The image vertices of D′ if the preimage is rotated 90° counterclockwise is: A. D′(−2, −4).
What is a rotation?In Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).
By applying a rotation of 90° counterclockwise to the vertices for D, the coordinates of the vertices of the image point D' are as follows:
(x, y) → (-y, x)
Ordered pair D = (-4, 2) → Ordered pair D' = (-2, -4)
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3. numbers with square roots that are whole
they are called perfect square numbers
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 heres some hope that helps!
5. A regular hexagon is a six-sided figure with congruent sides and
congruent angles. A regular hexagon can be composed of six equilateral
triangles. Write and solve an equation to determine the value of x on the
regular hexagon shown. Then determine the sum of the six interior angles
of a regular hexagon.
I need this answer ASAP
The sum of the interior angles of a regular hexagon is S = 720° and each angle of the regular hexagon x = 120°
What is the sum of the interior angles of a polygon?The sum of the interior angles of a polygon is given by the formula
Sum of Interior angles of a polygon with n sides is
nθ = 180 ( n - 2 )
where n is the number of sides
θ = angle in degrees
Given data ,
Let the sum of the interior angles of the hexagon be S
Now , the number of sides for a hexagon = 6
And ,Sum of Interior angles of a polygon with n sides is
nθ = 180 ( n - 2 )
Substituting the values in the equation , we get
6θ = 180 ( 6 - 2 )
On simplifying the equation , we get
6θ = 180 x 4
6θ = 720°
Therefore , the sum of angles is 720°
And , the value of x = 720 / 6 = 120°
Hence , the sum of interior angles of hexagon is 720°
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The associative property states that when adding or multiplying three or more terms, you can group the terms in any order and still get the same answer. The expression must use only one operation. For example, 4 × 3 × 2 can be thought of as 4·(3 · 2), which is 4 · 6, or as (4 · 3) · 2, which is 12 · 2.
The associative property works with addition and multiplication, but not division and subtraction.
The associative property works with addition and multiplication, but not division and subtraction because the grouping of terms affects the result of the operation.
A mathematical concept known as the associative property governs the addition and multiplication of three or more terms. It claims that the way the phrases are grouped has no bearing on how the operation turns out.
This implies that the same result may be obtained by adding or multiplying three or more terms in any sequence. Nevertheless, division and subtraction are not covered by the associative property.
We should take a gander at the accompanying cases to show how expansion and increase are connected:
Example 1: Addition
(2 + 3) + 4 = 2 + (3 + 4)
5 + 4 = 2 + 7
9 = 9
Example 2: Multiplication
(2 × 3) × 4 = 2 × (3 × 4)
6 × 4 = 2 × 12
24 = 24
We can observe from both cases that the word grouping has no impact on the operation's outcome. The associative characteristic, therefore, applies to both addition and multiplication.
The associative characteristic, however, does not apply to division and subtraction. For instance:
(8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2)
2 ÷ 2 ≠ 8 ÷ 2
1 ≠ 4
We can observe from this example that the term grouping has an impact on the operation's outcome. The associative property does not, therefore, apply to division.
The associative property does not also hold for subtraction. For instance:
(8 - 4) - 2 ≠ 8 - (4 - 2)
4 - 2 ≠ 8 - 2
2 ≠ 6
In this example, we can again see that the grouping of terms affects the result of the operation. Therefore, the associative property does not apply to subtraction.
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A mason needs to purchase mortar mix for a retaining wall consisting of 640 blocks. The guidelines for this mix suggest that 12 bags are required for every 100 blocks. Assuming that he cannot purchase part of a bag, how many bags will he need?
The mason will need 77 bags of mortar mix for the retaining wall.
What is purchase?Purchasing is the process a business or organization uses to acquire goods.
To find out how many bags of mortar mix the mason needs
we need to calculate the number of bags required for 640 blocks.
The guidelines suggest that 12 bags are required for every 100 blocks, so for 640 blocks, we need:
12 bags * (640 blocks / 100 blocks) = 12 * 6.4 = 76.8
Since the mason cannot purchase part of a bag, he will need to round up to the nearest whole number of bags. In this case, 77 bags will be required.
So, the mason will need 77 bags of mortar mix for the retaining wall.
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A person invests 7000 dollars in a bank. The bank pays 6. 25% interest compounded semi-annually. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 11300 dollars?
It takes 3.9 years long the person have to leave the money in the bank until it reaches 11300 dollars
Suppose that the initial amount of something is P.
Let after one unit of time, it increases by R% [per unit time] end compounds on the resultant total of that quantity, then, after T such units of time, then the quantity would increase to:
[tex]A=P\left(1+\frac{R}{100}\right)^{nT}[/tex]
A person invests 7000 dollars in a bank. The bank pays 6.25 % interest compounded semi-annually.
Given that:
Principal =7000
Interest (r) = 6.25 % compounded semi annually
As compounded semi annually n = 2
Calculate time, t, if final amount A = 11300
Using the compound interest formula
[tex]A=P\left(1+\frac{R}{100}\right)^{nT}[/tex]
A = Final amount
n = number of times interest is applied per period
11300 = 7000 [tex](1+0.0625)^{2t[/tex]
113 = 70 [tex](1.0625)^{2t[/tex]
1.614 = [tex](1.0625)^{2t[/tex]
Take the log of both sides
log 1.614 = 2t log 1.0625
0.2079 = 2t (0.0263)
0.2079 = 0.0526 t
t = 3.9
Time period will be 3.9 years.
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A science class planned a field trip for which the total costs were $80. When four additional students decided to go on the trip, the cost per student was reduced by $1. How many students went on the trip?
Answer:
There were 8 student who went on a trip.
Step-by-step explanation:
Let's call the original number of students "x". The total cost of the trip was $80, so the cost per student was $80/x.
When four more students decided to go, the new number of students became x + 4, and the cost per student was reduced by $1, to $80/x - $1.
We can set up an equation to solve for x:
$80/x = $80/(x + 4) - $1
Expanding the right side:
$80/x = $80/x + $80/4 - $1
Combining like terms:
$80/x = $80/4 + $80/x - $1
Solving for x:
$80/4 = $80/x - $1
$80/4 + $1 = $80/x
$20 + $1 = $80/x
$21 = $80/x
Finally, solving for x:
x = 80/$21 = 80/21 = 3.81 (rounded to nearest whole number)
So, there were 4 students on the original trip.
There were 4 students on the original trip and 4 more students joined, so in total, there were 4 + 4 = 8 students who went on the trip.
Answer: 8 student
Step-by-step explanation:
Calculate the tax for each taxable income of a head of household.
The income tax for each household are:
Household 1 = $113,775Household 2 = $1,09540Household 3 = $18,904.88Household 4 = $47,222.50Let's evaluate each household income tax cases:
Household 1
Taxable income = $400,000
Tax rate = $45,210 + 35% over $204,100
Income tax = $45,210 + 35% ($400,000 - $204,100)
Income tax = $45,210 + $68,565
Income tax = $113,775
Household 2
Taxable income = $10,954
Tax rate = 10% income
Income tax = 10% ($10,954)
Income tax = $1,095.40
Household 3
Taxable income = $108,962
Tax rate = $12,962 + 24% over $84,200
Income tax = $12,963 + 24%($108,962 - $84,200)
Income tax = $12,963 + $5,942.88
Income tax = $19,804.88
Household 4
Taxable income = $209,850
Tax rate = $45,210 + 35% over $204,100
Income tax = $45,210 + 35% ($209,850 - $204,100)
Income tax = $45,210 + $2,012.50
Income tax = $47,222.50
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PLEASE HELP ASAP WILL GIVE BRAINLEST!
The value of the x is 228.57 units.
What is the angle bisector theorem?
The angle bisector theorem states that a triangle's angle divides the opposing side into two portions that are proportionate to one another.
We have given,
Triangle, Let's consider it ΔABC,
then,
Measures of Angles ∠B= 37, ∠C=32, and side AD = 82
To find x =?
In the triangle side AD is the bisector, it forms a new triangle ΔADC.
According to the angle bisector theorem, the angle bisector of a triangle divides the opposite side into two parts that are proportional to the other two sides of the triangle.
As per the Angle bisector theorem, the ratio of the line segment BD to DC equals the ratio of the length of the side AB to AC.
To find AB & AC:
sin θ = side opposite to angle θ / hypotenuse
cos θ = side adjacent to angle θ / hypotenuse
sin θ = AD / AB
sin 37 = 82 / AB
AB = 82/ sin 37
AB = 136.25
cos θ = BD / AB
cos θ = BD/136.25
BD = cos37 * 136.25
BD = 104.9
sin θ = AD / AC
sin32 = 82 / AC
AC = 149
cos32 = DC / AC
cos32 = DC / 149
DC = 123.67
So, x = BD + DC
x = 104.9 + 123.67
x = 228.57
Hence, the value of the x is 228.57 units.
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02: Pretest CR Algebra 2 A / 2:Equations and Functions
option B is correct, the equation or function of the graph is y≥3/4x-3.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
The line passing through points (4,0) and (0,3)
Slope=3/-4=-3/4
Now let us find y intercept
0=-3/4(4)+b
0=-3+b
b=3
Hence, option B is correct, the equation or function of the graph is y≥3/4x-3.
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Classify each statement about the function f(x)=2x3+8 as true or false.
The domain of the function is all real numbers.
The range of the function is (0, ∞).
The y-intercept is (0, 0).
The center of symmetry is (0, 8).
Answer:
The answer to the classification of the statements is:
1.True
2.False
3.False
4.True
Answer:
True
False
False
True
Step-by-step explanation:
can someone help me with this ik its not math, but there wasn’t a category for band/music. this is due tmr and i’m struggling. i’ll give 50 points!!
1. a break in the music
2. how loud or soft to play
3. bar lines organize the staff into these boxes of music
4. perform two or more notes as one note for the combined sum of all the notes
5. noteheads are placed on the five horizontal lines and four spaces
6. Signifies the end of a piece of music show me your mom
7. Divide the staff in the measures
8. Medium soft
9. Holding note longer than its normal value
10. Fast
11. The style used to begin or end a note
12. A medium tempo
13. Stretch the note to hold it for full value
14. a slow tempo
15. Perform the note with emphasis
16. Gradually getting louder
17. Gradually getting softer same as decrescendo
18. Top number is the number of beats in a measure bottom number is the type of note that receives one beat
19. Gradually getting faster
20. Smoothly connected
the blacked out ones are the ones i’ve already done!!
Answer:
1. rest
2. dynamics
3. measure
4. chord
5. staff
6. double bar line
7. bar line
8. mezzo-piano
9. fermata
10. allegro
11. articulation
12. moderato
13. tenuto
14. larghetto?
15. accent
16. crescendo
17. diminuendo
18. time signature
19. accelerando
20. slur
To graph the shape on the coordinate grid, click on a point that represents a vertex, then repeat this process for each vertex point of the shape. To complete the shape, click on the first vertex point. The shape will be shaded in once it is completed.
Check the picture below.
You want to buy a pet hamster that costs $42. You already have $15 and plan to save $8 per week. If w represents the number of weeks until you have enough money to buy the hamster, what equation represents your plan to afford the hamster.
I need help I can't think of what the equation would be!
Answer: 42 = 8w + 15
Step-by-step explanation:
A social media website had 500,000 followers in 2012. The number y of followers increases by 19% each
year.
a. Write a function that represents the number of followers t years after 2012.
b. How many people will be following the website in 2019? Round your answer to the nearest thousand
Function representing number of followers for different situation are:
a. Number of followers in 't' years N = 500,000 ( 1 + 19/100 )^t.
b. Number of people following in 2019 are 1,690,000 ( nearest thousand )
Total number of followers of social media website in 2012 = 500,000
Followers increases by 19% every year.
let 'N' represents the number of followers.
a. Function to represents the number of follower in 't' years after 2012 is:
N = 500,000 ( 1 + 19/100 )^t
b. Number of people following website in 2019 is:
t = 2019 -2012
= 7
N = 500,000 ( 1 + 19/100 )^7
⇒ N = 500,000 ( 1.19 )^7
⇒ N = 500,000 × 3.379
⇒ N= 1689657.7
⇒ N = 1,690,000 ( nearest thousand )
Therefore, the number of followers represented for different condition are:
a. Function to represent number of followers in 't' years is
N = 500,000 ( 1 + 19/100 )^t
b. Number of followers in 2019 are 1,690,000 ( nearest thousand )
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The regression line for this data set has a slope close to m = , and the correlation coefficient is close to. Based on this information, we can conclude that employees' annual earnings are to their tenure. Employee earnings tenure
1) The slope of the regression line is 35
2) The correlation coefficient is 0.95
3) Employees’ annual earnings are strongly related to their tenure
4) Employee earnings increase with tenure.
The slope of the regression line in this data set is close to 35. In order to determine the slope, two points must be identified that are either on or near the line.
We have the following two points;
(1, 175) and (2.5, 225)
slope = (change in y) / (change in x)
= (225-175)/(2.5-1) = 33.33
This is close to 35.
2) The correlation coefficient of the annual earnings and tenures of employees at Stan & Earl Corp is almost 0.95. This value, known as the coefficient of correlation, shows the level of connection between two variables and the direction of their linear relationship. The scatter-plot demonstrates a strong positive relationship between the employees' annual earnings and their tenure.
Hence the correlation coefficient is close to 0.95.
3) The data suggests a strong connection between an employee's annual earnings and their tenure. The correlation coefficient was estimated to be close to 0.95, which represents a significant positive relationship between the two variables. As a result, it can be concluded that there is a clear relationship between an employee's annual earnings and the length of time they have been with the company.
4) As employees' tenure with the company increases, their annual earnings also increase. This can be deduced from the slope of the regression line which was calculated to be close to 35, indicating a positive relationship between the two variables. The positive correlation coefficient supports this
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The regression line for this data set has a slope close to m = (5.5,35,-50,-2.5) , and the correlation coefficient is close to (0.20,0.40,0.75,0.95) .
Based on this information, we can conclude that employees’ annual earnings are (strongly related, moderately related, weakly related, not related) to their tenure. Employee earnings (increase with, decrease with, are not affected by) tenure.
A parabola opening up or down has vertex (0, 0) and passes through (12, 18). Write its equation in
vertex form.
Simplify any fractions.
Answer:
[tex]y=\dfrac{1}{8}x^2[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Vertex form of a quadratic equation}\\\\$y=a(x-h)^2+k$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}[/tex]
Given that the vertex of the parabola is (0, 0):
h = 0k = 0Substitute the values of h and k into the formula:
[tex]y=a(x-0)^2+0[/tex]
[tex]y=ax^2[/tex]
Given that the parabola passes through the point (12, 18), substitute this into the equation and solve for a:
[tex]18=a(12)^2[/tex]
[tex]18=144a[/tex]
[tex]a=\dfrac{18}{144}[/tex]
[tex]a=\dfrac{1}{8}[/tex]
Therefore, the equation of the parabola in vertex form is:
[tex]y=\dfrac{1}{8}x^2[/tex]
Grace rakes led s every day for one week. She fills the same number of bags with leaves o each of the 5 weekdays. She fills 11 bags over the weekend. At the end of the week, Grace has filled a totallof 26 bags. She wants to know how many bags she filled each weekday.
bags filled on weekdays + bags filled on weekend = total bags filled
Let b equal the number of bags Grace filled each weekday.
5b + 11 = 26
How many bags of leaves did Grace fill each weekday?
Answer:
3 bags of leaves each weekday
Step-by-step explanation:
5b + 11 = 26
Subtract 11 from both sides.
5b = 15
Divide 5 on both sides.
b = 3
Grace filled 3 bags of leaves each weekday.
Attached imagine to answer.
Answer:
A)7y-4xy
Algebraic expressions are just that expressions. No inequality signs or equations. That's what makes them special
Step-by-step explanation:
How many liters are in 9 quarts
Answer: 8.5 L
Step-by-step explanation: I looked it up l o l
Answer:
Your answer is: 8.51718
Step-by-step explanation:
There's 1.057 quarts in a liter. You multiply that by 9 to get 8.51718.
The distance between two towns, A and B, is 100 km. Mr Jones drove from A to B at an average speed of v km/h. (a) Write down an expression, in terms of v, for the time, in hours, that he took to complete the journey from A to B. [1]
The expression giving the time that he took to complete the journey is given as follows:
t = 100/d.
What is the relation between velocity, distance and time?Velocity is distance divided by time, hence the following equation is built to model the relationship between these variables:
v = d/t.
For this problem, the distance is given as follows:
d = 100 km.
Hence the expression to obtain the time needed is given as follows:
v = 100/t
vt = 100
t= 100/v.
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Hide Sample Answer
The area formula is A = 7². Substitute the values of r (3.5 inches) and TT (3.14) in the formula to find A:
A-3.14x (3.5)² square inches
A = 38.465 square inches.
The area of each light is about 38.465 square inches.
Part C
What is the total area of all three circular lights?
The total area of all three circle lights is:
115.395 square inches.
How do you find the area of a circle?The area of the circle formula helps measure the space occupied by a circular field or plot. Assuming you have land to the fence, the area formula will help you determine how much fencing you will need.
The area of a circle is the area occupied by the circle in the two-dimensional plane. It can be easily determined by the formula A = [tex]\pi r^{2}[/tex], (Pi r-square) where r is the radius of the circle. Area units are square units such as For example m², cm², etc.
The total area of all three circular lights can be calculated by multiplying the area of one light (38.465 square inches) by 3.
Total area = 3 x 38.465 square inches
Total area = 115.395 square inches.
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Can someone please explain the answer with steps and formula I have a test tomorrow
Answer:
sorry but it not possible here i can explain on another platform
grade 4N usually has physical education on a Wednesday at 1:43pm and ends at 2:13 pm. with the new mandated change, at what time will grade 4N PE ends?
Grade 4N usually has physical education on a Wednesday at 1:43 pm and ends at 2:13 pm. with the new mandated change, at 2:28 pm Grade 4N PE will now end
The new regulations require physical education classes to be a minimum of 30 minutes long. Allowing for an extra 15 minutes to the existing PE class brings the end time to 2:28 pm. This is an important change because it ensures that all students receive the recommended amount of physical activity each week.
Physical education is a vital part of a student’s development, as it provides the opportunity for students to become more active, and socially engaged, and learn about the importance of a healthy lifestyle. The extra 15 minutes of physical education also allows for more time for students to learn about safety procedures and the proper way to use the equipment, which is important for the overall well-being of the students.
Additionally, the extra time allows for more time for individual instruction, which can be beneficial for students who need extra help or who may be struggling with a certain skill.
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Chris says that he can find the surface area faster by
thinking of the box as a triangular prism on top of a
rectangular prism. He finds the surface area of each one
and adds them together. Does his method make sense?
Chris's method of finding the surface area of a box by thinking of it as a combination of a triangular prism and a rectangular prism is a valid and useful approach.
What is Surface area?
The surface area of a solid item is a measurement of the total area occupied by the object's surface.
Yes, Chris's method of finding the surface area of a box by thinking of it as a combination of a triangular prism and a rectangular prism makes sense.
A box can be thought of as a combination of a triangular prism and a rectangular prism because a box is made up of six faces, with three rectangular faces and three triangular faces. By finding the surface area of the triangular prism and the rectangular prism separately, and then adding them together, one can find the total surface area of the box.
This method can make finding the surface area of a box faster and easier because one can use formulas that are specifically designed for finding the surface area of triangular prisms and rectangular prisms. By breaking down the box into its constituent parts, one can simplify the calculation and avoid having to use a more complex formula for finding the surface area of a box.
In summary, Chris's method of finding the surface area of a box by thinking of it as a combination of a triangular prism and a rectangular prism is a valid and useful approach.
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find the value of y in the proportion (4y+5):3=(y+7):4
The value of y in the given proportion is 1/13.
In mathematics, a proportion is a statement that two ratios are equal. A ratio is a comparison of two quantities, typically written as a fraction. For example, if we have two quantities, a and b, then the ratio of a to b is written as a/b.
A proportion is written in the form a/b = c/d, where a, b, c, and d are numbers. This means that the ratio of a to b is equal to the ratio of c to d. We can also write this as a:b = c:d.
To solve the proportion:
(4y+5):3=(y+7):4
We can cross-multiply and simplify:
4(4y+5) = 3(y+7)
16y + 20 = 3y + 21
Subtracting 3y from both sides:
3y + 20 = 21
Subtracting 20 from both sides:
13y = 1
Dividing both sides by 13:
y = 1/13
Therefore, the value of y that satisfies the proportion is y = 1/13.
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what is the tempature change of 15 to -9
Answer:
change in temperature = final - temperature
= -9°-15° = -24°C
a rectangle is meters long and meters wide. the area of the rectangle is square meters. the solution is
Multiplying the length and width together, we get 3m x 4m = 12m². This means that the area of the rectangle is 12 square meters.
Area = Length x Width
Area = 3m x 4m
Area = 12m²
To solve for the area of a rectangle, you will need to use the formula Area = Length x Width. In this example, the length of the rectangle is 3 meters and the width is 4 meters. To calculate the area of the rectangle, you will need to multiply the length and width together. 3m x 4m = 12m². Therefore, the area of the rectangle is 12 square meters.
The area of a rectangle is calculated by multiplying the length and width of the rectangle together. The formula used to calculate the area of a rectangle is Area = Length x Width. To put this formula into practice, lets use the example of a rectangle that is 3 meters long and 4 meters wide. Multiplying the length and width together, we get 3m x 4m = 12m². This means that the area of the rectangle is 12 square meters. This formula can be used for any rectangle, as long as the length and width are known. Knowing the area of a rectangle can be useful for many purposes, such as calculating the amount of paint or carpeting needed to cover the surface area of the rectangle.
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solid shapes are regular when all the faces are eaqual which 3d shapes is regular?
a) cylinder
b) cube
c) cuboid
A solid shape is considered to be regular if all of its faces are congruent, meaning they have the same shape and size. In the context of three-dimensional shapes, there are several examples of regular solid shapes, including the cylinder, cube, and cuboid.
A) Cylinder: A cylinder is a solid shape with two parallel, congruent circular bases connected by a lateral surface. All of the lateral faces of a cylinder are parallelograms, and the circular bases have the same shape and size, making the cylinder a regular solid shape.
B) Cube: A cube is a regular solid shape with six congruent square faces. All of the faces meet at right angles, and each face is parallel to the opposite face, making the cube a regular solid shape.
C) Cuboid: A cuboid is a solid shape with six rectangular faces. The opposite faces of a cuboid are congruent rectangles, and all of the faces meet at right angles, making the cuboid a regular solid shape.
In conclusion, the cube and the cuboid are regular solid shapes because all of their faces are congruent, while the cylinder is a regular solid shape because all of its lateral faces are congruent parallelograms and its bases have the same shape and size. These regular solid shapes are useful in geometry and provide a foundation for understanding the properties of more complex solid shapes.
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A total of
330
330 children and adults attended a school play. There were
21
21 times as many children in attendance as there were adults.
This situation is modeled by the given system of equations, where a represents the number of adults and c represents the number of children.
As a result, 315 pupils attended the school play.
What is equation?In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical phrase with two equal sides separated by an equal sign is called an equation. An example of an equation is 4 + 6 = 10.
Here,
The first equation in the system is given by the total number of attendees:
c + a = 330
The second equation is given by the number of children being 21 times the number of adults:
c = 21a
We can use substitution to find the value of a and then find the value of c. Substituting the second equation into the first equation, we get:
21a + a = 330
Combining the two terms on the left-hand side, we have:
22a = 330
Dividing both sides by 22, we have:
a = 15
So there were 15 adults in attendance at the school play. To find the number of children, we can use the second equation:
c = 21a = 21 * 15 = 315
So there were 315 children in attendance at the school play.
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in a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 4 hourse. after 12 hours, it is estimated that there are 1 million bacteria in the culture. what is the doubling time for the population?
The bacteria's population increases exponentially, and its equation is
P(t) = P[tex]_{0}[/tex]e[tex]^{kt}[/tex]
Where,
P[tex]_{o}[/tex] is the initial population t is the number of hours.
k is the growth rate.
It is given that the population of the bacteria after one hour triples itself. i.e
P(1)=3P[tex]_{o}[/tex]
3P[tex]_{o}[/tex]=Pe[tex]k^{(l)}[/tex]
3=e[tex]^{k}[/tex]
In 3 = In e[tex]^{k}[/tex]
In 3=k ln e
In 3 = k(1)
k = ln (3)
Thus, the rate of growth is k= ln (3)
Substitute k = ln (3) in the equation P(t) = P[tex]_{o}[/tex]e[tex]^{kt}[/tex].
P(t) = P[tex]_{o}[/tex]e[tex]tln^{(3)}[/tex]
It is given that there are 1 million bacteria after 5 hours. i.e. P(5)=10,00,000
P(5) = P[tex]_{o}[/tex]e[tex]^{(5)} ln^{(3)}[/tex]
10,00,000 Pe[tex]^{(5)}ln^{(3)}[/tex]
10,00,000 = P[tex]_{o}[/tex]e[tex]ln^{(3)^{5} }[/tex]
10,00,000 = P[tex]_{o}[/tex][tex](3)^{5}[/tex]
P[tex]_{o}[/tex] = 10,00,000 / [tex]3^{5}[/tex]
P[tex]_{o}[/tex] = 10,00,000 / 243
Thus, the formula for any t is P(t) = (10,00,000 / 243) e[tex]ln^{(3)^{t} }[/tex]
Find the doubling time for the bacteria population.
Let X= 10,00,000 / 243 then the equation for the doubling time for the bacteria is
2X = Xe[tex]ln^{(3)t[/tex]
2 = e[tex]ln^{(3)t}[/tex]
t= ln (2) / ln (3) ≈ 0.63094 hours
Or t=37.8564 minutes.
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