Answer:
1) y² - 5y = 750
2) 750 -y(y -5) = 0
3) (y + 25)(y -30) = 0
Step-by-step explanation:
Area of rectangle = 750 ft²
length = y ft
width = (y - 5) ft
Area of rectangle = 750
length * width = 750
y (y -5) = 750
Expand the equation,y*y - 5*y = 750
y² - 5y = 750
y(y - 5 )= 7500 = 750 - y(y -5)
750 - y(y-5) = 0
y² - 5y = 750y² - 5y - 750 = 0
Sum = -5
Product = -750
Factors = -30 , 25 {-30 + 25 = -5 & (-30)*25 = -750}
y² - 30y + 25y - 750 = 0 {Rewrite the middle term using the factors}
y(y - 30) +25(y - 30) = 0
(y - 30)(y + 25) = 0
A scientist calculates that a test at the significance level has probability 0.23 of making a type ii error when a specific alternative is true.What is the power of the test against this alternative?O 0.05O 0.23O 0.72O 0.95O 0.77
Answer option 5) 0.77 is correct. The power of the test against this alternative is 0.77, given the calculations of the Scientist.
The power of a statistical test is the probability of correctly rejecting a false null hypothesis. In this case, the null hypothesis is rejected in favor of a specific alternative, and the scientist has calculated that the probability of making a type II error (failing to reject the null hypothesis when the alternative is actually true) is 0.23.
Therefore, the power of the test is the complement of the type II error probability, which is 1 - 0.23 = 0.77. This means that the test has a 77% chance of correctly rejecting the null hypothesis when the alternative is actually true. Answer option 5) 0.77 is correct.
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A farmer packed some potatoes and carrots into
bags. Of these bags, 3/5 were bags of potatoes and
the rest were bags of carrots. After selling 45 bags of
potatoes and 3/4 of the bags of carrots, the farmer
had 1/4 of the bags left. How many bags of potatoes
and carrots did the farmer sell in all?
The number of bags of potatoes and carrots did the farmer sell in all is 75 bags.
What is word problem?A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.
Given that, a farmer is selling potatoes and carrots into bags, 3/5 were bags of potatoes and the rest were bags of carrots. He sold 45 bags of potatoes and 3/4 of the bags of carrots, and had 1/4 of the bags left.
The bags of potato = 3x/5
The bags of carrot = 2x/5
Therefore, according to question,
x-45-3x/5×2x/5 = x/4
7x/10-x/4 = 45
18x/40 = 45
x = 45×40/18
x = 100
Total bags sold = 100×2/5+3/5+45 = 75
Hence, the number of bags of potatoes and carrots did the farmer sell in all is 75 bags.
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The step function f(x) is graphed.
What is the value of f(0)?
• -2
• -1
• 0
• 1
The value of f(0) is -2.
Option A is the correct answer.
What is a function?A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
From the graph, we see that,
f(x) = -1 for -1 ≤ x < 0
Similarly,
f(x) = -2 for 0 ≤ x < 1 _____(1)
Now,
The value of f(x) at x = 0.
From (1),
f(0) = -2
Thus,
The value of f(0) is -2.
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Can someone help me
The correct statement about the quadratic function is given as follows:
C. The second difference would be constant.
How to model a quadratic function?A quadratic function is modeled as follows:
y = ax² + bx + c.
Then the first difference, which is the first derivative of the quadratic function, is obtained as follows:
y = 2ax + b.
The second difference, which is the second derivative of the quadratic function, is obtained as follows:
y = 2a.
The coefficient a is a constant, hence the second difference would be constant and the correct option is given by option C.
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Can you find out the area of these shapes
1) The Area of the first compound shape is given as = 52cm
2) The area of the second shape is given as = 47cm
Note that the method for deriving this area of the shapes is by dividing them into "regular" shapes as shown in the attached images.
A compound form is created by combining simple shapes. It's also known as a "composite" or "complex" form.
For Shape A:
We have a triangle and a rectangle when the compound shape is de-regularized.
The dimensions of the triangle are:
Base 8cm
Height 3cm
Since the area of a triangle is given as:
1/2 B * H,
The area of the triangle is = 1/2 * 8 * 3cm
= 12cm
For the rectangle, the area of a rectangle is given as L * B. Since
L = 8cm and
B = 5cm
The area = 8* 5 = 40cm
The total area of Shape A = 40cm + 12cm
= 52cm
For Shape B:
We have two rectangles and one right-angled triangle when the shape is "converted".
The dimensions of rectangle 1 are:
L = 3
B = 2
Area = 6
The dimensions of rectangle 2 =
L = 8cm
B = 4cm
Area = 32cm
The dimension of the Right angled triangle is:
Base = 6cm
Height = 3cm (that is 7-4)
Thus:
The area of the triangle is:
1/2 * 6* 3
= 9cm
Thus the total area of the second shape is:
= 32 + 6 + 9
= 47cm
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For a given distribution, the range is 60. Assuming the distribution is bell-shaped, the estimated standard deviation is = ____
Assuming the distribution is bell-shaped, the estimated standard deviation is 10.
Since the range is 60 and the distribution is bell shaped the N or Number will be 6
Thus, to calculate the standard deviation:
60/6 = 10
The standard deviation is a statistic that expresses the degree of variation or dispersion among a set of values. A low standard deviation suggests that values are typically close to the set's mean, whereas a high standard deviation suggests that values are dispersed over a wider range.
None of the standard deviations are "good" or "bad.". They serve as gauges of your data's dispersion. Sometimes, when using ratings scales, we want a wide spread because it shows that the range of the group we are rating is covered by our questions and ratings.
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Figure B is a scaled copy of Figure A. What is the scale factor from Figure A to Figure B?
Answer:
2:1
Step-by-step explanation:
If we examine the two figures we see that the length of each side of Figure B is twice the length of the corresponding side of Figure A
For example side of length 44 in Figure B corresponds to side of length 22 in Figure A
Scale Factor: Final Dimension/Initial Dimension =
Length of Figure B side/Length of Figure A corresponding side
44/22 = 2/1 or 2:1
So the scale factor from A to B is 2 or 2:1
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the data and then (e) answer the given questions.
Listed below are the highest amounts of net worth (in millions of dollars) of all celebrities. What do the results tell us about the population of all celebrities? Based on the nature of the amounts, what can be inferred about their precision?
Look at pic for numbers
Question content area bottom
Part 1
a. Find the mean.
Please don’t answer with the wrong answer! I’m tried of getting it wrong!!
According to the information, the mean of these values would be: 169
How to calculate the mean of this series of numbers?To calculate the mena of this series of numbers we must add all the values and multiply them by the number of values available. In this case, we would have to divide it by 10. The procedure is shown below:
290 + 190 + 175 + 165 + 155 + 155 + 140 + 140 + 140 + 140 = 1,6901,690 / 10 = 169According to the above, the mean of this operation would be 169.
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Circle thermos a) Work out the size of angle AED. b) Work out x.
Answer:
x=2⁰ AED=78⁰
Step-by-step explanation:
180⁰-108⁰= 78⁰
angle AED= 78⁰
108⁰+50⁰= 158⁰
180⁰-158⁰=22⁰
22⁰+78⁰=100⁰
100⁰+ 78⁰=178⁰
180⁰-178⁰= 2⁰
x=2⁰
Bethany has a photo website where she places photos that she has taken so others can see them. Below is how many likes each of her current photos were given: 10, 13, 15, 15, 17, 29 What is the average amount of likes each of her photos got? Hint: The average is your mean. 22 15 15.5 16.5
The average number of likes will be 16.5. the correct option is D.
What is a mean?Mean is defined as the ratio of the sum of the number of data sets to the total number of data. The mean is calculated by adding all the data points and dividing it by the total number of the data.
Given that Bethany has a photo website where she places photos that she has taken so others can see them. Below is how many likes each of her current photos was given: 10, 13, 15, 15, 17, 29.
The average will be calculated as:-
Mean = ( 10 + 13 + 15 + 15 + 17 + 29 ) / 6
Mean = 99 / 6
Mean = 16.5
Therefore, the average number of likes will be 16.5. the correct option is D.
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Math Question...............
Answer:12.90
Step-by-step explanation:
grace started reading her chemistry book 3 hours and 42 minutes before 19:12. What time did she start reading her book? Give your answer in 24 hour time.
Answer:
15:30 or 3:30
Step-by-step explanation:
19:12=18:72
18:72-3:42= 15:30
15:30 or 3:30
The time when Grace started reading her chemistry book is calculated as 15:30 in 24-hour format.
To find out what time Grace started reading her chemistry book in 24-hour format,
Simply subtract 3 hours and 42 minutes from 19:12.
Convert 19:12 to minutes: 19 hours × 60 minutes/hour + 12 minutes
= 1152 minutes.
Subtract 3 hours and 42 minutes from 1152 minutes:
1152 minutes - (3 hours × 60 minutes/hour + 42 minutes)
= 1152 - (180 + 42)
= 930 minutes.
Convert the remaining minutes back to hours and minutes:
930 minutes ÷ 60 minutes/hour
= 15 hours with a remainder of 30 minutes.
Combine the results:
Grace started reading her chemistry book at 15:30 in 24-hour format.
Therefore, time when Grace start her reading of chemistry book is 15:30 in 24-hour format.
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Does anyone know the answer to these?
The slopes of the lines are listed below:
Case 1: m = 1
Case 2: m = - 4 / 5
Case 3: m = - 4
Case 4: m = 5
Case 5: m = 2 / 7
Case 6: m = - 1
Case 7: m = 0
Case 8: m = NaN (Not a Number)
Case 9: m = 3 / 5
Case 10: m = - 8 / 3
How to determine the slope of a line
In this question we find ten cases of lines, whose slopes must be determined according to the following rules:
If the line is parallel with the x-axis, then its slope is zero.If the line is parallel with the y-axis, then its slope is undefined. If the line is oblique, then the slope is determined by secant line formula:Now we proceed to determine the slope for each case:
Case 1
m = (5 - 0) / [0 - (- 5)]
m = 1
Case 2
m = (- 3 - 1) / [0 - (- 5)]
m = - 4 / 5
Case 3
m = (- 3 - 1) / (1 - 0)
m = - 4
Case 4
m = [1 - (- 4)] / (1 - 0)
m = 5
Case 5
m = (3 - 1) / [0 - (- 7)]
m = 2 / 7
Case 6
m = (0 - 3) / (3 - 0)
m = - 1
Case 7
m = 0
Case 8
m = NaN (Not a Number)
Case 9
m = [- 1 - (- 4)] / (5 - 0)
m = 3 / 5
Case 10
m = (- 4 - 0) / (1.5 - 0)
m = - 4 / 1.5
m = - 8 / 3
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Which one is faster pulley with belt or just gears?
In both cases, pulley C is the fastest one since its radius is the smallest one.
Which pulley does rotate faster?
In this problem we find two cases of power transmition, the first example consists in a rigid transmition system formed only by pulleys and the second example consisting in a flexible transmition system formed by belts and pulleys.
Ideally, power is conserved throughout every system and rotational velocity of the pulley is inversely proportional to its radius. We need to determine does rotate faster in each case.
n ∝ 1 / R
Thus, we find the following results:
Case 1: Pulley C is the fastest pulley since it has the smallest radius.
Case 2: Pulley C is the fastest pulley since its has the smallest radius.
RemarkThere is no additional information (i.e. any angular speed of pulley, geometric dimensions in the second case) that may allows us to determine whether if rigid transmition system or a flexible transmition system is faster.
The representation only allows us to determine what pulley is the fastest within a system.
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the functions f, g, and h are defined by
f(x)=3x-4, g(x)=x-2, and h(x) = 5x²
find each of the following:
(f + g)(2)
The solution is, the answers to f(g(5)), g(f(78)), and h(g(f(2))) are 8, 60, and 83 respectively.
What is function?Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.
here, we have,
Given the expressions, f(x)=√x+5, g(x)=3x-7, and h(x)=5x, the answer to f(g(5)) is 8.
This is because when g(5) is substituted into f(x) we get f(g(5))=f(3x-7)=√(3x-7)+5. When 5 is substituted for x, this simplifies to f(g(5))=√(-2)+5 which equals 8.
The answer to g(f(78)) is 60. This is because when 78 is substituted into f(x) we get f(78)=√78+5 which simplifies to f(78)=9. When 9 is substituted into g(x) we get g(f(78))=3x-7 which simplifies to g(f(78))=3(9)-7 which equals 60.
Finally, the answer to h(g(f(2))) is 83. This is because when 2 is substituted into f(x) we get f(2)=√2+5 which simplifies to f(2)=3. When 3 is substituted into g(x) we get g(f(2))=3x-7 which simplifies to g(f(2))=3(3)-7 which equals 8. When 8 is substituted into h(x) we get h(g(f(2)))=5x which simplifies to h(g(f(2)))=5(8) which equals 83.
In summary, the answers to f(g(5)), g(f(78)), and h(g(f(2))) are 8, 60, and 83 respectively.
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If a = 4 units, b = 6 units, and c = 10 units, what is the volume of the three prisms above?
Does anyone know the answer to this?
Answer:
m<B = 58 degrees, m<C = 63 degrees
Step-by-step explanation:
let's find y!
angles B and A are complementary, so they should equal 90
therefore, B + A = 90
substitute
4y + 8y - 6 =90
12y = 96
y=8, and m<B = 8y-6; so m<B = 58
angles C and D are equal because of vertical angles
therefore, C = D
substitute
7x=5x+18
2x=18
x=9
m<C = 7x, so m<C = 63
A large data set contains information on all registered republicans in the United States. The following information is recorded.- Name
- Age
- Gender
- Home address
- Whether they voted or not in 2016 presidential election
Which of the following questions could not be answered based solely on the information in this data set?answer choicesa. What percent of registered republican voters are male?b. Which state had the most registered republican voters vote in the 2016 presidential election?c. What is the average age of registered republican voters?d. How many registered republicans voted for Gary Johnson in the 2016 presidential election.
The question which could not be answered based solely on the information in this data set is How many registered republicans voted for Gary Johnson in the 2016 presidential election? So, correct answer is option (d).
We have a large data consists of information on all registered republicans in the United States. The information of voters which present in data are Name, Gender, Age, Home address, Whether they voted or not in 2016 presidential election. We have to determine the question which answer is not possible on basis of this data. Let's consider the options one by one.
a) Here, gender and voters who vote or not in 2016 election, so it is easy to determine the percent of registered republican voters are male.
b) Address details and the information of voters who vote or not, helps to determine the state which had the most registered republican voters vote in the 2016 election.
c) The Age and registered voter information that he vote or not is used to determine the average age of registered republican voters.
d) Now, there is no information regarding the voters choice for voting their votes. So, it not determined that how many voters voted for Gary Johnson in the 2016 presidential election. This is write choice here.
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PLEASE HELP:
light travels at a speed of 1.86 x 10^5 miles per second. it takes about 15,000 seconds for the light to reach Neptune. write your answer in scientific notation.
The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.
What is speed?Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.
We know that the speed formula
Speed = Distance/Time
The light travels at a speed of 1.86 × 10⁵ miles per second. And it takes about 15,000 seconds for the light to reach Neptune.
Then the number of miles is given as,
1.86 × 10⁵ = d / 15,000
d = 1.86 × 10⁵ × 15,000
d = 2.79 × 10⁹ miles
The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.
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The students in Colleen's class got to choose whether to visit the zoo or the aquarium. 24 students went to the zoo and 3 students went to the aquarium. What is the ratio of the number of students who went to the aquarium to the total number of students?
The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.
What is ratio?The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.
here, we have,
24 students went to the zoo and 3 students went to the aquarium.
i.e. we get,
24 students went to the zoo
3 went to the aquarium
so, total no. of students = 24+ 3
= 27
then, we have,
the ratio of the number of students who went to the aquarium to the total number of students is = 3 : 27
= 1: 9
Hence, The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.
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Find the directional derivative of f(x,y,z)=xy+z^3 at the point (2,3,1) in the direction of a vector making an angle of 3π/4 with ∇f(2,3,1). Expert Answer.
The directional derivative of f(x, y, z) = [tex]xy+z^{3}[/tex] at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point is √2 (3cos(3π/4) + 2sin(3π/4)).
To find the directional derivative of f(x, y, z) = xy + z^3 at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point, we need to calculate the gradient of f at that point and then use it to find the dot product with the given direction vector.
The gradient of f at the point (2, 3, 1) is given by ∇f(2, 3, 1) = (y, x, 3z^2) = (3, 2, 3).
The direction vector in the direction of 3π/4 with the gradient can be found as follows:
Let [tex]d = (cos(3\pi /4), sin(3\pi /4), 0)[/tex]
The directional derivative of f in the direction d at the point (2, 3, 1) is given by
∇f(2, 3, 1) . d = [tex](3, 2,3).(cos(3\pi /4), sin(3\pi /4),0)[/tex]
= [tex]3cos(3\pi /4)+2sin(3\pi /4)[/tex]
= √2 (3cos(3π/4) + 2sin(3π/4))
So the directional derivative of f in the direction of a vector making an angle of 3π/4 with the gradient of f at (2, 3, 1) is √2 (3cos(3π/4) + 2sin(3π/4)).
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What’s the answer for these two
The amount freedome ride charge more than Envo car in d miles is $20.05
What is word problem?word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem.
The total charge for Enzo car =$50.50 +8¢ = $50.50 + $0.08 = $50.58
The total charge for Freedom = $70.50 + 12¢
= $70.50 + $0.12
= $70.62
therefore in d miles ;
Enzo will charge $50.58d
freedome will charge
$70.62d
therefore the amount freedome will charge more than Enzo car is $70.62 - $50.58
= $20.05
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Mrs. Ross the couselor, is helping plan a field at her middle school and is trying to decide what yard games to buy. She randomly surveyed 80 students at lunchtime anf asked about the yard game each one preferref. The results of the survey are shown in the table. Bean bag toss 14 checkers 16 connect 4 28 jenga 14 tic tac toe 8, based on the results in the table, which statement is true? A a student is more likely to prefer tic tac toe or connect 4 than checker or jenga. B a student is four times as likely to prefer connect 4 as tic tac toe. C a student is twice as likely to prefer tic tac toe as checkers. D a student is less likely to prefer checkers or bean bag toss than jenga or tic tac toe.
To determine which statement is true, we need to compare the number of students who prefer each game relative to the total number of students surveyed.
The total number of students surveyed is:
14 + 16 + 28 + 14 + 8 = 80
A) To find the probability of a student preferring tic tac toe or connect 4 versus checker or jenga, we add the number of students who prefer tic tac toe or connect 4:
28 + 8 = 36
And add the number of students who prefer checker or jenga:
14 + 16 + 14 = 44
Therefore, statement A is false.
B) To find the probability of a student preferring connect 4 versus tic tac toe, we divide the number of students who prefer connect 4 by the number of students who prefer tic tac toe:
28/8 = 3.5
Therefore, statement B is false.
C) To find the probability of a student preferring tic tac toe versus checkers, we divide the number of students who prefer tic tac toe by the number of students who prefer checkers:
8/16 = 0.5
Therefore, statement C is true.
D) To find the probability of a student preferring jenga or tic tac toe versus checkers or bean bag toss, we add the number of students who prefer jenga or tic tac toe:
28 + 8 = 36
And add the number of students who prefer checkers or bean bag toss:
14 + 16 = 30
Therefore, statement D is true.
Therefore, the correct answer is option D: a student is less likely to prefer checkers or bean bag toss than jenga or tic tac toe.
the cost of a new car is $32,000 it depreciates at a rate of 15% per year . this means that it loses 15% of each value each year
The mathematical function describing the cost of car the next year is →
C{n + 1} = C{n} (1 - 3/20).
What is depreciation rate?The depreciation rate is a rate at which the price of a quantity decreases every year.
Given is that the cost of a new car is $32,000 it depreciates at a rate of 15% per year.
We can understand it as follows.
After one year the cost would be -
{x} = 32000 - {15% of 32000}
{x} = 32000 - 4800
{x} = $27200
After second year the cost would be -
{x} = 27200 - {15% of 27200}
{x} = 27200 - 4080
{x} = $23120
This means that after every year, the value decrease by 15% of what it was a year before. We can write -
C{n + 1} = C{n} - (15/100) C{n}
C{n + 1} = C{n} (1 - 3/20)
Therefore, the mathematical function describing the cost of car the next year is →
C{n + 1} = C{n} (1 - 3/20).
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I need helps with this please
The missing statement from his process;
RM = RL and MA = LA
Radii of the constructed arc are some length.
What are angle bisectors?Angle bisectors are lines that cut across the angle under consideration. The term "bisect" means to divide into two equal portions. The considered angle is therefore divided in half by the bisected halves.
Given:
Oleg is writing proof of the construction of an angle bisector.
The missing statement from his process;
RM = RL and MA = LA
Radii of the constructed arc are some length.
From the reflexive property;
RA ≅ RA.
And ΔRMA ≅ ΔRLA, by SSS criterion.
And corresponding angles of congruent angles
∠MRA ≅ ∠LRA.
Therefore, ∠MRA ≅ ∠LRA.
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The number of bacteria in a culture is increasing according to the law of exponential growth. There are 135 bacteria in the culture after 2 hours and 385 bacteria after 4 hours.. use the model to determine the number of bacteria after 8 hours. (round your answer to the nearest whole number.)
The number of bacteria after 8 hours is approximately 2,713.
Let's apply the exponential growth formula:
N = N0 * e(rt),
N is the number of bacteria at a particular period. N0 is the initial bacterial population, e is the mathematical constant (about equal to 2.71828), r is the growth rate, and t is the amount of time that has passed.
The information provided in the problem can be used to determine the growth rate:
N1 = N0 * e(r*2)
N2 = N0 * e(r*4)
To get eliminate N0, divide the second equation by the first equation:
N2/N1 = e(r4) / e(r2)
385/135 = e(r4) / e(r2)
If we condense this expression, we get:
e(r*2) = 385/135
e(r*2) = 2.8519
When we take the natural logarithm of both sides, we get:
r*2 = ln(2.8519)
r = ln(2.8519)/2
r = 0.5457 (rounded to 4 decimal places)
The number of bacteria after 8 hours can now be determined using the exponential growth algorithm once more:
N = N0 * e(rt)
N = 135 * e(0.5457*8)
N = 2,713 (rounded to the nearest whole number)
Therefore, the number of bacteria after 8 hours is approximately 2,713.
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Complete the input-output table:
n 4n-5
Answer:
4(1) - 5 = -1
4(2) - 5 = 3
4(3) - 5 = 7
Step-by-step explanation:
Replace variable "n" with values of "n".
4n-5 when n = 1 is 4(1) - 5
4(1) - 5 = -1
4(2) - 5 = 3
4(3) - 5 = 7
z₁ = 3 cis(pi) and z₂ =5 cis(pi/2) If z₁ x z₂ = a+bi. then a = _ and b= _
The solution is, a = 0 & b = -15.
What is complex number?A complex number is a number of the form a + bi, where a and b are real numbers ; This way, a complex number is defined as a polynomial.
here, we have,
z₁ = 3 cis(pi) and z₂ =5 cis(pi/2)
we know, cisx = cosx + i sinx
so, we get,
z₁ = 3 cis(pi) = 3(-1)
and z₂ =5 cis(pi/2) = 5( i.1)
If z₁ x z₂ = a+bi.
or, -3 * 5i = a+ ib
or, -15i = a +ib
Hence, The solution is, a = 0 & b = -15.
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Write ____ as a complex number in the form a + bi, where a and b are real numbers.
[tex] \boxed{\sf Option \: A\text{:} \: \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1}{4} + \sqrt{2}i} [/tex]
[tex] \\ \\ [/tex]
Explanation:We are asked to express the given number in the form a + ib, also known as the algebraic form.
[tex] \\ \\ [/tex]
Algebraic form of a complex number[tex] \\ \\ [/tex]
[tex] \sf z = a + ib \\[/tex]
Where:
✰ a and b are real numbers and are the complex number's real part and imaginary part, respectively.
[tex] \\ \\ [/tex]
✰ i is the imaginary unit number and can be expressed as follows:
[tex] \\ \sf i= \sqrt{-1}[/tex]
[tex] \\ \\ [/tex]
Simplify the number's expression[tex] \\ \\ [/tex]
[tex] \sf z = \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1 + \sqrt{32 \times (-1)}}{4} \\ \\ \\ \sf \diamond \: We \: know \: that \: \sqrt{ab} = \sqrt{a} \times \sqrt{b}. \\ \\ \implies \dfrac{1 + \sqrt{32 \times (-1)}}{4} = \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4}[/tex]
[tex] \\ \\ \sf \implies \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4} = \dfrac{1 + \sqrt{16 \times 2} \times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times \sqrt{-1}}{4}[/tex]
[tex] \\ \\[/tex]
[tex] \diamond \: \sf Substitute \: \sqrt{-1} = i. \\ \\ \implies \sf \dfrac{1 + 4\sqrt{2}\times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times i}{4} = \dfrac{1 + 4i\sqrt{2}}{4} [/tex]
[tex] \\ \\ \: \sf \diamond \: Simplify \: knowing \: that \: \dfrac{a + b}{c} = \dfrac{a}{c}+\dfrac{b}{c}. [/tex]
[tex] \\ \\ \\ \implies \: \sf \dfrac{1 + 4i\sqrt{2}}{4} = \dfrac{1}{4} + \dfrac{4i\sqrt{2}}{4} = \boxed{\sf \dfrac{1}{4} + \sqrt{2}i} [/tex]
[tex] \\ [/tex]
Therefore, the first option is correct.
[tex] \\ \\ \\ [/tex]
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For one-pair poker hands, why is the number of denominations for the three single cards (12 3) rather than (12 1) (11 1) (10 1)?
The number of denominations for the three single cards is 12 3 because there are 220 possible combinations of three cards from a deck, each combination representing a unique poker hand.
The number of denominations for the three single cards is 12 3 because it is impossible to have a poker hand with only two cards. The two-card hands can contain only one pair or two different cards, and neither of these hands are possible with only two cards. Therefore, the number of possible combinations of three cards is 12 3.
We can calculate this using the combination formula: nCr = n! / r!(n-r)!
For a hand of three cards, n = 12 (there are 12 denominations of cards) and r = 3. Therefore, 12 3 = 12! / 3!(12-3)! = 220. This means there are 220 possible combinations of three cards from a deck, each combination representing a unique poker hand.
The number of denominations for the three single cards is 12 3 because there are 220 possible combinations of three cards from a deck, each combination representing a unique poker hand.
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