At any time t ≥ 0, in hours, the rate of growth of a population of bacteria is given by dy/dt = 0.5y. Initially, there are 200 bacteria.
(a) Solve for y, the number of bacteria present, at any time t ≥ 0.
(b) Write and evaluate an expression to find the average number of bacteria in the population for 0 ≤ t ≤ 10.
(c) Write an expression that gives the average rate of bacteria growth over the first 10 hours of growth. Indicate units of measure.
Answer:
(a) The equation for the number of bacteria at time t ≥ 0 is [tex]y = 200\cdot e^{0.5 \cdot t }[/tex]
(b) The average population in the bacteria for 0 ≤ t ≤ 10 is 29,482 bacteria
(c) The growth rate of the bacteria in the first 10 hours is 14,841 Bacteria/hour
Step-by-step explanation:
(a) The given rate of growth of the bacteria population is;
[tex]\dfrac{dy}{dt} = 0.5 \cdot y[/tex]
The initial amount of bacteria = 200
At time, t ≥ 0, we have;
[tex]\dfrac{dy}{y} = 0.5 \cdot dt[/tex]
[tex]\int\limits {\dfrac{dy}{y} } = \int\limits {0.5} \, dt[/tex]
ln(y) = 0.5·t + C
[tex]y = e^{0.5 \cdot t + C}[/tex]
[tex]y = C_1 \cdot e^{0.5 \cdot t }[/tex]
When, t = 0, y = 200, we have;
[tex]200 = C_1 \cdot e^{0.5 \times 0 } = C_1[/tex]
C₁ = 200
The equation for the number of bacteria at time t ≥ 0 is therefore given as follows;
[tex]y = 200\cdot e^{0.5 \cdot t }[/tex]
(b) The expression for the average number of bacteria in the population for 0 ≤ t ≤ 10 is given as follows;
[tex]y = \int\limits^t_0 {0.5 \cdot y} \, dt[/tex]
[tex]y = \int\limits^{10}_0 {0.5 \cdot 200\cdot e^{0.5 \cdot t }} \, dt = \left[200\cdot e^{0.5 \cdot t }\right]^{10}_0 \approx 2,9682.6 - 200 = 29,482.6[/tex]
The average population in the bacteria for 0 ≤ t ≤ 10 is therefore, y = 29,482
(c) The average rate of growth after 10 hours of growth is given as follows;
[tex]\dfrac{dy}{dt} = 0.5 \cdot y[/tex]
[tex]\dfrac{dy}{dt} = 0.5 \cdot 200\cdot e^{0.5 \cdot 10 } = 14,841.32 \ Bacteria/hour[/tex]
The growth rate of the bacteria in the first 10 hours is 14,841 Bacteria/hour
Population functions can either represent growth or decays.
The given parameter is:
[tex]\frac{dy}{dt}= 0.5y[/tex]
(a) Solve for y
We have:
[tex]\frac{dy}{dt}= 0.5y[/tex]
Integrate both sides
[tex]\int \frac{dy}{dt}= \int 0.5y[/tex]
Divide both sides by y, and multiply both sides by dt
[tex]\int \frac{dy}{y}= \int 0.5\ dt[/tex]
Integrate both sides of the equation
[tex]\ln(y) = 0.5t +c[/tex]
Take the exponent of both sides
[tex]y = e^{0.5t +c}[/tex]
This gives
[tex]y = ce^{0.5t}[/tex]
When t = 0, and y = 200.
So, we have:
[tex]200 = ce^{0.5 \times 0}[/tex]
[tex]200 = ce^{0}[/tex]
Evaluate the exponent
[tex]200 = c[/tex]
Rewrite as:
[tex]c = 200[/tex]
Substitute 200 for c in [tex]y = ce^{0.5t}[/tex]
[tex]y = 200e^{0.5t}[/tex]
Hence, the equation for y is [tex]y = 200e^{0.5t}[/tex]
(b) The average number of bacterial for 0 ≤ t ≤ 10.
The average value of the function is calculated as:
[tex]y' = y(10) - y(0)[/tex]
Calculate y(0) and y(10)
[tex]y = 200e^{0.5t}[/tex]
[tex]y(0) = 200e^{0.5(0)}= 200[/tex]
[tex]y(10) = 200e^{0.5\times 10} =29682.6318205[/tex]
So, we have
[tex]y' = y(10) - y(0)[/tex]
[tex]y' = 29682.6318205 - 200[/tex]
[tex]y' = 29482.6318205}[/tex]
[tex]y' = 29483[/tex]
Hence, The average number of bacterial for 0 ≤ t ≤ 10 is 29483
(c) The average rate over the first 10 hours of growth
We have:
[tex]\frac{dy}{dt}= 0.5y[/tex]
Substitute [tex]y = 200e^{0.5t}[/tex]
[tex]\frac{dy}{dt}= 0.5 \times 200e^{0.5t}[/tex]
[tex]\frac{dy}{dt}= 100e^{0.5t}[/tex]
When t = 10, we have:
[tex]\frac{dy}{dt}= 100e^{0.5 \times 10}[/tex]
[tex]\frac{dy}{dt}= 100e^{5}[/tex]
[tex]\frac{dy}{dt}= 14841[/tex]
Hence, the average rate over the first 10 hours of growth is 14841
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8x + 5y = 27.15
what is x and y?
Answer:
Step-by-step explanation:
Simplifying
8x + 5y = 27.15
Solving
8x + 5y = 27.15
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-5y' to each side of the equation.
8x + 5y + -5y = 27.15 + -5y
Combine like terms: 5y + -5y = 0
8x + 0 = 27.15 + -5y
8x = 27.15 + -5y
Divide each side by '8'.
x = 3.39375 + -0.625y
Simplifying
x = 3.39375 + -0.625y
Answer:
Infinite solutions
Step-by-step explanation:
There is no one right answer, plug in numbers for x and you will see
y = -8/5(x) + 27.15/5
plz amswer this for me
Answer:
$5
Step-by-step explanation:
$14 x 30% (0.30) = $4.2
$14 + $4.2 = $18.2
[tex]\frac{18.2}{4}[/tex] = 4.55
Round to the nearest dollar:
4.55 = 5
Take 85 POINTS!!!! PLEASE HELP JUST LOOK THE PICTURE
Answer: the person above has it
Step-by-step explanation:
NEED HELP!! ASAPPP PLEASEEE 20 POINT Which values are the solutions to the inequality below?
Answer:
81 is the right answer of your question
A rectangle’s length is three times its width, w. Its area is 243 square units. Which equation can be used to find the width of the rectangle?
w2 = 3(243)
3w2 = 243
4w2 = 243
3w2 = 3(243)
Answer:
3w² = 243
Step-by-step explanation:
Width = w
Length = 3w
Area = 243 square units
length *width = 243
3w * w = 243
3w² = 243
The equation that you can used to determine the width of the rectangle with the given area is: 3w² = 243
What is the Area of a Rectangle?Area = (length)(width)
The given dimensions are:
Width = w unitsLength = 3w unitsArea of rectangle = 343 sq. unitsThe equation to se in finding the width (w) of the rectangle will be:
3w(w) = 243
3w² = 243
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Evaluate the algebraic expression 3f + g,when f = 6, g = 8, h = 12 and j = 2. A. 30 B. 26 C. 17 D. 13
Answer: A.26
Step-by-step explanation:
= 3f+g
= 3 (6)+8
=26
2. For a school project, Terence is taking a survey of students' favorite type
of shoe. Which sampling method would likely provide the representative
sample of the students? *
O Survey 10 students in a math class.
O
Survey only the students wearing sandals.
Survey 20 students who play basketball
Survey every 4th student who walks into the cafeteria.
Survey every 4th student who walks into the cafeteria.
if p is the smallest of four consecutive even integers what is their sum interms of p.
Answer:
4p + 12
Step-by-step explanation:
p + p+2 + p+4 + p+6 = 4p + 12
if 24 golf balls weigh 2 1/4 pounds, how much would 60 golf balls weigh
Answer:
5.625 pounds or 5 5/8 pounds
Step-by-step explanation:
(2 1/4)/24 = weight of one golf ball = 0.09375
0.09375 x 60 = 5.625 pounds or 5 5/8 pounds
Answer:
5.625 pounds or 5 5/8 pounds as a fraction
Step-by-step explanation:
PLZZZZ HELP ME WITH THIS QUESTION WITH AN EXPLANATION
Answer :
Given : A and B are complementary angles. B and C are complementary angles.
Prove : A C
I have attached a picture of me doing it. Hope this helps, thank you :) !!
XYZ is a triangle in which XY = 11 centimeters and YZ = 15 centimeters. The length of XZ is in whole centimeters and is greater than 20 centimeters. What are the possible lengths of XZ?
Answer:
Possible lengths of [tex]XZ[/tex] are [tex]21\,\,cm,22\,\,cm,23\,\,cm,24\,\,cm,25\,\,cm[/tex]
Step-by-step explanation:
According to triangle inequality theorem,
length of one side is less than the sum of remaining two sides.
Also,
[tex]XY=11\,\,cm\\YZ=15\,\,cm\\XZ>20\,\,cm[/tex]
Therefore,
[tex]20<XZ<15+11\\20<XZ<26[/tex]
The length of [tex]XZ[/tex] is in whole centimeters.
Therefore, possible lengths of [tex]XZ[/tex] are [tex]21\,\,cm,22\,\,cm,23\,\,cm,24\,\,cm,25\,\,cm[/tex]
Jim and Jennifer have excellent credit, and they purchased property with a mortgage. However, they experienced financial difficulty and missed more than three months of payments. Which of the following is the most probable consequence of Jim and Jennifer's late payments?
Answer:
The bank could repossess the property.
Step-by-step explanation:
Took the exam.
Answer:
The bank will reposses
Step-by-step explanation:
What is the answer this question is hard answer fast plz
Answer:
Step-by-step explanation:
I believe it is b
Carlo puts tins into small boxes and into large boxes. He puts 6 tins into each small box. He puts 20 tins into each large box. Carlo puts a total of 3000 tins into the boxes so that number of tins in small boxes : number of tins in large boxes = 2:3 Carlo says that less than 30% of the boxes filled with tins are large boxes. Is Carlo correct? You must show all your working.
Answer: Carlo is incorrect
Step-by-step explanation:
3000/(2+3) = 600 (each ratio)
2:3 = 2(600):3(600) = 1200:1800
1200/6 = 200 small boxes
1800/20 = 90 large boxes
290 boxes altogether (large and small)
(large boxes/boxes altogether)x100
(90/290)x100 = 31%
31%>30%
Therefore Carlo is incorrect
the mean of a group of students in 60 . if their marks 85 ,65,36 ,48 , X ,75 ,39,72 then find the value of x
Answer: 60
Step-by-step explanation:
Find the total number of students.
In this case, there are 8 total students. Multiply 60 * 8 = 480
Add all the numbers up.
85 + 65 + 36 + 48 + 75 + 39 + 72 = 420
480 - 420 = 60
x = 60
D) 9. What is the solution to this equation?
2x + 10 = 28
Answer:
X=9
Step-by-step explanation:
I know this because you can find the answer by doing 28 - 10 which equals 18 and 2 times 9 gets you 18 and would make the equation
( 2 times 9) + 10= 28
18 + 10=28
Hi i have this class work of my cousing
but i need some help beacause i forgot this
the first question is 5_8
Answer:
5 < 8
-4 < 0
0 < 4
-5 < 5
-9 > -19
-5 > -8
Step-by-step explanation:
Remember, the "Alligator" or better known as the "Packman" symbol ( > < )
It will always want to "eat" the bigger number. so, 8 is bigger than five, so the "Packman" will want to go after the bigger number. 5 < 8
please help, fill in the 5 blanks
A manufacturer earns Rs4500 in the first month and Rs 6000 in second month. On plotting these points a manufacturer observes a linear function. Find the linear function that fits in data?
Answer:
[tex]f(x) = 1500x +3000[/tex]
Step-by-step explanation:
Given
Represent the months with x and the earnings with y.
So:
[tex](x_1,y_1) = (1,4500)[/tex]
[tex](x_2,y_2) = (2,6000)[/tex]
Required
Determine the linear function
First, calculate slope (m)
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]m = \frac{6000 - 4500}{2 - 1}[/tex]
[tex]m = \frac{1500}{1}[/tex]
[tex]m = 1500[/tex]
The equation is then calculated using:
[tex]y = m(x - x_1) + y_1[/tex]
[tex]y = 1500(x - 1) + 4500[/tex]
[tex]y = 1500x - 1500 + 4500[/tex]
[tex]y = 1500x +3000[/tex]
As a function, we have:
[tex]f(x) = 1500x +3000[/tex]
Two numbers have a sum of 42. One number is 2 more than 3 times the other number. Determine both of the numbers using an algebraic solution.
Answer:
8 and 4
Step-by-step explanation: have a good day g
I need help with these questions (see image). Please show workings.
Answer:
Inscribed angle is half of the intercepted arc.Central angle is the same size as the intercepted arc.As per above two statements we have:
y = 2*58° = 116°and
z = 90°/2 = 45°Answer:
y=58×2=116 central angle of the cyclic quadrilateral is double if inscribed angle
z=90/2=45
inscribed angle is half of central angle of the cyclic quadrilateral
There are 10 sweets in a bag
4 are red, 2 are green, 3 are yellow,1 is purple
The table shows the cost of a vacation package for a given number of people. The rate of change is constant in the table. Find the rate of change. Explain what the rate of change means for the situation.
Answer:
A. The cost is $45 per person.
Step-by-step explanation:
Divide $90 by 2 persons. It gives you $45 and so you could multiply $45 by x number of persons to see the total cost.
In ANOP, the measure of ZP=90°, NO = 72 feet, and PN = 43 feet. Find the measure
of 20 to the nearest degree.
Answer:
37
Step-by-step explanation:
Sin O = Opposite/Hypotenuse = 43/72
O = [tex]Sin^{-1}[/tex](43/72)
O = 36.671 ≅ 37
Answer:
37
Step-by-step explanation:
Can someone help me please
Another word one! Please help only if you know, thank you!
Answer:
Step-by-step explanation:
if C is the right angle,
AB is the hypotenuse
AB^2=AC^2+BC^2
AC^2=13^2-5^2
AC^2=169-25=144
AC=[tex]\sqrt{144}[/tex]=12
AC=12
Here is an inequality: -2x>10.
List some values for x that would make this inequality true.
How are the solutions to the inequality -2x10 different from the solutions to -2x>10? Explain your reasoning.
Answer:
The difference is -2x10 is basically-2 times x times 10.
Step-by-step explanation:
Cuz i said so
The x values that makes the given inequality true are -10, -9, -8, -7 and -6.
What are inequalities?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
The given inequality is -2x>10.
Divide -2 on both the sides of an inequality, we get
x<-5
So, the solution is {.......-10, -9, -8, -7, -6}
Hence, the x values that makes the given inequality true are -10, -9, -8, -7 and -6.
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help please i rlly need it
A stack of paper contains 400 sheets. The volume of the stack is 140.25 cubic inches. Each sheet of paper is identical, with a length of 11 inches and a width of 8.5 inches. Find the height of each sheet of paper. Please put the process.
Answer:
[tex]0.00375\ \text{inches}[/tex]
Step-by-step explanation:
Volume of the stack of paper is 140.25 cubic inches
The length and width of the stack will be equal to that of a single sheet of paper so
[tex]l=11\ \text{inches}[/tex]
[tex]w=8.5\ \text{inches}[/tex]
Let [tex]h[/tex] be the height of the stack.
Volume of the stack would be
[tex]lwh=140.25\\\Rightarrow h=\dfrac{140.25}{lw}\\\Rightarrow h=\dfrac{140.25}{11\times 8.5}\\\Rightarrow h=1.5\ \text{inches}[/tex]
So the stack has a height of 1.5 inches. Now there are 400 sheet of paper in the stack.
Thickness of one sheet of paper is given by
[tex]\dfrac{1.5}{400}=0.00375\ \text{inches}[/tex]
Hence, the height of each sheet of paper is [tex]0.00375\ \text{inches}[/tex].