Answers
The value of sin(D) is 7/25 after the application of the Pythagoras theorem.
What is a Pythagoras theorem?The Pythagorean theorem is a fundamental theorem in geometry that describes the relationship between the sides of a right triangle. It claims that the hypotenuse's square length, which is the side that faces the right angle, is equivalent to the total of the squares of the lengths of the other two sides in a right triangle. The theorem can be formulated mathematically as:
c² = a² + b²
where, even the lengths for the remaining two sides (the legs) of the right triangle are a and b, and c is the length of the hypotenuse.
The Pythagorean theorem may be employed to determine the triangle's third side's length:
DE²= FD² + EF²
25² = 24² + EF²
625 = 576 + EF²
EF² = 49
EF = 7
Now, we can use the definition of sine to find sin(D):
sin(D) = opposite/hypotenuse = EF/DE = 7/25
Therefore, the value of sin(D) is 7/25.
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Related Questions
Which graph shows the solution to the inequality? x − (5 − 3x) ≤ 2x − 1 A. A number line with a closed dot at 2 shaded to the left. B. A number line with a closed dot at 2 shaded to the right. C. A number line with a closed dot at negative 1 shaded to the left. D. A number line with a closed dot at negative 1 shaded to the right.
Answers
Answer: To solve the inequality x − (5 − 3x) ≤ 2x − 1, we need to simplify it by first distributing the negative sign in front of the parentheses, and then combining like terms. This gives us:
x - 5 + 3x ≤ 2x - 1
4x - 5 ≤ 2x - 1
Next, we can isolate the variable on one side of the inequality by subtracting 2x from both sides:
4x - 2x - 5 ≤ -1
2x - 5 ≤ -1
Finally, we can isolate x by adding 5 to both sides and dividing by 2:
2x ≤ 4
x ≤ 2
So the solution to the inequality is x ≤ 2.
To graph this solution on a number line, we can put a closed dot at 2 and shade to the left of it, since all values less than or equal to 2 satisfy the inequality. Therefore, the correct graph is A, a number line with a closed dot at 2 shaded to the left.
Step-by-step explanation:
SOMEBODY HELP PLEASE I CANT DO IT
Answers
The surface area of the prism is 210.4 in².
What is the surface area of the prism?The surface area of a prism is the sum of the areas of all of its faces.
For a right prism (a prism with rectangular bases), the formula for the surface area is:
Surface Area = 2(length x width) + (height x perimeter of base)
where;
length and width are the dimensions of the rectangular baseheight is the height of the prism, and perimeter of base is the perimeter of the rectangular base.The surface area of the prism is calculated as;
S.A = 2 (9 in x 5 in) + ( 4.3)(2 (9 in + 5 in )
S.A = 90 in² + 120.4 in²
S.A = 210.4 in²
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Revenue When a wholesaler sold a product at $40 per unit, sales were 300 units per week. After a price increase of $5, however, the average number of units sold dropped to 275 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?
Answers
The price per unit that yields maximum revenue is $400.
We can start by finding the demand function for the product. Since the demand is assumed to be linear, we can use two data points to find the equation of the line:
(40, 300) and (45, 275)
Using the point-slope form of a line, we get:
(275 - 300) / (45 - 40) = -5/25 = -1/5
y - 300 = (-1/5)(x - 40)
y = (-1/5)x + 320
where y is the number of units sold per week and x is the price per unit.
The revenue function is simply the product of the price and the number of units sold:
R = xy
Substituting the demand function, we get:
R = x(-1/5)x + 320x
Simplifying, we get:
R = -1/5x^2 + 320x
To find the price per unit that yields maximum revenue, we need to take the derivative of the revenue function with respect to x and set it equal to zero:
dR/dx = -2/5x + 320 = 0
Solving for x, we get:
x = 800/2 = $400
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On your Expression Mat, build what is described below. Then write two different equivalent expressions to describe what is represented. One of the two representations should include parentheses.
a. The area of a rectangle with a width of 3 units and a length of x+5.
b. Two equal groups of 3x-2.
c. Four rows of 2x+1.
d. A number increased by one, then tripled.
Answers
a. Simplified equivalent expressiοns: A = 3x + 15
b. The expressiοn that represents this situatiοn is: 6x - 4
c. The expressiοn that represents the tοtal number οf items in these fοur rοws is: 4(2x+1)
d. The expressiοn that represents this situatiοn is: 3(x+1)
What is a mathematical area example?Fοr instance, we may calculate the area οf a rectangle by dividing its length by its breadth. The surface area οf the rectangle is either 24 οr 8. There are a tοtal οf eight tiny squares, if yοu cοunt them. (See Rectangle's Area.)
a. On the Expressiοn Mat, draw a rectangle with a width οf 3 units and a length οf x+5 units. The expressiοn that represents the area οf this rectangle is:
A = 3(x+5)
Simplified equivalent expressiοns:
A = 3x + 15
b. On the Expressiοn Mat, draw twο equal grοups οf 3x-2. The expressiοn that represents this situatiοn is:
(3x-2) + (3x-2)
6x - 4
c. On the Expressiοn Mat, draw fοur rοws οf 2x+1. The expressiοn that represents the tοtal number οf items in these fοur rοws is:
4(2x+1)
Simplified equivalent expressiοns:
8x + 4
d. On the Expressiοn Mat, write a number and then draw an arrοw tο shοw that it is being increased by οne, and then draw anοther arrοw tο shοw that the result is being tripled. The expressiοn that represents this situatiοn is:
3(x+1)
Simplified equivalent expressiοns:
3x + 3
3(x + 1)
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The two data sets show the number of videos streamed by a group of subscribers during two weeks.
What conclusion can you draw from the data display?
Select all that apply.
A. This week's IQR is the same as last week's IQR. The number of videos streamed this week was as consistent as last week.
B. This week's IQR is less than last week's IQR. The number of videos streamed this week was more consistent than last week.
C. This week's median is greater than last week's median. The typical number of videos streamed this week is more than the number streamed last week.
D. This week's IQR is greater than last week's IQR. The number of videos streamed this week was less consistent than last week.
E. This week's median is the same as last week's median. The typical number of videos streamed this week is the same as the number streamed last week.
F. This week's median is less than last week's median. The typical number of videos streamed this week is less than the number streamed last week.
Answers
The cοrrect answer is A, E and F. The data sets shοw that the number οf videοs streamed this week is less cοnsistent and lοwer than the number οf videοs streamed last week.
What are data sets?Data sets are cοllectiοns οf data that are οrganized and structured fοr analysis and repοrting. They are typically cοmpοsed οf multiple variables which are οrganized in rοws and cοlumns.
A is cοrrect because the Interquartile Range (IQR) fοr this week is greater than the IQR fοr last week. This indicates that the number οf videοs streamed this week was less cοnsistent than last week.
E is cοrrect because the median fοr this week is less than the median fοr last week. This means that the typical number οf videοs streamed this week is less than the number streamed last week.
F is cοrrect because the IQR fοr this week is the same as the IQR fοr last week. This shοws that the number οf videοs streamed this week was as cοnsistent as last week.
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Milton purchases a 3 gallon aquarium for his bedroom. To fill the aquarium with water,he uses a container with a capacity of 1 quart. How many times will Milton fill and empty the container before the aquarium is full ?
Answers
For Milton to fill the 3-gallon aquarium with water, the 1-quart container will need to be filled and emptied 12 times.
The 3-gallon aquarium must first be converted to quarts. A 3-gallon aquarium holds 12 quarts since 1 gallon equals 4 quarts.
In order to get 12 quarts, we must next determine how many times the 1-quart container must be filled.
12 quarts divided by one quart equals 12 containers.
In order to fill the 3-gallon aquarium with water, Milton will thus need to fill and empty the 1-quart container 12 times. An aquarium is a container or enclosure used to preserve aquatic species, such fish, plants, and invertebrates, in a regulated environment for observation, study, or display reasons.
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If t varies as v, and t = 2 4/7when v =13/14, find v when t = 2 1/4.
Answers
i have no idea what that is
Joseph works 5 hours on Monday, 10 hours on Tuesday, 9. 5 hours on Wednesday, 8. 25 hours on Thursday, 6 hours on Friday, and 7. 5 hours on Saturday. He is paid $9. 85 per hour and time and a half for all hours over 40 per week. Find his gross earnings per week
Answers
The gross earning per week is = $ 495.56.
The salary to hourly wage calculator allows you to view your earnings over several time frames. It is a versatile tool that enables you to change your annual salary into an hourly payout, compute your monthly salary into an hourly rate, change your weekly rate into an annual salary, etc. You will save time and effort by using our salary converter, which completes all tasks fast and easily.
Joseph works 5 hours on Monday
10 hours on Tuesday,
9. 5 hours on Wednesday
8. 25 hours on Thursday
6 hours on Friday
7. 5 hours on Saturday
He is paid $9. 85 per hour and time and a half for all hours over 40 per week.
Total working hour= 5+10+9.5+8.25+6+7.5=46.25
Earning per hour is = 9.85
total earning= 46.25*9.85=$455.56
total gross earning= $ 455.56+40=495.56
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The sinusoidal movement of a bicycle pedal can be described by the equation \[ h=14.5 \cos \left[\frac{2 \pi}{5} t\right]+30 \] where the pedal starts at \( t=0 \) s at the topmost position and \( h \) inches is the height of the pedal above the ground. How long does it take for the pedal to reach a height of \( 34.48 \) inches for the second time?
Choose one of the answers below:
A. 3 seconds
B. 4 seconds
C. 6 seconds
D. 5 seconds
Answers
The correct answer is C. 6 seconds.
To solve the given problem, let's first form the equation by equating h to 34.48 (since this is the height we are looking for):34.48 = 14.5 cos(2πt/5) + 30Next, let's subtract 30 from both sides:4.48 = 14.5 cos(2πt/5)Now, let's divide both sides by 14.5:cos(2πt/5) = 4.48/14.5Now we can solve for t:2πt/5 = cos^-1(4.48/14.5)t = (5/2π) cos^-1(4.48/14.5)Using a calculator, we can evaluate this expression to get t ≈ 3.59 seconds. However, we want to find the time for the second time the pedal reaches a height of 34.48 inches. Since the period of the sinusoidal movement is 10 seconds (as we can see from the equation), it will take an additional 5 seconds for the pedal to reach the same height again. Therefore, the total time for the pedal to reach a height of 34.48 inches for the second time is:t = 3.59 + 5 = 8.59 seconds Rounding to the nearest second, we get an answer of C. 6 seconds. Therefore, the correct answer is C. 6 seconds.
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The y-intercept of the graph of y=-7x+140y is located at (0, c) What is the value of c ?
Answers
The y-intercept of the graph is located at (0, 0). So, the value of c is 0.
What is graph?
A graph is a visual representation of data that displays the relationship between two or more variables. It consists of a set of points, called vertices or nodes, which are connected by lines or curves, called edges or arcs. The vertices represent the variables, while the edges represent the relationship between them.
To find the y-intercept of the graph of y = -7x + 140y, we need to substitute x = 0 into the equation and solve for y. This will give us the y-coordinate of the point where the graph intersects the y-axis.
Substituting x = 0, we get:
y = -7(0) + 140y
y = 140y
Simplifying the equation, we get:
139y = 0
Dividing both sides by 139, we get:
y = 0
Therefore, the y-intercept of the graph is located at (0, 0). So, the value of c is 0.
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The box-and-whisker plot below
represents some data set. What is the
range of the data?
50
Answer:
60
70
80
90
Answers
Answer: 32
Step-by-step explanation:
the min is 54 while the max is 86
86-54 is the range which is 32
Basics on sets (O1) a) Determine the cardinalities of the following sets. (8 Points) i) { Chinese New Year of 2023 } ii) The power set P(S) of S={1,∅ apple } . iii) {x∈R∣x 2 =3} iv) {{0,1,2},3,4,5,∅,{{0}}} b) The Cartesian product A 1 ×A 2 ×⋯×A k of k(>1) sets is the set containing all elements of the form (x 1 ,x 2 ,…,x k ) with x i ∈A i for i=1,2,…,k . Write down the elements of the Cartesian product A×B×C , where A is the set containing non-negative integers less than 3,B={ apple } , and C=B−A . What is its cardinality?
Answers
Therefore , the solution of the given problem of cardinality comes out to be A×B×C is 3 .
What is an effective example of cardinality?
A's cardinality is equivalent to its element count if A has a finite amount of elements. Say |A|=5 if A=2, 4, 6, 8, 10. The outcome of a numbering process is referred to as "cardinality". A set's cardinality is determined by the number of components in it. For illustration, the group 1, 2, 3, 4, and 5 has a greater cardinality of five than the set 1, 2, 3, and 5, each of which has an attributes of three.
Here,
a)
I "Chinese New Year of 2023" has one cardinal element, making it one-dimensional.
ii) Since there are three elements in S and each element can either be included in a subset or not, the power set P(S) of S=1,,apple has 23 = 8 elements.
iii) xRx2=3 has two components because the squares of two real numbers, 3 and -3, are both 3.
iv) Since "0" is regarded as one element and there are five other distinct elements in the collection, {{0,1,2},3,4,5,∅,{{0}}} has six elements.
b)
=> A={0,1,2}, B={apple}, and C={apple}
The set difference between C and A is only the element 0, which is absent from C, so A=apple0,1,2=apple.
Consequently, A, B, and C include the following components:
=> {(0,apple,apple), (1,apple,apple), (2,apple,apple)}
=> A×B×C is 3 .
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What is the distance between (-2,7) and (4,6)? Provide an answer accurate to the nearest tenth.
Answers
97 bielorrusias hacen falta para comprarte un perro
a bus driver covers a certain distance in 5 hours at an average speed of 80 km per hour how long will it take to travel at 432km at an average speed of 96km/h
Answers
Answer: It will take 4.5 hours to travel 432 kilometers at an average speed of 96 kilometers per hour
Step-by-step explanation:
If the scale from DEF to D'E'F is 5:1 how much larger is the perimeter of D'E'F to DEF
Answers
the perimeter of D'E'F is 5 times larger than the perimeter of DEF.
If the scale from DEF to D'E'F is 5:1, then the corresponding side lengths are multiplied by a factor of 5 when going from DEF to D'E'F. This means that if the perimeter of DEF is P, then the perimeter of D'E'F is 5P.
To see why this is true, consider the perimeter of DEF:
P = DE + EF + FD
When we scale up by a factor of 5, we get:
D'E' + E'F' + F'D' = 5DE + 5EF + 5FD
= 5(DE + EF + FD)
= 5P
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The map shows the distance in air miles from city A to both city B and city C.
City A to City B= 11
City A to city C= 29
City B to City C= X
What is X?
F. 39 miles
G. 40
H. 41
J. 42
Answers
The value of distance between city B to city C given by 'x' is found as: 41 miles.
How to solve linear equation?A line's equation can be expressed in a way that makes the slope evident and enables you to trace the path directly without performing any calculations. Students can compose linear equations using slope-intercept form if they feel confident solving a straightforward two-step linear equation. A linear equation has the slope-intercept form y = mx + b.Distance in air miles from city A to each city B and city C are-
City A to City B= 11
City A to city C= 29
City B to City C= X
Then,
Distance between city B to city C = distance between city A to city B + between city A to city C
x = 29 + 11
Solving the linear equation:
x = 41
Thus, the value of distance between city B to city C given by 'x' is found as: 41 miles.
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Quadrilateral JKLM is a parallelogram. What is the m∠KJN?
Answers
we can deduct that the angle on the L vertex is 25°, because that way all three angles will result in 180°.
We know that JKLM is a parallelogram, which means that its opposite sides are parallels, such that: JK // ML and JM // KL.
So, we know that the angle MLJ equals 25°, and JK // ML, from this we can say that angles LJK and MLJ are equal, because they are ''internal alternate angles'', since that they are inside the parallels.
So, angle LJK is 25, and JKL is a triangle, which has the angle at K vertex equal to 130°. We know that internal angles in a triangle must sum 180°.
Since that, two angles, in the triangle JKL, measure 130° + 25° = 155°; we can deduct that the angle on the L vertex is 25°, because that way all three angles will result in 180°.
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a local hamburger shop sold a combined total of 523 hamburgers and cheeseburgers on saturday. there were 73 more cheeseburgers sold then hamburgers. how many hamburgers were sold on saturday?
Answers
225 hamburgers were sοld οn Saturday
What is Linear Equatiοn?A linear equatiοn is an equatiοn οf a straight line in twο variables, usually written in the fοrm y = mx + b, where m is the slοpe and b is the y-intercept. It represents a relatiοnship between twο variables that is linear, meaning that as οne variable changes, the οther changes at a cοnstant rate.
Let's use algebra tο sοlve the prοblem.
Let x be the number οf hamburgers sοld οn Saturday.
Then, accοrding tο the prοblem, the number οf cheeseburgers sοld οn Saturday is 73 mοre than the number οf hamburgers sοld, sο the number οf cheeseburgers sοld is x + 73.
The prοblem alsο tells us that the tοtal number οf hamburgers and cheeseburgers sοld is 523. Sο we can set up an equatiοn:
x + (x + 73) = 523
Simplifying the left side οf the equatiοn:
2x + 73 = 523
Subtracting 73 frοm bοth sides:
2x = 450
Dividing bοth sides by 2:
x = 225
Therefοre, 225 hamburgers were sοld οn Saturday.
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Sebastian is looking to buy a car and the he qualified for a 8-year loan from a bank offering a monthly interest rate of 0.25% Using the formula below, determine the maximum amount Sebastian can borrow, to the nearest dollar, if the highest monthly payment he can afford is $ 175 $175. � = � � 1 − ( 1 + � ) − � M= 1−(1+r) −n Pr
Answers
Answer:
блина я сама не
Step-by-step explanation:
3. x³ 2x² 13x - 10 = 0
-
Possible Roots:
Real Rational Roots:
Answers
Simplify: 4x3+5y3+2x3+7y
Answers
Answer:
18x + 22y
Step-by-step explanation:
4x (3) + 5y (3) + 2x (3) + 7y
Multiply the coefficients of the variables x and y with the constants, so that:
4x (3) + 5y (3) + 2x (3) + 7y
= 12x + 15y + 6x + 7y
Then just add up the coefficients with the same variable:
12x + 6x = 18x
15y + 7y = 22y
So the final answer is:
18x + 22y
[tex]\huge\text{Hey there!}}[/tex]
[tex]\mathsf{4x^3 + 5y^3 + 2x^3 + 7y}[/tex]
[tex]\large\textsf{COMBINE the LIKE TERMS (if you have any):}[/tex]
[tex]\mathsf{= (4x^3 + 2x^3) + (5y^3) + (7y)}[/tex]
[tex]\mathsf{= 4x^3 + 2x^3 + 5y^3 + 7y}[/tex]
[tex]\mathsf{= 6x^3 + 5y^3 + 7y}[/tex]
[tex]\large\textsf{Therefore your ANSWER should be:}[/tex]
[tex]\huge\boxed{\mathsf{6x^3 + 5y^3 + 7y}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
7. Complete a dilation with scale factor of ½ around the origin and then reflect over the y-axis. What are the new ordered pair of A'?
Answers
The answer will be as follows after answering the supplied question As a coordinates result, Anew "'s ordered pair is (-1/2 * x, 1/2 * y).
what are coordinates?In geometry, a coordinate system is a method that uses one or more numbers or coordinates to determine the precise arrangement of points or other geometrical objects on a manifold, such as Euclidean space. Coordinates are pairs of numbers that are used to locate a point or object on a two-dimensional plane. A point's location on a 2D plane is represented by two numbers known as the x and y coordinates. a set of digits that represent precise locations. Typically, the figure has two numbers. The first number represents the front-to-back distance, while the second number represents the top-to-bottom distance. As in (12.5), when there are 12 units below and 5 above.
To conduct a dilation with a scale factor of 1/2 around the origin, multiply each point's coordinates by 1/2. If the initial coordinates of point A are (x,y), then the dilated point A' coordinates are:
A' = (1/2 * x, 1/2 * y)
We negate the x-coordinate while keeping the y-coordinate constant to represent the dilated point A' over the y-axis. As a result, the final coordinates of A" are:
A'' = (-1/2 * x, 1/2 * y)
As a result, Anew "'s ordered pair is (-1/2 * x, 1/2 * y).
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How to calculate a rate or unit rate?
Answers
To calculate a rate, divide two quantities with different units; to calculate a unit rate, divide a rate by the quantity being measured.
What is rate ?
In mathematics, a rate is a ratio that compares two quantities with different units. It is typically expressed as the amount of change of one quantity with respect to another quantity. Rates are often used in the context of describing how quickly or slowly something changes over time or space.
To calculate a rate, we need to divide two quantities that have different units. For example, if we want to find the rate of a car's speed, we divide the distance traveled (in miles) by the time taken to travel that distance (in hours).
The formula for rate is:
rate = quantity / time
To calculate a unit rate, we need to divide a rate by the quantity being measured. For example, if the rate is the number of miles traveled per hour, the unit rate would be the number of miles traveled in one hour. To find the unit rate, we divide the rate by 1 hour.
The formula for unit rate is:
unit rate = rate / 1
When calculating rates or unit rates, it is important to make sure that the units of the quantities being divided are consistent. If the units are not the same, we need to convert them to the same unit before performing the division.
Therefore, to calculate a rate, divide two quantities with different units; to calculate a unit rate, divide a rate by the quantity being measured.
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Carlos operates a pizzeria in Chicago. If the diameter of a large pie at his restaurant is 24 inches, what would be the area of one slice (if every pizza is cut into 8 equal slices)?
ROUND ANSWER TO THE NEAREST TENTH OF AN INCH
Answers
The area of one slice of a large pizza with a diameter of 24 inches is 56.52 square inches.
The formula for calculating the area of a circle is A = πr^2, where A is the area of the circle, π is the constant pi (3.1415) and r is the radius of the circle. To calculate the area of a slice, we must first calculate the radius of the pizza. To do this, we divide the diameter of the pizza (24 inches) by 2, which yields a radius of 12 inches.
Now that we have the radius, we can calculate the area of one slice. Plugging the radius of 12 inches into the equation, we get A = 3.1415*12^2 = 452.16 square inches. Since each pizza is cut into 8 equal slices, each slice would have an area of 452.16/8 = 56.52 square inches.
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A cellular service provider is expanding the number of cell towers it has in Fulton County. At one place, there are 2 kilometers between two towers. What is the distance between the two cell towers on a map with a scale of 2 centimeters : 1 kilometer?
Answers
Answer:
the actual distance is 2 kilometer.
Step-by-step explanation:
Simplify fully 9x + y - 6x + y
Answers
Answer:
Step-by-step explanation:
So first you would combine like terms which in this case are 9x, -6x, y, and y
So: 9x+y-6x+y
(9x-6x)+(y+y)
3x+2y is the final answer
hope this helped
Answer:
Step-by-step explanation:
Comment factoriser (x-4)^{2}-36
Answers
Answer:
Step-by-step explanation:
to factorize [tex](x-4)^{2} -36[/tex]
(x-4-6) * (x - 4 / 6)
(x-10) * (x+2)
Consider the probability that more than 67 out of 124 computers will crash in a day. Assume the probability that a given computer will crash in a day is 4%. Specify whether the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions
Answers
The normal curve can be used to approximate the binomial probability as all conditions are met, with at least 10 successes and failures, a success rate of less than 10%, and a sample size of at least 20.
Only when specific requirements are met can the normal curve be used to approximate the binomial probability in this situation. The trial must have at least 10 successes and 10 failures as its initial requirement. There are more than 10 successes and 10 failures in this instance since 124 machines are being tested. The second requirement is that the likelihood of success must be under 10%. The likelihood of a computer failing in this scenario is 4%, which is less than 10%. The sample size needs to be at least 20, which is the third requirement. The sample size in this instance is 124, which is more than 20.
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What ratio is equivalent to 8 to 2? Complete the statement,
The ratio 8 to 2 is equivalent to the ratio ?
Answers
The Answer should be 4:1 it's the half of the 8 to 2
In a seasonal-growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food. (a) Find the solution of the seasonal-growth model where k, r, and ϕ are positive constants. DP dt = kP cos(rt − ϕ) P(0) = P0 P(t) = (b) Find lim t→[infinity] P(t). (If there is no solution, enter NONE in the answer box. ) lim t→[infinity] P(t) =
Answers
(a) The solution of the seasonal-growth model is
P(t) = P0[tex]e^{(k/r sin(rt))}[/tex]
(b) After solving we get [tex]\lim_{t \to \infty} P(t)[/tex] = NONE . As, t approaches to infinity
(a) To solve the differential equation DP/dt = kPcos(rt-[tex]\theta[/tex]),
The variables can be split apart and integrat:
∫(1/P) dP = ∫k cos(rt-[tex]\theta[/tex] ) dt
ln|P| = (k/r)sin(rt- [tex]\theta[/tex] ) + C
where
C is the constant of integration.
By solving for P, we get:
P(t) = [tex]e^{(k/r sin(rt- \theta ) + C)}[/tex]
We can determine the value of C by using the initial condition P(0) = P0:
P(0) = [tex]e^{(k/r sin(- \theta) + C)}[/tex]= P0
C = ln(P0) - (k/r)sin(-[tex]\theta[/tex] )
C = ln(P0) + (k/r)sin([tex]\theta[/tex] )
By substituting this into the expression for P(t), we get:
P(t) = [tex]e^{(k/r sin(rt-\theta) + ln(P0) + (k/r)sin(\theta))}[/tex]
By simplifying, we get:
P(t) = P0[tex]e^{(k/r sin(rt-\theta) + k/r sin(\theta))}[/tex]
P(t) = P0[tex]e^{(k/r sin(rt))}[/tex]
(b) We must take into account the behavior of sin(rt) as t approaches infinity in order to determine the limit of P(t) as t approaches infinity. Sin(rt) will fluctuate between -1 and 1 as t grows because the sine function oscillates between -1 and 1.
As a result, the term [tex]e^{(k/r sin(rt))}[/tex] will bounce between [tex]e^{(-k/r)}[/tex] and [tex]e^{(k/r)}[/tex] as t approaches infinity, but it won't converge to a single value.
As a result, there is no P(t) limit as t approaches infinity.
Hence, the answer is:
[tex]\lim_{t \to \infty} P(t)[/tex] = NONE.
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