The p-value method yields P (T > 2.119) = 0.0359. It may be said that, at a 1% level of significance, there has been no decline in crime since the program's inception.
Explain about the Paired T-Test:The observations for the paired t-test are drawn from a single random pair sample. There are the same number of observations in both samples. Each respondent has two observations for the experimental variable recorded.
The alternative is:
[tex]H_{0} =[/tex]: Since the program's launch, there has been no drop in the number of crimes.
The alternate theory is that:
[tex]H_{a} =[/tex] Since the start of the programme, fewer crimes have been committed.
Mathematically,
[tex]H_{0} =[/tex] μo = 0
[tex]H_{a} =[/tex] μd > 0
Given table:
Number of Crimes Before After d = (before - after) d²
A 14 2 14-2 = 12 14² = 144
B 7 7 17 - 7 = 0 0² = 0
C 4 3 4 - 3 = 1 1² = 1
D 5 6 5 - 6 = -1 -1² = 1
E 17 8 17-8 = 9 9² = 81
F 12 13 12 - 13 = -1 -1² = 1
G 8 3 8 - 3= 5 5² = 25
H 9 5 9 -5 = 4 4² = 16
Total 8 29 269
The formula for the t statistic is:
t = ∑d / √[n(∑d²) - (∑d)²] / (n - 1)
t = 29 / √[8*269 - 29²] / (8 - 1)
t = 2.119
As a result, the test statistic's value is 2.119.
8 – 1 Equals 7 degrees of freedom.The p-value method yields P (T > 2.119) = 0.0359.The p-value is inconsequential since it exceeds the specified significance level of 0.01 in this case.
This indicates that at a 1% level of significance, the null hypothesis cannot be ruled out.
As a result, it may be said that, at a 1% level of significance, there has been no crime decline in since the program's inception.
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Brandon read the book each day until he finishedit. Explain how you would use the data to label and plot the dotson a dot plot. What is the difference between the longest time andshortest time Brandon spent reading the book?q TEKS 4.9.B ©
1 _ 4 ,1 _ 4 , 1,1 _ 4 ,1 _ 2 ,3 _ 4 ,1 _ 2 ,1 _ 4
the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.
How to solve the problem?
To label and plot the data on a dot plot, we need to first determine the range of values. In this case, we have the following data:
1/4, 1/4, 1, 1/4, 1/2, 3/4, 1/2, 1/4
The range of values is from 1/4 to 1, and we can use a dot plot to represent the data. On the dot plot, we will place a dot above each value on the number line. For example, we will place four dots above the value 1/4 since it appears four times in the data.
The dot plot would look like this:
markdown
Copy code
|
0.2 | x
|
0.4 | x x x x
|
0.6 | x x
|
0.8 | x x x
|
1.0 | x
|
From the dot plot, we can see that Brandon spent the most time reading the book when he read for the full day, which is represented by the dot at 1. The shortest time he spent reading was when he read for 1/4 of a day, which is represented by the dots at 1/4.
To find the difference between the longest and shortest time, we simply subtract the shortest time from the longest time. In this case, the longest time was 1 day, and the shortest time was 1/4 of a day. So the difference is:
1 - 1/4 = 3/4 of a day
Therefore, the difference between the longest and shortest time Brandon spent reading the book was 3/4 of a day.
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Please help with question #2
Answer:
50 in
Step-by-step explanation:
Split into 4 shapes. One on each side with 5 in x 2 in, one on the top with 12 in x 2 in, and one in the middle with 3 in x 2 in. We add all together:
2(5×2)+(12×2)+(3×2) = 50 in
a tree in lamar’s backyard has been growing for 4 1/2 years. the tree is 4/5 meters tall. what is the unit rate in meters per year? write your answer as a fraction or pixies number in simplest form.
As the tree is 4/5 meters tall the unit rate in meters per year is 8/45 or approximately 0.177 meters per year.
To find the unit rate in meters per year, we need to divide the height of the tree by the time it took to grow.
Height of the tree = 4/5 meters
Time taken to grow = 4 1/2 years = 9/2 years
Unit rate = Height/Time = (4/5)/(9/2) meters per year
To divide by a fraction, we can multiply by its reciprocal:
Unit rate = (4/5) * (2/9) meters per year
Unit rate = 8/45 meters per year
Therefore, the unit rate of the tree in meters per year is 8/45 or approximately 0.177 meters per year.
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Given this equation what is the value of y at the indicated point?
Answer:
y = 4
Step-by-step explanation:
given equation of line is
y = - 3x - 2 and (- 2, y ) is a point on the line, then substitute x = - 2 into the equation for y , that is
y = - 3(- 2) - 2 = 6 - 2 = 4
A weather station on the top of a mountain reports that the temperature is currently -10*C and has been falling at a constant rate of 3*C per hour. If it continues to fall at this rate, find each indicated temperature.
What will the temperature be in 5 hours?
What was the temperature 8 hours ago?
PLEASE HELP ME!!!!
Answer:
The temperature in 5 hours will be -25°C.
The temperature 8 hours ago was 14°C.
Step-by-step explanation:
To find the temperature, y, in degrees Celsius at any given number of hours, x, from the current time, we can create a linear equation in the form y = mx + b, where m is the rate of change and b is the current temperature.
Given the current temperature is -10°C, then b = -10.
Given the temperature falls at a constant rate of 3°C per hour, then m = -3.
Substitute the values of m and b into the formula:
[tex]\boxed{y = -3x - 10}[/tex]
where:
y is the temperature in °C.x is the number of hours.To calculate the temperature in 5 hours time, substitute x = 5 into the equation:
[tex]\begin{aligned}\implies y&=-3(5)-10\\&=-15-10\\&=-25^{\circ}\text{C}\end{aligned}[/tex]
Therefore, the temperature in 5 hours will be -25°C.
To calculate the temperature 8 hours ago, substitute x = -8 into the equation. The value of x is negative since we want to find the temperature 8 hours before the current time.
[tex]\begin{aligned}\implies y&=-3(-8)-10\\&=24-10\\&=14^{\circ}\text{C}\end{aligned}[/tex]
Therefore, the temperature 8 hours ago was 14°C.
HELP PLSSS. turns out I got the last problem wrong :(
The amount invested at 4% interest rate that is $3900 and
the amount invested at 3% interest rate is $4300.
Last year Sarah invested money in two accounts. The first account had an interest rate of 3% and the second account had an interest rate of 4%. If she invested $400 more in the first account than the second and her total interest income was $285,
the equations are:
s= f + 400 (1)
0.03s + 0.04f = 285 (2)
modifying (1) and (2) for simpler calculations
3s - 3f = 3 * 400 (3)
3s + 4f = 100 * 285 (4)
by solving eqns (3) and (4) we get
we get f=3900 and s=4300
Here, f is the amount invested at 4% interest rate that is $3900 and
s refers to the amount invested at 3% interest rate is $4300.
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Suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000. Use the 68-95-99.7 rule to find the percentage of buyers who paid:
between $150,000 and $156,000 if the standard deviation is $2000.
The percentage of buyers who paid between $150,000 and $156,000 if the standard deviation is $2000 is 49.9%
Calculating the percent of data of values from the the z-scoresFrom the question, we have the following parameters that can be used in our computation:
Mean = 150000
Standard deviation = 2000
Scores = from $150,000 and $156,000
So, we have
z = (150000 - 150000)/2000 = 0
z = (156000 - 150000)/2000 = 3
This means that it is between a z-score of 0 and a z-score of 3
This is represented as
Probability = (0 < z < 3)
Using a graphing calculator, we have
Probability = 0.49865
Express as percentage
Probability = 49.9%
Hence, the probability is 49.9%
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Please help
Question in image
Answer:
C 86
Step-by-step explanation:
You want the measure of the angle marked x° where chords cross. The angle marked x° intercepts an arc of 104° and one that is unmarked. The remaining arcs of the circle are marked 111° and 77°.
Measure of xThe measure of angle x is half the sum of the arcs it intercepts. One of those is given as 104°. The other will be ...
360° -111° -104° -77° = 68°
Then the value of x is ...
(104 +68)/2 = 172/2 = 86
The value of x is 86, choice C.
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a skateboard ramp is 11 in high and raises at an angle of 22 in how long is the base of the ramp? Round to the nearest inch.
Answer: The length of the base of the ramp is approximately 27 inches when rounded to the nearest inch.
Step-by-step explanation:
We can use trigonometry to solve this problem.
Let's call the length of the base of the ramp "x". Then, we can use the tangent function to find x:
tan(22) = 11/x
To solve for x, we can multiply both sides by x and divide by tan(22):
x = 11 / tan(22)
Using a calculator, we can find that:
x ≈ 27 inches
Therefore, the length of the base of the ramp is approximately 27 inches when rounded to the nearest inch.
Greetings! ZenZebra at your service, hope it helps! <33
What is the inverse of the function, f(x)=2x−1‾‾‾‾‾‾√3−3?
Responses
f−1(x)=(x+3)3−12
f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator open parentheses x plus 3 close parentheses cubed minus 1 over denominator 2 end fraction
f−1(x)=(x−3)3−12
f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator open parentheses x minus 3 close parentheses cubed minus 1 over denominator 2 end fraction
f−1(x)=(x−3)3+12
f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator open parentheses x minus 3 close parentheses cubed plus 1 over denominator 2 end fraction
f−1(x)=(x+3)3+12
Answer:
its the third please mark me as a brainliest
I need help plssss i domt get it
Answer:
V = 5747.02 ft^3
Step-by-step explanation:
We're given the general formula for volume (V) and we must use this general formula and the information we're given to find the volume of the hemisphere.r stands for radius, which is 1/2 the distance between the center of a circle and the edge. The 14 in the diagram represents the radiusIn the formula r^3 means that the radius is cubed and thus 14 must be multiplied by itself three times (14 * 14 * 14)
Therefore, to solve for volume, we plug in 14 for r in the general formula:
[tex]V=2/3\pi (14)^3\\V=2/3\pi (14)(14)(14)\\V=2/3\pi *2744\\V=5747.020161\\V=5747.02[/tex]
Lastly, volume is always in units cubed (e.g., ft^3 or mi^3 since volume deals with the three-dimensional space of an object
[64+{23-(-14+5x12/2-6)}]
Answer:
77
Step-by-step explanation:
First, let's simplify the expression inside the innermost parentheses:
5x12 = 60
60/2 = 30
-14 + 30 - 6 = 10
Now, we have:
[64 + {23 - 10}] = [64 + 13] = 77
The body temperatures in degrees Fahrenheit of a sample of adults in one small town are:
98 97.9 99 97.3 98.9 99.6 97.8 99.8 99.1 98.4 98.7 97.6 96.5
Assume body temperatures of adults are normally distributed. Based on this data, find the 95% confidence
interval of the mean body temperature of adults in the town. Enter your answer as an open-interval (i.e.,
parentheses) accurate to 3 decimal places. Assume the data is from a normally distributed population.
95% C.I. =
The 95% confidence interval of the mean body temperature of adults in the town is (97.584, 98.876)°F (open-interval notation), accurate to 3 decimal places.
What is the 95% confidence interval of the mean body temperature of adults in the town?To find the 95% confidence interval (C.I.) of the mean body temperature, we need to use the formula:
C.I. = x ± z*(σ/√n)
Where:
x is the sample mean
σ is the population standard deviation (unknown, but we can estimate it using the sample standard deviation)
n is the sample size
z is the critical value for the desired confidence level (95% in this case)
First, we need to calculate the sample mean, sample standard deviation, and sample size:
x = (98 + 97.9 + 99 + 97.3 + 98.9 + 99.6 + 97.8 + 99.8 + 99.1 + 98.4 + 98.7 + 97.6 + 96.5) / 13 = 98.23
s = √((1/(n-1)) * Σ(xi - x)²)
s = 1.339
n = 13
Next, we need to find the critical value, z, for a 95% confidence level. We can look this up in a standard normal distribution table or use a calculator. For a 95% confidence level, z = 1.96.
Now we can plug in the values into the formula to get the 95% confidence interval:
C.I. = 98.23 ± 1.96*(1.339/√13) = (97.584, 98.876)
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there are 105 days until basketball season starts. Melanie says the season starts in 15 weeks. Is this reasonable? Use estimation to justify your answer.
Answer:
Yes, Melanie's statement that the basketball season starts in 15 weeks is reasonable.
To see why, we can estimate the number of days in 15 weeks. Since 1 week is equal to 7 days, 15 weeks would be approximately equal to:
15 weeks x 7 days/week = 105 days
What is the meter reading, in ccf, indicated by each of the gas meters shown? cengage
The meter reading, in ccf, indicated by each of the gas meters would be 8.83 ccf.
Required to find the meter reading in ccf
Let x be [tex]m^{2}[/tex] - meter readings.
The readings in ccf is:
Reading = x/ 2.832
Now, Assume the value of x is 25; The readings will be:
Reading = 25/ 2.832 ccf
Reading = 8.83 ccf
The complete question is
Manuel works for the water company as a meter reader. The meter below is the Jansen's water meter. What is the reading, in ccf, on the meter shown?
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Which statements are true about circle Q? Select three options.
The ratio of the measure of central angle PQR to the measure of the entire circle is One-eighth.
The area of the shaded sector is 4 units2.
The area of the shaded sector depends on the length of the radius.
The area of the shaded sector depends on the area of the circle.
The ratio of the area of the shaded sector to the area of the circle is equal to the ratio of the length of the arc to the area of the circle.
Answer:
Step-by-step explanation:
The three true statements about circle Q are:
1. The ratio of the measure of central angle PQR to the measure of the entire circle is One-eighth.
2. The area of the shaded sector is 4 units 2.
3. The ratio of the area of the shaded sector to the area of the circle is equal to the ratio of the length of the arc to the circumference of the circle.
The True statements are:
The ratio of the measure of central angle PQR to the measure of the entire circle is 1/8.
The area of the shaded sector depends on the length of the radius.
The area of the shaded sector depends on the area of the circle.
What is Circumference?The measurement of the circle's perimeter, also known as its circumference, is called the circle's boundary. The region a circle occupies is determined by its area.
First, The measure of the central angle is 45°.
So, ratio of the central angle to the entire circle
= 45/360
= 9/72
= 1/8
Now, area of shaded sector
A = πr² = π(6²) = 36π
or, 1/8(36π)
= 36π/8
= 18π/4
= 9π/2
= 4.5π square units.
Since, we use the circle's radius to determine area, the shaded sector's area is dependent on the circle's radius.
Given that the sector is a portion of the circle, its size is also influenced by the size of the circle.
and, The ratio of the shaded sector's area to the circle's area wouldn't be the same as the ratio of the arc's length to the circle's area.
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given h(x)=2x^2-7x+4, find h(-7)
The output value of h(-7) in the function h(x) = 2x² - 7x + 4 is 151.
What is the output value of h(-7) in the given function?A function is simply a relationship that maps one input to one output.
Given the function in the question
h(x) = 2x² - 7x + 4,
Find h(-7)
To find h(-7), we substitute -7 for x in the equation of h(x) and simplify:
h(x) = 2x² - 7x + 4
Plug in x = -7
h(-7) = 2(-7)² - 7(-7) + 4
h(-7) = 2(49) + 49 + 4
h(-7) = 98 + 49 + 4
h(-7) = 151
Therefore,the output value of h(-7) is equal to 151.
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Which monomial is a perWhich is the completely factored form of 4x2 + 28x + 49?fect cube?
The completely factored form of the expression 4x² + 28x + 49 is (x + 7)(4x + 7) (option a).
In this case, we need to factor the expression 4x² + 28x + 49. One way to do this is by using the quadratic formula, which is a formula used to find the roots or solutions of a quadratic equation.
However, since we are only looking for the completely factored form, we can use a simpler method known as "factoring by grouping." This method involves grouping terms with common factors and factoring them separately.
Starting with the given expression:
4x² + 28x + 49
We notice that the first two terms have a common factor of 4x:
4x(x + 7) + 49
Now, we can factor out the common factor of (x + 7) from the first two terms:
4x(x + 7) + 49 = (x + 7)(4x + 49/7)
Simplifying the expression in the second set of parentheses:
4x + 49/7 = 4x + 7
Hence the correct option is (a).
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Complete Question:
Which is the completely factored form of 4x2 + 28x + 49?
a) (x + 7)(4x + 7)
b) 4(x + 7)(x + 7)
c) (2x + 7)(2x + 7)
d) 2(x+7)(x + 7)
What is the surface area of this cylinder?
Use ≈ 3.14 and round your answer to the nearest hundredth.
10 mm
5 mm
square millimeters
The surface area of the cylinder is 471 mm².
Given is a cylinder of height 5 mm and radius 10 mm, we need to find the surface area of the cylinder,
So, the surface area of the cylinder is = 2π × r (h+r)
= 2 × 3.14 × 5 (5+10)
= 2 × 3.14 × 75
= 471 mm²
Hence, the surface area of the cylinder is 471 mm².
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Eight cards with one letter on each card spell out the word SURPRISE. If you choose one card at random, what is the probability of this event?
P(R) =
The tractor will cost £64,950 the compound interest rate is 4% per annum. You will pay back the loan in 36 equal monthly payments. What will you pay each month?
Answer: You will pay £1,912.54 each month.
Step-by-step explanation:
We can use the formula for the monthly payment of a loan with compound interest:
monthly payment = (principal * monthly interest rate) / (1 - (1 + monthly interest rate)^(-n))
where principal is the amount borrowed, monthly interest rate is the annual interest rate divided by 12, and n is the total number of monthly payments.
In this case, the principal is £64,950, the annual interest rate is 4%, so the monthly interest rate is 4%/12 = 0.00333, and the number of monthly payments is 36.
Plugging in these values, we get:
monthly payment = (64950 * 0.00333) / (1 - (1 + 0.00333)^(-36)) = £1,912.54
Therefore, you will pay £1,912.54 each month.
5x+3y=4 -2x-8y=6 Which strategy can you use to eliminate a variable.
The strategy that could be used to eliminate a variable is:
Multiply the first equation by 2 and multiply the second equation by 5. Then, add the resulting equations
Solving system of linear equations using the elimination method: Determining a strategy to eliminateFrom the question, we are to determine a strategy that could be used to eliminate a variable in the given system of equations
From the given information,
The system of linear equations is
5x + 3y = 4
-2x - 8y = 6
To eliminate the variable x, we can multiply the first equation by 2 and then multiply the second equation by 5. After that, we will add the resulting equations
2 × [ 5x + 3y = 4
5 × [-2x - 8y = 6
10x + 6y = 8
-10x - 40y = 30
Add the resulting equations
10x + 6y = 8
+ (-10x - 40y = 30
------------------------
-34y = 38
This way we have eliminated x
Hence,
The above strategy can be used to eliminate a variable
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Leslie deposits $1,600 into an account that earns 3.7% annual interest compounded quarterly. What will be
the total value of her investment after 4 years?
Answer:
A = $1853.96
Step-by-step explanation:
The formula for compound interest is
[tex]A(t)=P(1+r/n)^n^t[/tex], where A(t) is the amount, P is the principal (amount invested or deposited), r is the interest rate, n is the number of compound periods per year, and t is the time in years.
For the problem, our P value is $1600Our r value is 0.037 (we must convert the percent to a decimal)Our n value is 4 (quarterly means 4 so the money is compounded once every 3 months since there 3 months make up a quarter in a year)Our t value is 4We must solve for A(t)[tex]A(t)=1600(1+0.037/4)^(^4^*^4^)\\A(t)=1600(1.00925)^1^6\\A(t)=1853.958942\\A(t)=1853.96[/tex]
a bookshelf holds 24 mysteries, 12 nonfiction, and 4 biographies. what is the probability of selecting a mystery book, then without replacing it, selecting a non-fiction book?
The probability of selecting a mystery book, then without replacing it, selecting a nonfiction book is approximately 0.1462.
How to determine the probability of selecting a mystery book, then without replacing it, selecting a non-fiction bookOn the bookcase, there are 24 + 12 + 4 = 40 volumes.
The odds of picking a mystery book on the first draw are 24/40.
There are 39 books remaining after the first is drawn, including 12 nonfiction works. Because the initial book was never replaced, there are currently only 11 nonfiction volumes left.
Given that a mystery book was chosen without replacement in the first draw, the probability of selecting a nonfiction book in the second draw is 12/39.
When we multiply these probability together, we get:
P(nonfiction after mystery, not replacing) * P(mystery, not replacing) = (24/40) * (12/39) = 0.14615...
P(nonfiction after mystery, not replacing) * P(mystery, not replacing) = 0.1462
So, the probability of selecting a mystery book, followed by a nonfiction book without replacing it is roughly 0.1462.
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Find the surface area of a sphere with the circumference of 13mm
well, we know the sphere's "great circle" has a circumference of 13mm, so
[tex]\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=13 \end{cases}\implies 13=2\pi r\implies \cfrac{13}{2\pi }=r \\\\[-0.35em] ~\dotfill\\\\ \textit{surface area of a sphere}\\\\ SA=4\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=\frac{13}{2\pi } \end{cases}\implies SA=4\pi \left( \cfrac{13}{2\pi } \right)^2 \\\\\\ SA=\cfrac{169}{\pi } \implies SA\approx 53.79~mm^2[/tex]
Callie spent 1 1/2 hours outside with her sister. They spent half of that time jumping on the trampoline. How long did they jump on the trampoline?
Answer:
45 Minutes
Step-by-step explanation:
1.5 Hours = 90 Minutes (1.5*60)
90/2 = 45
45 Minutes on the trampoline.
let A=B=R. define R=(x, y)E A×B{x²+y²=1, |x-y|=1}. Find R
R is the set containing the two points (1, 0) and (-1, 0).
Describe Sets?In mathematics, a set is a collection of distinct objects, called elements or members, that share a common property. The objects in a set can be anything, such as numbers, letters, shapes, or even other sets. Sets are denoted by braces {} and the elements are separated by commas.
Set theory is an important branch of mathematics and is used in various fields such as algebra, geometry, logic, and computer science. Sets provide a foundation for mathematical concepts such as functions, relations, and number systems. They are also used in the study of probability and statistics, and in the design and analysis of algorithms.
To find R, we need to determine the set of all points (x, y) that satisfy the conditions given:
x² + y² = 1 and |x - y| = 1
Let's consider the two cases of |x - y| = 1:
Case 1: x - y = 1
In this case, we can substitute y = x - 1 into the equation x² + y² = 1 to get:
x² + (x - 1)² = 1
2x² - 2x = 0
2x(x - 1) = 0
So x = 0 or x = 1. If x = 0, then y = -1, which does not satisfy the condition x² + y² = 1. If x = 1, then y = 0, which does satisfy the condition. Therefore, we have one point that satisfies the conditions in this case:
R1 = {(1, 0)}
Case 2: x - y = -1
In this case, we can substitute y = x + 1 into the equation x² + y² = 1 to get:
x² + (x + 1)² = 1
2x² + 2x = 0
2x(x + 1) = 0
So x = 0 or x = -1. If x = 0, then y = 1, which does not satisfy the condition x² + y² = 1. If x = -1, then y = 0, which does satisfy the condition. Therefore, we have one more point that satisfies the conditions in this case:
R2 = {(-1, 0)}
Putting both cases together, we have:
R = R1 ∪ R2 = {(1, 0), (-1, 0)}
Therefore, R is the set containing the two points (1, 0) and (-1, 0).
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$44,000 a 6% for 2 years for simple interest
The total interest that will accrue on the principal amount of $44,000 over a period of 2 years at a simple interest rate of 6% is $5,280.
The given problem involves calculating the amount of interest that will accrue on a principal amount of $44,000, which is invested at a simple interest rate of 6% for a period of 2 years.
To solve the problem, we can use the formula for simple interest, which is:
Interest = Principal x Rate x Time
In this case, the principal amount is $44,000, the rate of interest is 6%, and the time period is 2 years. So, plugging in these values, we get:
Interest = $44,000 x 0.06 x 2
Interest = $5,280
Unlike compound interest, simple interest is calculated only on the principal amount, and not on the accumulated interest. This means that the interest earned each year remains constant, which can make it easier to calculate and understand compared to compound interest.
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A farmer ships oranges in wooden crates. Suppose each orange weighs the same amount. The total weight of a crate filled with g oranges is 24.5 pounds. Write an equation that represents the relationship between the weight of the crate and the number of oranges it contains.
Empty crate:15 ib
Orange:0.38 ib
24.5=___+___×___
In the diagram shown, ABCD is a square. Suppose AB = 6. Find:
17). AC
18). AO
19). BO
Suppose AO = 4. Find:
20). BO
21). Area of Δ AOB
22). Area of ABCD
Suppose DO = 5. Find:
23). DC
24). Area of Δ DOC
25). Area of ABCD
ABCD is square, AC = 6, AO = 3√2, BO = 3√2, BO = 4, DC = 6, OC = 2√13 and area is 36 square units.
To solve the problems, we can use the following properties of squares:
The diagonals of a square bisect each other and are equal in length.
The diagonals of a square form two congruent right triangles.
The center of a square is equidistant from all four vertices of the square.
Since ABCD is a square, we have AC = AB = 6.
Since O is the center of the square, AO is half the length of the diagonal of the square. Using the Pythagorean theorem, we can find the diagonal of the square:
diagonal² = AB² + BC²
= 6² + 6²
= 72
diagonal = √72 = 6√2
Therefore, AO = (1/2) diagonal = 3√2
Since O is the center of the square, BO is equal in length to AO, which we found in part (18). Therefore, BO = 3√2.
If AO = 4 and BO = AO, then BO = 4.
The area of ΔAOB is half the area of the square ABCD. The area of a square is given by side², so the area of ABCD is 6² = 36. Therefore, the area of ΔAOB is (1/2) × 36 = 18 square units.
The area of a square is given by side², so the area of ABCD is 6² = 36 square units.
Since the diagonals of a square bisect each other, DO is half the length of the diagonal of the square. Using the Pythagorean theorem, we can find the diagonal of the square:
diagonal² = AB² + BC²
= 6² + 6²
= 72
diagonal = √72 = 6√2
Therefore, DO = (1/2) diagonal = 3√2
Since ABCD is a square, DC = AB = 6.
The area of ΔDOC is half the area of the square ABCD. Using the Pythagorean theorem, we can find the length of OC:
OC² = AO² + AC²
= 4² + 6²
= 52
OC = √52 = 2√13
Therefore, the area of ΔDOC is (1/2) × 6 × 2√13 = 6√13 square units.
The area of a square is given by side², so the area of ABCD is 6² = 36 square units.
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