The correct answer is d)C(x)=4x+60. This is because an equation should have an equal sign to show that both sides of the equation are equal.
Additionally, the variable "x" should be on one side of the equation and the constants should be on the other side. In this case, the variable "x" is multiplied by 4 and the constant is 60, making the equation C(x)=4x+60. This equation shows the relationship between the variable "x" and the function "C(x)".
It is important to note that the other options are not correct because they do not have an equal sign or the variable and constants are not on the correct sides of the equation. Option a)C(x)−60x+4 does not have an equal sign, option b)C(x)=−60x+4 has the variable and constants on the wrong sides of the equation, option c)C(x)−−4x+60 does not have an equal sign, and option e)C(x)−4x+4 does not have an equal sign.
Therefore, the correct answer is d)C(x)=4x+60.
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Gemma makes a conjecture that in the sum of an eye and the jersey, and is selfish, always an even integer. Which choice is the best proof of her conjecture
As per the concept of integer, the best proof of her conjecture is Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even. (option b)
Now, let's look at the given choices to find the best proof for Gemma's conjecture.
Choice B) Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even.
This choice provides a proof by using algebraic equations. It begins by defining an odd integer as 2n + 1, where n is any integer. Then, it uses algebraic manipulation to show that the sum of this odd integer and itself results in an even integer. Specifically, (2n + 1) + (2n + 1) simplifies to 4n + 2, which is equal to 2(2n + 1). This proves that the sum of an odd integer and itself is always even.
This choice is a tautology, which means that it's always true, but it doesn't provide any evidence or proof to support Gemma's conjecture.
The best proof for Gemma's conjecture is choice B.
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Complete Question:
Gemma makes a conjecture that the sum of an odd integer and itself is always an even integer. Which choice is the best proof of her conjecture?
A) Look at these different examples: 7 + 7 = 14, 13 + 13 = 26, 23 + 23 = 46. So the sum of an odd number and itself must be even
B) Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even.
C) Let n represent an odd number, and let n + n be an even number. Therefore, n + n = n + n, which shows that the sum of an odd number and itself is even.
D) Every time you add an odd number and itself, the sum is an even number.
The yearly income of a family is Rs. 500000. The ratio of the expenditure and saving of the family is 4 : 1. Find the amount of expenditure and saving.
Answer:
Let's assume that the amount of saving is x.
According to the problem, the ratio of expenditure to saving is 4:1, so the amount of expenditure can be expressed as 4x.
The total income of the family is Rs. 500000, and it can be expressed as the sum of expenditure and saving:
Expenditure + Saving = 500000
Substituting the values of expenditure and saving, we get:
4x + x = 500000
Simplifying this equation, we get:
5x = 500000
Dividing both sides by 5, we get:
x = 100000
Therefore, the amount of saving is Rs. 100000, and the amount of expenditure is 4 times this value, which is Rs. 400000.
If the transformation from f to g is such that f(x)=e^(x) is verticall shrunk by a factor of ( 1)/(3) to make g(x), what is g(x) ?
After the transformation from f to g, g(x) = eˣ/3.
Vertical shrink refers to a transformation of a function that causes it to be compressed vertically. To perform a vertical shrink, you must multiply the output (y) values of the function by a constant value between 0 and 1.
The transformation of f(x) to g(x) involves a vertical shrink by a factor of 1/3. This means that the value of g(x) will be one third of the value of f(x). We can write this transformation as: g(x) = 1/3 · f(x).
Since f(x) = eˣ, we can substitute this into the equation for g(x) to find the final expression for g(x):
g(x) = (1/3)eˣ
Therefore, the function g(x) after the vertical shrink by a factor of 1/3 is g(x) = (1/3)eˣ or g(x) = eˣ/3.
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Leon asks his classmates how many mystery books and how many adventure books they own. He states that because the mean number of mystery books, 5, is less than the mean number of adventure books, 10, there is less variability in the number of mysstery books. Do you agree?
To make any conclusions about the variability of the two datasets.
What is variable?a variable is a symbol or letter that represents a value that can change or vary. Variables are used to express mathematical relationships and equations, and they can take on different values depending on the context.
by question.
No, we cannot conclude that there is less variability in the number of mystery books based solely on the mean number of mystery books being less than the mean number of adventure books. The variability of the data is captured by the standard deviation or variance, and we would need this information to make any conclusion about the variability of the two types of books. It's possible that there is less variability in the number of mystery books,
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QUESTION 4 A and B are events with P(A) = 0.5 , P(BA)=0.6, and P(BIA) = = 0.8. Find P(AB) (if rounding is necessary, give answers to within 1% of the exact value). QUESTION 5 A and B are events with P(A) = 0.2, P(B) = 0.6 and P( BA) = 0.1. Find P(B|Ā), giving answers to within 1% of the exact value. QUESTION 6 + = X is a continuous random variable with probability density function 17/(x + 1), X>0 f(x)= 0, X50. Which of the following is the cumulative distribution function of X? O F(x) = 1-1/(x+1)^7, x>0 O F(x) = 56/(x+1)^9, x>0 O F(x) = -1/(x+1)^7, x>0 =
The probability of both events A and B occurring is 0.3 or 30%.
The cumulative distribution function of X is F(x) = 17 * ln(x + 1) for x > 0.
The probability of event B occurring given that event A does not occur is 0.125 or 12.5%.
ANSWER 4:
We can use the formula P(AB) = P(A) * P(B|A) to find the probability of both events A and B occurring.
P(AB) = P(A) * P(B|A)
P(AB) = 0.5 * 0.6
P(AB) = 0.3
So the probability of both events A and B occurring is 0.3 or 30%.
ANSWER 5:
We can use the formula P(B|A) = P(AB) / P(A) to find the probability of event B occurring given that event A does not occur. We can also use the formula P(A') = 1 - P(A) to find the probability of event A not occurring.
P(A') = 1 - P(A)
P(A') = 1 - 0.2
P(A') = 0.8
P(B|A') = P(BA') / P(A')
P(B|A') = 0.1 / 0.8
P(B|A') = 0.125
So the probability of event B occurring given that event A does not occur is 0.125 or 12.5%.
ANSWER 6:
The cumulative distribution function (CDF) of a continuous random variable X is defined as F(x) = P(X <= x). To find the CDF of X, we need to integrate the probability density function (PDF) of X from 0 to x.
F(x) = ∫ f(t) dt from 0 to x
F(x) = ∫ (17/(t + 1)) dt from 0 to x
F(x) = 17 * ln(t + 1) from 0 to x
F(x) = 17 * ln(x + 1) - 17 * ln(0 + 1)
F(x) = 17 * ln(x + 1)
So the cumulative distribution function of X is F(x) = 17 * ln(x + 1) for x > 0.
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Si al salario por semana se le rebaja el 9. 5% para ccss y el salario es de 50. 000 ¿cuánto me pagan?
If the weekly salary is reduced by 9.5% for CCSS and the salary is 50000, then they would pay $45250.
The percent reduction in the weekly salary is = 9.5% = 0.095,
So, the amount of the reduction is;
⇒ reduction = 9.5% of 50,000
⇒ reduction = 0.095 x 50,000
⇒ reduction = 4750
So, the reduction in the weekly salary is 4750,
Now, we subtract the reduction from the original salary to get the new salary after the reduction,
⇒ new salary = 50000 - 4750
⇒ new salary = 45250
Therefore, the new salary after the 9.5% reduction for CCSS is $45250.
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Please refer to this output. Method DE H-Value P-value
Not adjusted for ties 2 0.01 0.994
Adjusted for ties 2 0.01 0.994 Ten cities are selected, and the number of daily passenger trips (in thousands) for subways and commuter rail service is obtained. At alpha equal to 0.05, the researcher wants to check the strength of relationship between the variables. Assume that data is normally distributed For the next questions, please refer to this problem.
What does the test for significant relationship tell you? 1 point a. There is a significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service? b. The simple linear regression equation conforms to the strength of relationship defined. c. It is good to proceed with defining the simple linear regression equation, d. There is no significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service?
There is a significant relationship between the variables (H-value = 2 and P-value = 0.994)
The test for significant relationship is used to determine whether there is a statistically significant relationship between two or more variables. In this case, the test can be used to determine whether there is a statistically significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service. Based on the H-value and P-value, the result of this test indicates that there is a significant relationship between the variables (H-value = 2 and P-value = 0.994). Therefore, option (a) is correct and option (d) is incorrect.
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calculate the areas of the rectangles having the following measurements
Perimeter(p)=220cm and breadth(b)=45cm
Answer:
it is 2,475
Step-by-step explanation:
In triangle FGH shown below, what is the measure of ∠
F in degrees?
Record your answer on the keypad.
Answer:
∠F = 19.5
Step-by-step explanation:
180 - 127 - 33.5 = 19.5
For A-C, choose Yes or No to indicate whether or not each expression has a value greater than 6.
29
√9
B. 4+ √3
A.
C. 6.4 -
D. 27
18
√27
OYes ONO
OYes No
OYes No
OYes No
Plot the points A(-6,-4), B(3, -8), C(7, 1) on the coordinate axes below. State the coordinates of point � D such that � A, � B, � C, and � D would form a rectangle.
The coordinates of point D are (-2,5) and the points given have been plotted. The solution has been obtained by using the distance formula.
What is distance formula?
The distance between two points can be calculated using the xy-plane distance formula.
We are given that ABCD should form a rectangle.
We know that opposite sides of a rectangle are equal.
So, AB = CD
Using the distance formula which is
d = √(x₂ - x₁)² + (y₂ - y₁)²
We get
⇒AB = √(3 - (-6))² + (-8 - (-4))²
⇒AB = √(3 + 6)² + (-8 + 4)²
⇒AB = √(9)² + (-4)²
⇒AB = √81 + 16
⇒AB = √97 ...(1)
Let the coordinates of Point D be (x, y)
So, similarly
⇒CD = √(x - 7)² + (y - 1)² ...(2)
On equating (1`) and (2), we get
⇒√97 = √(x - 7)² + (y - 1)²
On squaring both sides, we get
⇒97 = (x - 7)² + (y - 1)²
⇒97 = x² + 49 - 14x + y² + 1 - 2y
⇒97 = x² + 50 - 14x + y² - 2y
⇒47 = x² - 14x + y² - 2y ...(3)
Also, AC = BD as these are the opposite sides of the rectangle.
Using the distance formula, we get
⇒AC = √(7 - (-6))² + (1 - (-4))²
⇒AC = √(7 + 6)² + (1 + 4)²
⇒AC = √(13)² + (5)²
⇒AC = √169 + 25
⇒AC = √194 ...(4)
Similarly,
⇒BD = √(x - 3)² + (y - (-8))²
⇒BD = √(x - 3)² + (y + 8)² ...(5)
On equating (4`) and (5), we get
⇒√194 = √(x - 3)² + (y + 8)²
On squaring both sides, we get
⇒194 = (x - 3)² + (y + 8)²
⇒194 = x² + 9 - 6x + y² + 64 + 16y
⇒194 = x² + 73 - 6x + y² + 16y
⇒121 = x² - 6x + y² + 16y ...(6)
On subtracting (3) and (6), we get
8x + 18y = 74
On dividing by 2, we get
4x + 9y = 37
From this, we get
x = (37 - 9y) / 4
Substituting this in (3), we get
⇒47 = [(37 - 9y) / 4]² - [14 * [(37 - 9y) / 4]] + y² - 2y
On solving this, we get
y = 5
On substituting this in x, we get
x = (37 - 9(5)) / 4
x = (37 - 45) / 4
x = -8 / 4
x = -2
Hence, the coordinates of point D are (-2,5) and the points given have been plotted.
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Fluffy, Spot, and Shampy have a combined age in dog years of 82. Spot is 14 years younger than Fluffy. Shampy is 6 years older than Fluffy. What is Fluffy's age, f, in dog years?
The age of fluffy is 30 in dog years. The solution has been obtained by using linear equation.
What is a linear equation?One is the degree of the linear equation. The absence of variables in linear equations with exponents greater than one is obvious. The graph's equation results in a straight line.
We are given that combined age of 3 dogs in dog years is 82.
Let Fluffy's age be 'f'.
Age of Spot = (f - 14)
Age of Shampy = (f + 6)
From this, we get
f + (f - 14) + (f + 6) = 82
On solving this, we get
⇒f + f - 14 + f + 6 = 82
⇒3f - 8 = 82
⇒3f = 90
⇒f = 30
Hence, the age of fluffy is 30 in dog years.
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What is the measure of "AC"?
Enter your answer in the box.
Will give Brainiest if right. and to help other people thx.
Answer: 21 degrees
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Graph the solution to the following system of inequalities.
2x +7vs - 14
-3x +5y> 5
Then give the coordinates of one point in the solution set.
Point in the solution set: (П.П)
A solution to the given system of linear inequalities is (-6, -1).
How to graph the solution to this system of inequalities?In order to to graph the solution to the given system of inequalities on a coordinate plane, we would use an online graphing calculator to plot the given system of inequalities and then take note of the point of intersection;
2x + 7y ≤ -14 .....equation 1.
-3x + 5y > 5 .....equation 2.
Based on the graph (see attachment), we can logically deduce that the solution to the given system of inequalities is the shaded region below the solid and dashed line, and the point of intersection of the lines on the graph representing each, which is given by the ordered pair (-6, -1).
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explain and develop this
quote
Human suffering anywhere concerns men and women
everywhere.
Elie Wiesel, Night
Elie Wiesel's quote speaks to the idea that human suffering is a universal concern and that we all have a responsibility to respond to it. It is a call to action for all of us to be aware of the suffering of others and to do what we can to alleviate it.
The quote "Human suffering anywhere concerns men and women everywhere" by Elie Wiesel in his book Night speaks to the idea that suffering is not an isolated event. Rather, it is something that affects all of humanity, regardless of where it occurs. This quote highlights the interconnectedness of humanity and the responsibility we all have to care for one another.
Wiesel, a Holocaust survivor, experienced unimaginable suffering during his time in concentration camps. Through his writing, he is able to convey the importance of recognizing and responding to the suffering of others. This quote serves as a reminder that we cannot turn a blind eye to the suffering of others, even if it is happening in a different part of the world.
In conclusion, Elie Wiesel's quote speaks to the idea that human suffering is a universal concern and that we all have a responsibility to respond to it. It is a call to action for all of us to be aware of the suffering of others and to do what we can to alleviate it.
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real zero, Including any repeated zeroes? Choose your answ f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4
The real zeroes of the function f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4 are -4, -1, and 1.
The real zeroes of the function f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4 can be found by using the Rational Zero Theorem and synthetic division.
The Rational Zero Theorem states that the possible rational zeroes of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is 4 and the leading coefficient is 1, so the possible rational zeroes are ±1, ±2, and ±4.
We can use synthetic division to test these possible zeroes and find the actual zeroes of the function. Synthetic division is a method of dividing a polynomial by a linear factor of the form x-a. The result of synthetic division is a quotient and a remainder, and if the remainder is 0, then x-a is a factor of the polynomial and a is a zero of the function.
Using synthetic division, we find that the real zeroes of the function are -4, -1, and 1. These are the values of x that make the function equal to 0. There are no repeated zeroes in this case.
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What is the least common denominator for his problem? A, b, c, or d
Answer:
the answer is "c"! :)
Step-by-step explanation:
Curt and Melanie are mixing blue and yellow paint to make seafoam green paint.use the percent equation to find how much yellow paint they should use
The percentage of yellow paint that has to be used is given as 0.45
How to find the amount of yellow paint that is to be used hereThe amount of seafoam paint is given as 1.5 quartz
The percentage of yellow paint in the seafoam paint is 30 percent
Hence the amount that would be in it that would be made of yellow paint is given as
1.5 x 30%
1.5 x 0.30
= 0.45
Hence we would conclude by saying that the amount of yellow paint that has to be used in order to make the paint should be 0.45
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Curt and Melanie are mixing blue and yellow paint to make seafoam green paint. 1.5 quarts of seafoam paint is made of 70% blue, 30% yellow. Use the percent equation to find how much yellow paint they should use.
Find the area of the trapezoid. Leave your answer in simplest radical form.
a) 156 ft^2
b) 78 ft^2
c) 78 √ 2 ft^2
d) 13 ft^2
The area of the trapezoid in simplest radical form is 78 feet².
Given a trapezoid.
Length of the bases are 10 feet and 16 feet.
We have to find the height.
Consider the smaller right triangle formed by the height of the trapezoid.
Triangle base length or one leg = 16 - 10 = 6 feet
Since one of the angle is 45°, the other angle in the right triangle is also 45°.
Since it is isosceles, opposite sides for 45° angles are same.
Other leg = 6 feet, which is the height.
Area of the trapezoid = 1/2 (a + b)h, where a and b are bases and h is the height.
A = 1/2 (10 + 16) 6
= 78 feet²
Hence the correct option is b.
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1- Indicate the transformations to f(x)
a) y = −0.5f(−0.5x) − 1
b) y = 2f(x − 5) + 4
c) y = 3f (−2(x + 1))
2- Indicate the transformations to f(x) = √x
It is a vertical stretching by a factor of 2 followed by a vertical shift of 1 unit.
1- To transform f(x) in the given equations, the following transformations need to be done:
a) y = −0.5f(−0.5x) − 1 is a transformation by vertical stretching by a factor of 0.5, followed by a horizontal compression by a factor of 0.5, and then a vertical shift of 1 unit.
b) y = 2f(x − 5) + 4 is a transformation by horizontal shifting of 5 units to the left, followed by a vertical stretching by a factor of 2, and then a vertical shift of 4 units.
c) y = 3f (−2(x + 1)) is a transformation by horizontal shifting of 1 unit to the right, followed by a horizontal compression by a factor of 2, and then a vertical stretching by a factor of 3.
2- To transform f(x) = √x, it is a vertical stretching by a factor of 2 followed by a vertical shift of 1 unit.
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4. Show how you find the average rate of change off(x)=2 x^{2}-9over the interval [4,6] .
The average rate of f(x), in [4,6], is equal to 20.
To find the average rate of change of a function over an interval, we use the formula:
Average rate of change = (f(b) - f(a))/(b - a)
Where f(x) is the function, a and b are the endpoints of the interval, and f(a) and f(b) are the values of the function at those endpoints.
In this case, our function is f(x) = 2x² - 9, and our interval is [4, 6]. So, we plug in the values into the formula:
Average rate of change = (f(6) - f(4))/(6 - 4)
First, we find f(6) and f(4):
f(6) = 2(6²) - 9 = 2(36) - 9 = 72 - 9 = 63
f(4) = 2(4²) - 9 = 2(16) - 9 = 32 - 9 = 23
Now we plug these values back into the formula:
Average rate of change = (63 - 23)/(6 - 4) = 40/2 = 20
In conclusion, the average rate is worth 20.
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Aisling runs 2km on Monday and three times that many on Tuesday. If she wants to run 20km this week. How many km does she need to run
Answer:12km
Step-by-step explanation:
She ran 2km on Monday, and three times as much on Tuesday so we do 2 x 3 which equals 6. So 6km, plus the original 2km is 8km (6+2) is she wants to run 20km this week, then she needs to subtract what shes already ran to find out how many more km she needs to run. 20-8=12 so 12km.
3)(9pts)Use Cramer's rule to find the values of the variablesx,y, andz( 3 points each). Show the determinants involved clearly and expand the numerator determinants as follows to get any credit: For variablex, expand along the first column; for variabley, expand along the second column; for variablez, expand along the third column. You will receive 0 points if you expand these determinants any other way, even if your answer is correct.−3x+0y−6z8x−2y+3z2x−y−4z=11=17=3
The values of the variables x, y, and z obtained by using Cramer's rule are 83/105, -277/105, and -99/105, respectively.
To find the values of the variables x, y, and z using Cramer's rule, we need to find the determinant of the coefficient matrix and the determinants of the matrices obtained by replacing the first, second, and third columns of the coefficient matrix with the constant terms.
The coefficient matrix is:
|−3 0 −6|
| 8 −2 3|
| 2 −1 −4|
The determinant of the coefficient matrix is:
|−3 0 −6| = (−3)(−2)(−4) − (0)(3)(2) − (−6)(8)(−1) − (−3)(−1)(3) − (0)(−4)(8) − (−6)(−2)(2)
= −24 − 0 − 48 − 9 − 0 − 24
= −105
The matrix obtained by replacing the first column with the constant terms is:
|11 0 −6|
|17 −2 3|
| 3 −1 −4|
The determinant of this matrix is:
|11 0 −6| = (11)(−2)(−4) − (0)(3)(3) − (−6)(17)(−1) − (11)(−1)(3) − (0)(−4)(17) − (−6)(−2)(3)
= 88 − 0 − 102 − 33 − 0 − 36
= −83
The matrix obtained by replacing the second column with the constant terms is:
|−3 11 −6|
| 8 17 3|
| 2 3 −4|
The determinant of this matrix is:
|−3 11 −6| = (−3)(17)(−4) − (11)(3)(2) − (−6)(8)(3) − (−3)(3)(3) − (11)(−4)(2) − (−6)(17)(2)
= 204 − 66 − 144 − 9 + 88 + 204
= 277
The matrix obtained by replacing the third column with the constant terms is:
|−3 0 11|
| 8 −2 17|
| 2 −1 3|
The determinant of this matrix is:
|−3 0 11| = (−3)(−2)(3) − (0)(17)(2) − (11)(8)(−1) − (−3)(−1)(17) − (0)(3)(8) − (11)(−2)(2)
= 18 − 0 + 88 − 51 − 0 + 44
= 99
Now we can use Cramer's rule to find the values of the variables x, y, and z:
x = (−83)/(-105) = 83/105
y = (277)/(-105) = -277/105
z = (99)/(-105) = -99/105
83/105, -277/105, and -99/105 are the values of x,y and z respectively.
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What should be reason 8 in the following proof?
Answer: I think putting Prop of ║lines might work. I haven't done this in a long time, so I'm most likely wrong, but that might help. I'm so sorry.
3
The drama club at Hawthorne Middle School is selling tickets to their spring musical. Student tickets
cost $10 and adult tickets cost $15. Last week, they sold 120 tickets for the Sunday matinee show. If those
ticket sales totaled to $1,400, how many adult tickets were sold? How many student tickets were sold?
40 adult tickets were sold and 80 student tickets were sold.
What is an equation?
An equation is a mathematical statement that asserts that two expressions are equal. It typically consists of variables, constants, and mathematical operations.
Let's use algebra to solve this problem.
Let's call the number of student tickets sold "s" and the number of adult tickets sold "a".
From the problem, we know two things:
The total number of tickets sold was 120:
s + a = 120
The total amount of money made from ticket sales was $1,400:
10s + 15a = 1400
Now we have two equations with two variables, so we can solve for "a" and "s".
Let's start by solving for "s" in the first equation:
s + a = 120
s = 120 - a
Now we can substitute this expression for "s" into the second equation:
10s + 15a = 1400
10(120 - a) + 15a = 1400
1200 - 10a + 15a = 1400
5a = 200
a = 40
So 40 adult tickets were sold.
Now we can substitute this value of "a" into the first equation to find "s":
s + a = 120
s + 40 = 120
s = 80
So 80 student tickets were sold.
Therefore, 40 adult tickets were sold and 80 student tickets were sold.
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Big ideas 7.5 (question)
1) A perpendicular bisector in kite ABCD is; BD
2) An isosceles triangle in kite ABCD is; ΔABC
3) A right triangle in kite ABCD is; ΔABM
What is a Kite?In geometry, a kite is defined as a quadrilateral with reflection symmetry across a diagonal.
here, we have,
1) A perpendicular bisector is defined as a line segment which bisects another line segment at 90 degrees.
Looking at the diagram, Line BD bisects Line AC and as such BD is the perpendicular bisector.
2) An Isosceles triangle is defined as a a triangle in which two sides have the same length.
In this case, in Triangle ABC, AB and BC have the same length and as such Triangle ABC is the Isosceles Triangle.
3) A right triangle is defined as a triangle, that has one of its interior angles equal to 90 degrees or any one angle is a right angle.
In this case if Mis the intersection of BD and AC, then the right angle triangle is ABM
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(3 points) Problem 5: Determine the value(s) ofasuch that [1a], [aa+2] are linearly independent.
To determine the value(s) of a such that [1 a], [a a+2] are linearly independent, we need to find the values of a that make the determinant of the matrix non-zero. The determinant of a 2x2 matrix is given by:
|1 a|
|a a+2| = (1)(a+2) - (a)(a) = a + 2 - a^2
To make the determinant non-zero, we need to solve the equation:
a + 2 - a^2 ≠ 0
Rearranging the equation, we get:
a^2 - a - 2 ≠ 0
Factoring the equation, we get:
(a - 2)(a + 1) ≠ 0
Therefore, the values of a that make the determinant non-zero are a ≠ 2 and a ≠ -1. These are the values of a that make the vectors [1 a], [a a+2] linearly independent.
So, the value(s) of a such that [1 a], [a a+2] are linearly independent are a ≠ 2 and a ≠ -1.
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In his motorboat, Bill Ruhberg travels upstream at top speed to his favorite fishing spot, a distance of 90 mi, in 3 hr. Returning, he finds that the trip downstream, still at top speed, takes only 2.5 hr. Find the rate of Bill's boat and the speed of the current. Let x = the rate of the boat in still water and y = the rate of the current.
The rate of Bill's boat in still water is 33 mph and the speed of the current is 3 mph.
To find the rate of Bill's boat and the speed of the current, we can use the distance formula, which states that distance = rate × time. Since we know the distance and time for both the upstream and downstream trips, we can set up two equations and solve for the rate of the boat and the speed of the current.
Let x = the rate of the boat in still water and y = the rate of the current.
For the upstream trip:
90 = (x - y) × 3
For the downstream trip:
90 = (x + y) × 2.5
Simplifying the equations gives us:
90 = 3x - 3y
90 = 2.5x + 2.5y
Multiplying the first equation by 2.5 and the second equation by 3 gives us:
225 = 7.5x - 7.5y
270 = 7.5x + 7.5y
Adding the two equations together eliminates the y variable:
495 = 15x
Solving for x gives us:
x = 33
Substituting x back into the first equation gives us:
90 = (33 - y) × 3
90 = 99 - 3y
3y = 9
y = 3
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Let a, b, and c be elements of a field F.Prove:
(1) If a + b = c + b, then a = c.
(2) If ab = cb and b doesn't equal 0, then a = c.
Answer:
The given statements can be proved as follows:
(1) If a + b = c + b, then a = c.
Proof:
- Start with the given equation: a + b = c + b
- Subtract b from both sides of the equation: a + b - b = c + b - b
- Simplify the equation: a = c
- Therefore, if a + b = c + b, then a = c.
(2) If ab = cb and b doesn't equal 0, then a = c.
Proof:
- Start with the given equation: ab = cb
- Divide both sides of the equation by b: ab/b = cb/b
- Simplify the equation: a = c
- Therefore, if ab = cb and b doesn't equal 0, then a = c.
In both cases, the elements a and c are equal when the given conditions are met.
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help :(( What should be reason 5 in the following proof?
Answer:
Reason 5 should be "Alternate Interior Angles Converse".
Step-by-step explanation: