The p-value when the test statistic is equal to 2.30 is 0.033.
What is the p-value?A statistical measurement known as a p-value is employed to check a hypothesis' validity against actual data.
A p-value calculates the likelihood of getting the outcomes that were observed, presuming that the null hypothesis is correct.
The statistical significance of the difference that was found increases with decreasing p-value.
The p-value, used in null-hypothesis significance testing, represents the likelihood that the test findings will be at least as extreme as the result actually observed, assuming that the null hypothesis is true.
So, the p-value will be:
The t-value for (n-1 = 20-1 = 19) degrees of freedom serves as the test statistic in this experiment.
Using the formula: T.DIST.2T(2.3,19)
The p-value will be 0.033.
Therefore, the p-value when the test statistic is equal to 2.30 is 0.033.
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Complete question:
A restaurant decides to test their oven's thermostat to see if it is working properly, that is if the actual temperature inside the oven is the same as the temperature to which the thermostat was set. Twenty times, the oven was set to 350 degrees and then the temperature was measured with a thermometer.
H o: μ=350 and Ha: μ≠350
The test statistic is equal to 2.30. What is the p-value _____?
Use the Laplace transform to solve the following initial value problem: y + 10y + 25y = 0 y(0) = - 3, y'(0) = 3 First, using Y for the Laplace transform of y(t), i.e., Y = L{y(t)} find the equation you get by taking the Laplace transform of the differential equation = 0 Now solve for Y (s) = and write the above answer in its partial fraction decomposition, Y(s) = A/s + a + B/(s+a)2 Y (s) = + Now by inverting the transform, find y(t) =
On solving the given question, by the Help of LaPlace Inverse form we got - [tex]y(t) = \frac{-7}{2} + \frac{1}{2}e^{-8t}[/tex]
What is The Laplace transform?In terms of its usefulness in resolving physical issues, the Laplace transform is an important transform that is probably second only to the Fourier transform. The Laplace transform is especially helpful in solving linear ordinary differential equations, such as those that arise in the analysis of electronic circuits.
Take Laplace for DE we get -
[tex]L(y)" + 8L(y)' =0\\s^2Y (s) - sy(0) - y'(0) 8(sY(s) - y(0))=0\\Y(s)(s^2 + 8s) + 3s 4 + 24 =0\\Y(s)(s^2 + 8s) = -3s - 28\\Y(s) = \frac{-3s - 28}{(s^2 + 8s) } \\Right Side = \\\frac{-7}{2s} + \frac{1}{2(s+8)} \\ Now taking inverse LaPlace - \\y(t) = \frac{-7}{2} + \frac{1}{2}e^{-8t}[/tex]
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Ms. Lopez had some drinking glasses. The ratio of the number of blue glasses to the number of white glasses was 5 : 6. Her children broke 8 blue glasses and the ratio became 13 : 18. How many white glasses did she have?
Answer:
12
Step-by-step explanation:
(5x-8)/6x
use the same regression results where you already found r square and other results for this regression. what is the standard error for the estimated slope of the regression line? answer to four decimal places.
A regression model can only predict values that are lower or higher than the actual value. As a result, the only way to determine the model's accuracy is through residuals.
What is regression ?
Regression is a statistical technique used in the fields of finance, investing, and other disciplines that aims to establish the nature and strength of the relationship between a single dependent variable (often represented by Y) and a number of independent variables (known as independent variables).
The most popular variation of this method is linear regression, which is also known as simple regression or ordinary least squares (OLS).
Based on a line of best fit, linear regression determines the linear relationship between two variables.
The slope of a straight line used to represent linear regression thus indicates how changing one variable affects changing another..
In a linear regression connection, the value of one variable when the value of the other is zero is represented by the y-intercept. Regression without linearity.
Hence, A regression model can only predict values that are lower or higher than the actual value.
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There are 36 children and 6 adult at the prechool. To keep the ame child-to-adult ratio, how many adult are needed for 108 children?
To maintain the same child-to-adult ratio, the number of adult needed for 108 children are 18.
Explain the term ratio of the number?The amounts (numbers) m as well as n are referred to as the ratio's terms in the ratio m: n. Both the first term (i.e., m) and also the second term (i.e., is n) are referred to as antecedent and consequent, respectively.As the stated question-
At the preschool, there's many 36 children and 6 adults.
Thus, this is solved by using unitary method.
36 children -----> 6 adults.
1 children -----> 6/36 adults.
For 108 children, multiply both sides by 108.
108 children -----> 6 x 108/36 adults.
108 children -----> 18 adults.
Thus, to maintain the same child-to-adult ratio, the number of adult needed for 108 children are 18.
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Using the t Distribution Table, find the critical value(s) for the t test for a right-tailed test with n=8 and a =0.01. Enter the answers separated by a comma if needed. Critical value(s) - Using The Distribution Table, find the critical value(s) for the test for a left-tailed test with n=23 and a =0.05. Enter the answers separated by a comma if needed. Critical value(s)
The critical value for right tailed test is 2.998, the critical value for left tailed test is - 1.717.
How to using t distribution table?First, we must know the following data,
df = degree of freedom = n - 1
α = significance level
Next, t test have three type of test which is
Two or Both tailed test, for this test we use negative and positive value.One right tailed test, for this test we use positive value.One left tailed test, for this test we use negative value.Then, to find critical value using the t distribution table search df value in left column of table and search the α in the top row of the table (depending on the type of test, use one tail row for this case). For example, see attached image.
Thus, the right tailed test critical value is 2.998, and the left tailed test critical value is - 1.717.
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how is factoring symbolically and factoring with algebra tiles similar
The two methods are identical because by both using algebra tiles and factoring symbolically, one need to think of the factors.
What is factoring?Factoring simply has to do with the process of writing a number or another mathematical object as a product of different factors, and this is usually smaller or simpler objects that are of the same kind.
Factoring is a common mathematical process that's used for separating the factors, or numbers, which multiply to form another number. Some numbers are affected by multiple factors.
For example, in a situation when you multiply the factors 6 and 4, 8 and 3, 12 and 2, and 24 and 1, you get the number 24.
Algebra tiles are rectangular and square tiles that represent numbers and variables. Using algebra tiles allows us to solve our problems in a more visual manner. It allows us to see exactly what quantities we're dealing with. It's as if we're using building blocks to assist us.
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How many solutions does this equation have. QUESTION BELOW please help
Answer:
one solution
Step-by-step explanation:
the solution to a system of equations given graphically is at the point of intersection of the 2 lines, that is (1, 2 )
Since there is only 1 point of intersection then there is only one solution.
Can anyone help me with this?
8^5/8^3
Answer:
64
Step-by-step explanation:
do multiplication :)
mr banda is buying chocolate bars and hot chips for his students who scored meets and masters on the semester exam. chocolate bars cost $1.1 you each and hot chips 0.40 each Mr banda spent $87 on buying 130 chocolate bars and hot chips. how many chocolate bars did Mr banda buy?
The number of chocolate bars which Mr Banda bought is equal to 50 chocolate bars.
How to determine the number of chocolate bars?In order to determine the number of chocolate bars and the number of hot chips that Carter bought, we would have to write a system of linear equation and assign variables to the number of chocolate bars and number of candies respectively, and then translate the word problem into algebraic equation as follows:
Let the x indicate the number of chocolate bars.Let the y indicate the number of hot chips.Since chocolate bars cost $1.1 each and hot chips 0.40 each and Mr Banda spent $87 on buying 130 chocolate bars and hot chips, we would have the following system of linear equation:
1.1x + 0.40y = 87 ......equation 1
x + y = 130 ......equation 2
From equation 2, we have:
y = 130 - x ......equation 3
Substituting equation 3 into equation 1, we have:
1.1x + 0.40(130 - x) = 87
1.1x + 52 - 0.4x = 87
0.7x + 52 = 87
0.7x = 87 - 52
0.7x = 35
x = 35/0.7
x = 50 chocolate bars.
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Solve six times three-sixths equals blank.
Leave your answer as an improper fraction.
thirty-six thirds
eighteen-sixths
eighteen-sixteenths
a cylinder shaped can needs to be constructed to hold 400 cubic centimeters of soup. the material for the sides of the can costs 0.04 cents per square centimeter. the material for the top and bottom of the can needs to be thicker, and costs 0.08 cents per square centimeter. find the dimensions for the can that will minimize production cost. helpful information: h : height of can, r : radius of can volume of a cylinder: v
The dimensions of the can are radius of can = 4cm and
height of can = 78.6cm
What is the volume of cylinder?
A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up.
Thus, the volume of the cylinder can be given by the product of the area of base and height.
Volume of cylinder = πr²h
According to the given question:
We have V cylinder = V = 400cc
We will first find the areas involved
Let x = radius of base then area of the base = π*x² and this is the area of the top too
For lateral area we need to get h the height of the cylinder as function of x
V = πx²h ⇒ h= v/πx² ⇒ h = 400/πx²
Now the total area of the cylinder is:
A(x) = Area of the base + area of the top + lateral area
A(x) = 2*π*x² + 2πx h ⇒ A(x) = 2*π*x² + 2πx (V/πx² )
A(x) = 2*π*x² + 800/x
Taking derivatives:
A´(x) = 4πx - 800/x²
A´(x) = 0 4πx - 800/x² =0 ⇒ πx - 200/x² = 0
( πx³ - 200 )/ x² = 0
πx³ - 200 = 0
x = 4 cm
and h = 400/πx² ⇒ h = 78.6 cm
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cos x = - 0.809, find the values of x. ( - 180≤ x ≥ 180 )
Answer:
cos x = - 0.809, find the values of x. ( - 180≤ x ≥ 180 )
x = - 135° and x = 45°
Given the functions below, find f(2)-g(-7).
f(x)=-|3-8x| g(x)=x^2+4x
The value of f(2)-g(-7) is -34
What is function?A function is defined as a relation between a set of inputs having one output each.
Given are two functions, f(x) = -|3-8x| and g(x) = x²+4x
f(2) = -|3-8x2|
= -|3-16|
= -13
g(x) = (-7)²+4(-7)
= 49-28
= 21
f(2)-g(-7) = -13-21 = -34
Hence, The value of f(2)-g(-7) is -34
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when she chooses a password, eloise always uses exactly ten different characters: five letters (a, b, c, d, and e) and five numbers (2, 3, 4, 5, and 6). additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive. how many different passwords conform to these rules?
These rules of five letters (a, b, c, d, and e) and five numbers allow for 28,800 different passwords. Additionally, she makes sure that no two letters or two numbers are ever placed back-to-back.
What is counting principle?The basic counting concept is a method for keeping track of all the potential outcomes in a scenario. According to this statement, there are n*m methods to carry out both of these acts if there are n ways to execute one activity and m ways to conduct another action after that.
Here,
Eloise has 10 alternatives since the password's initial character can be any letter or number. She has five alternatives, and her second selection must be from the set (letter or number) that has not previously been utilized. She has four choices remaining, and Choice 3 is from the same group as Choice 1. She gets four possibilities since Choice 4 is the second item in the same set as Choice 2. The choices 5 and 6 come from separate sets, each of which has three possibilities; the choices 7 and 8 come from separate sets, each of which has two options; and the choices 9 and 10 come from separate sets, each of which has only one option left.
10 5 4 4 3 3 2 2 1 1
10 ×5 ×4 ×4 ×3 ×3 ×2 ×2 ×1 ×1 = 28,800
28,800 different passwords conform to these rules of five letters (a, b, c, d, and e) and five numbers (2, 3, 4, 5, and 6). additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive.
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As a project for his business statistics class, a student examined the factors that determined parking meter rates throughout the campus area. Data were collected for the price ($) per hour of parking, blocks to the quadrangle, and whether the parking is on or off campus.
The population regression model hypothesized is Yi = α + β 1 X 1i + β 2 X 2i + ε
where
Y is the meter price per hour
X 1 is the number of blocks to the quad
X 2 is a dummy variable that takes the value 1 if the meter is located on campus and 0 otherwise
The following Excel results are obtained.
Statistics Regression
Multiple R 0.5536
R Square 0.3064
Adjusted R Square 0.2812
Standard Error 0.4492
Observations 58
ANOVA
df SS MS F Significance F
Regression 2 4.9035 2.4518 12.1501 0.0000
Residual 55 11.0984 0.2018
Total 57 16.0019
Coefficients Standard Error t Stat P-value Lower 99% Upper 99%
Intercept 1.6500 0.2028 8.1359 0.0000 1.1089 2.1912
Block -0.2504 0.0529 -4.7355 0.0000 -0.3915 -0.1093
Campus 0.1552 0.1297 1.1966 0.2366 -0.1908 0.5011
Referring to Scenario 13-12, what is the correct interpretation for the estimated coefficient for X 2?
Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus.
Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus for each additional block away from the quad.
Holding the effect of the distance from the quad constant, the estimated mean costs for parking off campus is $0.16 per hour more than parking on campus for each additional block away from the quad.
Holding the effect of the distance from the quad constant, the estimated mean costs for parking off campus is $0.16 per hour more than parking on campus.
Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus.
According to given data determined parking meter rates throughout the campus area.
What is the hypothesis for regression?
For simple linear regression, the chief null hypothesis is H0 : β1 = 0, and the corresponding alternative hypothesis is H1 : β1 = 0. If this null hypothesis is true, then, from E(Y ) = β0 + β1x we can see that the population mean of Y is β0 for every x value, which tells us that x has no effect on Y .
Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus.
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Write the equation of the parabola in vertex form.
vertex (3,1), point (2,-5)
f(x) = ___
Step-by-step explanation:
vertex form :
y = a(x - h)² + k
(h, k) are the coordinates of the vertex.
"a" is the orientation (up-down) and widening factor.
we use the given h, k and x, y (the given point) to calculate "a".
-5 = a(2 - 3)² + 1 = a(-1)² + 1 = a + 1
a = -6
and the full vertex form is then
y = -6(x - 3)² + 1
just a quick addition to the superb reply by "reinhard10158" above
[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\begin{cases} h=3\\ k=1 \end{cases}\implies y=a(x-3)^2 + 1\hspace{5em}\textit{we also know that} \begin{cases} x=2\\ y=-5 \end{cases} \\\\\\ -5=a(2-3)^2 + 1\implies -6=a(-1)^2\implies -6=a \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} y=-6(x-3)^2 + 1 \end{array}} ~\hfill[/tex]
A truck that can carry no more than 7200 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 200 lb and each piano weighs 425 lb. Write and graph an inequality to show how many refrigerators and how many pianos the truck could carry.Will 18 refrigerators and 11 pianos overload the truck? Explain.
Answer: Yes, it will overload the truck.
Step-by-step explanation:
To begin, form an algebraic equation.
200(r) + 425(f)
Second, substitute your numbers for the variables.
200(18) + 425(11) = ?
Third, multiply your numbers of fridges by the weight of each.
3600 + 4675 = 8275 is the total
Finally, determine whether or not if it is greater than the weight limit the truck has.
7200 < 8275
Therefore, the total weight is greater than the weight limit.
Which best describes the solution for the
system of linear equations graphed below?
A. (3, 0) only
B. (6,0) only
C. no solution
D. infinitely many solutions
Answer:
C no solution.
Step-by-step explanation:
The lines are parallel, they will never intersect.
endure all, a manufacturer of batteries claims that the lifetime of their batteries is normally distributed with a mean of 500 hours and a standard deviation of 40 hours. what is the probability that an endure all battery selected at random will last more than 550 hours? a) 0.8944 b) 0.1151 c) 0.8849 d) 0.1056 e) 0.1587 f) none of the above.
The probability that an Endure All battery selected at random will last more than 610 hours is 0.1056.
What is meant by the term normal distribution?A probability distribution that is symmetric around the mean is the normal distribution, sometimes referred . Eventsare approximately normal happen more often than information that are far the mean.
The normal distribution appears as a "bell curve" on a graph.
A probability bell curve is more properly described as the normal distribution.
In a normal distribution, the mean is 0 and the standard deviation is 1. The kurtosis is 3, and there is no skew.All normal distributions are symmetrical.
Natural occurrences frequently resemble the usual distribution.
However, in finance, most price distributions are not fully normal.
How to solve?
Given that μ = 500, σ = 40, for x > 550
z = 550-500/40
z = 1.25
From the normal distribution table, P(z > 1.25) = 1 - P(z < 1.25) = 1 - 0.8944 = 0.1056
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What is the slope of a line perpendicular to the line whose equation is x - 6y = 24.
Fully simplify your answer.
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]x-6y=24\implies -6y=-x+24\implies y=\cfrac{-x+24}{-6} \\\\\\ y=\cfrac{-x}{-6}+\cfrac{24}{-6}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{6}} x-4\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{\cfrac{1}{6}} ~\hfill \stackrel{reciprocal}{\cfrac{6}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{6}{1}\implies {\Large \begin{array}{llll} -6 \end{array}}}}[/tex]
Answer:
-6
Step-by-step explanation:
i just got this question and it wasn’t 1/6 it was -6
suppose you have a collection of 24 candles. there are 6 candles each of 4 different colors, and they are identical in every other way. how many different ways can 6 candles be chosen?
The different ways in which 6 candles be chosen 84 ways.
What is a combination?
The combination is a method of choosing things from a collection where the order of the choices is irrelevant.
Given, When we have 'n' types of elements and choose 'r' elements from them, then we have C(n+r-1, r) combinations.
n=4(4 types of candles), r=6(no of candles to pick)
Choose 6 different candles, then
=C(4+6-1,6)
=C(9,6)
= 9! / (9-6)! (6!)
= 9!/ 6! 3!
=84
The different ways in which 6 candles can be chosen are 84 ways.
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Kennedy has $0.81 worth of pennies and nickels. She has 9 more nickels than pennies. Write a system of equations that could be used to determine the number of pennies and the number of nickels that Kennedy has. Define the variables that you use to write the system of equations.
Answer:
Step-by-step explanation:
kwik trip
Answer:
Let x be the number of pennies, and y be the number of nickels.
1 penny = 0.01 dollar
1 nickel = 0.05 dollar.
The equations that will describe the given problem will be,
0.01 x + 0.05 y = 0.81
y = x + 9
Plugging the second equation into the first gives us,
0.01 x + 0.05(x+ 9) = 0.81
0.06 x + 0.45 = 0.81
0.06 x = 0.36
y = 15
So, Kennedy has 6 pennies, and 15 nickels.
Step-by-step explanation:
the scores on a standardized test are normally distributed with a mean of 100 and standard deviation of 20. what test score is 1.5 standard deviations below the mean?
The 1.5 standard deviations below the mean implies that the Z-score value will be equal to -1.5.
The test score 70 is 1.5 standard deviations below the mean .
We have given that,
The Score standardized test are Normally distributed,
Mean ,μ = 100
Standard deviations, s.d = 20
we have to calculate test score is 1.5 standard deviations below mean.
so, Z- score = -1.5
This is the z-score formula or z-score equation.
Z = (x − μ )/σ
Where, Z--> z-score
x---> the value in question
μ--> mean of the distribution
σ --> standard deviation of the distribution
plugging all known the value in formula
-1.5 = (x- 100)/20
=> -1.5 × 20 = x - 100
=> x = -30 + 100 = 70
So, 70 is the test score that is 1.5 standard deviations from the mean.
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ASAP!!!
Determine the perimeter of the right triangle shown. Round your final answer to the nearest whole number, if necessary.
5 units
12 units
16 units
25 units
The perimeter of the triangle ABC is 12 units. Hence, option (B) is correct.
What is a triangle?A triangle is a geometric figure with three edges, three angles and three vertices. It is a basic figure in geometry.
The sum of the angles of a triangle is always 180°
In the given right angle triangle,
The vertices are A(-3, 2), B(-3, -1) and C(1, -1)
The distance between AB = 3,
And the distance between BC = 4
To find the distance of AC, use Pythagoreans theorem,
AC² = AB² + BC²
= 3² + 4²
= 9 + 16
AC² = 25
AC = 5
The perimeter of triangle ABC = AB + BC + AC
= 3 + 4 + 5
= 12
The perimeter of the triangle ABC is 12 unit.
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A census was conducted at a school. All of the students were asked how
many cars their family owns. The results are below.
Number of Cars 0 1 2 3
Probability 0.005 0.120 0.125 0.500 0.250
For the questions below enter the value only. Round to the nearest hundredth, if necessary.
1. What is the expected number of cars?
2. What is the standard deviation?
1. The expected number of cars is given as follows: 2.87.
2. The standard deviation of the number of cars is given as follows: 0.94.
How to obtain the expected value and the standard deviation of a discrete distribution?The discrete distribution is given from the table in this problem, as follows:
P(X = 0) = 0.005.P(X = 1) = 0.12.P(X = 2) = 0.125.P(X = 3) = 0.5.P(X = 4) = 0.25.The expected value is given by the sum of each outcome multiplied by it's probability, as follows:
E(X) = 0 x 0.005 + 1 x 0.12 + 2 x 0.125 + 3 x 0.5 + 4 x 0.25 = 2.87.
The standard deviation is given by the square root of the sum of the differences squared between each observation and the mean, as follows:
S(X) = sqrt((0-2.87)² x 0.005 + (1-2.87)² x 0.12 + (2-2.87)² x 0.125 + (3-2.87)² x 0.5 + (4-2.87)² x 0.25) = 0.94.
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the temperature of a duck breast is checked during cooking. according to the operation's policy, the duck breast must be cooked for 16 minutes to allow the internal temperature to reach 165 degrees f. what haccp principle is addressed by cooking the duck breast to 165 degrees f?
The HAACP principle addressed by cooking the duck breast to 165 degrees F is a 'critical limit'.
What is the critical limit?
A critical limit is a value to which a biological, chemical or physical factor must be controlled to prevent food safety hazards. These are the conditions under which potential hazards can be controlled, reduced, or eliminated. They are monitored by observation or measurement, such as measuring the temperature of food storage to avoid spoilage.
Critical limits are based on factors such as:
Temperature, Humidity, Moisture level, pH, Water activity (Aw), Time, Salt concentration, and Preservatives.
Production processes that fail to stay within the bounds of the critical limits must be stopped or the products must be discarded.
Hence the answer is critical limit.
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A large conical tank is positioned so that its vertex is pointed downward. Water drains out from a hole at the vertex. As the water drains, the height of the water (measured from the vertex to the top surface) is always twice the radius of the waters surface.
a. draw and label a diagram of the situation, defining any variables you use
b. write a function that gives the volume of the water in the cone as a function of radius of the water at the top surface, using no other variables/parameters.
c. using variable t to represent the time aspect of these changes, what rate of change notation can we use for how the volume of water changes with respect to time and how the radius of the water at the top surface changes with respect to time? (notation for the two quantities only, not an equation)
d. Starting from the original quantity relation in (b) determine the equation that relates their rates of change with respect to time, that is "find the related rates equation"
e.suppose that the water drains from the cone at a constant rate of 15 cm^3 per second. how fast is the radius of the top surface of the water changing when the radius measures 75 cm? write your result in a complete sentence
2. use implicit differentiation to determine dy/dx for the following relation: sin(2x^2y^3)-3x^3=1
1. Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.
2.The implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)
1. a) check the image attached
r= radius
h=height
b) Volume of conical tank ,V = (1/3)πr²h
= (1/3)πr²(2r)
V = 2/3πr³
c) Rate of change of volume with respect to time is denoted as dV/dt
Rate of change of radius is denoted as dr/dt
d) V = 2/3πr³
dV/dt = 2/3π (3r²) dr/dt
dV/dt = 2πr² dr/dt
e) Rate of change in volume, dV/dt = 15cm³ / sec , r =75cm
∴ 15 = 2π(75)² . dr/dt
dr/dt = 15 / (2π(75)²)
dr/dt = 15/35342.92
dr/dt = 0.00042 cm/sec
∴Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.
2. sin(2x² y³) - 3x³ = 1
Differentiating :
cos(2x² y³) [2x² y³dy/dt + 2y³ 2x] - 9x² = 0
6x²y² dy/dx . cos(2x² y³) + 4xy³ cos(2x² y³) - 9x² = 0
dy/dx = 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)
Hence, the implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)
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1. use the equation to answer these questions. (a) what is the solution to the equation? (b) what is the extraneous solution? why? (c) in general, what is an extraneous solution?
a. The solution of the equation x + 2 = √3x + 10 is x = -3 or x =2.
b. Extraneous solution is x= -3.
c. Extraneous solution does not satisfy original equation.
As given in the question,
Given equation :
x + 2 = √3x + 10
a. Squaring both the sides
( x + 2 )² = ( √3x + 10 )²
⇒ x² + 4x + 4 = 3x + 10
⇒ x² + x - 6 =0
⇒ x² + 3x -2x -6 =0
⇒ ( x+ 3) ( x -2) =0
⇒ x= -3 or x = 2
b. Extraneous solution:
x = -3
-3 + 2 = √3(-3)+ 10
⇒ -1 ≠ 1 extraneous solution.
x= 2
2+ 2 =√3(2) +10
⇒4 = 4 not extraneous.
c. Extraneous solution does not satisfy original equation
Therefore, the solution of the equation is x = -3 or x=2and its extraneous solution is x =-3. Extraneous solution does not satisfy original equation.
The complete question is :
Use the equation below to answer these questions.
x + 2 = √3x + 10
a) What is the solution to the equation?
b) What is the extraneous solution? Why?
c) In general, what is an extraneous solution?
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consider the initial value problem take the laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. denote the laplace transform of by . do not move any terms from one side of the equation to the other (until you get to part (b) below). help (formulas) solve your equation for . take the inverse laplace transform of both sides of the previous equation to solve for .]
Initial value that take the LaPlace transform of both sides of the given differential equation to create the corresponding algebraic equation is [tex]y(t) = 3t + sin4t + 3cos4t[/tex]
What is The Laplace transform?In terms of its usefulness in resolving physical issues, the Laplace transform is an important transform that is probably second only to the Fourier transform. The Laplace transform is especially helpful in solving linear ordinary differential equations, such as those that arise in the analysis of electronic circuits.
The initial value-
[tex]y" + 16y = 48t\\y(0)=3,y'(0)=7\\L( f^{n}(t) ) = s^{n}L(f(t)) - s^{n - 1}(f(0)) - s^{n - 2}(f(0)).........................................f^{n - 1} (0)\\[/tex]
By the formula,
[tex]s^{2}Y(s) - sy(0) - y'(0) + 16Y(s) = 48L(t)\\L(t^{n}) = \frac{n!}{s^{n+1} } \\\\On substitutinf the values,\\Y(s)(s^{2} + 16) - 3s - 7 = \frac{48}{s^{2} }\\Y(s)(s^{2} + 16) = \frac{48}{s^{2} } +3s + 7\\Y(s) = \frac{48 + s^{3} -7s^{2} }{s^{2}((s^{2} + 16) }\\Y(s) = \frac{3}{s^{2} } + \frac{4}{(s^{2} + 16} + \frac{3s}{(s^{2} + 16}[/tex]
On taking LaPlae inverse both side,
[tex]Y(t) = L( \frac{3}{s^{2} } ) + L( \frac{4}{(s^{2} + 16)} ) + L( \frac{3s}{(s^{2} + 16)} )\\Y(t) = 3t + sin4t + 3cos4t\\[/tex]
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25 Points
The total cost to rent and drive a car for one day is listed below for two car rental agencies.
Blue Car Rental charges $27.60 to rent a car plus $0.18 per mile driven.
Red Car Rental charges $40.40 to rent a car plus $0.14 per mile driven.
• Write a variable expression that shows the total cost to rent and drive a car for one day at Blue Car Rental. Write a variable expression that shows the total cost to rent and drive a car for one day at Red Car Rental. Define the variable.
• Calculate the total cost at each agency to rent a car for one day and drive it 280 miles. Show all your work.
• Write and solve an inequality that shows how many miles must be driven before the total daily cost at Red Car Rental is less than the total daily cost at Blue Car Rental. Show all your work.
• If Mr. Kao rents a car for one day and drives it 400 miles, which company has a lower total cost?
The variable expression are
y = $0.18x + $27.60 - Blue Car Rentaly = $0.14x + $40.40 - Red Car Rentalcost at each agency to rent a car for 280 miles
y = $78.00 - Blue Car Rentaly = $79.60 - Red Car RentalThe company that has a lower cost for 400 miles
Red Car RentalHow to write the expressionThe required expression is a linear function
A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.
y = mx + c
Definition of variable to suit the problem
y = output variable (total cost/)
m = slope (cost per mile driven)
x = input variable (number of miles)
c = y intercept (Car Rental charges)
The equation as described in the problem is represented using linear function as as follows
y = $0.18x + $27.60 - Blue Car Rental
y = $0.14x + $40.40 - Red Car Rental
the total cost at each agency to rent a car for one day and drive it 280 miles
y = $0.18x + $27.60 - Blue Car Rental
y = 0.18 * 280 + 27.60
y = $78
y = $0.14x + $40.40 - Red Car Rental
y = 0.14 * 280 + 40.40
y = $79.60
Write and solve an inequality
$0.18x + $27.60 < $0.14x + $40.40
0.18x + 27.60 < 0.14x + 40.40
0.18x - 0.14x < 40.40 - 27.60
0.04x < 12.80
x < 320 miles
Mr. Kao rents a car for one day
y = $0.18x + $27.60 - Blue Car Rental
y = 0.18 * 400 + 27.60
y = $99.60
y = $0.14x + $40.40 - Red Car Rental
y = 0.14 * 400 + 40.40
y = $96.40
comparing the both shows that Red Car Rental has a lower cost
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