Answer:
"t3" was replaced by "t^3".
1 more similar replacement(s).Step by step solution :
STEP 1:Equation at the end of step 1 (((9 • (t6)) - (2•7t3)) + 4t) - 1 = 0
STEP 2 :Equation at the end of step 2: ((32t6 - (2•7t3)) + 4t) - 1 = 0
STEP 3:Checking for a perfect cube 3.1 9t6-14t3+4t-1 is not a perfect cube
Trying to factor by pulling out : 3.2 Factoring: 9t6-14t3+4t-1
Thoughtfully split the expression at hand into groups, each group having two terms:
Group 1: 4t-1
Group 2: -14t3+9t6
Pull out from each group separately:
Group 1: (4t-1)
(1)Group 2: (9t3-14) • (t3)
Factoring by pulling out fails : The groups have no common factor and can not be added up to form a multiplication.Polynomial Roots Calculator : 3.3
Find roots (zeroes) of : F(t) = 9t6-14t3+4t-1
Polynomial Roots Calculator is a set of methods aimed at finding values of t for which F(t)=0 Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 9 and the Trailing Constant is -1. The factor(s) are: of the Leading Coefficient : 1,3 ,9 of the Trailing Constant : 1 Let us test .... P Q P/Q F(P/Q) Divisor -1 1 -1.00 18.00 -1 3 -0.33 -1.80 -1 9 -0.11 -1.43 1 1 1.00 -2.00 1 3 0.33 -0.17 1 9 0.11 -0.57 Polynomial Roots Calculator found no rational roots Equation at the end of step 3: 9t6 - 14t3 + 4t - 1 = 0
STEP 4:Equations of order 5 or higher: 4.1 Solve 9t6-14t3+4t-1 = 0In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.Method of search: Calculate polynomial values for all integer points between t=-20 and t=+20 Found change of sign between t= -1.00 and t= 0.00 Approximating a root using the Bisection Method :We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).The function is F(t) = 9t6 - 14t3 + 4t - 1At t= 0.00 F(t) is equal to -1.00 At t= -1.00 F(t) is equal to 18.00 Intuitively we feel, and justly so, that since F(t) is negative on one side of the interval, andpositive on the other side then, somewhere inside this interval, F(t) is zero Procedure :(1) Find a point "Left" where F(Left) < 0(2) Find a point 'Right' where F(Right) > 0(3) Compute 'Middle' the middle point of the interval [Left,Right](4) Calculate Value = F(Middle)(5) If Value is close enough to zero goto Step (7)Else : If Value < 0 then : Left <- MiddleIf Value > 0 then : Right <- Middle(6) Loop back to Step (3)(7) Done!! The approximation found is MiddleFollow Middle movements to understand how it works : Left Value(Left) Right Value(Right)
0.000000000 -1.000000000 -1.000000000 18.000000000
0.000000000 -1.000000000 -1.000000000 18.000000000
-0.500000000 -1.109375000 -1.000000000 18.000000000
-0.500000000 -1.109375000 -0.750000000 3.508056641
-0.500000000 -1.109375000 -0.625000000 0.454410553
-0.562500000 -0.473213613 -0.625000000 0.454410553
-0.593750000 -0.050185024 -0.625000000 0.454410553
-0.593750000 -0.050185024 -0.609375000 0.191316528
-0.593750000 -0.050185024 -0.601562500 0.067944440
-0.593750000 -0.050185024 -0.597656250 0.008233857
-0.595703125 -0.021135877 -0.597656250 0.008233857
-0.596679688 -0.006491229 -0.597656250 0.008233857
-0.596679688 -0.006491229 -0.597167969 0.000861241
-0.596923828 -0.002817510 -0.597167969 0.000861241
-0.597045898 -0.000978764 -0.597167969 0.000861241
-0.597106934 -0.000058919 -0.597167969 0.000861241
-0.597106934 -0.000058919 -0.597137451 0.000401122
-0.597106934 -0.000058919 -0.597122192 0.000171092
-0.597106934 -0.000058919 -0.597114563 0.000056084
-0.597110748 -0.000001418 -0.597114563 0.000056084
-0.597110748 -0.000001418 -0.597112656 0.000027333
-0.597110748 -0.000001418 -0.597111702 0.000012957
-0.597110748 -0.000001418 -0.597111225 0.000005770 Next Middle will get us close enough to zero: F( -0.597110868 ) is 0.000000379 The desired approximation of the solution is: t ≓ -0.597110868
Note, ≓ is the approximation symbol One solution was found : t ≓ -0.597110868
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There is 1 solution to this equation.
What is equation?An equation is a mathematical statement that two expressions are equal. It consists of two expressions separated by an equals sign (=), and expresses a relationship between the two expressions. Equations can be used to represent a variety of problems and can help solve real-world problems. For example, an equation can be used to calculate the amount of money someone will have after a certain period of time, or to calculate the distance between two points.
In order to answer this question, we must first solve the equation. 9t6-14+³ +4+- 1 = 0 can be written as 9t6 - 11 + 4 - 1 = 0. After simplifying, this equation becomes 9t6 - 7 = 0, which can be further simplified to t6 = 7. Taking the sixth root of both sides, we can determine that t = √7. Therefore, there is 1 solution to this equation.
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Write the equation in general form of a parabola with zeros of 3 and 9 and goes through the point \( (6,-18) \). \[ \begin{array}{l} y=-2 x^{2}-24 x+54 \\ y=-2 x^{2}+24 x+54 \\ y=2 x^{2}-24 x+54 \\ y=
The equation of a parabola with zeros at 3 and 9 and goes through the point \( (6,-18) \) is
y = 2x^2 - 24x + 54.
To write the equation in general form of a parabola with zeros of 3 and 9 and goes through the point (6,-18), we can use the fact that the equation of a parabola can be written in the form y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola and a determines the width of the parabola.
First, we can use the zeros to find the vertex of the parabola. The vertex is located halfway between the zeros, so the x-coordinate of the vertex is (3 + 9)/2 = 6. We can plug this value into the equation to find the y-coordinate of the vertex:
y = a(6 - 6)^2 + k = k
Since the parabola goes through the point (6,-18), we know that k = -18.
Now we can plug in the zeros and the vertex into the equation to find the value of a:
0 = a(3 - 6)^2 - 18
0 = a(9) - 18
18 = 9a
a = 2
So the equation of the parabola is y = 2(x - 6)^2 - 18.
To write this equation in general form, we can expand the squared term and simplify:
y = 2(x^2 - 12x + 36) - 18
y = 2x^2 - 24x + 72 - 18
y = 2x^2 - 24x + 54
So the equation in general form is y = 2x^2 - 24x + 54.
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If g(z)=28z^(4)-9z^(3)-26z^(2)+10z+34, use synthetic division to find g(1). Submit
Using synthetic division, the value of g(1) is 37.
To find g(1) using synthetic division, we need to divide the polynomial g(z) by the binomial (z - 1). The process of synthetic division is as follows:
1. Write down the coefficients of the polynomial g(z): 28, -9, -26, 10, 34
2. Write down the value of z that we are plugging in, which is 1, in the leftmost column.
3. Bring down the first coefficient, which is 28, to the bottom row.
4. Multiply the value in the bottom row by the value of z, which is 1, and write the result in the next column.
5. Add the value in the top row to the value in the bottom row and write the result in the bottom row.
6. Repeat steps 4 and 5 until all the columns are filled.
7. The last value in the bottom row is the remainder, and the values in the bottom row before the remainder are the coefficients of the quotient polynomial.
The synthetic division table looks like this:
1 | 28 -9 -26 10 34
| 28 19 -7 3
| 28 19 -7 3 37
The remainder is 37, so g(1) = 37.
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4) Let U = {x:6≤ x ≤40, x is a positive whole number} ,A = {1, 2, 3, 4, 5} , B = {2, 3, 4, 5} C = {x|x is an integer and 2x + 1 = 5 }and D = {x|x is a positive integer and x²/2 = even numbers less than 10}. Provide answer to the following: a) AUA =
The set that is resulting of the union operation between set A and it's complementary A' is given as follows:
U = {x:6≤ x ≤40, x is a positive whole number}.
How to obtain the union between two sets?The union operation of two sets is a mathematical operation that combines all the elements from two sets into a single set, without any duplicates. The union of two sets A and B is denoted as A ∪ B, and is defined as the set that contains all elements that are in at least one of the sets A and B.
The set A is given as follows:
A = {1, 2, 3, 4, 5}.
The complement of it's set is the set containing all the elements that are in the universe set and are not in A.
The union operation between a set and it's complement is always the universe set, hence it is given as follows:
U = {x:6≤ x ≤40, x is a positive whole number}.
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The number 52m is a perfect cube. Find the smallest positive integer value of m.
The smallest positive integer value of m is 676.
To find the smallest positive integer value of m, we need to factor 52m into its prime factors and find the smallest value of m that makes 52m a perfect cube.
First, let's factor 52m into its prime factors:
52m = 2 * 2 * 13 * m
A perfect cube has all of its prime factors raised to the power of 3. So, in order for 52m to be a perfect cube, we need to have two more 2's, two more 13's, and two more m's.
This means that m must be equal to 2 * 2 * 13 * 13 = 676.
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Show that the associationA \mapsto g_{A}is an isomorphism between the space of m x n matrices with coefficients in K and the space of bilinear forms in Km x Kn
The associationA \mapsto g_{A} is an isomorphism between the space of m x n matrices with coefficients in K and the space of bilinear forms in Km x Kn.
The associationA \mapsto g_{A} is an isomorphism between the space of m x n matrices with coefficients in K and the space of bilinear forms in Km x Kn if it satisfies the following conditions:
1. It is a one-to-one correspondence, meaning that for every matrix A there is a unique bilinear form g_{A} and vice versa.
2. It preserves the structure of the spaces, meaning that the operations of addition and scalar multiplication are preserved.
To show that the associationA \mapsto g_{A} is a one-to-one correspondence, we can start by assuming that g_{A} = g_{B} for two matrices A and B. Then, for any vectors u \in Km and v \in Kn, we have:
g_{A}(u,v) = g_{B}(u,v)
A \cdot (u \otimes v) = B \cdot (u \otimes v)
(A - B) \cdot (u \otimes v) = 0
Since this is true for all u and v, we can conclude that A - B = 0, or A = B. This means that the associationA \mapsto g_{A} is a one-to-one correspondence.
To show that the associationA \mapsto g_{A} preserves the structure of the spaces, we can start by considering the addition of two matrices A and B and the scalar multiplication of a matrix A by a scalar c. Then, for any vectors u \in Km and v \in Kn, we have:
g_{A + B}(u,v) = (A + B) \cdot (u \otimes v) = A \cdot (u \otimes v) + B \cdot (u \otimes v) = g_{A}(u,v) + g_{B}(u,v)
g_{cA}(u,v) = (cA) \cdot (u \otimes v) = c(A \cdot (u \otimes v)) = c g_{A}(u,v)
This means that the associationA \mapsto g_{A} preserves the operations of addition and scalar multiplication.
Therefore, the associationA \mapsto g_{A} is an isomorphism between the space of m x n matrices with coefficients in K and the space of bilinear forms in Km x Kn.
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the equation of L is y=5x+1
the equation of L is 2y-10x+3=0
show that these two lines are parallel
The two lines are parallel because both have equal slope.
What area parallel lines?
If two lines do not cross one another anywhere on the plane, they are considered to be parallel. All parallel lines are spaced equally apart from one another.
Numerous forms have parallel lines running along their edges. In the rectangle below, the single and double arrow lines, as well as their junction points, are parallel to one another. Additionally, it is possible to create both horizontal and vertical shapes.
Given : line 1 : y=5x+1
line 2 : 2y-10x+3=0
2y = 10x - 3
We know that, parallel lines always have equal slope.
The slope can be found by using formula :
coefficient of y / coefficient of x
So, here, line 1 has slope = 2/10 = 1/5
and, line 2 has slope = 1/5
So, both have same slopes hence, both are parallel lines.
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Ben reads 6 books in ten days what fraction of a book did Ben read each day
There's a screen shot thank you so much have a good day! <3
Answer: 6 and 12
Step-by-step explanation:
In triangle XYZ the measure of angle YXZ is 50°, and the measure of angle XYZ is 95°. What is the measure of angle XZY in degrees?
Substituting the values into the formula gives us Angle XZY = 180° - angle XYZ - angle YXZ = 180° - 95° - 50° = 35°. Thus, the measure of angle XZY in triangle XYZ is 35 degrees
What is the measure of angle XZY in degrees?We use the knowledge that the total of the angles in any triangle is always 180 degrees to determine the size of angle XZY in triangle XYZ. Angle YXZ is known to measure 50 degrees, while angle XYZ is known to measure 95 degrees. In order to determine the measure of angle XZY, we can subtraction the measurements of these two angles from 180 degrees. We obtain Angle XZY = 180° - angle XYZ - angle YXZ = 180° - 95° - 50° = 35° by substituting the values into the formula. As a result, the angle XZY in triangle XYZ has a measure of 35 degrees.
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A coin with P[H]=0. 3 is flipped 3 times. Define N to be the total number of heads. Find the CDF FN(n), the cumulative distribution function for N. (please provide an explanation for the answer)
The cumulative distribution function for N when a coin is flipped 3 times with P[H]=0. 3 is given by ,
F(N≤ 0) = 0.343 , F(N≤1) = 0.784 , F(N≤2) = 0.973 , F(N≤3) = 1.000.
Cumulative distribution function (CDF) of N,
Probability of getting n or fewer heads in three coin flips,
(All values of n from 0 to 3)
Probability of getting exactly n heads in three flips.
Apply binomial distribution,
P(N = n) = ³Cₙ ×(0.3)ⁿ × (0.7)³⁻ⁿ
For cumulative distribution function, add up the probabilities of getting 0, 1, 2, or 3 heads in three flips,
F(N≤ 0)
=P(N=0)
=³C₀ × (0.3)⁰ × (0.7)³
= 1 × 1 × 0.343
= 0.343
For F(N≤1) is equal to,
⇒F(N≤1)
= P(N=0) + P(N=1)
= ³C₀ × (0.3)⁰ × (0.7)³ + ³C₁ × (0.3)¹ × (0.7)²
= 0.343 + 0.441
= 0.784
F(N ≤ 2)
= P(N=0) + P(N=1) + P(N=2)
= ³C₀ × (0.3)⁰ × (0.7)³
+ ³C₁ × (0.3)¹ × (0.7)²
+³C₂ × (0.3)² × (0.7)¹
= 0.343 + 0.441 + 0.189
= 0.973
This implies CDF for N is equal to,
F(N ≤n) = [tex]\sum_{0}^{x}[/tex]³Cₓ × (0.3)ˣ × (0.7)³⁻ˣ
And F(N≤3) is equal to,
⇒F(N≤3)
= P(N=0) + P(N=1) + P(N=2) + P(N=3)
= ³C₀ × (0.3)⁰ × (0.7)³
+ ³C₁ × (0.3)¹ × (0.7)²
+ ³C₂ × (0.3)² × (0.7)¹
+ ³C₃× (0.3)³ × (0.7)⁰
= 0.343 + 0.441 + 0.189 + 0.027
= 1.000
Therefore, the cumulative distribution function for N is equal to,
F(N≤ 0) = 0.343
F(N≤1) = 0.784
F(N≤2) = 0.973
F(N≤3) = 1.000
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On average, a clothing store gets 120 customers per day. What is the probability that the store will have fewer than 12 customers in the first two hours? (Assume the store is open 12 hours each day. Round your answer to four decimal places. )
The probability that clothing store have fewer than 12 customers in first 2 hours is 0.0003.
We know that, the store is open for 12 hours and gets an average of 120 customers per day, the expected number of customers in a 2-hour period can be calculated as:
The Expected number of customers in 2 hours = 120×(2/12) = 20 customers,
We, use the Poisson distribution to find the probability of having fewer than 12 customers in the first two hours.
The Poisson distribution is given by : P(X = k) = (e^(-λ) × λ^k)/k! ;
where X = random variable (the number of customers), λ = expected value of X (in this case, λ = 20), k = number of customers we want to calculate the probability for, and k! is the factorial of k.
To find the probability of having fewer than 12 customers in the first 2 hours, we need to calculate the probability for k = 0, 1, 2, ..., 11 and add up the probabilities.
So, we have,
⇒ P(X < 12) = e⁻²⁰×(20⁰/0!) + e⁻²⁰×(20¹/1!) + ... + e⁻²⁰×(20¹¹/11!);
On solving, We get,
⇒ P(X < 12) = 0.0003,
Therefore, the required probability is 0.0003.
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The distribution of SAT scores of all college-bound seniors taking the SAT in 2014 was approximately normal with a mean of
1497
14971497 and standard deviation of
322
322322. Let
�
XX represent the score of a randomly selected tester from this group.
Find
�
(
1497
<
�
<
1819
)
P(1497
The probability of a randomly selected tester from this group having a score between 1497 and 1819 is approximately 0.68.
What is Probability?Probability is the measure of how likely a certain event is to occur. It is a mathematical concept that is used to quantify the likelihood of a certain outcome from a given set of circumstances. Probability is expressed as a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain. Probability is used in many fields, including mathematics, finance, and decision making.
This is because approximately 68% of the data is within one standard deviation of the mean, and the data is normally distributed. The area between the mean and the upper limit of 1819 is 0.68, which means that the graph probability of a randomly selected tester from this group having a score between 1497 and 1819 is 0.68.
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what is the line into the slope intercept from of 5y-6x=5
7. 05 circles discussion-based assessment discussion-based assessment a student using a computer to study once you have completed the lesson and assignments, please contact your instructor to complete your discussion-based assessment. You and your instructor will discuss what you have learned up to this point in the course to make sure you're ready to move on
The discussion-based assessment is meant to provide you with an opportunity to reflect on the topics and skills you have learned, as well as identify any areas you may need additional support in.
The conversation should be guided by your instructor, who will ask questions to ensure you understand the material, as well as provide feedback and advice on how to improve your understanding and performance.
At the end of the assessment, you and your instructor should have a clear understanding of your progress and where you need to focus your efforts moving forward.
The instructor should also provide feedback on your performance and suggest strategies for improvement. Additionally, you should have an understanding of the topics you need to work on and the resources available to you to help you improve.
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The complete question is:
Discussion-based Assessment
A student using a computer to study once you have completed the lesson and assignments, please contact your instructor to complete your discussion-based assessment. You and your instructor will discuss what you have learned up to this point in the course to make sure you're ready to move on to the next lesson. What does it mean?
Keith will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of $40 and costs an additional $0. 60 per mile driven. The second plan has no initial fee but costs $0. 80 per mile driven
200 miles is covered by Keith when the plans have the same price. When both plans have the same price, the cost is $160.
Two rents plans and the costs, the miles traveled by Keith when the two plans cost the same and the cost when the two plans cost the same, quadratic equations will be
Let the total miles covered for each plan be
The cost of the first plan would be
cost of the plan A=40+0.60x
The cost of the second plan would be
Cost of second plan=0.80x
If the two cost is the same, then
[tex]0.80x=40+0.60x[/tex]
Solve for x by collecting like terms
[tex]0.80x-0.60x=40[/tex]
[tex]0.20x=40[/tex]
[tex]x=\frac{40}{0.20}[/tex]
[tex]x=200[/tex]
The cost when the two plans cost the same would be
[tex]0.80x=0.80(200)=160[/tex]
or
[tex]40+0.60x=40+0.60(200)=160[/tex]
Hence, the miles covered when the plans cost the same is 200 miles.
The cost when the two plans cost the same is $160.
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Mr. Ling is adding a pond in the shape of a semicircle in his backyard. What is the area of the pond? Use 3.14 for n. Round to the nearest
hundredth if necessary.
Next Question
ft²
Check Answer
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The area of the pond after rounding to the nearest hundredth is 39 feet²
What is the area?An area is total space occupied by two-dimensional or flat surfaces. In other words we can say that it is a number of unit squares present in a closed figure. We use various units for measurement of area like, cm², m², ft², mm².
To find the area of a semicircle, we need to first find the radius of the pond.
Let's assume the diameter of the pond is 10 feet.
Since a semicircle is half of a circle, the radius of the pond is half the diameter, which is 5 feet.
Now we can use the formula for the area of a semicircle, which is:
A = (1/2)πr²
where A is the area and r is the radius.
Plugging in the values, we get:
A = (1/2) × 3.14 × 5²
A = 39.25
Rounding to the nearest whole number, the area of the pond is approximately 39 square feet.
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[tex]3\sqrt{x} -\sqrt{14x-10 =0[/tex]
The value of the variable x is 2 in the given expression [tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex].
What is square root?Square rοοt οf a number is a value, which οn multiplicatiοn by itself, gives the οriginal number. The square rοοt is an inverse methοd οf squaring a number. Hence, squares and square rοοts are related cοncepts.
Suppοse x is the square rοοt οf y, then it is represented as x=√y, οr we can express the same equatiοn as x² = y. Here, ‘√’ is the radical symbοl used tο represent the rοοt οf numbers. The pοsitive number, when multiplied by itself, represents the square οf the number. The square rοοt οf the square οf a pοsitive number gives the οriginal number.
Find the value of x in the expression given below.
[tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex]
Square both sides
[tex](3\sqrt{x} - \sqrt{14x - 10} )^2= 0^2[/tex]
Use (a - b)²
9x + 14x - 10 - 2([tex]3\sqrt{x} \times \sqrt{14x - 10}[/tex]) = 0
9x + 14x - 10 - 2([tex]3\sqrt{x} \times \sqrt{14x - 10}[/tex]) = 0
Use distributive property
23x - 10 - 43x + 50 = 0
-20x + 40 = 0
40 = 20x
x = 40/20
x = 2
Thus, the value of the variable x is 2 in the given expression [tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex].
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10% of 700 the simplest to do it
Answer:
70
Step-by-step explanation:
When we want to find 10% of a number, we move it's current decimal to the left one time.
700.
←
70.
10% of 700 is 70.
Your favorite restaurant is at the intersection of Second Street and Washington Boulevard.
Second Street is on y =
-4
x + 4.
3
Washington Boulevard is on 7y = 3x -9. With the help of this map, can you figure out the
name of this eatery?
19-8
Answer:
To find the name of the restaurant, we need to find the intersection point of the two streets.
The equation of Second Street is y = (-4/3)x + 4/3.
The equation of Washington Boulevard is y = (3/7)x - 3/7.
To find the intersection point, we can set the two equations equal to each other:
(-4/3)x + 4/3 = (3/7)x - 3/7
Multiplying both sides by 21 (the least common multiple of 3 and 7) to eliminate fractions, we get:
-28x + 28 = 9x - 9
Bringing all the x terms to one side and all the constants to the other side, we get:
-37x = -37
Dividing both sides by -37, we get:
x = 1
Substituting this value of x into either equation, we can find y:
y = (-4/3)(1) + 4/3 = 0
Therefore, the intersection point of Second Street and Washington Boulevard is (1,0), which means the restaurant is located at that point.
Step-by-step explanation:
help please! giving brainliest and reward for this is 100 points. and if you dont mind giving a explanation to it that is what i mainly need but thanks help please!!1
not completely sure
3x25=75
answer 75
i think so bye
Exercise 3. Let \( n \in \mathbb{N} \) with \( n>2 \). Prove that there exists \( a \in \mathbb{Z}_{n} \) such that \( a^{2}=[1] \in \mathbb{Z}_{n} \) and \( a \neq[1] \).
\(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\)
Let \(n \in \mathbb{N}\) with \(n>2\). We consider the set \( S = \{a \in \mathbb{Z}_{n} \ | \ a^{2} = [1] \in \mathbb{Z}_{n}\} \). We have to prove that \( S \neq \emptyset \).
We prove by contradiction. Suppose \( S = \emptyset \). This implies that for all \( a \in \mathbb{Z}_{n}, \ a^{2} \neq [1] \in \mathbb{Z}_{n}\). Thus, \( [1] \) is not a square in \(\mathbb{Z}_{n}\). But since \(n >2\), \([1]\) has at least two square roots in \(\mathbb{Z}_{n}\) which implies that \( S \neq \emptyset \).
Therefore, \(S \neq \emptyset\) and thus there exists \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).
This proves that there exists an \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).
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A triangle ABC has a perimeter of 59cm. AB is twice the length of AC and 6cm longer than BC. Find the length of AB.
Answer: 6 cm
solution let the length of AB= x cm the length of BC = (2x-2) cm, and the length of AC = (x+10) cm The perimeter of ABC=32 cm
x+2x-2+x+0=32
4x+8=32
4x=24
x=6
Find A - B Enter the element in row 3 column 2. A=[[-1,0,2],[4,1,-1],[2,0,1]],B=[[2,1,0],[-1,0,2],[4,-3,-1]]
A - B is 5.
To find A - B, subtract each corresponding element in B from the corresponding element in A.
For the element in row 3, column 2, it is 2 - (-3) = 5.
Therefore,
the element in row 3, column 2 of A - B is 5.
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40% of the books on a bookshelf were non-
fiction.
Kian removed 6 of the non-fiction books and
there are now 22 non-fiction books on the
bookshelf.
How many books were there in total to start
with?
The number of total books there to start with were 70
What is the percentage?Percentage is a way to express a number as a fraction of 100. It is often used to represent ratios and proportions in a more convenient and understandable form, especially in financial and statistical contexts. For example, 50% means 50 per 100, or half of a given quantity. It is denoted using the symbol "%".
Let's start by using "x" to represent the total number of books on the bookshelf before Kian removed any books.
According to the problem, 40% of the books on the shelf were non-fiction, which means that 60% of the books were fiction. We can express this as an equation:
0.4x = number of non-fiction books
0.6x = number of fiction books
If Kian removed 6 non-fiction books, there are now 22 non-fiction books remaining, so we can write:
0.4x - 6 = 22
Solving for x, we can add 6 to both sides and then divide by 0.4:
0.4x = 28
x = 70
Therefore, there were 70 books in total on the bookshelf before Kian removed any books.
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Compute the area of triangle, if x equals 3 less than 6
The correct option is C, the area of the triangle is 9 square units.
How to get the area of the triangle?For a triangle of base B and height H the area is:
A = B*H/2
Here we can see that:
B = x
H = 2x
And we know that x = 6 - 3 = 3
Then we can use the value of x to find the values of H and B.
B = 3
H = 2*3 = 6
Now we can replace these two values in the formula for the area, then we will get the area of the triangle:
A = 3*6/2 = 3*3 = 9 square units.
When x = 3, the area is 9 square units.
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What is the volume (in cubic units) of a sphere with a radius of 15 units?
Answer:
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
Substituting r = 15, we get:
V = (4/3)π(15)^3
V = (4/3)π(3375)
V = 4π(1125)
V = 4500π
Therefore, the volume of the sphere with a radius of 15 units is 4500π cubic units. This can also be approximated as 14,137.17 cubic units by using a value of 3.14 for π and rounding to the nearest hundredth.
Step-by-step explanation:
The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 600 people. By selling tickets, the members would like to raise $3,300 every night to cover all expenses. Let d represent the number of adult tickets sold at $7.50. Let s represent the number of student tickets sold at $4.50 each. If all 600 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $3,300? At one performance there were three times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of $3,300 did ticket sales fall?
Ticket sales fell $960 below the goal.
What is system of equations?
A system of linear equations can be solved graphically, by substitution, by elimination, and by the use of matrices.
Since we know that the goal is to raise $3,300 each night and that the price of an adult ticket is $7.50 and the price of a student ticket is $4.50, we can write:
7.5d + 4.5s = 3300
We also know that the auditorium holds 600 people, so the total number of tickets sold must be:
d + s = 600
total number of tickets sold was 480. We can use this information to set up another system of equations:
s = 3d (since there were three times as many student tickets sold as adult tickets)
d + s = 480 (since the total number of tickets sold was 480)
Now we can solve the first system of equations to find the values of d and s that satisfy the constraints:
7.5d + 4.5s = 3300
d + s = 600
Multiplying the second equation by 4.5 and subtracting it from the first equation, we get:
3d = 1650
So, d = 550. Substituting this value back into the equation d + s = 600, we get:
550 + s = 600
s = 50
Therefore, 550 adult tickets and 50 student tickets must have been sold to raise exactly $3,300.
To answer the second part of the question, we can use the second system of equations to find the values of d and s for that performance:
s = 3d
d + s = 480
Substituting the first equation into the second equation, we get:
d + 3d = 480
So, 4d = 480 and d = 120. Substituting this value back into the first equation, we get:
s = 3d = 360
Therefore, 120 adult tickets and 360 student tickets were sold at that performance.
To calculate how much below the goal of $3,300 ticket sales fell, we can plug in the values for d and s from this performance into the equation:
7.5d + 4.5s = revenue
7.5(120) + 4.5(360) = $2,340
So, ticket sales fell $960 ($3,300 - $2,340) below the goal.
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What inequality whose solution set is represented by this graph???
The inequality that represents the set of solutions above the straight line is: x-3y<5
What is a linear inequality?
A linear inequality is an inequality in which one side is a linear expression in one or more variables, and the other side is a constant, or another linear expression. The solution set of a linear inequality is a region in the coordinate plane that satisfies the inequality.
Calculate the inequality to represent the set of solution by shaded region:
A linear inequality can be represented by a shaded region on the coordinate plane. The boundary of the region is a straight line, and the shaded region represents all the points in the plane that satisfy the inequality. In this case, the line that intersects the x-axis at (5,0) and the y-axis at (0, -5/3) represents the boundary of the region.
To find the inequality that represents the set of solutions above the line, we first need to find the equation of the line. We can use the point-slope form of the equation of a line, which relates the slope of the line and a point on the line to the equation of the line.
We know that the line passes through the points (5,0) and (0, -5/3), so we can calculate the slope of the line by using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points through which the line passes.
m = (-5/3 - 0) / (0 - 5) = 1/3
Now we can use the point-slope form of the equation of the line to write the equation of the line:
y - y1 = m(x - x1)
where (x1, y1) is one of the two points through which the line passes.
Using the point (5,0) as (x1, y1), we get:
y - 0 = (1/3)(x - 5)
y = (1/3)x - 5/3
3y = x - 5
To represent the set of solutions above the line, we need to use the greater than symbol. So the inequality that represents the set of solutions above the straight line is:
x-3y<5
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Given that the measurement is in centimeters, find the circumference of the circle to the nearest tenth. (use 3.14 for π)
The circumference of the circle is approximately 31.4 cm to the nearest tenth.
Use this formula to get a circle's circumference:
C = 2πr
where C is the circumference, pi (roughly 3.14), is a mathematical constant, and r is the circle's radius.
If you have the diameter of the circle, you can find the radius by dividing the diameter by 2.
Once you have the radius, you can plug it into the formula to find the circumference.
For example, if the radius is 5 cm:
C = 2πr
C = 2 x 3.14 x 5
C = 31.4
So the circumference of the circle is approximately 31.4 cm to the nearest tenth.
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3. A school has to be guarded 24 hours a day. Four safety guards are ordered to split
each day's safety guard duty equally. How long will each guard spend on guard duty
in one day?
Answer:
6
Step-by-step explanation:
24h/6g= 4 hours per gaurd
