So the base of the triangle is 60 feet.
What Does a Triangle's Area Mean?
The total area that is bounded by a triangle's three sides is referred to as the triangle's area. In essence, it is equal to 1/2 of the height times the base, or A = 1/2 b h. So, we need to know the triangular polygon's base (b) and height (h) in order to calculate its area. Any triangle kinds, including scalene, isosceles, and equilateral, can use it. It should be observed that the triangle's base and height are parallel to one another. Square units are used to measure the area unit (m2, cm2).
Let’s call the height of the triangle “h” and the base “b”. We know that the area of the triangle is 450 square feet. We also know that the base is 4 times the height. So we can write:
b = 4h
We can use this information to write an equation for the area of the triangle:
(1/2)b×h = 450
Substituting b = 4h, we get:
(1/2)(4h)(h) = 450
Simplifying this equation gives us:
2h²= 450
Dividing both sides by 2 gives us:
h² = 225
Taking the square root of both sides gives us:
h = 15
So the height of the triangle is 15 feet. We can use this to find the base:
b = 4h = 4(15) = 60
So the base of the triangle is 60 feet.
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A new virus test was recently developed by a pharmaceutical company. It returns a
positive result for 99.5% of individuals who have the virus. However, it also returns a
positive result for 0.5% of those who do not have the virus. To evaluate the test, a large number of people were tested at random. Overall, 1% of people tested positive. What percentage of those testing positive actually had the virus?
"3.1" is the answer. Is the answer correct? If not, why? What's the right answer
Using probability, the percentage of people who test positive for the virus who actually have the virus is approximately 99.85%.
What is probability?
Probability is a measure of the likelihood or chance that an event will occur. It is a numerical value between 0 and 1, where 0 represents an impossible event (i.e., an event that will never occur) and 1 represents a certain event (i.e., an event that will always occur).
Now,
To determine the percentage of people who test positive for the virus who actually have the virus, we can use Bayes' theorem.
Let A be the event that an individual has the virus, and B be the event that an individual tests positive for the virus. Then, we want to calculate the conditional probability of A given B, denoted as P(A|B), which is the probability that an individual has the virus given that they test positive.
Using Bayes' theorem, we can write:
P(A|B) = P(B|A) * P(A) / P(B)
where P(B|A) is the probability of testing positive given that an individual has the virus, P(A) is the overall probability of having the virus, and P(B) is the overall probability of testing positive.
We are given that the test returns a positive result for 99.5% of individuals who have the virus, so P(B|A) = 0.995. We are also given that the test returns a positive result for 0.5% of those who do not have the virus, so the probability of testing positive given that an individual does not have the virus is 1 - 0.005 = 0.995.
We are given that overall, 1% of people tested positive, so:
P(B) = 0.01
We can calculate the overall probability of having the virus as:
P(A) = (number of people with the virus) / (total number of people tested)
We do not have enough information to calculate these values directly, but we can use the fact that the test returns a positive result for 99.5% of individuals who have the virus to estimate the number of true positives.
Let n be the total number of people tested. Then, we can estimate the number of true positives as:
(number of true positives) = 0.995 * (number of people with the virus)
We know that overall, 1% of people tested positive, so:
(number of true positives) / n = 0.01
Substituting the estimate for the number of true positives, we get:
0.995 * (number of people with the virus) / n = 0.01
Solving for the number of people with the virus, we get:
(number of people with the virus) = 0.01 * n / 0.995
Substituting this into the expression for P(A), we get:
P(A) = (0.01 * n / 0.995) / n = 0.01005
So the overall probability of having the virus is approximately 0.01005.
Finally, substituting all the values into the expression for P(A|B), we get:
P(A|B) = 0.995 * 0.01005 / 0.01 ≈ 0.9985
Therefore, the percentage of people who test positive for the virus who actually have the virus is approximately 99.85%.
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Which method can be used to prove the triangles are congruent?
Develop a for a random variate generator for a random variable X with p.d.f f(x)={█(ⅇ^(2x:)-∞
The inverse transform method can be used to produce a random variate for a random variable X with p.d.f[tex]f(x) = e^{2x}[/tex].
How are random variates developed?The cumulative distribution function (CDF) of X is given by [tex]F(x) = 1 - e[/tex], and we must first determine its inverse (-2x). When x is solved for, the answer is x = -(1/2)ln(1 - u), where u is an evenly spaced random number between 0 and 1.
As a result, we can take the following actions to create a random variate for X:
Create an even, random number between 0 and 1, u.
Determine[tex]x = -(1/2)ln (1 - u)[/tex].
A random variate for X is the produced value of x.
The generated values will adhere to the specified probability distribution thanks to this procedure.
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Correct question is:
Develop a random variate generator for a random variable X with the probability density function: [tex]f(x) = e^2x x < 0 e^ -2x x > 0[/tex] Use the generator to generate two random variates using the following uniform random numbers: U1= 0.95 and U2= 0.35.
For each of the following relations, i) state whether the relation is an equivalence relation and ii) if it is an equivalence relation, list the blocks of its corresponding partition, and if it is not, list all of the properties (reflexive, symmetric, transitive) it violates. In all cases, the relations are on the set A = {1, 2, 3, 4}
a. {(1, 1), (1,4), (2, 2), (3, 3), (4, 1), (4, 4)}
b. A x A
c. {(1, 1), (2, 2), (3, 1), (1, 3), (4, 4)
d. ({2, 3} x A) U ((1, 1), (4, 4)}
a. The relation is an equivalence relation, and its corresponding partition has blocks {1, 4}, {2}, and {3}.
b. The relation is not specified.
c. The relation is not an equivalence relation because it is not transitive.
d. The relation is not an equivalence relation because it is not reflexive.
a. The relation is an equivalence relation because it is reflexive, symmetric, and transitive. The blocks of its corresponding partition are {1, 4} and {2} and {3}.
b. The relation is not specified, so it is impossible to determine whether it is an equivalence relation or not.
c. The relation is not an equivalence relation because it is not transitive. Specifically, (3, 1) and (1, 3) are in the relation, but (3, 3) is not.
d. The relation is not an equivalence relation because it is not reflexive (i.e., 1 and 4 are not related to themselves).
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PLEASEEEEE HELP MEEE
A car was valued at $45,000 in the year 1991. The value depreciated to $12,000 by the year 2000.
A) What was the annual rate of change between 1991 and 2000?
r=-------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2003 ?
value = $----------------Round to the nearest 50 dollars.
The rates and the car value are 86.3% and 7679.66 respecyivelt
A) The annual rate
The function is represented as
y = ab^x
Where
a = 45000 i.e. value in 1991
So, we have
y = 45000b^x
In 2000, we have
45000b^9 = 12000
This gives
b^9 = 0.266
Solving for b, we have
9ln(b) = ln(0.266)
So, we have
ln(b) = ln(0.266)/9
This gives
b = e^(ln(0.266)/9)
Evaluate
b = 0.863
B) To convert the rate to a percentage, we multiply by 100 and add the percent symbol:
b = 0.863
So, we have
b = 86.3%
C) If the car value continues to drop by the same percentage, we can use the formula:
Value = 45000* (0.863)^12
Evaluate
Value = 7679.66
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For this question, I thought the correct way to lay out the problem was by finding the probability for each prize and then adding up the products. I am getting it wrong.
The expected value of the raffle if you buy 1 ticket is -$0.56, which means, you can expect to lose $0.56 for each ticket you buy.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
To calculate the expected value of the raffle if you buy 1 ticket, we need to find the expected monetary value of the prizes you could win, and then subtract the cost of the ticket, which is $1.
The probability of winning each prize can be calculated as follows:
1 prize of $500: 1/7000 chance of winning
3 prizes of $100: 3/7000 chance of winning each prize
5 prizes of $20: 5/7000 chance of winning each prize
19 prizes of $5: 19/7000 chance of winning each prize
To calculate the expected value of each prize, we multiply the probability of winning by the amount of the prize:
$500 prize: (1/7000) x $500 = $0.07
$100 prize: (3/7000) x $100 = $0.04
$20 prize: (5/7000) x $20 = $0.01
$5 prize: (19/7000) x $5 = $0.01
To find the expected value of the raffle if you buy 1 ticket, we add up the expected values of all the possible prizes:
$0.07 + ($0.04 x 3) + ($0.01 x 5) + ($0.01 x 19) = $0.44
Therefore, the expected value of the raffle if you buy 1 ticket is $0.44 - $1 = -$0.56, which means that on average, you can expect to lose $0.56 for each ticket you buy.
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please i need it thanks
I think it is <YTZ, because they are corresponding angles.
A spinner with 4 colored sections (red, blue, yellow, green) is spun two times in a row. Using your sample space from the previous question,
what is the probability of getting red on at least one spin?
a
1/16
b
1/4
c
7/16
d
1/2
the probability of getting red on at least one spin is 1 - 9/16 = 7/16 or 0.4375 in decimal form.
hope that helps!
A pizza shop in Montgomery wants to determine how often it delivers to three different neighborhoods. The table shows the areas from a random sample of 80 deliveries. Neighborhood Number of Deliveries 18 24 38 Cloverdale Forest Park Ridgefield Based on these data, if the driver makes 200 deliveries, how many deliveries will be to the Forest Park neighborhood?
It can be estimated that 60 deliveries will be to the Forest Park neighborhood if the driver makes 200 deliveries.
How do we calculate?We can deduct that from the sample of 80 deliveries, the Forest Park neighborhood received 24 deliveries.
We then estimate how many deliveries will be to the Forest Park neighborhood if the driver makes 200 deliveries by applying proportional reasoning:
The sample of 80 deliveries represents 100% of the deliveries in the sample.The Forest Park neighborhood received 24 deliveries in the sample, which represents 30% of the deliveries in the sample (24/80 = 0.3).If the driver makes 200 deliveries, we can estimate that 30% of those deliveries will be to the Forest Park neighborhood.In conclusion, the estimate is that the driver will make 0.3 x 200 = 60 deliveries to the Forest Park neighborhood.
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John is 6 feet tall and casts a shadow of 5 feet. He is standing against a tree that casts a shadow of 15 feet. How tall is the tree?
Answer:
18
Step-by-step explanation:
6 divided by 5 is 1.2. 1.2 times 15 is 18. Via your answer.
Find all integer solutions of -
1/x +1/y =1/19
Answer: To solve the equation 1/x + 1/y = 1/19 for integer solutions, we can use a common method called "diophantine equations" which involves finding solutions to equations with integer variables.
Multiplying both sides by the least common multiple of x, y, and 19, we get:
19y + 19x = xy
Bringing all the terms to one side, we have:
xy - 19x - 19y = 0
Using Simon's Favorite Factoring Trick, we can add 361 to both sides of the equation, which gives:
xy - 19x - 19y + 361 = 361
(x - 19)(y - 19) = 361
Now, we need to find all the pairs of integers (x, y) such that (x - 19)(y - 19) = 361.
The factors of 361 are 1, 19, and 361 itself. So, we can solve the equation by setting x - 19 equal to each factor and finding the corresponding value of y:
x - 19 = 1, y - 19 = 361, then x = 20, y = 380
x - 19 = 19, y - 19 = 19, then x = 38, y = 38
x - 19 = 361, y - 19 = 1, then x = 380, y = 20
Therefore, the integer solutions to the equation 1/x + 1/y = 1/19 are (x, y) = (20, 380), (38, 38), and (380, 20).
Step-by-step explanation:
Among 2302 respondents, 1750 said that they only played hockey. What percentage said that they only play hockey ?
Answer:
The percentage of respondents who said that they only play hockey is 40.75.
Approximately 75.92% of the respondents said that they only played hockey.
To find the percentage of respondents who said that they only played hockey, we can use the following formula:
percentage = (part / whole) x 100
where "part" is the number of respondents who only played hockey, and "whole" is the total number of respondents.
In this case, we have:
part = 1750
whole = 2302
So the percentage of respondents who said that they only played hockey is:
percentage = (1750 / 2302) x 100 = 75.92%
Therefore, approximately 75.92% of the respondents said that they only played hockey.
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50 POINTS!!!!!! PLEASE HELP ASAP!!!!!!
Example of the law of contraposition?
Premise:
Premise:
Conclusion:
law of contraposition = if p, then q. if not q, then not p.
example: if it's snowing, then there's no school. if there is school, then it's not snowing.
Find local minimum and maximum values and saddle point(s) of the function f(x,y)=ycosx.
After answering the provided question, we can conclude that Therefore, derivatives the function f(x, y) = ycos(x) has no local minimum or maximum points, only two saddle points at (π/2, 0) and (3π/2, 0).
what is derivative?In mathematics, the derivative of a function with real variables measures how sensitively the function's value varies in reaction to changes in its parameters. Derivatives are the fundamental tools of calculus. Differentiation (the rate of change of a function with respect to a variable in mathematics) (in mathematics, the rate of change of a function with respect to a variable). The use of derivatives is essential in the solution of calculus and differential equation problems. The definition of "derivative" or "taking a derivative" in calculus is finding the "slope" of a certain function. Because it is frequently the slope of a straight line, it should be enclosed in quotation marks. Derivatives are rate of change metrics that apply to almost any function.
∂f/∂x = -ysin(x)
∂f/∂y = cos(x)
-ysin(x) = 0 (Equation 1)
cos(x) = 0 (Equation 2)
Therefore, the critical points are:
(π/2, 0) (saddle point)
(3π/2, 0) (saddle point)
∂²f/∂x² = -ycos(x)
∂²f/∂y² = 0
∂²f/∂x∂y = cos(x)
At the point (π/2, 0), ∂²f/∂x² = 0 and ∂²f/∂x∂y = 1, so it is a saddle point.
At the point (3π/2, 0), ∂²f/∂x² = 0 and ∂²f/∂x∂y = -1, so it is also a saddle point.
Therefore, the function f(x, y) = ycos(x) has no local minimum or maximum points, only two saddle points at (π/2, 0) and (3π/2, 0).
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En una empresa fabricante de productos A, B ,C y D, cuyos costos de producción, están por el orden de 480, 650, 790 y 940 bolívares respectivamente. Dicha empresa obtiene un beneficio del resultado de su ingreso por la venta del 60% sobre el costo durante un cierto periodo estipulado de 85 productos A, 68 productos B, 43 productos C y 27 productos D. Si el beneficio es igual al ingreso menos el costo, entonces: ¿Cuál será el beneficio de la empresa en ese periodo expresado en dólares y bolívares? (1 $ al cambio 25 Bs)
Answer:
Step-by-step explanation:
At the time of her grandson's birth, a grandmother deposits $ 11000 in an account that pays 2.5 % compounded monthly. What will be the value of the account at the child's twenty-first birthday, assuming that no other deposits or withdrawals are made during this period? The value of the account will be $= (Round to the nearest dollar as needed.)
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = final amount
P = principal (initial amount)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = time (in years)
In this case:
P = $11,000
r = 2.5% = 0.025 (since it's compounded monthly, we need to divide by 12 to get the monthly rate)
n = 12 (compounded monthly)
t = 21 years
Plugging in these values, we get:
A = 11000(1 + 0.025/12)^(12*21)
A ≈ $16,180.64
Therefore, the value of the account at the child's twenty-first birthday will be approximately $16,181.
What it mean by “what is the unit rate”
Answer:
3
Step-by-step explanation:
unit rate means slope
y2-y1/x2-x1
12-9/4-3
3/1
3
pls help!! brainliest!!! Find the equation of a line perpendicular to 3x=−20−2y that passes through the point (−6,−8).
The equation of the line is 2y+3x=-38
To find the equation of a line perpendicular to 3x= -20-2y that passes through the point (-6,-8), we can use the point-slope form of a line. The point-slope form of a line is[tex]y - y_1 = m(x - x_1)[/tex], where m is the slope of the line and [tex](x_1,y_1)[/tex] is an arbitrary point on the line. Since the given line has a slope of -2/3, the perpendicular line has a slope of 3/2. Therefore, the equation of the line can be written as y - (-8) = (3/2)(x - (-6)). Solving for y, we get 2y + 3x = -38. Therefore, the equation of the line is 2y + 3x = -38.
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I need help with this please don’t skip if you know it
The image of the composite figure is composed of 11 smaller cubes
How to find the sum of the composite shapesThe sum of the composite shape is solved by dividing the shape into two parts and courting using multiplication then adding the two parts
The first part which is the upper part is solved as follows
= length * width
= 2 * 2
= 4
The second part which is the base is solved as follows
= length * width
= 3 * 3
= 9
Sum of the counting
= 9 + 4
= 11
We can say that the composite figure is made up of 11 smaller cubes
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Math step by step answer
Answer:
1) 165.7 cm²
2) 736 cm²
3) 155.5 cm²
4) 169 cm²
Step-by-step explanation:
1) To find the area of shaded region, subtract the area of circle from the area of triangle.
r = 4 cm
[tex]\boxed{\text{\bf Area of circle = $\pi r^2$}}[/tex]
[tex]= 3.14 * 4 * 4 \\\\= 50.27 \ cm^2[/tex]
Triangle :
base = b = 18 cm
height = h = 24 cm
[tex]\boxed{\text{\bf Area of triangle = $\dfrac{1}{2}b*h$}}[/tex]
[tex]= \dfrac{1}{2}*18*24\\\\= 9 * 24\\\\= 216 \ cm^2[/tex]
Area of shaded part = area of triangle - area of circle
= 216 - 50.27
= 165.73
= 165.7 cm²
3) To find the area of shaded part, subtract the area of semicircle from the area of square.
side = 16 cm
[tex]\boxed{\text{\bf Area of square = side * side}}[/tex]
[tex]= 16 * 16\\\\= 256 cm^2[/tex]
diameter of semicircle = side of the square
d = 16 cm
r = 16÷ 2
r = 8 cm
[tex]\boxed{\text{\bf area of semicircle = $\dfrac{1}{2}\pi r^2$}}[/tex]
[tex]= \dfrac{1}{2}*3.14*8*8\\\\= 100.53 \ cm^2[/tex]
Area of shaded part = area of square - area of semicircle
= 256 - 100.53
= 155.47
= 155.5 cm²
2) To find the area of shaded part, add the area of rectangle I and the area of rectangle II.
Rectangle I:
l = 32 - 8 = 24 cm
w = 24 cm
So, it is a square.
Area of rectangle I = 24 * 24
= 576 cm²
Rectangle II:
l = 8 cm
w = 24 - 4 = 20 cm
Area of rectangle II = 8 * 20
= 160 cm²
Area of shaded part = area of rectangle I + area of rectangle 2
= 576 + 160
= 736 cm²
4) length = l = 26 cm
width = w = 13 cm
Rectangle can be divided into two congruent triangles through its diagonal. So, the area of shaded part can be found by dividing the area of rectangle by 2.
Area of rectangle = l * w
= 26 * 13
= 338 cm²
Area of shaded part = area of rectangle ÷ 2
= 338 ÷ 2
= 169 cm²
cuanto sale
-2/3 + 1/33 + 2/4= ?
The sum of the given fractions is equal to -3/22.
How to solve the sum?Here we have the sum of 3 fractions:
-2/3 + 1/33 + 2/4
To solve this we needto find common denominators, we can write:
3*44 = 132
33*4 = 132
4*33 = 132
Then we can rewrite each of the fractions as:
-2/3 = -88/132
1/33 = 4/132
2/4 = 66/132
Then the sum becomes:
-88/132 + 4/132 + 66/132 = -18/132 = -9/66 = -3/22
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A soccer player stands at one corner of the field and kicks a soccer ball 100 yards to the opposite corner if the field is 80 yards long what is the width
The width οf the field is 80 yards.
What is the Pythagοrean theοrem?Pythagοras Theοrem is the way in which yοu can find the missing length οf a right angled triangle.
Assuming the field is rectangular, we can use the Pythagοrean theοrem tο sοlve fοr the width:
Let the width οf the field be x. Then, we have a right triangle with legs x/2 and 80/2 = 40 and hypοtenuse 100:
[tex](x/2)^2 + 40^2 = 100^2[/tex]
[tex]x^2/4 + 1600 = 10000[/tex]
[tex]x^2 = 6400[/tex]
[tex]x = 80[/tex]
Hence, the width οf the field is 80 yards.
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Can someone tell me if I got this geometry question right? Find the value of x and write answers in simplest radical form using Pythagorean theorem.
Answer:
[tex]2\sqrt{21}[/tex]
Step-by-step explanation:
I'm not sure if you know how to simplify radicals or not, but the [tex]\sqrt{84}[/tex] can be further simplified to [tex]2\sqrt{21}[/tex]
To simplify a radical, you just need to divide by squares you already know
In this case, divide 84 by 4 (which is the square of 2)
This will give you 21. Leave that inside the radical. Outside the root, leave the 2 as multiplier to [tex]\sqrt{21}[/tex].
give the coordinate of the fourth vertex
They estimate that their current fridge, which they got for free cost him about $60 a month in electricity charges to operate. They found a new fridge that cost 1140 to purchase it would cost only around five dollars a month in electricity charges. The new fridge comes with a 10 year warranty so they assume the new fridge to last for 10 years without any extra repair or replacement fees if they keep their old fridge, did they assume they will need to pay around $170 sometimes over the next 10 years and maintenance cost to keep it running
Based on the analysis it will be less expensive to get the new fridge, hence buying the new fridge is worth it.
Given that the current fridge
electricity charges= $60 monthly,
maintenance costs=$ 170
let the number of months be = x
and the total cost is y
the expression for the cost of the old fridge is
y=170+60x ...say (1)
new fridge
electricity charges=$5
costs= $1140
let the number of months be x,
and the total cost be y'
the expression for the cost of the new fridge is
y'=1140+5x, ... say(2)
For 10 year , there are 12*10=120 months
put x=120 in both equations,
y=170+60x
y=170+60*120
y=170+7200
y=$7370
y'=1140+5*120
y'=1140+600
y'=$1740
Based on the analysis it will be less expensive to get the new fridge, hence buying the new fridge is worth it.
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Several scientists decided to travel to South America each year beginning in 2001 and record the number of insect species they encountered on each trip. The table shows the values coding 2001 as 1, 2002 as 2, and so on. Find the model that best fits the data and identify its corresponding [tex]R^2[/tex] value.
A. quadratic model, 0.840
B. linear model, 0.527
C. exponential model, 0.488
D. power model, 0.257
After answering the provided question, we can conclude that As a result, the scatter plot correct answer is A. quadratic model, 0.840.
What exactly is a scatter plot?"Scatter plots are graphs that depict the relationship between two is that variables in a data set. Data points are represented using a two-dimensional plane or a Cartesian system. The independent variable or characteristic is represented by the X-axis, while the dependent variable is represented by the Y-axis. These plots are often known as scatter graphs or scatter diagrams."
To find the best-fit model, we must examine the pattern of the data points. To visualise the trend, we can make a scatter plot of the data.
Using a regression tool, we obtain the quadratic model shown below:
[tex]y = -0.027x^2 + 0.33x + 2.19[/tex]
where y represents the number of insect species and x represents the year (coded as 1 for 2001, 2 for 2002, and so on).
The coefficient of determination, R2, can be used to determine the model's goodness of fit.
The quadratic model has an R value of 0.840, which is relatively high and indicates a decent match.
As a result, the correct answer is A. quadratic model, 0.840.
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Today, the sum of Hailey's age and Jordan's age is 48. 3 years ago Jordan was 6 times older than Hailey.
Answer: J=39 H=9
Step-by-step explanation:
J+H=48
J-3=6(H-3)
Use the substitution method and isolate one variable to do so.
J-3=6h-18
J=6h-15
plug J into first equation
6h+h-15=48
7h-15=48
7h=63
h=9
J+H=48
J+9=48
48-9=39
WILL GIVE 20 POINTS, NEED HELP! Match the description table or graph or equation with the correct type of function.
1, D /non linear function table/
2, F /linear function table/
3, C /linear function equation/
4, B /non linear function equation/
5, A /linear function graph/
6, E /non linear function graph/
What quantity of 65% acid solution must be mixed with 40% to produce 200 ml of 60% acid solution
To produce 200 ml of 60% acid solution from 65% and 40% acid solutions, approximately 715.8 ml of the 65% acid solution must be mixed with approximately 484.2 ml of the 40% acid solution.
Since the resulting solution is 200 ml and is a 60% acid solution, we know that the amount of acid in the solution will be:
0.6 * 200 ml = 120 ml
For the quantity 65% acid solution, the amount of acid is 0.65x ml.
For the quantity 40% acid solution, the amount of acid is 0.4y ml.
Since we want to end up with a total of 200 ml of solution, we know that:
x + y = 200
We also know that the amount of acid in the final solution is the sum of the amounts of acid in the two solutions we're mixing together:
0.65x + 0.4y = 120
Solve the second equation for y:
0.4y = 120 - 0.65x
y = (120 - 0.65x) / 0.4
x + (120 - 0.65x) / 0.4 = 200
Solve for x:
1.6x + 120 - 0.65x = 800
0.95x = 680
x = 715.8 ml
So we need to mix approximately 715.8 ml of the 65% acid solution with approximately 484.2 ml of the 40% acid solution to get 200 ml of 60% acid solution.
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A recent Time magazine reported the following information about a sample of workers in Germany and the United States.
United States
United States average length of workweek is 42 hours, sample standard deviation is 5, sample size is 600. Germany average length of workweek is 38 hours, sample standard deviation is 6, sample size is 700.
We want to determine whether or not there is a significant difference between the average workweek in the United States and the average workweek in Germany.
a. State the null and the alternative hypotheses.
b. Compute the test statistic.
c. Compute the p-value. What is your conclusion?
a. The hypotheses are coded as: H0: μ1 = μ2; Ha: μ1 ≠ μ2; b. The test statistic is: 9.35; c. The p-value is less than 0.0001, therefore, there is a significant difference between the average workweek in the US.
What is a Null Hypothesis?A null hypothesis is a statement that there is no significant difference or relationship between two variables in a population.
a. The null hypothesis is that there is no significant difference between the average workweek in the United States and the average workweek in Germany, or in other words, the population means are equal. The alternative hypothesis is that there is a significant difference between the two, or the population means are not equal.
H0: μ1 = μ2
Ha: μ1 ≠ μ2
Where:
μ1 = population mean of workweek in United States
μ2 = population mean of workweek in Germany
b. To compute the test statistic, we can use the two-sample t-test formula:
t = (x1 - x2) / sqrt((s1²/n1) + (s2²/n2))
Where:
x1 = sample mean of workweek in United States
x2 = sample mean of workweek in Germany
s1 = sample standard deviation of workweek in United States
s2 = sample standard deviation of workweek in Germany
n1 = sample size of United States
n2 = sample size of Germany
Plugging in the values, we get:
t = (42 - 38) / sqrt((5²/600) + (6²/700)) = 9.35
c. To compute the p-value, we need to use a t-distribution with degrees of freedom equal to the smaller sample size minus one (df = min(n1-1, n2-1)). In this case, df = 599. Using a two-tailed test with alpha level of 0.05, the p-value is less than 0.0001.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is a significant difference between the average workweek in the United States and the average workweek in Germany.
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