30% percent of the students passed both Mathematics and English.
What is Percentage?A rate, number, or amount in each hundred is known as a percentage
Let's use a Venn diagram to represent the information given in the problem. Let M be the set of students who passed Mathematics, E be the set of students who passed English, and F be the set of students who failed both.
We know that there are 40 students in the class, and 90% passed Mathematics, so the number of students who passed Mathematics is 0.9 × 40 = 36. Similarly, 75% passed English, so the number of students who passed English is 0.75 × 40 = 30.
We also know that 2 students failed both Mathematics and English, so we can label the F section with 2.
where the number in each section represents the number of students who passed the respective exam.
To find the percentage of students who passed both examinations (i.e., the intersection of M and E), we need to add the number of students in the M and E intersection to the F section, then subtract that from the total number of students (40), and finally divide by 40 to get the percentage. That is:
percentage of students who passed both exams = (M ∩ E + F) / 40 × 100%
= (28 + 2) / 40 × 100%
= 30%
Therefore, 30% of the students passed both Mathematics and English.
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Drag the correct equation for each graph into the blank spaces. Remember you should convert to slope intercept
form first!
In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.
What is graph?Graph is a data structure that consists of nodes (vertices) and edges (connections between nodes). Graphs are used to represent networks, such as social networks, transportation networks, and communication networks. Graphs provide a way to visualize relationships between data points and can be used to discover patterns in data and make predictions. Graphs are a powerful tool in data science and can be used to analyze complex systems.
This is happening because the equations represent the linear relationships between the variables of the graphs. The first equation is in slope-intercept form, and can be used to calculate the y-value of a given x-value. The equation for the second graph is also in slope-intercept form, and can be used to calculate the y-value of a given x-value.
In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.
These equations represent the linear relationships between the variables of the graphs. The slope of the line is the rate of change between the x- and y-values, while the y-intercept is the starting point of the line. By using the equations, we can calculate the y-value of any given x-value on the graph, and thus accurately represent the linear relationships between the variables.
The diagram is given below.
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Complete questions as follows-
what is the area of the parallelogram
The area of the parallelogram is 324 square yards.
What is a parallelogram ?
A parallelogram is a four-sided geometric shape that has two pairs of parallel sides. It isdefined by its four vertices, four sides, and two diagonals that intersect at their midpoint. The opposite sides of a parallelogram are congruent, and the opposite angles are also congruent. The adjacent angles are supplementary and add up to 180 degrees.
To find the area of a parallelogram, we need to multiply the length of the base by the height, which is the perpendicular distance from the base to the opposite side. This formula is similar to finding the area of a rectangle, where the base is one side of the rectangle and the height is the distance from that side to the opposite side.
Calculating the area of the given parallelogram :
The area of a parallelogram is A = bh, where b is the base and h is the height.
Given the height of the parallelogram is 12 yards and the base is 27 yards. Using the formula for the area of a parallelogram, we can calculate the area as follows:
A = bh
A = 27 yards × 12 yards
A = 324 square yards
Therefore, the area of the parallelogram is 324 square yards.
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The monthly rent for a pizza parlor is $1,200. the average production cost per pizza is $6.75. the monthly expenses for the pizza parlor are given by the function E(x) = 1,200 + 6.75x, where x is the number of pizzas sold. for x pizzas sold, the pizza parlor's revenue is given by the function R(x) = 12.5x. the monthly profit of the pizza parlor is the difference between its revenue and its expenses. which function represents the monthly profit, P(x)?
The monthly profit, P(x), of a pizza parlor is given by the function P(x) = 5.75x - 1,200, where x is the number of pizzas sold, given revenue and expense functions.
The monthly profit, P(x), is the difference between the revenue and the expenses, so:
P(x) = R(x) - E(x)
From the given information:
R(x) = 12.5x (revenue per pizza is $12.5, and x is the number of pizzas sold)
E(x) = 1,200 + 6.75x (expenses, which include the monthly rent of $1,200 and the production cost of $6.75 per pizza, multiplied by the number of pizzas sold)
Substituting the values of R(x) and E(x) into the equation for P(x):
P(x) = 12.5x - (1,200 + 6.75x)
Simplifying the equation:
P(x) = 5.75x - 1,200
Therefore, the function representing the monthly profit, P(x), is P(x) = 5.75x - 1,200.
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Answer:
P(x)= 5.75x -1,200
Step-by-step explanation:
Plato/Edmentum
Write an equation for an ellipse centered at the origin, which has foci at (\pm\sqrt{12},0)(± 12 ,0)left parenthesis, plus minus, square root of, 12, end square root, comma, 0, right parenthesis and vertices at (\pm\sqrt{37},0)(± 37 ,0)left parenthesis, plus minus, square root of, 37, end square root, comma, 0, right parenthesis
The equation for the ellipse is: [tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]= 1$$[/tex]
The standard equation for an ellipse centered at the origin is:
[tex]$\frac{x^2}{a^2}[/tex] + [tex]$\frac{y^2}{b^2}[/tex] [tex]= 1$$[/tex]
where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.
In this case, the vertices are located at [tex]$(\pm\sqrt{37}, 0)$[/tex], which means [tex]$a=\sqrt{37}$[/tex]. The distance between the foci is [tex]$2c=2\sqrt{12}=2\sqrt{3\times 4}=2\sqrt{3}\times 2=4\sqrt{3}$[/tex], which means [tex]$c=2\sqrt{3}$[/tex].
The value of b can be found using the relationship between a, b, and c in an ellipse:
[tex]$$a^2 = b^2 + c^2$$[/tex]
Substituting the values we know, we get:
[tex]$$37 = b^2 + (2\sqrt{3})^2$$[/tex]
Simplifying:
[tex]$$37 = b^2 + 12$$[/tex]
[tex]$$b^2 = 37 - 12 = 25$$[/tex]
Taking the square root of both sides, we get:
[tex]$$b = \pm 5$$[/tex]
Since the co-vertices are located at [tex]$(0,\pm b)$[/tex], we can see that [tex]$b=5$[/tex] (and not -5, since the ellipse is centered at the origin).
Therefore, the equation for the ellipse is:
[tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]$ = 1$$[/tex]
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seven less than the product of a number n and 1/4 is no more than 95
[tex]\frac{1}{4} n - 7 \leq 95[/tex]
Grace wants to buy a jump rope that costs $7, a board game that costs $10, and a playground ball that costs $4. She has saved $10 from her allowance, and her uncle gave her $3. How much more money does Grace need to buy the jump rope, the game, and the ball?
Grace need to buy the jump rope, the game, and the ball $8.
$7 will get you a jump rope.
$10 will get you a board game.
$4 will get you a playground ball.
total amount to be spent: $7 + $10 + $4 = $21
She has ten dollars.
$3 was all her uncle gave her.
13 dollars are all she has.
She needed $8, therefore 21 - 13 = $8.
a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.
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*9. The consultancy Imagination Inc. Is working with its manufacturing client Parts-R-Us to improve their on-time
performance. The firm can earn a bonus of up to $1,000,000 based on how much the on-time performance actually
improves. It's current (baseline) on-time performance is 90%.
The company typically completes approximately 1,000 orders per month, with approximately 100 orders delayed. The
bonus payment is prorated according to the following criteria:
• The on-time performance improvement is calculated based on a reduction in late events or an improvement in on-
time performance.
• No bonus is earned for the first 25% reduction in late events, say from 100 to 75. Maximum bonus is earned once
Parts-R-Us achieves 95% on-time performance.
Please answer the following:
Write down a formula to
determine the total bonus
amount to be received
Using your formula, show
how much bonus would be
paid if Parts-R-Us achieves
94% on-time performance
Parts-R-Us could be eligible for a $0 bonus if it achieves an on-time performance rate of 94% or more.
The company would have to maintain an order delivery rate of 95% in order to qualify for the bonus. If achieved, the bonus would be as follows: Bonus = 94% * (1 - 100).
The following is the calculation for the bonus:
Bonus = P * (1 - D)
Where:
P represents the proportion of orders that are delivered on time.
D represents the total number of orders placed late.
In the event that Parts-R-Us achieves an on-time performance rate of 94%, the bonus would be as follows:
Bonus = 94% * (1 - 100)
Bonus = $0
If the company only managed to complete its tasks 94% of the time, it will not be eligible for a bonus.
In order to qualify for the maximum bonus, Parts-R-Us would have to maintain an on-time delivery rate of 95%.
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17 Troy scored 945 points playing 3 games of pinball. He scored
312 points in the first game and 356 points in the second game.
How many points did Troy score in the third game?
Answer:
Troy scored 277 points in the third game of pinball.
Step-by-step explanation:
Let x be the number of points Troy scored in the third game.
We know that Troy scored a total of 945 points in 3 games, so we can set up an equation:
312 + 356 + x = 945
Simplifying this equation, we get:
668 + x = 945
Subtracting 668 from both sides, we get:
x = 277
Therefore, Troy scored 277 points in the third game of pinball.
indicate the method you would use to prove the two triangles ~=. if no method applies, enter none
The method we would be using here to prove the triangle's congruency is LA, the leg-acute theorem.
Define congruency?The dimensions of the sides and angles of two or more triangles determine whether they are congruent or not. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are equal, two triangles are said to be congruent. They are the exact same size and form.
Here in the given figures, we can use the LA theorem also known as leg acute. It says that a right triangle is congruent if its leg and any acute angle are congruent with the corresponding leg and any acute angle of another right triangle.
The hypotenuse of the first triangle is corresponding to the hypotenuse of the second triangle.
The angle opposite to the right angle in both the triangles are corresponding to each other.
So as per the LA theorem we can prove that the triangle is congruent.
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PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
The pink triangle
Step-by-step explanation:
IMPORTANT NOTE: Make sure all the units are the same and consistent
Perimeter of a figure = Total length of the outer boundary
Shape of each figure in this question = Isosceles Triangle
Perimeter of triangle = Sum of all three sides
Perimeter of pink triangle = 36m + 36m + 20m
= 92m
Perimeter of green triangle = 25m + 25m + 35m
= 85m
∴Comparing the two values calculated above, it can be observed that the pink triangle has a greater perimeter
Kids's Kingdom, a retail toy chain, placed a seasonal order for stuffed animals from Stuffed Stuff, a distributor. Large animals cost $20, and small ones cost $14.
If the total cost of the order was $7,320 for 450 pieces, how many of each size were ordered? What was the dollar amount of each size ordered?
Answer:
Kids's Kingdom ordered 170 large stuffed animals and 280 small stuffed animals. The dollar amount of each size ordered was $3,400 for the large stuffed animals and $3,920 for the small stuffed animals.
Step-by-step explanation:
Let's use the following variables:
L for the number of large stuffed animals
S for the number of small stuffed animals
We can set up a system of two equations to represent the given information:
L + S = 450 (equation 1)
20L + 14S = 7320 (equation 2)
We can solve this system of equations using substitution or elimination. Let's use substitution.
From equation 1, we can solve for L:
L = 450 - S
Substitute this expression for L into equation 2:
20(450 - S) + 14S = 7320
Distribute the 20:
9000 - 20S + 14S = 7320
Simplify and solve for S:
6S = 1680
S = 280
So, Kids's Kingdom ordered 280 small stuffed animals. We can use equation 1 to find the number of large stuffed animals:
L + 280 = 450
L = 170
Therefore, Kids's Kingdom ordered 170 large stuffed animals.
To find the dollar amount of each size ordered, we can multiply the number of each size by the cost per item:
170 large stuffed animals at $20 each: 170 * $20 = $3,400
280 small stuffed animals at $14 each: 280 * $14 = $3,920
So, Kids's Kingdom spent $3,400 on large stuffed animals and $3,920 on small stuffed animals for a total cost of $7,320.
Can anyone solve this ???
The result (recurrent value), A = sum j=1 to 89 ln(j), is true for every n. This is the desired result.
How do you depict a relationship of recurrence?As in T(n) = T(n/2) + n, T(0) = T(1) = 1, a recurrence or recurrence relation specifies an infinite sequence by explaining how to calculate the nth element of the sequence given the values of smaller members.
We can start by proving the base case in order to demonstrate the first portion through recurrence. Let n = 1. Next, we have:
Being true, ln(a1) = ln(a1). If n = k, let's suppose the formula is accurate:
Sum j=1 to k ln = ln(prod j=1 to k aj) (aj)
Prod j=1 to k aj * ak+1 = ln(prod j=1 to k+1 aj)
(Using the logarithmic scale) = ln(prod j=1 to k aj) + ln(ak+1)
Using the inductive hypothesis, the property ln(ab) = ln(a) + ln(b)) = sum j=1 to k ln(aj) + ln(ak+1) = sum j=1 to k+1 ln (aj)
(b), we can use the just-proven formula:
A = ln(1, 2,...) + ln + ln (89)
= ln(j=1 to 89) prod
sum j=1 to 89 ln = ln(prod j=1 to 89 j) (j).
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find two positive real numbers such that the sum of the first number and the second number is 48 and their product is a maximum
Answer:
x = 24 and y = 24
Step-by-step explanation:
Let's use algebra to solve this optimization problem.
Let x be the first number, and y be the second number. Then we have the following two equations based on the problem statement:
x + y = 48 (sum of the two numbers is 48)
xy = ? (product of the two numbers, which we want to maximize)
To solve for x and y in terms of each other, we can use the fact that:
(x + y)^2 = x^2 + 2xy + y^2
Expanding the left side of the equation gives:
x^2 + 2xy + y^2 = 2304
And substituting xy for its value in terms of x and y gives:
x^2 + 2xy + y^2 = x^2 + 2(48 - x)y + y^2 = 2304
Simplifying this equation gives:
2y^2 - 96y + x^2 - 2304 = 0
To maximize the product xy, we need to maximize the value of xy = x(48 - x) = 48x - x^2. This function is a quadratic that opens downwards, and therefore, its maximum value occurs at the vertex of the parabola, which is located at x = -b/2a = -48/(2*-1) = 24.
Thus, the two positive real numbers that sum up to 48 and their product is a maximum are x = 24 and y = 24.
If John had 3 apples then Droped 2 then found 4 then gave one to his friend how many apples does he have now
Answer:
4 apples
Step-by-step explanation:
We know
John had 3 apples, then Dropped 2, found 4, then gave 1 to his friend.
How many apples does he have now?
3 - 2 + 4 - 1 = 4 apples
So, he has 4 apples now.
Answer:
4 apples
Step-by-step explanation:
We know that John had 3 apples
but then he had dropped 2 of them
he then found 4
then he gave 1 to his friend
3-2=1
1+4=5
5-1=4
The answer is 4
Hope this helps!
PLEASE HELP ME
(Answer these four questions please)
3. The statement is true, the correlation coefficient is close to -1. 4. temperature for a city with a latitude of 48 is 43. 5. The statement is false. 6. cannot make a reasonable estimate
Describe Equation?Equations can be used to model real-world situations and solve problems in many fields, including science, engineering, finance, and more. They are an essential tool in mathematics and are used extensively in algebra, calculus, and other advanced branches of math.
Equations can involve various mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and others.
Question 3:
The statement is true. We can check this by calculating the correlation coefficient between the latitude and temperature data points, which should be close to -1. The calculated line of best fit is also consistent with the given data.
Question 4:
To estimate the temperature for a city with a latitude of 48, we can use the equation of the line of best fit:
y = -1.07x + 92.87
Substituting x = 48, we get:
y = -1.07(48) + 92.87
y = 42.79
Rounding to the nearest whole number, the estimated temperature for a city with a latitude of 48 is 43.
Question 5:
The statement is false. We can check this by calculating the correlation coefficient between the passengers and suitcases data points, which should be close to 1. The given line of best fit has a negative slope, which is inconsistent with the positive correlation between the variables.
Question 6:
To estimate the number of suitcases for a flight carrying 250 people, we can use the equation of the line of best fit:
y = -1.98x + 7.97
Substituting x = 250, we get:
y = -1.98(250) + 7.97
y = -485.03
However, it does not make sense for the number of suitcases to be negative. Therefore, we cannot make a reasonable estimate for the number of suitcases on a flight carrying 250 people using this line of best fit.
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A triangle has a base that measures 6.4 feet. Its height measures 5 feet. How many square feet is the area of the triangle?
Answer:
16 feet
Step-by-step explanation:
Area of triangle = 1/2(a x b)
A = 1/2(6.4 x 5)
A = 32/2
A = 16 feet
You pick a card at random. 1 2 3 What is P(not even)?
An 1 οr perhaps an even integer will be drawn back 75% οf the times frοm the set οf 1, 2, 3 and 4. Thus, Prοbability P(nοt even) is 75%.
Hοw simple is prοbability?Prοbability is the likelihοοd that sοmething will οccur οr the prοbability that sοmething will happen. Prοbability is the measure οf hοw prοbable it is that a cοin will land heads up after being tοssed intο the air.
As prοbabilistic arguments sοmetimes prοduce οutcοmes that appear cοntradictοry οr illοgical, prοbability is usually regarded as amοng the mοst challenging tοpics οf mathematics.
P(1) = 1/4(there is one card with a 1)
P(even) = 2/4 = 1/2 (there are 2 cards with even numbers out of 4)
Therefore,
P( 1 or even) =P(1) + P(even)
= 1/4 + 1/2
= 3/4
To express the answer as a percentage, we can multiply by 100:
P(1 or even) = 3/4 × 100%
=> 75%
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You pick a card at random. card 1, card 2 and card 3
What is P(not even)?
Write your answer as a fraction or percentage.
Emma and Cooper went to Tico’s tacos for lunch. Emma ordered three tacos and one burrito and Cooper ordered one taco and two burritos Emmas order total was $3.65 and Cooper’s bill was $3.30. Write and solve a system of equations to model the situation above. Explain the solution in the context of this problem. Explain, or show your work in the box below.
In the given system of equations one taco costs $0.80 and one burrito costs $0.72.
What is a system of equations?An equation system is a finite collection of equations for which we searched for the common solutions. It is sometimes referred to as a set of simultaneous equations or an equation set. The classification of a system of equations is similar to that of a single equation. In modelling issues where the unknown values may be expressed in the form of variables, a system of equations finds use in everyday life.
Let us suppose the cost of one taco = x.
Let us suppose the cost of one burrito = y.
Then, for Emma we have:
3x + y = 3.65
For Cooper we have:
x + 2y = 3.30
Using elimination, multiply the first equation by 2 and subtract it from the second equation:
(2)(3x + y = 3.65)
6x + 2y = 7.30
x + 2y = 3.30
-5x = -4
x = 4/5
Substituting this value of x into either equation:
3(4/5) + y = 3.65
y = 2.15/3 ≈ 0.72
Therefore, one taco costs $0.80 and one burrito costs $0.72.
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What x-values are a solution to the System?
Select EACH correct answer.
Question 2 options:
x = -2
x = -1
x = 0
x = 1
x = 2
The solutions to the system equations are x = -1 and x = 0.
What is a system of equations?When two or more variables are related to one another and equations are constructed to determine each variable's value, the result is a system of equations. an equation is a balance scale, and for the equation to stay true, both sides must be equal.
Given equations are:
[tex]y=2^x - 1\\y=\frac{1}{2} x[/tex]
The solutions of the system for which; [tex]y_{1} =y_{2}[/tex]
According to the given table, the solutions to the system of both equations at x = -1 and x = 0.
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When solving the equation 3(x-2)=-12, what is a possible first step?
*
1)Distributing the 3 into each term of the parenthesis.
2)She could either divide by 3 or distribute 3 into the parenthesis
3)Adding 2 to each side of the equation
4) Subtracting 2 on each side of the equation
5) Dividing each side of the equation by 3
6)none of the answer choices tell what the first step could be.
Answer:
2)She could either divide by 3 or distribute 3 into the parenthesis
What is the perimeter of the trapezoid below?
Answer:
82
Step-by-step explanation:
divide this shape into a triangle and rectangle and use 24 as the base of the triangle using the Pythagoras theorem you get that the perpendicular is 7 and since 13 is also parallel to that side, 13+7=20 so one side equals 20 one side equals 25 one side equals 24 and one side equals 13 now add all of these to get 82
Step-by-step explanation:
See image below ..... perimeter = 25 + 13 + 24 + 13 + x
x is found using pythagorean theorem for right triangles
x = 7 units
total perimeter is then 82 units
a red cap fire hydrant provides 1700 liters per minute of water. how long (in minutes to the nearest minute) will it take to fill a water truck with a tank of the following dimensions: 68 inches diameter and 24 feet long? when the tank is full of water, how heavy will the water load be in pounds (lbm) to the nearest pound?
It will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.
How to find out how long it will take to fill the tank.First, we need to convert the dimensions of the tank from inches to feet:
68 inches diameter = 68/12 feet = 5.67 feet diameter
24 feet long = 24 feet long
Next, we can calculate the volume of the tank in cubic feet:
Volume = [tex]pi x (diameter/2)^2 x length[/tex]Volume = [tex]3.14 x (5.67/2)^2 x 24[/tex]Volume = [tex]485.15 cubic feet[/tex]Since 1 cubic foot of water weighs 62.4 pounds, we can calculate the weight of the water in the tank in pounds:
Weight = Volume x DensityWeight = 485.15 x 62.4Weight = 30,296.16 poundsTo find out how long it will take to fill the tank, we can use the flow rate of the fire hydrant:
Flow rate = 1700 liters per minute1 liter = 0.264172 gallonsFlow rate = 1700 x 0.264172 = 449.10 gallons per minute1 gallon = 0.133681 cubic feetFlow rate = 449.10 x 0.133681 = 60.05 cubic feet per minuteFinally, we can divide the volume of the tank by the flow rate to find out how long it will take to fill the tank:
Time = Volume / Flow rateTime = 485.15 / 60.05Time = 8.08 minutesTherefore, it will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.
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a pastry chef accidentally inoculated a cream pie with six s. aureus cells. if s. aureus has a generation time of 60 minutes, how many cells would be in the cream pie after 7 hours?
After the time of seven hours, the cream pie would have approximately 768 S. aureus cells after 7 hours with a generation time of 60 minutes.
How many cells would be in the cream pie after 7 hours?Six S. aureus cells have been accidentally inoculated into a cream pie. S. aureus has a generation time of 60 minutes. S. aureus is a pathogenic bacterium found in the environment, as well as on the skin, and in the upper respiratory tract.
The generation time of this bacterium is 60 minutes, meaning that a single bacterium can produce two new cells in 60 minutes.
If there are 6 S. aureus cells in a cream pie, the number of bacteria will continue to increase as time passes.
The number of generations (n) in seven hours is calculated as:
n = t/g
n = 7 hours × 60 minutes/hour/60 minutes/generation = 7 generations
The number of cells in the cream pie after 7 hours is calculated as :
N = N₀ × 2ⁿ
N = 6 cells × 2⁷
N = 768 cells
Therefore, after seven hours, the cream pie would have approximately 768 S. aureus cells.
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A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters
Using Hooke's law the maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
To find the spring constant k, we use Hooke's law:
F = -ky
where F is the weight of the object, and y is the distance it is stretched from its rest position. At equilibrium, F = mg = 10 × 9.81 = 98.1 N. Thus,
98.1 = -k × 0.098
k = -1000 N/m
The equation of motion for the system is given by:
my'' + ky = F(t)
Substituting the given values, we get:
10y'' + (-1000)y = 100cos(8t)
y'' - 100y = 10cos(8t)
with initial conditions y(0) = 0 and y'(0) = 0.
The characteristic equation is r² - 100 = 0, with roots r = ±10i. The complementary solution is therefore y_c(t) = c1cos(10t) + c2sin(10t).
For the particular solution, we assume a form of yp(t) = Acos(8t) + Bsin(8t), and substitute it in the differential equation to get:
-64Acos(8t) - 64Bsin(8t) - 100(Acos(8t) + Bsin(8t)) = 10cos(8t)
Solving for A and B, we get A = -1/6 and B = 0. Thus, the particular solution is yp(t) = (-1/6) × cos(8t).
The general solution is therefore y(t) = c1cos(10t) + c2sin(10t) - (1/6)*cos(8t). Applying the initial conditions, we get c1 = 0 and c2 = 0, so the particular solution is simply y(t) = (-1/6) × cos(8t).
The maximum excursion from equilibrium can be found by taking the absolute value of y(t) and finding its maximum value. We have:
|y(t)| = (1/6) × |cos(8t)|
The maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
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In an introductory psychology class with n = 50 students, there are 9 freshman males, 15 freshman females, 8 sophomore males, 12 sophomore females, and 6 junior females. A random sample of n = 2 students is selected from the class. If the first student in the sample is a male, what is the probability that the second student will also be a male?
a. 8/14
b. 12/20
c. 20/44
d. 20/50
Answer: Total number of males= C
Step-by-step explanation:
2x^3+15x^2+24x+k has 3 linear factors, two of which are identical. find k, and hence factorise the cubic
The value of k in the cubic equation is 2 and factorization of the equation is 2(x-1)^2(x+1)
What is the value of k?If 2x^3 + 15x^2 + 24x + k has three linear factors, then it must be in the form:
[tex]2x^3 + 15x^2 + 24x + k = a(x-b)(x-c)^2[/tex]
where a, b, and c are constants, and (x-c)^2 means that (x-c) is a repeated factor.
Expanding the right side, we get:
[tex]2x^3 + 15x^2 + 24x + k = a(x^3 - 2cx^2 + cx^2 - 2cx + c^2x - c^2)[/tex]
Simplifying, we get:
[tex]2x^3 + 15x^2 + 24x + k = ax^3 - 2acx^2 + ac^2x - 2acx + ac^2 + a(-c^2)\\2x^3 + 15x^2 + 24x + k = ax^3 + (-2ac + ac^2)x^2 + (-2ac + ac^2)x - ac^2[/tex]
Equating the coefficients of each power of x on both sides, we get:
[tex]a = 2\\-2ac + ac^2 = 15\\-2ac + ac^2 = 24\\-ac^2 = k\\\\[/tex]
From the second and third equations, we can see that -2ac + ac^2 = 15 and -2ac + ac^2 = 24. These equations are equivalent, so we can set them equal to each other:
[tex]-2ac + ac^2 = 15 = 24[/tex]
Simplifying, we get:
ac^2 - 2ac + 9 = 0
This is a quadratic equation in ac. Solving for ac using the quadratic formula, we get:
[tex]ac = [2 \± \sqrt(4 - 4(1)(9))] / 2[/tex]
[tex]ac = 1 \± \sqrt(-5)[/tex]
Since ac is the product of two real numbers (b and c), we know that the square root of -5 must cancel out. This is only possible if ac = 1 and c = -1. Therefore, b = -c = 1, and we can find k by setting ac^2 = -k:
[tex]1 = 2a\\15 = -2ac + ac^2 = -2a + a^2\\24 = -2ac + ac^2 = -2a + a^2\\-k = -ac^2 = 2(-1)^2 = 2\\[/tex]
Therefore, k = -2, and the factorization of 2x^3 + 15x^2 + 24x - 2 is:
2x^3 + 15x^2 + 24x - 2 = 2(x-1)^2(x+1)
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Help!
I need your help.
Find the missing side of the triangle below
The value of y in the given triangle is 7.44 units.
What is tangent in trigonometry?The trigonometric ratio between the opposing and adjacent sides of a right triangle that contains an angle is its tangent. The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths. All geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more, utilise them extensively.
The given triangle is a right-triangle.
The trigonometric identity that gives the relationship between opposite side and adjacent side is tan.
Thus,
tan (28) = opposite / adjacent = y / 14
y = 0.53 (14)
y = 7.44
Hence, the value of y in the given triangle is 7.44 units.
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1. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15°
The length of the arc is approximately 5.51 meters when a circle with a radius of 21 meters is subtended by a central angle of 15 degrees.
The length of an arc of a circle with radius 21m that subtends a central angle of 15° can be calculated using the formula:
Arc length = (central angle/360°) x 2πr
where r is the radius of the circle, and π is the mathematical constant pi.
Substituting the given values, we get:
Arc length = (15/360) x 2π x 21
Arc length = (1/24) x 2 x 3.14 x 21
Arc length = (1/12) x 3.14 x 21
Arc length = 5.51 meters (rounded to two decimal places).
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We need to apply the formula to determine the length of a circle's arc: (Central angle / 360°) x (2 x x radius) is the formula for arc length. the radius is distance from the circle center to any point on its perimeter,
and the central angle is the angle subtended by the arc at its center. The radius in this instance is stated as 21 meters, while the arc's center angle is provided as 15 degrees. When these values are added to the formula, we obtain: arc length is equal to (15°/360°) x (2x x 21m) 3.68 m. As a result, the arc measures around 3.68 meters in length. As a result, the radius is the distance from the circle's center to any point on its perimeter, if we were to sketch an arc of The arc's length would be around 3.68 meters for a circle with a radius of 21 meters and a center angle of 15 degrees.
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in a heptagon, the degree measures of the interior angles are $x, ~x, ~x-2, ~x-2, ~x 2, ~x 2$ and $x 4$ degrees. what is the degree measure of the largest interior angle?
The degree measure of the largest interior angle of the given heptagon is 132.57 degrees.
The largest interior angle of the given heptagon. The degree measures of the interior angles of a heptagon are 7, with 7 sides or vertices, are $x, ~x, ~x-2, ~x-2, ~x+2, ~x+2,$ and $x+4$ degrees.
The sum of the degree measures of the interior angles of a polygon with n sides is given by $S = (n-2) \c dot 180^\circ$. The sum of the interior angles of a heptagon is given by $S = (7-2) \cdot 180^\circ = 900^\circ$.
The sum of the degree measures of the interior angles of the heptagon is equal to $x+x+x-2+x-2+x+2+x+2+x+4 = 7x+4$. To find the value of x, we will set this equation equal to the total sum of the interior angles:$7x+4 = 900^\circ$. Solving for x, we get$x = 128.57$
We may now substitute the value of x to get the degree measures of each of the angles in the heptagon:$x = 128.57^\circ$$x = 128.57^\circ$$x - 2 = 126.57^\circ$$x - 2 = 126.57^\circ$$x + 2 = 130.57^\circ$$x + 2 = 130.57^\circ$$x + 4 = 132.57^\circ$
To find the degree measure of the largest interior angle, we must look for the angle with the largest value. We can see that the largest angle measures $132.57^\circ$.
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