Answer:
4.67 (nearest hundredth)
Step-by-step explanation:
In 3 weeks, Landscape Elementary uses 14 reams of paper.
In 1 week, Landscape Elementary uses 14/3 reams of paper. = 4.66666...
The digit at thousandth place is 6 and it is more than 5. Thus add 1 with hundredth digit.
4.66666... = 4.67 (nearest hundredth)
a) Work out the minimum number of dogs that could have a mass of more than 24 kg
b) Work out the maximum number of dogs that could have a mass of more than 24 kg}.
Mass(kg) Frequency
0≤x<10 2
10≤x<20 7
20≤x<30 12
30≤x<40 6
a) The minimum number of dogs that could have a mass of more than 24 kg is 6.
b) The maximum number of dogs that could have a mass of more than 18 kg is 20.
How to find the minimum and maximum value mean?We will first of all have to rearrange the function with the aid of fundamental algebraic concepts that makes us calculate the value of x in the event that the derivative is equal to 0.
Now, the response above will give us the x-coordinate of the function's vertex. This vertex x-coordinate is the point at which the maximum or minimum will occur. To find the minimum or maximum, we will start by rewriting the solution into the original function.
From the given figures in the question, it is clear that the minimum of the values present in the interval 20 to 40 is 6
From the given figures in the question, it is clear that the maximum of the values present in the interval 20 to 40 is;
(12 + 6) = 18
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Describe and correct error
The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 80.2% of their carbon-14. How old were the bones at the time they were discovered?
The age of the bones when they were discovered is 13439.66 years
What is half-life of an element?
The half-life of a radioactive isotope is the amount of time it takes for one-half of the radioactive isotope to decay. The half-life of a specific radioactive isotope is constant; it is unaffected by conditions and is independent of the initial amount of that isotope.
Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value
The decay constant λ is = 0.693/t½
where t½ is the half-life of the element
Given data ,
The radioactive element carbon-14 has a half-life of 5750 years
So , the value of t½ = 5750 years
Now , the decay constant λ = 0.693 / t½
Substituting the values in the equation , we get
Decay constant λ = 0.693 / t½
Decay constant λ = 0.693 / 5750
Decay constant λ = 1.205 x 10⁻⁴
N=N₀e ^ −λt
where N₀ = Mass of radioactive element initially
N = Mass of radioactive element after time t
Apply natural logarithm on both the sides of the equation , we get
ln N = ln N₀ - λt
Let N0 be 100 then N will be 100 - 80.2 = 19.8
Substituting the values in the equation , we get
ln 19.8 = ln 100 - 1.205 x 10⁻⁴ ( t )
ln [ 19.8 / 100 ] = - 1.205 x 10⁻⁴ ( t )
-1.61948 = -1.205 x 10⁻⁴ ( t )
Therefore , the value of t is
t = -1.61948 / -1.205 x 10⁻⁴
t = 1.343966 x 10⁴
t = 13439.66 years
Therefore , the value of t is 13439.66 years
Hence ,
The age of the bones when they were discovered is 13439.66 years
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Construct a confidence interval that will capture the true population mean with 95% confidence. Population standard deviation is unknown.
Data: 18
19,19,19,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,21,21,21, 22,22,22,22,23,23,23,23,23,23,23,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,26,26,26,26,26,26,26,26,,26,26,26,27,27,27,27,27,28,28,28,28,28,28,28,28,28,28,29,29,29,30,30,30,31
A confidence interval that will capture the true population mean with 95% confidence. The value is ( 23.6459 , 24.8293 ) .
What is population mean?
A population in statistics is a group of comparable objects or occurrences that are relevant to a particular topic or experiment. A statistical population can be a collection of real things or a hypothetical, possibly limitless collection of objects derived from experience.
Data: 18
for given data
[tex]$$\begin{aligned}\bar{X} & =\frac{\sum X i}{n} \\& =2448 / 101 \\& =24.2376\end{aligned}$$[/tex]
as population variance is unknown so we used sample variance
[tex]$$\begin{aligned}S^2 & =\frac{1}{n-1}\left(\sum X i^2-n \times \bar{X}^2\right) \\S^2 & =\frac{1}{100}\left(60232-101 \times 24.2376^2\right) \\& =8.9830\end{aligned}$$[/tex]
Now
[tex]\frac{\bar{X}-\mu}{S / \sqrt{n}} \sim t_{n-1}$$[/tex]
hence 95 % of population mean is
As n=101 which is large so critical value is 1.98397
[tex]$$\begin{aligned}& =\left(\bar{X}-\sqrt{\frac{S}{n}} \times t_{(n-1, \alpha / 2)}, \bar{X}+\sqrt{\frac{S}{n}} \times t_{(n-1, \alpha / 2)}\right) \\& =\left(24.2376-\sqrt{\frac{8.9830}{101}} \times 1.98397,24.2376+\sqrt{\frac{8.9830}{101}} \times 1.98397\right) \\& =(23.6459,24.8293)\end{aligned}$$[/tex]
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One fisherman and 6 his friends were fishing and they spent the night on
the island putting all fish in a box. Fishermen fell asleep, but one decided to go
home. He took 48 fish which was 3/7 of all fish in the box. Then he sees that
his friends did not get any fish, He put fish he took back into the box and tried
to divide fish evenly among his friends. He took his part, and went home. How many fish does each person get
Each person got 16 fish equally divided among 7 fishermen.
What is a numerical expression?A mathematical statement expressed as a string of numbers and unknowable variables is known as a numerical expression. Statements can be used to create numerical expressions.
Given, One fisherman and 6 his friends were fishing.
So, The total no. of people is (6 + 1) = 7.
one decided to go home so he took 48 fish which was 3/7 of all fish in the box.
Let, 'x' be the total no. of fish.
Therefore, (3/7)x = 48.
x = 48×(7/3).
x = 112.
Now, by dividing the total no. of fish by total no. of fishermen
we get, (112/7).
= 16.
So, Each person got 16 fish.
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Consider the following distribution: 5, 5, 8, 9, 10, 10, 11. Which of the following is a correct statement about this distribution?
The mode is 9.
The median is 7.
The distribution is bimodal.
The mean is 6.5.
The distribution is bimodal. - this is a correct statement.
According to the question, the following distribution is 5, 5, 8, 9, 10, 10, and 11.
In statistics, the mode is the value that is being occurred frequently in a given set.In the above distribution, there are two modes i,e 5 and 10.
In statistics and probability theory median is the value separating the higher half from the lower half of a data sample. It is the point above and below which half (50%) of the observed data falls and so represents the midpoint of the data.Median: Middle value
In the above distribution, the median is - 9.
A data set is called bimodal if it has two modes which means that there is not a single data value that occurs with the highest frequency. Instead, there are two data values that tie in having the highest frequency.In statistics mean is the sum of a collection of numbers divided by the count of the numbers.Mean = [tex]\frac{5 + 5 + 8 + 9 + 10 + 10 + 11}{7}[/tex] = [tex]\frac{58}{7}[/tex] = 8.28
Hence,
The distribution is bimodal. - this is a correct statement.
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"5. Mr. Roberson created this frequency table to record the favorite color of each of his students. Which of the following statements are true based on the frequency table?
More students prefer red than green.
Twice as many students prefer green than purple.
More students prefer yellow than blue.
Two more students prefer blue than purple.
The same number of students prefer yellow or purple as the number who prefer blue."
The following statements are true based on the frequency table:
Twice as many students prefer green than purple
And
Two more students prefer blue than purple
What is frequency table ?
The information that is gathered is called data. The quantity of occurrences of an event or value is referred to as its frequency. A frequency table is a table that lists items and displays how frequently each item happens.
From the frequency table,
Red= 6
yellow = 2
Blue = 5
Purple = 3
Green = 6
We can see that "Twice as many students prefer green than purple"
And
Two more students prefer blue than purple
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What are the coordinates of the point on a directed line segment from (-8,8) to (-2,-10) that partitions the segment into a ratio of 1 to 2
[tex]\textit{internal division of a line segment using ratios} \\\\\\ A(-8,8)\qquad B(-2,-10)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{2}\implies \cfrac{A}{B} = \cfrac{1}{2}\implies 2A=1B\implies 2(-8,8)=1(-2,-10)[/tex]
[tex](\stackrel{x}{-16}~~,~~ \stackrel{y}{16})=(\stackrel{x}{-2}~~,~~ \stackrel{y}{-10}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-16 -2}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{16 -10}}{1+2} \right)} \\\\\\ C=\left( \cfrac{ -18 }{ 3 }~~,~~\cfrac{ 6}{ 3 } \right)\implies {\Large \begin{array}{llll} C=(-6~~,~~2) \end{array}}[/tex]
I REALLY NEED HELP PLEASE
Which of the following is the dilation of scale factor 2 about the center (4, -2)?
If the scale factor is less than 1, the dilated figure is smaller than the original, if it is greater than 1 the dilated figure is larger than the original.
What is the dilation of scale factor?Image result for Which of the following is the dilation of scale factor Scale Factor (Dilation) The scale factor in the dilation of a mathematical object determines how much larger or smaller the image will be (compared to the original object). When the absolute value of the scale factor is greater than one, an expansion occurs. if we have a scale factor of 1/4, we just multiply each of those by 1/4. So instead of going to the left eight, we would go to the left two. Eight times 1/4 is two. Instead of going up four, we would go up one.A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).The location of the point (4, 2) is to the right of the triangle RST, therefore,A dilation from the left or a contraction from the right is required.The dilation of ΔRST that would result in a line segment with slope of 2That passes through (4, 2) is C. A dilation with a scale factor of 0.5 centered at
The given vertex of the line are;
R(-2, 6), T(-6, -2), S(6, -4)
Point on the lint RT that has the same y-coordinate as the point (4, 2) is (-4, 2)Taking the center of dilation as the point (12, 2), we have;Coordinates of the point (-4, 2) relative to the point (12, 2) is found asfollows;(-4 - 12, 2 - 2) = (-16, 0)Therefore, dilating (-16, 0) by a scale factor of 0.5, with a center at (12,2) gives;0.5 × (-16, 0) = (-8, 0)The actual location of the point (-8, 0) is (-8 + 12, 0 + 2 ) = (4, 2)Therefore, the dilation of ΔRST that would result in a line segment withslope of 2 that passes through (4, 2) is C. A dilation with a scale factor of 0.5 centered atTo learn more about scale refer to:
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URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
The relationship between the cone cylinder and sphere is sphere takes up two-thirds of the volume of the cylinder, the cone takes up one-third of the volume of the cylinder.
What is the properties of the cone cylinder and sphere?Similar to a prism, a cylinder has two bases, however they are circles rather than polygons. A cylinder's sides are also curved rather than flat. A cone has a vertex that is not on the base and a single circular base. The sphere is a form in space with equal distances between each of its points and the centre.Any sphere has a volume that is 2/3 the size of any cylinder with an equivalent radius and height.While CYL, or cylinder, is the amount of lens power required to correct astigmatism, SPH, or sphere, is the amount of lens power required to correct nearsightedness or farsightedness. People frequently allude to their SPH value while discussing their eye degree or "power."Answers of blank :
Blank 1 = 1/3
Blank 2 = Three
Blank 3 = Two
Blank 4 = 2/3
Blank 5 = 1/3
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What is the slope of a line perpendicular to the line whose equation is 4x-6y=-604x−6y=−60. Fully simplify your answer.
The slope of the line perpendicular to 4x - 6y = - 60 is (-3/2).
What is slope?The slope is the rate of change of the y-axis with respect to the x-axis.
The equation of a line in slope-intercept form is
y = mx + b, where
slope = m and y-intercept = b.
We know the greater the absolute value of a slope is the more steeper is it's graph or rate of change is large.
We know lines perpendicular to each other have slopes that are negative reciprocals of each other.
Given line is 4x - 6y = - 60 ⇒ - 6y = - 4x - 60 ⇒ y = (2/3)x + 10.
Now the line perpendicular to it will have a slope of (-3/2).
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For what values of a does the equation 2a-4x=4-a^2*x have a solution in the interval (-8,-2)?
The values of a such that the linear equation 2 · a - 4 · x = 4 - a² · x have a solution in the interval (- 8, - 2) is a = 2 and - 7 / 4 ≤ a ≤ - 1.
How to find the values of a variable associated to an equation
In this problem we find the case of a linear equation whose values of variable a must be determined in terms of an interval of values of variable x. First, write the defined equation:
2 · a - 4 · x = 4 - a² · x
Second, form the standard form of a quadratic equation in terms of a:
x · a² + 2 · a - 4 · (x + 1) = 0
Third, use the quadratic formula:
a = - 2 / (2 · x) ± [1 / (2 · x)] · √[4 - 4 · x · [- 4 · (x + 1)]]
a = - 1 / x ± (1 / x) · √[1 + 4 · x · (x + 1)]
a = - 1 / x ± (1 / x) · √(4 · x² + 4 · x + 1)
There are two cases for the given interval:
Case 1
a = - 1 / x + (1 / x) · √(4 · x² + 4 · x + 1)
x = - 8
a = - 1 / (- 8) + [1 / (- 8)] · √[4 · (- 8)² + 4 · (- 8) + 1]
a = - 7 / 4
a = - 1 / (- 2) + [1 / (- 2)] · √[4 · (- 2)² + 4 · (- 2) + 1]
a = - 1
Case 2
a = - 1 / x - (1 / x) · √(4 · x² + 4 · x + 1)
x = - 8
a = - 1 / (- 8) - [1 / (- 8)] · √[4 · (- 8)² + 4 · (- 8) + 1]
a = 2
a = - 1 / (- 2) - [1 / (- 2)] · √[4 · (- 2)² + 4 · (- 2) + 1]
a = 2
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The table shows the proportional relationship between the price for a certain number of buckets of golf balls at a driving range.
hurry pls
Buckets 3 5 7
Price (dollars) 28.50 47.50 66.50
Determine the constant of proportionality.
The constant of proportionality will be 9.5 when the table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.
Given that,
The table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.
We have to find the constant of proportionality.
We know that,
What are ratio and proportion?A collection of consecutively ordered numbers a and b expressed as a/b, where b is never equal to zero, is referred to as a ratio. A statement is considered proportionate when there is equality between two items.
The table displays the proportional relationship between the price for a particular quantity of golf ball cans at a driving range.
Buckets 3 5 7
Price (dollars) 28.50 47.50 66.50
The constant of proportionality is calculated as,
⇒ 25.8 / 3
⇒ 9.5
Therefore, The constant of proportionality will be 9.5 when the table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.
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Answer:
9.5
Step-by-step explanation:
I took the exam and got it right
multiply v by 2, then divide 8 by the result
2 * V is equal to 2 v you divide 2 from 8 and your answer is 4
ASAP!! Which graph could be used to find the distance between the points (−3, −13) and (11, 9)?
graph of a right triangle with hypotenuse beginning at negative 13 comma negative 3 and ending at 9 comma 11
graph of a right triangle with hypotenuse beginning at negative 3 comma negative 13 and ending at 11 comma 9
graph of a right triangle with hypotenuse beginning at negative 11 comma negative 9 and ending at 3 comma 13
graph of a right triangle with hypotenuse beginning at negative 9 comma negative 11 and ending at 13 comma 3
The graph that could be used to find the distance between the points
(-3, -13) and (11, 9) is given as follows:
B. graph of a right triangle with hypotenuse beginning at negative 3 comma negative 13 and ending at 11 comma 9.
What is distance between two points?
The distance between two points in a plane is the length of the line segment separating the two points. Typically, the equation
d=√((x2 - x1)^2 + (y2 - y1)^2) is used to calculate the distance between two points.
Suppose we have a pair of points, with the coordinates given as follows:
[tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex].
The two points form a right triangle, in which the sides are given as follows:
[tex]y_2-y_1\\x_2-x_1[/tex]
The distance between these two points is given by the hypotenuse of the right triangle.
Hence, applying the Pythagorean Theorem, the distance is obtained as follows:
[tex]d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]
In this problem, the coordinates of the points are given as follows:
(−3, −13) and (11, 9).
Hence the correct option is given by option B.
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Refer to the number line which inequality is true?
Answer: D
Step-by-step explanation:
B = -6.3
F = -5
M = -4.5
A.) -6.3 is not greater than -4.5
B.) -5 is not less than -6.3
C.) -4.5 is not less than -5
D.) -5 is less than -4.5
help meeeeeeeeeeeeeeeeeeeeeeeeeee pleaseeeeeeeeeeeeeeeeeeeeee!!!!!
Answer:
a.) The population is years measured after 2000 so 2000 is 0. Any number to the 0 power is 1 so the population is 18.5 (million).
b.) after 8.4 years, so 2008
Step-by-step explanation:
b.) 26.6 = [tex]18.5e^{0.1708t}[/tex]
1.43783784 ≈ [tex]e^{0.1708t}[/tex]
ln 1.43783784= 0.1708t
t=8.4... years
CAN SOMEONE HELP WITH THIS QUESTION?✨
The exponential function passes through the points (-1,4/5) and (1,20) will be y = 4 × [tex]5^{x}[/tex].
What is an exponential function?In mathematics, an exponential function is a relationship of the type y = ax, where x is an independent variable that spans the entire real number line and is expressed as the exponent of a positive number.
In other meaning, an exponential function is that function above which a variable x has imposed.
Suppose the exponential function as
y = a[tex]e^{bx}[/tex]
Substitute, (-1,4/5)
4/5 = a[tex]e^{-b}[/tex] (1)
Substitute, (1,20)
20 = a[tex]e^{b}[/tex] (2)
Multiply (1) with (2)
(4/5)20 = a[tex]e^{-b}[/tex] × a[tex]e^{b}[/tex]
4 × 4 = a²
a = 4
Substitute, a = 4 into (2)
20 = 4[tex]e^{b}[/tex]
[tex]e^{b}[/tex] = 5
Take log on both sides of the above equation,
ln [tex]e^{b}[/tex] = ln 5
b = ln 5
Substitute a = 4 and b = ln 5 into y = a[tex]e^{bx}[/tex]
y = 4 e^(x(ln 5)) = 4 e^(ln5^x)
y = 4 × [tex]5^{x}[/tex]
Hence "The exponential function passes through the points (-1,4/5) and (1,20) will be y = 4 × [tex]5^{x}[/tex]".
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Convert 5pi/7 radians to degrees.
(Use pi = 3.14 when necessary and round your final answer to the hundredths place.)
5pi/7radians=________ degrees
well, since we know there are 180° in π/2 radians, then, what's 5π/7 radians?
[tex]\begin{array}{ccll} degrees&radians\\ \cline{1-2} 180&\frac{\pi }{2}\\[1em] d&\frac{5\pi }{7} \end{array}\implies \cfrac{180}{d}~~ = ~~\cfrac{\frac{\pi }{2}}{ ~~ \frac{5\pi }{7} ~~ }\implies \cfrac{180}{d}=\cfrac{\pi }{2}\cdot \cfrac{7}{5\pi } \\\\\\ \cfrac{180}{d}=\cfrac{7}{10}\implies 1800=7d\implies \cfrac{1800}{7}=d\implies 257.14^o\approx d[/tex]
for this problem use the data in sheet wages in the excel file hw5data2020 to estimate a simple regression explaining
With the use of regression analysis, you may determine which of the independent factors actually has an effect statistically and learn how the dependent variable changes when one of the independent variables changes.
What is linear regression?Regression analysis is used in statistical modeling to determine the associations between two or more variables:
The fundamental element you are attempting to comprehend and anticipate is the dependent variable. The elements that could have an impact on the dependent variable are known as independent variables.
Regression analysis enables you to statistically ascertain which of the independent variables actually has an impact and explains how the dependent variable changes when one of the independent variables changes. There is a difference between a simple and multiple linear regression in statistics. A dependent variable and one independent variable are modeled using a linear function in simple linear regression. Multiple linear regression is used when there are two or more explanatory variables used to predict the dependent variable.
Here,
estimate a simple regression in excel:
On the Data tab, in the Analysis group, click the Data Analysis button.
Select Regression and click OK.
In the Regression dialog box, configure the following settings: Select the Input Y Range, which is your dependent variable.
Click OK and observe the regression analysis output created by Excel.
Regression analysis illustrates how the dependent variable changes when one of the independent factors changes and allows you to statistically determine which of the independent variables actually has an impact.
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Samuel and his children went into a movie theater and he bought $43.75 worth of bags of popcorn and candies. Each bag of popcorn costs $5 and each candy costs $4.75. He bought a total of 9 bags of popcorn and candies altogether. Write a system of equations that could be used to determine the number of bags of popcorn and the number of candies that Samuel bought. Define the variables that you use to write the system.
In linear equation, Samuel bought 7 bags of popcorn and 10 bags of candies.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
Let x represent the number of bags of popcorn and y represent the number of candies that Samuel bought.
She bought $43.75 worth of bags of popcorn and candies. Hence:
5x + 4.75y = 43.75 ........... (1)
She bought a total of 17 bags of popcorn and candies altogether. Hence:
x + y = 9 (2)
Solving equations 1 and 2 simultaneously gives:
x = 4, y = 5
Therefore Samuel bought 4 bags of popcorn and 5 bags of candies.
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-24 – x = -34 pls help
Answer:
x = 10
Step-by-step explanation:
Given equation,
→ -24 - x = -34
Now the value of x will be,
→ -24 - x = -34
→ -x = -34 + 24
→ -x = -10
→ [ x = 10 ]
Hence, the value of x is 10.
Answer: x=10
1. Rearrange the terms
-24 - x= -34
-x - 24= -34
2. Add 24 to both sides
-x - 24= -34
-x - 24= -34
+24 +24
3. Simplify(add the numbers)
-x=-10
4. Divide both sides by the same factor
-x=-10
-x/-1 = -10/-1
Simplify (cancel terms, divide numbers)
X=10
Step-by-step explanation:
A research department wants to test the hypothesis that caffeine from organic coffee more safely stimulates mental alertness than caffeine from nonorganic coffee. Which of the following would most effectively provide the desired primary data?
Experimentation, one form of primary research,can produce data suggesting cause and effect
The most effectively provide the desired primary data is experimentation
Experimentation is the step in the scientific method that helps people decide between two or more competing explanations—or hypotheses. These hypotheses suggest reasons to explain a phenomenon or predict the results of an action.
With an experiment, you run a test and see what the results are. If you don't get good results, you can try another option, and run another test. Then you can see what the outcomes of the choices are (the info you didn't have when first thinking about the decision), and you can make a better-informed decision now.
According to the question,
A research department wants to test the hypothesis that caffeine from organic coffee more safely stimulates mental alertness than caffeine from nonorganic coffee. Experimentation, one form of primary research, can produce data suggesting cause and effect
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A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 26∘ and that the angle of depression to the bottom of the tower is 20∘. How tall is the tower?
340 ft high
d = 400' distance from building to tower
26° from the window to the top of the tower and 20° from the window to the bottom of the tower
Tan 26 = h2/d
Tan 20 = h1/d
h2 = 400Tan26
h1 = 400Tan20
on solving the equations,
h=a+b
Tower = 340 ft high
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Add
900-200= 700
and then add 700 + 700 =
Answer =
Answer:1400
Step-by-step explanation:
What is the value of c for the parabola of x=1/10(y+6)^2+2?
Answer:
Parabolas
A quadratic function is a function that can be written in the form f(x)=ax2+bx+c
where a,b, and c are real numbers and a≠0
. This form is called the standard form of a quadratic function.
The graph of the quadratic function is a U-shaped curve is called a parabola.
The graph of the equation y=x2
, shown below, is a parabola. (Note that this is a quadratic function in standard form with a=1 and b=c=0
.)
In the graph, the highest or lowest point of a parabola is the vertex. The vertex of the graph of y=x2
is (0,0)
.
If a>0
in f(x)=ax2+bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0
makes the parabola wide.
If a<0
in f(x)=ax2+bx+c, the parabola opens downward. In this case the vertex is the maximum, or highest point, of the parabola. Again, a large negative value of a
makes the parabola narrow; a value close to zero makes it wide.
For an equation in standard form, the value of c
gives the y
-intercept of the graph.
The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry.
The equation of the axis of symmetry for the graph of y=ax2+bx+c
, where a≠0, is x=−b2a
In all the above graphs, the axis of symmetry is the y
-axis, x=0. In the graphs below, the axis of symmetry is different (marked in red.) Note that c still gives the y
-intercept.
If you write a quadratic function like x=f(y)=ay2+by+c
, where x is a function of y (instead of a y as a function of x
), you get a parabola where the axis of symmetry is horizontal.
Note that in this case, c
is the x-intercept. If a is positive, the graph opens to the right; if a
is negative, the graph opens to the left.
Example:
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the parabola y=−3x2−6x+4
.
The equation of the axis of symmetry for the graph of y=ax2+bx+c
.
x=−b2a
Substitute −3
for a and −5 for b
in the equation of the axis of symmetry.
x=−−62(−3) =−1
So, the equation of the axis of symmetry is x=−1
.
Since the equation of the axis of symmetry is x=−1
and the vertex lies on the axis, the x-coordinate of the vertex is −1
.
To find the y
-coordinate of the vertex, first substitute −1 for x
in the given equation.
y=−3(−1)2−6(−1)+4
Simplify.
y=−3+6+4 =7
Therefore, the coordinates of the vertex of the parabola is (−1,7)
.
Step-by-step explanation:
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Suppose Dariya is an avid reader and buys only reusable tote bagses. Dariya deposits $4,000 into a savings account that pays an annual nominal interest rate of 5%. Assume this interest rate is fixed, and so it will not change over time. On the day she makes her deposit, suppose that a reusable tote bags has a price of $10.00.
The price of reusable tote bags and the interest rate may change over time, which could affect the length of time that Dariya's savings will last.
What is the interest rate?
The proportion of a loan that is charged as interest to the borrower is typically expressed as an annual percentage of the loan outstanding.
If Dariya deposits $4,000 into a savings account with an annual nominal interest rate of 5%, the amount of interest she earns each year will be $4,000 * 5% = $200.
If the price of a reusable tote bag is $10.00 and Dariya wants to buy one every year, the cost of buying one reusable tote bag every year will be $10.00/year.
To determine how long Dariya's savings will last, you can divide the amount of her annual interest earnings by the cost of buying a reusable tote bag each year.
In this case, Dariya's savings will last for 200/10 = 20 years.
Keep in mind that this calculation assumes that the price of reusable tote bags remains constant at $10.00/year and that the nominal interest rate remains fixed at 5% per year.
Hence, the price of reusable tote bags and the interest rate may change over time, which could affect the length of time that Dariya's savings will last.
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what is the average rate of change
The required average rate of change in the amount of Kendall's account is $50 per month.
What is rate?Rate is a measurement to determine the change in one quantity with respect to another quantity.
Given that,
The amount of the end of the 8th month in Kendall's account = $450,
And at the end of the 11th month, the saved amount = $600.
To find the rate of change in saving amount
The increased amount = 600 - 450 = 150.
Since, In 3 months the amount increased by 150.
So, the average rate of change in amount
= 150 / 3.
= 50.
The average rate of change in the amount is $50 per month.
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occurs when three or more people work together interdependently for the purpose of accomplishing a task.
A small group communication occurs when three or more people work together interdependently for the purpose of accomplishing a task.
Small Group Communication
Small group communication is the term used to describe face-to-face conversation between more than two people. This kind of internal communication exists.
A small group is typically thought of as a collection of three or less people, with a maximum number of twelve to fifteen. A group would not be considered tiny if it had more than fifteen members or if it had just two.
Since almost all businesses and organisations employ people in small squads, there are advantages or even strengths to small group communication. When compared to one individual working alone on a project, it has been found to be more appropriate and produce better results.
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what is the slope of a line with a y intercept of 5 and an x intercept of -4
Answer:
slope = [tex]\frac{5}{4}[/tex]
Step-by-step explanation:
calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (0, 5 ) ← coordinates of y- intercept
and (x₂, y₂ ) = (- 4, 0 ) ← coordinates of x- intercept
m = [tex]\frac{0-5}{-4-0}[/tex] = [tex]\frac{-5}{-4}[/tex] = [tex]\frac{5}{4}[/tex]