The orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.
To find this, we can take the derivative of the original equation 3x^2 − y^2 = C with respect to x, which will give us 6x. Then, we can take the negative reciprocal of this derivative, which is -1/6x. This is the slope of the orthogonal trajectories.
We then can use the point-slope form of a line to get y - y1 = -1/6x (x - x1). Since the slope of the orthogonal trajectories is always -1/6x, we can just set the equation y - y1 = -1/6x (x - x1) equal to 0 and solve for x to get x1 = -y1/6.
We can then plug this value back into the point-slope equation to get y - y1 = -1/6(-y1/6) (x - (-y1/6)), which simplifies to y - y1 = y1/6 + x. Now, if we let y1 = -C/3, then we get y - (-C/3) = C/6 + x, which can be rewritten as x^2 + y^2 = C/3.
Therefore, the orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.
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Antonio le da cada dia a su perro siete decimos de bote de comida para perros ¿ para cuantos dias tiene con 7/2 de bote
La cantidad de comida de 7 / 2 de un bote sirve para un número de días de 5 días.
¿Cómo determinar la duración de un bote de comida para perros?
En este problemas tenemos que Antonio da una ración fija de comida a su perro, se indica que consume siete décimos (7 / 10) de un bote de comida cada día para alimentar a su perro. Tenemos aquí un situación de relación directa entre la cantidad de comida (y) y el número de días para provisiones (x).
A partir del enunciado, se deriva la siguiente ecuación:
y = (7 / 10) · x
Si tenemos que y = 7 / 2, entonces el número de días para provisiones es:
7 / 2 = (7 / 10) · x
x = 10 / 2
x = 5
La cantidad de comida de 7 / 2 de un bote sirve para un número de días de 5 días.
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Write the equation of the line that passes through the point (0, 1) and is perpendicular to the line y=1/5x+2
Answer:
the requisite equation for the given question is y=−5x
Step-by-step explanation:
firstly, select the coordinates (0, 0) through which the perpendicular line will cross (0,0)
then, locate the slope when y=15x+2y=15x+2.
The slope is 1/5 when expressed in the slope-intercept format.
therefore, m=1/5
since, a perpendicular line's equation must have a slope equal to the inverse reciprocal of its initial slope.
therefore, [tex]m_{perpendicular}[/tex]=−1/(1/5)
To determine the slope of the perpendicular line, simplify 1/(1/5).
[tex]m_{perpendicular}[/tex]=−5
By using point-slope method, determine the perpendicular line's equation.
y+0=−5.(x+0)
Solve for y.
y= -5x + 0
y=−5x
hence, the answer for the question is y=-5x
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Which of the following is a possible event?
A) Pulling a blue sock from a drawer with 5 red socks and 4 pink socks.
B) Pulling a blue sock from a drawer with 1 blue sock and 8 pink socks.
C) Pulling a blue sock from a drawer of 9 red socks.
D) Pulling 2 blue socks from a drawer with 1 blue sock and 4 pink socks.
A possibility to event pulling a blue sock from a drawer with 1 blue sock and 8 pink socks.
What is an event in probability?Events in probability are the results of an experiment. An event's probability is a gauge of its likelihood to transpire as a result of an experiment.
In this question
A) Pulling a blue sock from a drawer with 5 red socks and 4 pink socks. - This is a possible event. There is one blue sock in the drawer, so it is possible to pull it out.
B) Pulling a blue sock from a drawer with 1 blue sock and 8 pink socks. - This is a possible event. There is one blue sock in the drawer, so it is possible to pull it out.
C) Pulling a blue sock from a drawer of 9 red socks. - This is not a possible event. There are no blue socks in the drawer, so it is not possible to pull one out.
D) Pulling 2 blue socks from a drawer with 1 blue sock and 4 pink socks. - This is not a possible event. There is only one blue sock in the drawer, so it is not possible to pull two out.
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What is a name for the marked angle?
Maria was born on January 1, 2000. Her mother was born on January 1, 1975 and her father was born on January 1, 1970. In what year was the sum of their ages 100?
Choose matching term
2015
2017
2011
2016
In year 2015 the sum of the age of maria, her mother, and her father, will be 100 years.
As the date and month of the birth is same for maria, her mother, and her father. Lets say in year X, the sum of the age will be 100 years.
The age of maria in year X = (X - 2000)
The age of her mother in year X, = (X - 1975)
The age of her father in year X, = (X - 1970)
Sum of their age,
100 = (X - 2000) + (X - 1975) + (X - 1970)
100 = 3X - (2000 + 1975 + 1970)
100 = 3X - 5945
3X = 5945 + 100
3X = 6045
X = 2015
Hence option (a) is correct.
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show that each vertex of g can be assigned an integer in {1, 2, . . . , d 1} so that no two adjacent vertices are assigned the same integer. hint: use induction on the number of vertices.
Every vertex of a graph can be assigned an integer in {1, 2, ..., d1} so that no two adjacent vertices are assigned the same integer.
We can use induction to show that every vertex of a graph can be assigned an integer in {1, 2, ..., d1} so that no two adjacent vertices are assigned the same integer. Let G be a graph with d vertices, and assume that the result is true for all graphs with fewer than d vertices. Consider any vertex v in G and let N(v) be the set of neighbours of v. Since |N(v)| < d, by our induction hypothesis, there is an assignment of the integers 1,2, ..., d−1 to the vertices in N(v) so that no two adjacent vertices are assigned the same integer. Now assign the dth integer, i.e., d, to v and we are done. By our induction hypothesis, this assignment works for all graphs with d vertices. Thus, by induction, every vertex of a graph can be assigned an integer in {1, 2, ..., d1} so that no two adjacent vertices are assigned the same integer.
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arrange the following list of functions in increasing order of growth rate. that is, if function g(n) follows function f (n), then it should be the case that f (n) is o(g(n)) (the base of logarithms is 2). justify in simple words. (you do not have to show detailed calculations.)
The correct order of functions, in increasing order of growth rate, is g1, g4, g3, g5, g2, g7, and g6.
What is functions?A mathematical expression, rule, or law that establishes the relationship between an independent variable and a dependent variable (the dependent variable). In mathematics, functions are everywhere, and they are crucial for constructing physical relationships in the sciences. A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.
Here,
The list of function is attached in image.
g1, g4, g3, g5, g2, g7, g6 is the correct sequence of functions in increasing order of growth rate.
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Find all solutions for each triangle. If no solutions exist, write none. Round to the nearest tenth. A=76 degree, a=5, b=20
Utilize the fact that the sum of all triangles is 180 to determine that angle C is 42.
What is Pythagorean Theorem?According to the Pythagorean Theorem, the squares on a right triangle's hypotenuse, or the side that faces the right angle, are equal when added together.This is written as a2 + b2 = c2 in the usual algebraic notation. A fundamental relationship in Euclidean geometry between a right triangle's three sides is known as the Pythagorean theorem or Pythagoras' theorem. According to this rule, the area of the square with the hypotenuse side is equal to the sum of the areas of the squares with the other two sides.Therefore,
There is no other way to resolve this, thus you must employ the Law of Cosines. You cannot apply the Pythagorean Theorem since it is not a right triangle. We can apply the Law of Cosines to determine the missing side length, and then the Law of Sines to determine an additional angle.
The Triangle Angle-Sum theorem will then be used to complete it. Ready?
The Cosine Law to find side b is
[tex]$b^2=a^2+c^2-2 a c \cos B$[/tex]
and fill in the info we know, which is everything but the b.
[tex]$b^2=(21)^2+(29)^2-2(21)(29) \cos 109$[/tex]
Doing all that math gives us that side
b = 40.9 or 41 .
Now the Law of Sines to find missing angle
A or C. Let's find
[tex]$A \cdot \frac{\sin A}{21}=\frac{\sin 109}{41}$[/tex].
That gives us that angle A is 29 .
Now use the fact that all triangles add up to 180 to get that angle C is 42
The complete question is:
Solve each triangle. Round your answers to the nearest tenth
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An earthquake with a rating of 3.8 can
cause destruction in areas of up to 100
kilometers that are populated.
36,810,717.06
........ ... . ... . . took it on edge
Y=3x+5 function rule
3 is the slope of the given function y=3x+5.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The given function rule is y=3x+5.
Let us find some x and y values to plot the graph.
x=0, then y=3(0)+5=5
x=1, then y=3(1)+5=8
x=2, then y=3(2)+5=11
x=3, then y=3(3)+5=14
x=4, then y=3(4)+5=17
Now by taking these value we plot the graph.
In the given graph slope is 3.
Hence, 3 is the slope of the given function y=3x+5.
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For example, you can select the coordinates of the black point (plus symbol) you placed on the graph previously to see that the intersection of the curves occurs at the point representing a quantity of ________ boxes.
The black point (plus symbol) selected on the graph has coordinates of (4, 0.4).
This point represents the intersection of the two curves on the graph. To calculate the quantity of boxes represented by this point, we can use the equation y=mx+b, where m is the slope of the line, x is the x-coordinate, and b is the y-intercept.
In this case, the slope of the line is -2, the x-coordinate is 4, and the y-intercept is 0.4. We can substitute these values into the equation to solve for the quantity of boxes, which would be: y=-2x+0.4. Plugging in the x-coordinate of 4, we would get: 0.4=-2(4)+0.4, which simplifies to 0.4=-8. Solving for y, we would get 8 boxes. Therefore, the point (4, 0.4) on the graph represents a quantity of 8 boxes.
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Simplify 5/8x+1/2(1/4x+10)
The simplification 5/8(x )+ 1/2 ( 1/4( x) +10) is 3/4(x) + 5
What is simplification of fractions?The word simplify means to make something easier to do or understand. So, reducing or simplifying fractions means we make the fraction as simple as possible. We do this by dividing the numerator and the denominator by the largest number that can divide into both numbers exactly.
For example 25/100 can be simplified by cutting down 25 and 100 by their common factor which is 25.
25/100 = 1/4
⅝(x) + 1/8(x) + 5
= 6/8(x) +5
= 3/4(x) + 5
therefore the simplification of 5/8(x )+ 1/2 ( 1/4( x) +10) is 3/4(x) + 5
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25. At a cost of $1.41 per foot, what does 8 feet of air duct hose cost?
(Round the answer to the nearest cent)
let and let have critical numbers -2, 0, and 2. use the second derivative test to determine which critical numbers, if any, give a relative maximum
According to the second derivative test, whether or not there is a relative maximum or minimum at a crucial number depends on the sign of the second derivative. According to the second derivative test, in this situation, -2 and 0 result in a relative minimum, whereas 2 results in a relative maximum.
To ascertain if a crucial number belongs to a relative maximum or relative minimum, one can utilise the second derivative test. Calculate the function's second derivative before using the test. After that, calculate the second derivative at the threshold value. The critical number relates to a relative maximum if the second derivative is negative. The crucial number corresponds to a relative minimum if the second derivative is positive. Specifically, At -2 and 0, when the second derivative is negative, there is a relative minimum. The crucial number 2 corresponds to a relative maximum since the second derivative is positive at that value.
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Determine the measure of Angle B in the triangle below. thank you
Answer:
55°
Step-by-step explanation:
2x + 5 + 2x + 3x = 180 Combine like terms The sum of the interior angles of a triangle is 180
7x + 5 = 180 Subtract 5 from both sides
7x + 5 - 5 = 180 - 5
7x = 175 Divide both sides by 7
[tex]\frac{7x}{7}[/tex] = [tex]\frac{175}{7}[/tex]
x = 25
Substitute 25 for x
2x + 5
2(25) + 5
50 + 5
55 is the measurement of angle B
Zachary purchased a computer for $1,400 on a payment plan. Five months afyer he purchased the computer, his balance was $625. seven months after he purchased the computer, his balance was $315. what is the equationt that models the balance y after x months?
hi id torjwh3 tenthjge r t
please help me asp i dont got much time
The function are f+g(x)=f(x)+g(x), (f-g)x=f(x)-g(x), (g-f)(x)=g(x)-f(x), (f.g)(x)=f(x).g(x), f/g(x)=f(x)/g(x) and g/f(x)=g(x)/f(x)
What is a function?A relation is a function if it has only One y-value for each x-value.
We know that from properties of function,
f+g(x)=f(x)+g(x)
(f-g)x=f(x)-g(x)
(g-f)(x)=g(x)-f(x)
(f.g)(x)=f(x).g(x)
f/g(x)=f(x)/g(x)
g/f(x)=g(x)/f(x)
Hence, the function are f+g(x)=f(x)+g(x), (f-g)x=f(x)-g(x), (g-f)(x)=g(x)-f(x), (f.g)(x)=f(x).g(x), f/g(x)=f(x)/g(x) and g/f(x)=g(x)/f(x)
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True or False? In reference to different sampling methods, systematic sampling includes the steps: divide the population into groups; use simple random sampling to identify a proportionate number of individuals from each group. Select the correct answer below: O True O False
In reference to different sampling methods, systematic sampling includes the steps: divide the population into groups; use simple random sampling to identify a proportionate number of individuals from each group: [FALSE]
What is Systematic sampling?Systematic sampling is a method of sampling where the sample is selected using a fixed interval, rather than using random sampling. The sampling process begins by selecting a random starting point in the population and then selecting every kth element from the population.
For example, if the population has 100 elements and k=10, every 10th element would be selected for the sample.
Dividing the population into groups, and then using simple random sampling to identify a proportionate number of individuals from each group is a method called stratified sampling.
It is important to note that systematic sampling can be vulnerable to sampling bias if the sampling interval is not chosen carefully and the population is not randomly ordered, because in that case, it will not be a truly random sample.
Hence, the correct answer to this is False.
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which of the following gives the volume of the solid created by revolving a curve f(x) around the y-axis?
The volume of the solid created by revolving a curve f(x) around the y-axis can be calculated by formulae [tex]V = \pi \int\ {dc[F(y)]} \, 2dy[/tex] .
Given,
A solid is created by revolving a curve f(x) around the y-axis.
We know that,
The volume of a solid that is rotated around the y-axis is calculated using the "Disk Method"
The disk method is generally used when we rotate a particular curve around the x-axis or y-axis.
The volume of the solid that is formed by revolving the region bounded by the curve [tex]x = F(y)[/tex] and the y-axis between [tex]y=c[/tex] and [tex]y = d[/tex] about the y-axis is given by:
[tex]V = \pi \int\ {dc[F(y)]} \, 2dy[/tex]
Thus, the required formulae is [tex]V = \pi \int\ {dc[F(y)]} \, 2dy[/tex]
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The complete question is :
Which of the following gives the volume of the solid created by revolving a curve f(x) around the y-axis?
[tex]V = \pi \int\ {dc[F(y)]} \, 2dy[/tex][tex]V = \pi \int\ {dc[F(y)]} \, dy[/tex][tex]V = \pi^{2} \int\ {dc[F(y)]} \, 2dy[/tex][tex]V = \pi \int\ {dc[F(y)]^{2} } \, 2dy[/tex]This graph shows a proportional relationship.
What is the constant of proportionality?
y= 1/4 x= 3/8
Answer:
y = 1/4x
Step-by-step explanation:
Proportional Relationships:A proportional relationship is similar to a linear relationship, with one distinction, it's y-intercept is equal to zero, so it can be expressed in the form: [tex]y=kx[/tex], where "k" is some constant. More specifically, when we divide y by x, we always get a constant amount, denoted as "k": [tex]\frac{y}{x}=k[/tex].
Using this knowledge, we can kind of immediately tell it's [tex]y=\frac{1}{4}x[/tex] as it's the only one in the form: [tex]y=kx[/tex], but let's look at why [tex]x=\frac{3}{8}[/tex] is not a proportional relationship.
It's important to note that [tex]x=\frac{3}{8}[/tex] is a vertical line, since the x value is constant while the y value varies. This means "y/x" cannot be constant, since "y" changes, except what it's being divided by will remain the same, equal to 3/8.
The length of a rectangular part is 20 feet longer than the width of the park is the length of the park is 36 feet what is the width of the park
The width of the rectangular park is found to be 16 feet.
Define the properties of the shape rectangle?An enclosed 2-D shape called a rectangle has four sides, four corners, and four right angles (90°).There are 4 corners, 4 angles, plus 4 sides (vertices).There's many two dimensions to it: length and width.A rectangle has 90° angles on each side.The opposing sides are parallel and equal.It includes 2 equal-length diagonals.For the stated question-
Length of a park = 20 ft + width of a park
Width of a park = Length of a park - 20 ft
Width of a park = 36 - 20 = 16 ft
Thus, the width of park is found to be 16 ft.
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Identify the three triangle theorems explain what the theorem states
The three triangle theorems include the following:
The Pythagorean TheoremThe Law of CosinesThe Law of Sines What do the triangular theorems state?The positions of the theorems are as follows:
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse). This can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the legs.
The Law of Cosines states that in any triangle, the square of the length of a side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the length of one side times the cosine of the angle between the other two sides. This can be written as c^2 = a^2 + b^2 - 2ab * cos(C)
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. This can be written as a/sin(A) = b/sin(B) = c/sin(C)
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given ae=ce be=dc
prove m < acd=m < cab
pls help rsm problem
The proof of the angles is shown below by SAS
How to determine the proof of the trianglesThe complete question is added as an attachment
With the given information and the image attached below, we can state that the two triangles are congruent by SAS,
Then go ahead to use the CPCTC to show that any of their corresponding parts are congruent as well.
The proof is given as:
Statement Reason
1. AE ≅ CE, BF ≅ FC 1. Given2. AC ≅ AC 2. Reflexive property of congruence3. <BAC ≅ <BCA 3. Base angles of isosceles ∆BAC4. ∆AEC ≅ ∆AFC. 4. SAS5. <BAF ≅ <BCE. 5. CPCTCRead more about congruence proof at:
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SOMEONE, PLEASE HELP I WILL APPRECIATE SO MUCH
Answer: 20
Step-by-step explanation: Hello! The first thing you have to do is substitute m = 7 and n = 5 into the equation 4 + (m - n)^4. After substituting m and n into the equation, you will get: 4 + (7 - 5)^4. Use PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction). Parentheses is first, so solve everything inside of the parentheses first. 7 - 5 = 2. Now you have 4 + (2)^4. Next is exponents in pemdas, so solve the exponents. 2^4 = 2 x 2 x 2 x 2, or 4 x 4, or 16. Now you have 4 + 16. The last thing you have to do is add 4 + 16, which is 20. The answer is 20.
In the xy-plane, line m passes through the origin and is perpendicular to the line 6x+y=h, where h is a constant. If the two lines intersect at point (p, p+2), what is the value of p?
Using slope intercept equation the value of p is -2.4 .
What is slope intercept equation?
When the slope of the line being studied is known, and the provided point is also the y intercept, the slope intercept formula, y = mx + b, is utilised (0, b). It provides both a slope and an intercept, which is why the form is known as the slope-intercept form.
Let’s call line m y’ and we can deduce that y’ = ax where a is the slope of line y’. (In the standard line equation, y = mx + b, b is the y intercept which, in the case of line m, is zero since it passes through the origin (0,0).)
Then name the 2nd line y’’ = -6x + h
When straight lines are perpendicular, their slopes are negative inverses. So the slope of line y’, which I’ve called a, must equal +1/6
Now we know that y’ = (1/6)x and y’’ = -6x + h
If the lines intersect at a point (x , y) which we are told is (p, p+2), then at that point y’ = y’’, or p+2 = (1/6) and p = -6(p+2) + h.
From this we can solve for p,
=> p - p/6 = -2=> (5/6)p = -2 or
=>p = -12/5 = -2.4
Now we know that the intersection point (p,p+2) is (-2.4, -0.4)
Hence the value of p is -2.4 .
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What is the value of x in the equation 3/4(1/4x+7)-(1:2x+2)=3/8(4-x)-1/4x?
The value of x in the given equation is 6.4. The solution is obtained by solving the linear equation.
What is a linear equation?
A linear equation is the equation which has the highest degree as 1. This shows that a linear equation with an exponent greater than one has no variables. On the graph, such an equation results in a straight line.
We are given [tex]\frac{3}{4} (\frac{x}{4} +8) - (\frac{x}{2} +2)=\frac{3}{8} (4-x) - \frac{x}{4}[/tex]
Now, let's simplify the linear equation
⇒[tex]\frac{3x}{16} + 6 - \frac{x}{2} -2= 6 - \frac{3x}{8} - \frac{x}{4}[/tex]
Now, on combining the like terms, we get
⇒[tex]\frac{3x}{16} - \frac{x}{2} +\frac{3x}{8} + \frac{x}{4} = 2[/tex]
⇒[tex]\frac{3x-8x+6x+4x}{16} = 2[/tex]
⇒[tex]\frac{5x}{16} = 2[/tex]
⇒ 5x = 32
⇒ x = 6.4
Hence, the value of x in the given equation is 6.4.
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Manufacturing overhead is charged to three operating divisions on the basis of capital investment in the three divisions. If the investment is $10.8 million in the Eastern Division, $8.4 million in the Southern Division, and $17.4 million in the Far Eastern Division, how should manufacturing overhead of $888 000 be allocated to the three divisions?
Manufacturing overhead costs typically include the depreciation of equipment, wages provided to factory personnel, and power used to operate the equipment. $10.8 million, $8.4 million, and $17.4 million will make up the three divisions.
What are the costs of manufacturing overhead?Manufacturing overhead (MOH) cost is the sum of all indirect costs incurred during the production of a good.
It is included in the cost of the final good along with the costs of direct materials and direct labor.
In the manufacturing sector, the three major cost components are materials, labor, and overhead. There are no hidden expenses.
In other words, while the foreman's pay and supplies are covered, neither the office's nor the corporation's accountant's pay is. To calculate applicable production overhead, multiply the overhead allocation rate by the amount of labor or equipment used. Therefore, if your allocation rate is $25, your applied manufacturing overhead for this product is $25.
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Tracy is receiving lanoxin every 8.0 hours. She weighs 61 kg. The
dose range for this drug is 0.030 to 0.060 mg/kg every day. What is
the maximum safe dose (in mg) you could deliver every 8.0 hours?
The maximum safe dosage that can be delivered every 8 hours is 1.22 mg.
What is maximum value?A point in mathematics where the value of a function is greatest. It is an absolute maximum if the value is greater than or equal to all other function values.The smallest value in a given data set will be the minimum. The maximum is the most significant value in the data set.Here given that,
Tracy receives the drug lanoxin every 8.0 hours.
Her weight = 61 kg
The minimum dosage range = 0.03 mg/kg
The maximum dosage range = 0.060 mg/kg
To find,
Drug requirement for Tracy is :
0.060 x 61
= 3.66 mg per day
The maximum safe dose for each 8.0 hours is :
3.66 / 3 = 1.22mg
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8x-8}[/tex]...............
The solution to the expression when x = 2 is 8
How to determine the solution to the expressionFrom the question, we have the following parameters that can be used in our computation:
8x - 8
Also, we have
x = 2
substitute the known values in the above equation, so, we have the following representation
8 * 2 - 8
Evaluate the product
16 - 8
Evaluate the difference
8
Hence, the solution is 8
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Complete question
Given that 8x - 8, calculate the expression when x = 2
identify the graphs a (blue), b( red) and c (green) as the graphs of a function and its derivatives:
a) The blue graph is the graph of a function, f(x). This is because it is a continuous line with no breaks, and it is a smooth curve with no sharp corners.
The formula for this function is f(x) = x3 + 2x2 + 3x. To calculate the derivative of this function, we take the first derivative using the power rule. The derivative of f(x) is f'(x) = 3x2 + 4x + 3.
b) The red graph is the graph of the derivative of f(x), the first derivative. This can be seen because the red graph is a continuous line that is a smooth curve, with no sharp corners. The formula for this derivative is f'(x) = 3x2 + 4x + 3. To calculate the second derivative of f(x), we take the second derivative using the power rule. The second derivative of f(x) is f''(x) = 6x + 4.
c) The green graph is the graph of the second derivative of f(x), the second derivative. This can be seen because the green graph is a continuous line that is a smooth curve, with no sharp corners. The formula for this derivative is f''(x) = 6x + 4. To calculate the third derivative of f(x), we take the third derivative using the power rule. The third derivative of f(x) is f'''(x) = 6.
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