The coordinates of v with respect to the basis β are (cos(1), sin(1)).
The coordinates of a vector with respect to a given basis can be found by expressing the vector as a linear combination of the basis vectors and finding the coefficients of the linear combination. These coefficients are the coordinates of the vector with respect to the given basis.
(i) In the first case, we have V=R2, β={(1,1),(1,−1)}, and v=(1,2). We can express v as a linear combination of the basis vectors as follows:
v = a(1,1) + b(1,-1) = (a+b, a-b)
Equating the coordinates, we get:
a+b = 1
a-b = 2
Solving for a and b, we get a=3/2 and b=-1/2. Therefore, the coordinates of v with respect to the basis β are (3/2, -1/2).
(ii) In the second case, we have V=P2(R), β={1,1+x,1+x+x2}, and v=(x+1)2. We can express v as a linear combination of the basis vectors as follows:
v = a(1) + b(1+x) + c(1+x+x2) = a + (b+c)x + cx2
Equating the coefficients, we get:
a = 1
b+c = 2
c = 1
Solving for a, b, and c, we get a=1, b=1, and c=1. Therefore, the coordinates of v with respect to the basis β are (1, 1, 1).
(iii) In the third case, we have V=span{sin(x),cos(x)}, β={sin(x),cos(x)}, and v=sin(x+1). Using the trigonometric identity sin(x+1) = sin(x)cos(1) + cos(x)sin(1), we can express v as a linear combination of the basis vectors as follows:
v = a(sin(x)) + b(cos(x)) = (a)sin(x) + (b)cos(x)
Equating the coefficients, we get:
a = cos(1)
b = sin(1)
Therefore, the coordinates of v with respect to the basis β are (cos(1), sin(1)).
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Identify the transformations that have been applied to the parent function y=-(1)/(2)(x-3)^(2)-5. Select all that apply.
The transformations that have been applied to the parent function y=-(1)/(2)(x-3)^(2)-5 are: horizontal shift, vertical shift, and vertical stretch.
The transformations that have been applied to the parent function y=-(1)/(2)(x-3)^(2)-5 are:
- Horizontal shift: The parent function has been shifted to the right by 3 units. This is indicated by the subtraction of 3 from x in the equation.
- Vertical shift: The parent function has been shifted downward by 5 units. This is indicated by the subtraction of 5 from the entire equation.
- Vertical stretch: The parent function has been stretched vertically by a factor of (1)/(2). This is indicated by the multiplication of the entire equation by (1)/(2).
Therefore, the correct answers are horizontal shift, vertical shift, and vertical stretch.
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d. The height of the ball is represe x is time in seconds. Rewrite the -1(-6x^(2)+4x+2)
The height of the ball is represented by the equation -1(-6x^(2)+4x+2), where x is time in seconds.
We can rewrite this equation by distributing the -1 and simplifying.
First, we distribute the -1 to each term inside the parentheses:
-1(-6x^(2)+4x+2) = 6x^(2) - 4x - 2
Now, we can simplify by combining like terms:
6x^(2) - 4x - 2 = 6x^(2) - 4x - 2
Therefore, the rewritten equation for the height of the ball is:
h = 6x^(2) - 4x - 2
This equation shows the relationship between the height of the ball (h) and the time in seconds (x). As time increases, the height of the ball changes according to the equation.
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Find the final amount for an investment of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly.
The final amount for an investment of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly is $5386.42.
What is interest rate?Interest is the amount a lender or financial institution receives for lending money. Interest can also refer to a shareholder's level of ownership in a company, usually expressed as a percentage.
Interest rate indicates the cost of borrowing, or the reward for saving. So if you are a borrower, the interest rate is the amount charged to borrow money, expressed as a percentage of the total loan amount.
Solution according to the information given in the question :
Given, Principal(P) = $5,000
Compounding frequency(n) = 4
Time(in years)(T) = 5
Interest rate = 6%
Amount = P× (1 + r/n)^t
= 5,000 × (1 + 6/4×100)⁵
= 5,000 × (1 + 0.015)⁵
= 5,000 × (1.015)⁵
= 5000 ×1.077
= $5286.42
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Which statement is true about evaluating the expression 72÷12+9−5 on the basic calcul
On simplifying the given expression using simplification method BODMAS, we get the answer 10. Thus, in the calculator the calculation must be done from left to right.
What is simplification in mathematics?Simplification means using various operations to reduce the formula to a simpler form, and approximation simplifies the formula to the nearest, but not exactly correct, value.
What is BODMAS?BODMAS in simplification stands for Bracket, Ob, Division, Multiplication, Addition, and Subtraction. BODMAS is used to describe the order of operations in formulas. In some areas, BODMAS is also known as PEDMAS. This represents parentheses, exponents, division, multiplication, addition, and subtraction.
Solution using BODMAS and simplifying it:
72÷12+9−5 = (72/12) + 9 - 5
=6 + 9 - 5
= 15 - 5
⇒72÷12+9−5 = 10
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Need help with these questions please
1. The height of the pyramid is 3.7 inches.
2. The length of the crossbar is 2√2 feet.
What is a pyramid?A three-dimensional figure is a pyramid. Its base is a flat polygon. The remaining faces are all triangles and are referred to as lateral faces. The number of sides on its base is equal to the number of lateral faces. The line segments that two faces intersect to form its edges. The intersection of three or more edges forms a vertex.
1. We know, The height, slant height, and half of the base length form a
right-angled triangle
Therefore, h² = 4² - (3/2)².
h² = 13.75
h = 3.7 inches.
2. We know, In a 45°, 45° and 90° triangle, It is an isosceles right-angled triangle, The hypotenuse is √2 times the two similar sides.
Therefore, From the given information, The length of the crossbar is
2√2 feet.
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If the principal is 18000 the time is 54 months and the simple interest is 4252.50 what is the interest rate
Answer:
To find the interest rate when given the principal, time, and simple interest, we can use the formula:
Simple Interest = (Principal * Rate * Time) / 100
where:
Simple Interest is the given value of 4252.50
Principal is 18000
Time is 54 months
Rate is the unknown we want to solve for
Substituting the given values, we get:
4252.50 = (18000 * Rate * 54) / 100
Multiplying both sides by 100 and dividing by 18000 * 54, we get:
Rate = (4252.50 * 100) / (18000 * 54)
Rate = 4.398%
Therefore, the interest rate is 4.398% (rounded to three decimal places).
Step-by-step explanation:
Sure! Here is a step-by-step explanation of how to find the interest rate:
Given:
Principal = 18000
Time = 54 months
Simple Interest = 4252.50
Formula:
Simple Interest = (Principal * Rate * Time) / 100
We want to solve for Rate.
Substitute the given values into the formula:
4252.50 = (18000 * Rate * 54) / 100
Simplify by multiplying both sides by 100:
425250 = 18000 * Rate * 54
Divide both sides by 18000 * 54:
Rate = 425250 / (18000 * 54)
Simplify the expression on the right-hand side:
Rate = 0.04398
Convert the decimal to a percentage:
Rate = 4.398%
Therefore, the interest rate is 4.398%.
Using the formula for simple interest, the value for interest rate is obtained as 5.25%.
What is Simple Interest?
Simple interest is a way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time. Contrary to compound interest, where we add the interest of one year's principal to the next year's principal to compute interest, the principal amount under simple interest remains constant.
We can use the formula for simple interest -
I = P × r × t
where I is the interest, P is the principal, r is the interest rate, and t is the time in years.
In this case, the principal is $18,000, the time is 54 months which is 4.5 years, and the simple interest is $4,252.50.
Substituting these values in the formula, we get:
4252.50 = 18000 × r × 4.5
Solving for r, we get -
r = 4252.50 / (18000 × 4.5)
= 0.0525 or 5.25%
Therefore, the interest rate is 5.25%.
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The number e is the base of the ____________?
natural logarithm
decimal system
irrational number system
rational functions
Answer: C.
Step-by-step explanation:
Write the vector form of the general solution of the given system of linear equations. X1+x2+x4 = 0
X1 +2x2 +4x4 = 0
2x1 -4x4 = 0
The vector form of the general solution of the given system of linear equations is given by:
{x} = {x1, x2, x4} = s { -1, 1, 0} + t { -2, 0, 1}
where s and t are arbitrary constants.
To find the vector form of the general solution, we need to find the null space of the coefficient matrix of the system of linear equations. The coefficient matrix of the system is:
A = [ 1 1 0 1; 1 2 0 4; 2 0 0 -4]
The null space of A is the set of all vectors x such that Ax = 0. We can find the null space of A by reducing A to its reduced row echelon form:
A = [ 1 0 0 -2; 0 1 0 1; 0 0 0 0]
From the reduced row echelon form of A, we can see that x1 and x2 are the leading variables and x4 is the free variable. Therefore, the general solution of the system of linear equations is given by:
x1 = -2x4
x2 = x4
x4 = x4
We can write the general solution in vector form as:
{x} = {x1, x2, x4} = x4 { -2, 1, 1}
Since x4 is an arbitrary constant, we can write the general solution in terms of two arbitrary constants s and t as:
{x} = {x1, x2, x4} = s { -1, 1, 0} + t { -2, 0, 1}
Therefore, the vector form of the general solution of the given system of linear equations is:
{x} = {x1, x2, x4} = s { -1, 1, 0} + t { -2, 0, 1}
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The display summarizes the favorite sports among 9th graders.
Circle graph titled favorite sports with five pieces. Baseball or softball is labeled 44 percent, basketball is labeled 20 percent, lacrosse is labeled 16 percent, soccer is labeled 12 percent, and field hockey is labeled 8 percent.
Which of the following describes the data set?
The data is univariate and categorical.
The data is univariate and numerical.
The data is bivariate and categorical.
The data is bivariate and numerical.
The function distribution of this categorical variable is shown as percentages on the circle graph. As a result, the data is categorical and univariate (one variable).
what is function?Mathematicians research numbers, their variants, equations, associated structures, forms, and possible configurations of these. The word "function" describes the connection between a group of inputs, each of which has a corresponding output. A function is a connection between inputs and outputs where each input results in a single, distinct outcome. Each function has a domain, codomain, or scope assigned to it. Functions are usually denoted by the letter f. (x). An x is entered. On functions, one-to-one capabilities, so multiple capabilities, in capabilities, and on functions are the four main categories of accessible functions.
The right response is: The information is categorical and univariate.
The only variable in the data set is the preferred sports among ninth graders. As it includes several categories or groupings, the variable is categorical (i.e., baseball or softball, basketball, lacrosse, soccer, and field hockey). The distribution of this categorical variable is shown as percentages on the circle graph. As a result, the data is categorical and univariate (one variable).
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The mean selling price of new homes was $114,000. The population standard deviation was $5,000. A random sample of 100 new home sales was taken without doing the calculations, state in which of the following ranges the sample mean selling price is most likely to lie? a. $115,000 - $117,000 b. $116,000 - $118,000 c. $113,000 - $115,000 d. $114,000 -- $116,000
The mean selling price of new homes was $114,000. The population standard deviation was $5,000. A random sample of 100 new home sales was taken without doing the calculations, ranges the sample mean selling price is most likely to lie between C) $113,000 - $115,000.
The sample mean selling price is most likely to lie in the range of $113,000 to $115,000 because this range is within one standard deviation (i.e., $5,000) of the population mean of $114,000, and therefore it contains approximately 68% of the sample mean values. The other ranges are outside one standard deviation from the mean, and therefore, the probability of the sample mean being in those ranges is lower. However, it is important to note that the actual sample mean could fall outside this range as it is still subject to random variation.
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Im confused of what the answer should be
p please help I have no clue.
Answer:
11 meters, 4.1 centimeters
Step-by-step explanation:
If for every centimeter it equals 2 meters, we can use this expression to solve for the width of the building in real-life:
5.5 × 2 = 11Therefore, the width of the building is 11 meters in real life.
If the real-life height of the building is 8.2 m tall, then we can use this expression:
8.2 ÷ 2 = 4.1We multiplied the first time, so why are we dividing now?Because we were given the drawings width the first time, we needed to multiply by 2 to get the real-life height in meters. But now that we are given the real-life height, we now need to divide by 2 to get the height of the drawing in centimeters.
Therefore, the width, in meters, of the building in real life is 11 meters, and the height of the drawing is 4.1 centimeters.
The population of bears in a national forest is changing according to the function N(t) = 850(0.96)*, where t is the time in years and N(C) is the number of bears.
Which statement explains how the bear population is changing?
A. The population is decreasing by 4% per year.
B. The population is increasing by 96% per year.
C. The population is increasing by 4% per year.
D. The population is decreasing by 96% per year.
The population function is N(t) = 850(0.96)^t.
The base of the exponential function is (0.96), which is less than 1. This means that the function is decreasing over time.
Therefore, the correct answer is:
A. The population is decreasing by 4% per year.
packages of knives and packages of forks Two students evaluate the expression 17(4+15). Kode evaluates the expression by adding the product of 17 and 4 to the product of 17 and 15 . Elijah evaluates the expression by determining the product of 17 and 19.
Both methods are correct and either one can be used to evaluate the expression 17.
About distributive propertyBoth Kode and Elijah are using the distributive property to evaluate the expression 17(4+15).
The distributive property states that a(b+c) = ab + ac.
Kode's method:
17(4+15) = (17*4) + (17*15) = 68 + 255 = 323
Elijah's method:
17(4+15) = 17*19 = 323
Both students get the same answer of 323.
Therefore, both methods are correct and either one can be used to evaluate the expression 17(4+15).
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At this Sunday’s Super Bowl game, 150 out of thefirst 500 people who entered the main gate were not wearing team jerseys. Ifthis sample is representative of the 75,000 people attending the game, abouthow many of them will probably NOT be wearing team jerseys? 2,500 people 25,000 people 22,500 people 2,200 people
As a result, approximately 22,500 spectators at the Super Bowl will likely not be donning club jerseys. The correct response is (C) 22,500 people.
what is proportionality ?A mathematical concept known as proportionality describes the relationship between two quantities that change in a way that keeps their ratio constant. This means that if the size of one quantity increases or decreases by a certain amount, the size of the other quantity will do the same, maintaining their relative size. In other words, we can state that A is directly proportional to B if we have two quantities, A and B, and their ratio is constant. Frequently, this is expressed as: A ∝ B or A = k * B where k is a ratio constant.
given
The number of spectators who won't be sporting team jerseys at the Super Bowl can be estimated using proportions:
If 150 out of the first 500 individuals to enter through the main gate were not donning team uniforms, then the percentage of those without team uniforms is:
150/500 = 0.3
This indicates that 30% of those who entered through the main gate did not have team jerseys on.
We can determine the approximate number of spectators who won't be donning team jerseys as follows, assuming that this proportion is representative of the entire 75,000 people in attendance:
0.3 * 75,000 = 22,500
As a result, approximately 22,500 spectators at the Super Bowl will likely not be donning club jerseys. The correct response is (C) 22,500 people.
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NEED HELP ON THIS ASAP
The expression, h(x), in standard form that describes the length of the top surface of the rectangular boxes can be modelled by the function
[tex]h(x) = 6x + 66x^2 + 17[/tex].
What are expressions in standard form?A simplified way to write an expression is to use the expression in standard form. It entails expressing the expression in terms of its highest exponent, variables, and numerical coefficients. The use of standard form is intended to simplify the reading and comprehension of an expression. As opposed to writing an expression as, [tex]3x^2 + 4x +[/tex]7, it can be written in standard form as [tex]3x^2 + 4x + 7[/tex].
The two functions, f(x) and g(x), can be combined to form a single equation that expresses the length of the top surface of the rectangular boxes, which leads to the derivation of this expression.
The top surface's area is represented by the function f(x), which has the following form: [tex]f(x) = 66x^2 + 17x + 1[/tex]. With the formula g(x) = 6x + 1, the function g(x) denotes the width of the top surface. These two functions are combined to give us [tex]h(x) = 6x + 66x^2 + 17[/tex]. As a result, this expression describes how long the top surface of the rectangular boxes is.
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Emilia's lemon cookie recipe calls for 1 1/8 cups of sugar. How much sugar would Emilia use to make 3 5/6 batches of cookies?
To find how much sugar would Emilia use to make about 3 5/6 batches of cookies.
1 1/8 cups of sugar
Next, we need to convert the mixed number to an improper fraction:
1 1/8 = (8 + 1) / 8 = 9/8
So, the amount of sugar needed for one batch of cookies is:
9/8 cups of sugar
To find the total amount of sugar needed for 3 5/6 batches, we can multiply the amount of sugar per batch by the number of batches:
(9/8 cups of sugar) x (23/6 batches) = 69/48 cups of sugar
thus, Emilia would need 69/48 cups of sugar to make 3 5/6 batches of cookies.
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How much distance will a tein cover in 3⅓ hoursif it maintains a constant speed of 80¼ km/h?
Write an equation for the graph that passes through (0,9) and has a slope of -3.
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. We are given that the line passes through the point (0,9) and has a slope of -3, so we can substitute these values into the equation to find the y-intercept:
y = mx + b
9 = (-3)(0) + b
b = 9
Now we know that the y-intercept is 9, so we can write the equation of the line:
y = -3x + 9
Therefore, the equation of the graph that passes through (0,9) and has a slope of -3 is y = -3x + 9.
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For the following decision problem where the objective is to maximize the expected return, find a. The optimal policy and expected return. b. The value of knowing X, but not Y, before D. c. The value of knowing Y, but not X, before D. d. The value of knowing both X and Y before D. e. Draw the influence diagram that corresponds to this decision tree.
a. The optimal policy and expected return can be found by working backwards from the end nodes of the decision tree. For each decision node, choose the action that results in the highest expected return. The expected return is calculated by multiplying the probability of each outcome by the corresponding return and summing the results. The optimal policy is to choose A if X is high, and B if X is low.
The expected return is 0.7*30 + 0.3*10 = 24.
b. The value of knowing X, but not Y, before D is calculated by subtracting the expected return without knowing X from the expected return with knowing X.
The expected return without knowing X is 0.5*24 + 0.5*24 = 24.
The expected return with knowing X is 0.7*30 + 0.3*10 = 24.
Therefore, the value of knowing X is 24 - 24 = 0.
c. The value of knowing Y, but not X, before D is calculated by subtracting the expected return without knowing Y from the expected return with knowing Y.
The expected return without knowing Y is 0.5*24 + 0.5*24 = 24.
The expected return with knowing Y is 0.8*30 + 0.2*10 = 26.
Therefore, the value of knowing Y is 26 - 24 = 2.
d. The value of knowing both X and Y before D is calculated by subtracting the expected return without knowing X or Y from the expected return with knowing both X and Y.
The expected return without knowing X or Y is 0.5*24 + 0.5*24 = 24.
The expected return with knowing both X and Y is 0.7*0.8*30 + 0.7*0.2*10 + 0.3*0.8*10 + 0.3*0.2*30 = 25.2.
Therefore, the value of knowing both X and Y is 25.2 - 24 = 1.2.
e. The influence diagram that corresponds to this decision tree is shown below:
X -> D -> Y -> R
-> A -> R
-> B -> R
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need help with this question
The amount of element left after the decay is found as 43 grams.
Explain about the element's decay rate?The proportion of radioactive nuclei that decrease per unit of time is the rate of decay, or radioactivity, of a measurement of a radioactive substance. The amount of time needed for the substrate concentration to drop to half its initial value is known as the half-life of a reaction. One element spontaneously changes into another during radioactive decay. Only by altering the amount of protons within the nucleus is this possible.Decay rate is given as:
N = N₀(1 - r)ˣ
x - is the time in minutes
N₀ - initial mass (570 grams)
r - rate of decay (18%)
Put the values:
N = 570(1 - 0.18)¹³
N = 570*0.0757
N = 43
Thus, the amount of element left after the decay is found as 43 grams.
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Dierks made $13.50 per hour. He got a new job in which he made 8% more. How much did her her make per hour at his new job?
Answer: $14.58
Step-by-step explanation:
13.50 x .08 = 1.08
13.50 + 1.08 = 14.58
Solve the system by Elimination.
−6x+2y=−28
−4x+5y=18
(-8, -6)
(8, 6)
(8, 10)
(8, 5)
Answer:
x would be 8 and y is 10
so (8,10)
Step-by-step explanation:
hope it helps
Michael sang 12 songs. Every song was 3/8 minutes long. For how much time did he sing in total? Write your answer in simplest form.
Answer:
Total time = (length of one song) x (number of songs)
Length of one song = 3/8 minutes
Number of songs = 12
Total time = (3/8) x 12
Total time = 36/8
Simplifying the fraction, we get:
Total time = 4 and 4/8 minutes
Total time = 4 and 1/2 minutes
Therefore, Michael sang for a total of 4 and 1/2 minutes.
y>=-(7)/(6)x Alot points on the Boundary Ine. Select the Ene to swith between solif and dotted. Select a reglon to shade it.
The final solution to the inequality y>=-(7)/(6)x is the shaded region on the graph.
To solve the inequality y>=-(7)/(6)x, we will first graph the boundary line and then shade the appropriate region.
1. Begin by graphing the boundary line y=-(7)/(6)x. This line has a slope of -(7)/(6) and passes through the origin (0,0).
2. Since the inequality includes the "greater than or equal to" symbol (>=), the boundary line should be solid to indicate that points on the line are included in the solution.
3. Next, we need to determine which region to shade. To do this, we can choose a test point that is not on the boundary line. A common test point is (0,1), which is above the boundary line.
4. Substitute the coordinates of the test point into the inequality to see if it is true or false: 1>=-(7)/(6)(0)
5. Since the inequality is true, the region that includes the test point is the solution. This means we should shade the region above the boundary line.
6. The final graph should have a solid boundary line with the region above it shaded.
The solution to the inequality y>=-(7)/(6)x is the shaded region on the graph.
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2.) During recent seasons, Major League Baseball has been criticized for the length of games. A report indicated that the average game lasts 3 hours and 30 minutes. A sample of 17 games revealed the following time to completion. (Note that the minutes have been changed to fractions of hours, so that a game that lasted 2 hours and 24 minutes is reported as 2.40 hours.) Can we conclude that the mean time for a game is less than 3.50 hours?
2.98 2.40 2.70 2.25 3.23 3.17 2.93 3.18 2.80
2.38 3.75 3.20 3.27 2.52 2.58 4.45 2.45
a) What is the critical value of t for this question at the .05 significance level?
b) What is the calculated value of t for this question at the .05 significance level? (You need to calculate the standard deviation for the sample.)
Enter the value as - x.xx
c) Can we conclude that games are less than 3.5 hours long based upon the calculated t-value compared to the critical t-value?
Choose 1 answer for question c.
c1) Reject the null because the calculated t-value is less than the critical t-value.
c2) We fail to reject the null because the calculated t-value is greater than the critical t-value indicating that games are longer than 3.50 hours.
a) The critical value of t for this question at the .05 significance level is -2.11.
b) The calculated value of t for this question at the .05 significance level is -1.47, given a standard deviation of 0.63.
c) We can reject the null hypothesis because the calculated t-value (-1.47) is less than the critical t-value (-2.11). This indicates that games are less than 3.50 hours long.
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Find the volume of the earth dug out from a pit 6. 5 m long 2. 4 M wide and 1. 5 m deep
The volume of the earth dug out from a pit = 23.4 m³
Let us assume that l represents the length of the pit, 'w' represents the width and 'h' be the height of the pit.
Here, the length of the pit(l) = 6.5 m
the width of the pit (w) = 2.4 m
height of the pit(h) = 1.5 m
We need to find the volume of earth dug out from a pit.
We know that pit is of the cuboid shape.
Using the formula of volume of cuboid,
V = l × w × h
V = 6.5 × 2.4 × 1.5
V = 23.4 m³
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If you roll a fair number cube, what is the probability of landing on a 6? What is the probability of rolling a 7?
The probability of landing on a 6 is 1/6. The probability of rolling a 7 is zero.
What is probability?The simple definition of probability is the likelihood that something will occur. We can discuss the probabilities of different outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics refers to the study of events subject to probability.
If you roll a fair number cube, The probability of landing on a 6 is 1/6.
As, all the possible outcome is 6 and favourable outcome is 1 (6), Probability = 1/6 or 16.66%
The probability of rolling a 7 is zero as there is only 6 possible out come and 7 is the 7th outcome which is not possible, therefore 0.
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20 POINTSS!! HURRY!
Airplane A traveled 1662 miles in six hours.
For Airplane B, y =
922/
3
x represents its rate of speed over the same six hours.
Compare the two airplanes to determine which airplane traveled at a faster rate?
Group of answer choices
Airplane A
Airplane B
airplane A
To compare the rate of speed of Airplane A and Airplane B, we need to calculate the speed of Airplane B first.
The formula to calculate distance is:
distance = rate x time
For Airplane A, we know:
distance = 1662 miles
time = 6 hours
So, we can rearrange the formula to solve for the rate:
rate = distance / time = 1662 / 6 = 277 miles per hour
For Airplane B, we know:
y = 922/3
x = rate of speed over 6 hours
We can use the formula to solve for x:
distance = rate x time
922/3 = x * 6
x = (922/3) / 6 = 153.67 miles per hour
Comparing the rates of the two airplanes, we can see that Airplane A traveled at a faster rate with 277 miles per hour compared to Airplane B's rate of 153.67 miles per hour. Therefore, Airplane A traveled at a faster rate.
Plot the points A(-2,-3), B(1, 12), C(3, -4) on the coordinate axes below. State the
coordinates of point D such that A, B, C, and D would form a rectangle. (Plotting
point D is optional.)
The opposite sides of the quadrilateral should be parallel and the diagonals should be equal. [For a rectangle]
First, let's find the slope of AB and its perpendicular slope.
Slope of AB = (y2 - y1)/(x2 - x1) = (12 - (-3))/(1 - (-2)) = 5
Perpendicular slope of AB = -1/5
The midpoint of AB is ((-2+1)/2, (-3+12)/2) = (-0.5, 4.5)
Let D be the point on line segment AC such that AD is perpendicular to AC. Since AB is parallel to DC, the slope of DC is also 5.
Slope of AC = (-4 - (-3))/(3 - (-2)) = -1/5
The equation of line AC is y = (-1/5)x - (13/5)
The equation of line DC passing through (3,-4) with slope 5 is y - (-4) = 5(x - 3)
Solving these two equations, we get D(2,-14/5).
Therefore, the coordinates of point D are (2, -14/5).
We can plot the points on the coordinate axes and verify that A, B, C, and D indeed form a rectangle.