P(30) = 23,668,562 (population in 1980, 30 years later)
t = 30
P(0) = P0 * e^(0*r) = P0 (population in 1950)
So we have:
23,668,562 = 10,586,223 * e^(30r)
Dividing both sides by 10,586,223:
e^(30r) = 2.234
Taking the natural logarithm of both sides:
30r = ln(2.234)
r = ln(2.234)/30
Now we can use this value of r to find the exponential function:
P(t) = 10,586,223 * e^(t*ln(2.234)/30)
To predict the population in 2020, we can plug in t = 70 (since 2020 is 70 years after 1950) into the function we just found:
P(70) = 10,586,223 * e^(70*ln(2.234)/30) ≈ 43,301,300
Therefore, the predicted population of California in 2020 is approximately 43,301,300.
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Bria is a customer who would like to display her collection of soap carvings on top of her bookcase. The collection needs an area of 300 square inches. What should b equal for the top of the bookcase to have the correct area? Round your answer to the nearest tenth of an inch.
I need help D:please !!!!
If the top of Bria's bookcase has a length-to-width ratio of 3:1, the width of the top should be about 10 inches to provide an area of 300 square inches for her soap carving collection.
How to find the dimension of the top of the bookcaseThe top of the bookcase is a rectangle, the area of the top can be expressed as the product of its length and width:
Area of the top = length × width
Area of the top = 300 square inches
length × width = 300 square inches
width = 300 square inches / length
Assuming that the top of the bookcase is with a length-to-width ratio of
3 : 1
length = 3 × width
substituting the value gives
width = 300 / (3 × width)
Simplifying, we have:
width² = 100 square inches (the polynomial)
Taking the square root of both sides, we get:
width = √100 ≈ 10 inches
Then the length will be = 30 inches
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The functions a ( t ) = 8.75 ⋅ ( 1 + 0.01 ) t and b ( t ) = 8.75 ⋅ e ( 0.01 t ) each model the population of a city, in thousands, t years after 2010. Describe how each model predicts that the population in the town will grow.
Using the functions, we can predict that the population in the town will grow 't' times as t is directly proportional to the function.
Define a function?A function in mathematics is a connection between two x and y values that come from different sets. These four types of correspondence are all possible. Yet not every correspondence serves a purpose.
The following are the results of a survey conducted by the National Institute of Standards and Technology (NIST) on the effectiveness of the standardised testing process. So, we state that a function can only have one input and one output. If we are given any x, the only y that can be coupled with that x is the one and only y. Two outputs cannot be connected to the same function.
Here in the given question,
The functions a(t) = 8.75 × (1+0.01) t and
b(t) = 8.75 × e × (0.01t) here are dependent on the variable t, where t is the time in years after 2010.
So, as t will increase after 2010, the value of a(t) and b(t) will increase with t.
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