Answer:
y = 2(x + 1.5)² + 0.5
Step-by-step explanation:
"To convert the standard form y = ax² + bx + c to vertex form:
Extract a from the first two terms: y = a[x² + (b/a)x] + c.
Add and subtract (b/(2a))² inside the bracket: y = a[x² + (b/a)x + (b/(2a))² - (b/(2a))²] + c.
Use the short multiplication formula: y = a[(x + b/(2a))² - (b/(2a))²] + c.
Expand the bracket: y = a(x + b/(2a))² - b²/(4a) + c.
This is your vertex form with h = -b/(2a) and k = c - b²/(4a)."
^^ not my explanation!!
Will mark brainiest for good and real answers only!
Vector C is 3.5 units West and Vector D is 3.3 units South. Vector R is equal to Vector D - Vector C. Which of the following describes Vector R?
8.3 units 54
South of East
8.3 units 54circ South of East
4.8 units 47
East of South
4.8 units 47circ East of South
6.2 units 32
West of South
6.2 units 32circ West of South
5.9 units 52
South of West
The corresponding to Vector R is 4.8 units 47 East of South 4.8 units 47circ East of South
How to solve for the vectorVector C comprises a magnitude of -3.5i
Vector D is established as –3.3j (South bearing is thought to be negative along the y-axis).
Vector R = Vector D - Vector C
= (-3.3j) - (-3.5i)
= 3.5i - 3.3j
To evaluate the strength of Vector R, we must first compute its magnitude:
Magnitude of R = √((3.5)^2 + (-3.3)^2) ≈ 4.8 units.
determine the direction,
we shall need to calculate the angle θ with respect to the South direction (the negative y-axis):
tan(θ) = (3.5) / (3.3);
θ = arctan(3.5 / 3.3) ≈ 47°
Hence the answer is option 2
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Enter the correct answer in the box.
What is the factored form of this expression?
x² + 6x-16
Substitute numerical values into the expression for p and q.
(0) 101
√O
d'º
+
11
Vo 4
(x+p)(x+q)
<> ≤ 2
TT
aß
A HP
sin cos tan sin-¹ cos¹ tan-¹
CSC sec cot log logo In
The calculated factored form of the expression x² + 6x - 16 is (x - 2)(x + 8)
What is the factored form of the expression?From the question, we have the following parameters that can be used in our computation:
x² + 6x - 16
Expand the expression
So, we have
x² + 6x - 16 = x² + 8x - 2x - 16
Factorize the expression
So, we have the following representation
x² + 6x - 16 = x(x + 8) - 2(x + 8)
Factor out x + 8
So, the expression becomes
x² + 6x - 16 = (x - 2)(x + 8)
Hence, the factored form of the expression x² + 6x - 16 is (x - 2)(x + 8)
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PLSSSS LOOK AT ATTACHMENT
Answer:
Step-by-step explanation:
Since they are in a row, add all the measurements
4 ft 10 in
5 ft 6 in add ft with the ft and in with in
3 ft 3 in
12 ft 19 in
There are too many inches. So you can take away 12 of those inches and and turn it into 1 more foot. That would make 13 ft. You'd be left with 7 in.
12 ft + 1 ft 19 in - 12 in
13 ft 7in
The mass of an oxygen molecule is
5.31×10^-23 grams. The mass of an electron is 9.11×10^-28 grams. How many times greater is the mass of an oxygen molecule than the mass of an electron? Write your answer in standard notation, rounding to the nearest tenth.
The mass of an "oxygen-molecule" is approximately 5.82 × 10⁴ times greater than the mass of an electron, and written in "standard-notation", the answer is 58287.59.
The "Standard-Notation" is defined as a way of writing a number using digits, a decimal point.
The mass of the "oxygen-molecule" is = 5.31×10⁻²³ grams,
The mass of the "electron" is = 9.11×10⁻²⁸ grams,
To find how many times greater the mass of an oxygen molecule is than the mass of an electron, we need to divide the mass of the oxygen molecule by the mass of the electron:
⇒ (5.31 × 10⁻²³)/(9.11 × 10⁻²⁸) = 58287.59 ≈ 5.82 × 10⁴.
Therefore, The mass of oxygen is 5.9 × 10⁴, times greater than electron.
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The percent of fat calories an American consumes each day is normally distributed withy mean of 36 and SD of about 10. Suppose a random sample of 20 Americans are chosen. Let X be the mean (average) percent of fat calories in the sample.
The percent of fat calories an American consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10.
What is standard deviations?Standard deviation is a measure of the spread of a data set. It is calculated by taking the square root of the variance of the data set. It is a measure of how much the individual values in a data set are spread out from the mean. It is used to quantify the amount of variation or dispersion of a set of data values.
According to the Central Limit Theorem, the mean of the sample X will be normally distributed with a mean of 36 and a standard deviation of 10/sqrt(20). This means that the mean percent of fat calories in the sample will be normally distributed with a mean of 36 and a standard deviation of about 3.16. This means that 68% of the time, the mean percent of fat calories in the sample will be between 32.84 (36-3.16) and 39.16 (36+3.16), 95% of the time it will be between 29.68 (36-6.32) and 42.32 (36+6.32), and 99% of the time it will be between 26.52 (36-9.48) and 45.48 (36+9.48).
In other words, this is happening because the Central Limit Theorem states that the sample mean will be normally distributed, and that the standard deviation of the sample mean will be the standard deviation of the population divided by the square root of the sample size. By understanding this theorem, we can calculate the probability that the mean of the sample will fall within certain ranges.
In conclusion, the percent of fat calories an American consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10. By understanding the Central Limit Theorem, we can calculate the probability that the mean of the sample will fall within certain ranges. This allows us to make predictions about the sample mean and helps us understand why this is happening.
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Suppose that a researcher is interested in finding out whether a new math program increases students’ math skills. 36 students participate in the program. After the training, the student’s math skills are measured by an exam that is known to have a µ = 70 and σ = 12. The researcher finds out that students who participated in the program scored an average of 73.6. Can we conclude that the training was successful? Justify and explain your answer.
The hypothesis test conclude that the training program was successful in increasing students' math skills.
As the average exam score of students who participated in the program was significantly higher than the population mean.
Number of students participated = 36
Population mean µ = 70
and σ = 12
To determine whether the math program was successful in increasing students' math skills, we can conduct a hypothesis test.
The null hypothesis denoted as H₀,
There is no difference between mean exam scores of students who participated in math program and population mean µ = 70.
The alternative hypothesis denoted as H₁,
There is a difference between the mean exam scores of the two groups.
Mathematically, the hypotheses as follows,
H₀: µ = 70
There is no difference in the mean exam scores of students who participated in the math program and the population mean.
H₁: µ > 70
The mean exam scores of students who participated in the math program are greater than the population mean.
Use a one-sample t-test to test the hypotheses.
The formula for the t-test statistic is,
t = (X - µ) / (s / √n)
X is the sample mean = 73.6
µ is the population mean = 70
s is the sample standard deviation
s = 12 / √36
= 2
and n is the sample size= 36
Substituting these values into the formula, we get,
t = (73.6 - 70) / (2 / √36)
= 3.6 / (1/3)
≈10.8
To determine whether the t-value is significant,
Compare it to the critical t-value at a chosen significance level α. Assuming significance level of 0.05 and 35 degrees of freedom df=n - 1 .
Critical t-value to be approximately 1.69.
Since calculated t-value 10.8 is greater than the critical t-value 1.69.
Reject the null hypothesis .
Therefore, using hypothesis test conclude that there is sufficient evidence to suggest that the math program was successful in increasing students' math skills.
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ILL GIVE U BRAINLIST OR WHATEVER
The nationwide percentage of Americans who invest in the stock market is usually taken to be 55%. Is this percentage different in Georgia? A survey and analysis among Georgia residents led to a test statistic of 1.75.
(b) What is the p-value to test if the proportion of Georgians who invest in the stock market is different than the nationwide average? (Use 4 decimals.)
Okay, here are the steps to find the p-value:
1) The nationwide proportion of Americans who invest in the stock market is 55% (0.55)
2) The test statistic for Georgia is 1.75
3) To calculate the p-value, we need to know the degrees of freedom (df) and the critical value of the test statistic.
4) The df = 1, since we're comparing a single proportion (Georgia) to a fixed value (national average).
5) The critical value at df = 1 and 95% confidence is 3.84 (for a two-tailed test)
6) Since the test statistic of 1.75 is less than 3.84, we look up 1.75 in tables to find the p-value.
7) For a test statistic of 1.75 and df = 1, the p-value is 0.1860.
So the p-value to test if the proportion of Georgians who invest in the stock market is different than the nationwide average is 0.1860.
Let me know if you have any other questions!
Okay, here are the steps to find the p-value:
1) The nationwide proportion of Americans who invest in the stock market is 55% (0.55)
2) The test statistic for Georgia is 1.75
3) To calculate the p-value, we need to know the degrees of freedom (df) and the critical value of the test statistic.
4) The df = 1, since we're comparing a single proportion (Georgia) to a fixed value (national average).
5) The critical value at df = 1 and 95% confidence is 3.84 (for a two-tailed test)
6) Since the test statistic of 1.75 is less than 3.84, we look up 1.75 in tables to find the p-value.
7) For a test statistic of 1.75 and df = 1, the p-value is 0.1860.
So the p-value to test if the proportion of Georgians who invest in the stock market is different than the nationwide average is 0.1860.
Let me know if you have any other questions!
Talisa plans a 36-foot deep pond. While digging, she hits rock 30 feet down. How can Talisa modify the radius to maintain the original volume of the pond?
Talisa should make the radius ___ ft
The radius of the pond should be modified to 19.72ft to be able to maintain the original volume of the pond.
How to modify the radius to maintain the original volume?The original shape of the pond is the shape of a cylinder and the formula tye can be used to calculate the volume of a cylinder = πr²h
The volume of the original pond is calculated as follows;
π = 3.14
Radius = 18 ft
height = 36ft
Vol = 3.14×18×18×36
= 36,624.96ft³
The radius of the modified pond is calculated as follows; Volume = 36,624.96ft³
height = 30ft
radius = ?
That is;
36,624.96 = 3.14×r²× 30
r² = 36,624.96/94.2
= 388.8
r = √388.8
= 19.72 ft
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7 of 7
Expand:
f(x)=x²-
DONE
+
1
Expanding the function expression f(x) = x^2 - 1, we get f(x) = (x - 1)(x + 1)
Expanding the function expressionFrom the question, we have the following parameters that can be used in our computation:
f(x) = x^2 - 1
The above expression can be expanded using the difference of two squares
When this theorem is applied, we have
f(x) = (x - 1)(x + 1)
Hence, the function expression f(x) = x^2 - 1 when expanded is f(x) = (x - 1)(x + 1)
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Expanding the function expression f(x) = x^2 - 1, we get f(x) = (x - 1)(x + 1)
Expanding the function expressionFrom the question, we have the following parameters that can be used in our computation:
f(x) = x^2 - 1
The above expression can be expanded using the difference of two squares
When this theorem is applied, we have
f(x) = (x - 1)(x + 1)
Hence, the function expression f(x) = x^2 - 1 when expanded is f(x) = (x - 1)(x + 1)
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Terry bought a new television set for $450. She paid nothing down but agreed to payments of $40.03 per month for 12 months. Find the annual percentage rate for the loan using the APR table
The annual percentage rate is%
The annual percentage rate for Terry's loan is 19.21%.
What is statistics?
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.
To find the annual percentage rate (APR) for Terry's loan, we can use the following formula:
APR = (2 × Monthly Payment ÷ Loan Amount) × (12 ÷ Loan Term in Months + 1)
First, we need to calculate the total amount Terry will pay for the TV over the 12-month period:
Total payments = Monthly payment × Loan term in months
Total payments = $40.03 × 12
Total payments = $480.36
Next, we can use the formula to calculate the APR:
APR = (2 × $40.03 ÷ $450) × (12 ÷ 12 + 1)
APR = (0.1774) × (1.0833)
APR = 0.1921 or 19.21%
Therefore, the annual percentage rate for Terry's loan is 19.21%.
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One leg of a right triangle is 3 inches shorter than the other leg. If the hypotenuse is 7
inches, what are the lengths of the legs? Give the exact answers, not decimal
approximations.
Let x = 7/8 and y = -2/9. If x * z = y, then what is z?
Please help me figure this out, im not sure how i should be doing this
The table for the population of the colony at the different times of the study is found.
Explain about the exponential function:Calculating the exponential growth as well as decay of a given collection of data is done using an exponential function, which is a mathematical function. Exponential functions, for illustration, can be used to estimate population changes, loan interest rates, bacterial growth, radioactive decay, as well as disease spread.
Given functions are-
P(t) = 9.[tex](1.02)^{t}[/tex]
Q(t) = 9.[tex]e^{0.02t}[/tex]
Population is estimated in thousands.
Table for the population of the colony:
t (time in months) P(t) = 9.[tex](1.02)^{t}[/tex] Q(t) = 9.[tex]e^{0.02t}[/tex]
6 P(6) = 9.[tex](1.02)^{6}[/tex] = 10.13 Q(t) = 9.[tex]e^{0.02*6}[/tex]= 10.14
12 P(12) = 9.[tex](1.02)^{12}[/tex] = 11.41 Q(t) = 9.[tex]e^{0.02*12}[/tex] = 11.44
24 P(24) = 9.[tex](1.02)^{24}[/tex] = 38.60 Q(t) = 9.[tex]e^{0.02*24}[/tex] = 14.54
48 P(48) = 9.[tex](1.02)^{48}[/tex]= 23.28 Q(t) = 9.[tex]e^{0.02*48}[/tex]= 23.50
100 P(100) = 9.[tex](1.02)^{100}[/tex] = 65.20 Q(t) = 9.[tex]e^{0.02*100}[/tex]= 66.50
Thus, the table for the population of the colony at the different times of the study is found.
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What are the values of the "t" column when the rule of the table is 3n + 5.
Answer:
8,11,14,17
Step-by-step explanation:
3n + 5
When n = 1,
3(1) + 5
=> 3 + 5
=> 8
When n = 2,
3(2) + 5
=> 11
When n = 3,
3(3) + 5
=> 14
When n = 4,
3(4) + 5
=> 17
7
An acceleration of 2.5 miles per hour per second
mi/hr
S
feet per hour per second
is closest to which of the following values, in
(1 mile = 5,280 feet)
A) 13,200
B) 2,112
C) 220
D)
1/2,112
The converted acceleration in feet per hour per second is 13200 feet per hour per second
Converting the acceleration from miles per hour per second to feet per hour per secondFrom the question, we have the following parameters that can be used in our computation:
Acceleration = 2.5 miles per hour per second
From the question, we have
(1 mile = 5,280 feet)
Substitute the known values in the above equation, so, we have the following representation
Acceleration = 2.5 * 5,280 feet per hour per second
Evaluate the products
So, we have
Acceleration = 13200 feet per hour per second
Hence, the acceleration = 13200 feet per hour per second
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The diagram shows a field. 66 m 140 m 102 m Work out the area of the field.
The area of the field, given that it is a trapezium, would be 7, 986 m ² .
How to find the area ?The field is in the shape of a trapezium which means that the area of the field can be found by the formula :
= ( a + b ) / 2 x height
Side a = 102 m
Side b = 140 m
height = 66 m
The area of the field is therefore:
= ( 102 + 140 ) / 2 x 66
= 121 x 66
= 7, 986 m ²
In conclusion, the area of the field is 7, 986 m ².
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Supplementary angles
The angles that is supplementary to <HAE is <IAE
What are supplementary angles?Supplementary angles are simply described as a pair of angles whose sum is 180 degrees.
Some properties of supplementary angles are;
The two angles are said to be supplementary angles when they sum up to 180°The two angles together make a straight line, but the angles must not be together in the same place.In a cyclic quadrilateral, opposite angles are said to be supplementaryFrom the information given, we can see that;
<HAE is on a straight line with <IAE , making their sum to be 180 degrees
Then, <IAE is supplementary is <HAE
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Circle O shown below has a radius of 8 inches. To the nearest tenth of an inch, determine the length of the arc, x, subtended by an angle of 74 .
The length of the arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches is approximately 13.1 inches.
What is the central angle?A central angle is an angle with endpoints located on a circle's circumference and a vertex located at the circle's center (Rhoad et al. 1984, p. 420). A central angle in a circle determines an arc.
The length of an arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches can be found using the formula:
x = (θ/360) × 2πr
where θ is the central angle in degrees, r is the radius of the circle, and π is the mathematical constant pi.
Substituting the given values, we get:x = (74/360) × 2π(8)
x ≈ 4.17π
To get the value to the nearest tenth of an inch, we can use the approximation π ≈ 3.14:
x ≈ 4.17 × 3.14
x ≈ 13.1
Therefore, the length of the arc, x, subtended by an angle of 74° in a circle with a radius of 8 inches is approximately 13.1 inches.
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2.4.4 Quiz: Parabolas with Vertices Not at the Origin
The vertex of this parabola is at (2, -4). When the y-value is -3, the x-value is
-3. What is the coefficient of the squared term in the parabola's equation?
10
-10
OA. -1
B. 1
10-
C. 5
OD. -5
(2,-4)
10
Identify the law illustrated by the following: 4bracket(3x-7) is equivalent to 12x-28.
Compound interest on a certain sum of money for 2 years is Rs.2600 while the simple interest on the same sum for the same time period is Rs.2500. Find the rate of interest.
13.5%
15%
12.8%
8%
The rate of interest according to the given data is 12.8%.
To answer this problem, we may utilize the compound interest formula:
A = P(1 + r/n)^(nt)
where A denotes the ultimate amount, P the principle, r the yearly interest rate, n the number of times the interest is compounded each year, and t the period in years.
We know that the compound interest on a certain payment over two years is Rs. 2600. Let us refer to the principal as P. Then we may type:
2600 = P(1 + r/100)^2 - P
When we simplify this equation, we get:
2600 = P[(1 + r/100)^2 - 1]
Dividing both sides by P[(1 + r/100)^2 - 1], we get:
1 = 1/[(1 + r/100)^2 - 1]
Simplifying even further, we get:
1 + (1 + r/100)^2 - 1 = (1 + r/100)^2
Taking the square root of both sides yields:
1 + r/100 = 1 + sqrt(2600/2500)
1 + r/100 = 1 + 0.04
r/100 = 0.04
r = 4%
As a result, the yearly interest rate is 4%. This, however, is the basic interest rate. To get the compound interest rate, use the following formula:
CI = P(1 + r/100)^2 - P
2500 = P(1 + 4/100)^2 - P
Simplifying this equation, we get:
2500 = P(1.04^2 - 1)
2500 = P(0.0816)
P = 30637.25
So the principal is Rs. 30637.25, and the annual compound interest rate is:
r = 100[(30637.25/10000)^(1/2) - 1]
r = 12.8%
Therefore, the rate of interest is 12.8% (rounded off to one decimal place).
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Two cyclists, 64 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 2 hours later, what is the speed (in mi/h) of the faster cyclist? a. Write an equation using the information as it is given above that can be solved to answer this problem. Use the variable r to represent the speed of the slower cyclist. b. What is the speed of the faster cyclist?
Let ABC be any triangle .D is the midpoint of [BC]
1) Locate the points M, N and I such that :
MA+MB = 0
3NA+NC = 0
IM +2IN = 0
Answer:
We can use vectors to solve this problem. Let's assume that the position vector of point A is a, and the position vectors of points B and C are b and c respectively.
Since D is the midpoint of BC, we can find its position vector as:
d = (b + c)/2
Now, let's find the position vectors of M, N, and I using the given conditions:
MA + MB = 0
This means that the vector MA is equal in magnitude and opposite in direction to the vector MB. Since M is a point on the line segment AB, we can write:
MA = M - a
MB = M - b
So, MA + MB = 0 gives us:
M - a + M - b = 0
2M = a + b
M = (a + b)/2
Therefore, the position vector of M is:
m = (a + b)/2
Similarly, we can find the position vectors of N and I:
3NA + NC = 0
This means that the vector NA is three times the magnitude of the vector NC, and they are in opposite directions. Since N is a point on the line segment AC, we can write:
NA = N - a
NC = N - c
So, 3NA + NC = 0 gives us:
3(N - a) + (N - c) = 0
4N = 3a + c
N = (3a + c)/4
Therefore, the position vector of N is:
n = (3a + c)/4
IM + 2IN = 0
This means that the vector IM is twice the magnitude of the vector IN, and they are in opposite directions. Since I is a point on the line segment DM, we can write:
IM = I - m
IN = I - n
So, IM + 2IN = 0 gives us:
I - m + 2(I - n) = 0
3I = 2m + 2n
I = (2m + 2n)/3
Therefore, the position vector of I is:
i = (2m + 2n)/3
So, the points M, N, and I are located at:
M = (a + b)/2
N = (3a + c)/4
I = (2m + 2n)/3
where:
m = (a + b)/2
n = (3a + c)/4
(I NEED THIS RIGHT NOW!! PLEASE GIVE EXPLANATION!!) Millie and her brother, Jack, sold candy baes for the band fund-raiser. Millie sold three times as many candy bars as Jack. If Jack sold 30 candy bars, which equation and solution can be used to represent y, the number of candy bars Millie sold?
A. 3y = 30; y = 10
B. 30 = y/3; y = 103
C. 30 = y/3; y = 90
D. 3y = 30; y = 90
Answer:
C
Step-by-step explanation:
When we do 30=y/3 we get 90 because of the fraction symbol and its says in the problem 3 times as much so more not less so 3x30 equals 90 so we already take out A and B and when there is nothing but a simple equation like 30=3 we dived so it cant be D Ethier so its C
PLS GIVE BRAINLIST
Happy Easter
If 600 bolts were already examined and 12 were defective, determine the probability that next bolt found would be nondefective.
The probability that the next bolt found would be non-defective is 0.98 or 98%.
What is Probability Theory?A fundamental idea in mathematics and statistics, probability theory has several uses in the sciences, engineering, business, and social sciences. It is used to assess risks and uncertainties in diverse circumstances, analyse and forecast the possibility of events happening, and make decisions in the face of uncertainty.
Given:
Number of bolts already examined = 600
Number of defective bolts found = 12
Number of non-defective bolts = Total number of bolts examined - Number of defective bolts
= 600 - 12
= 588
Probability of finding a non-defective bolt = Number of non-defective bolts / Total number of bolts
= 588 / 600
= 0.98 or 98%
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1/x is undefined for which real numbers?
Answer:
0
Step-by-step explanation:
Don't know answer please help?
Answer:
104 square meters
Step-by-step explanation:
Make sure all the units are same and consistent!!
The shape of the shaded region = Trapezium
h = Perpendicular height of trapezium
= 16 m
∴Area of shaded region = Area of trapezium
= [tex]\frac{1}{2}[/tex] × (Sum of parallel sides) × h
= [tex]\frac{1}{2}[/tex] × [(9 + 4) m] × (16 m)
= [tex]\frac{1}{2}[/tex] × (13 m) × (16 m)
∴Area of shaded region = 104 [tex]m^{2}[/tex]
= 104 square meters
what is the equation that represents the linear function f for the points (1,2) and (4,1)
Answer: f(x)= -0.333x + 2,333
A television show conducted an experiment to study what happens
Answer:
Step-by-step explanation:
[tex]H_o: p = 0.5,H_1:p\neq 0.5[/tex]
[tex]p(\text{buttered side down})=\frac{19}{51}[/tex]
[tex]\pi _o=0.5\\n=51[/tex]
[tex]z=\frac{p-\pi _o}{\sqrt{\frac{\pi _o(1-\pi _o)}{n} } } =\frac{\frac{19}{51}-0.5 }\sqrt{{\frac{0.5(1-0.5)}{51} } }[/tex]
≈ [tex]-1.82[/tex]
a = 0.01 [tex]z_a/z=2.58\\|z| < |z_[ay}}_2|[/tex]
Thus, it cant reject H₀
The toast will land with the buttered side down 50% of the time.
{Hypothesis test}
1) Find AB
B
A 9 in
Use the Law of
Sines:
SinA
a
(0) 8 in
A
54°
11 in
D 7 in
5 in
SinB
b
12
31 in
SinC
C
Remember
Check the picture below.
[tex]\textit{Law of sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(54^o)}{31}=\cfrac{\sin(12^o)}{AB}\implies AB\sin(54^o)=31\sin(12^o) \\\\\\ AB=\cfrac{31\sin(12^o)}{\sin(54^o)}\implies AB\approx 8~in[/tex]
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