Answer:
a) The expression for the number of bacteria is [tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex].
b) There are 2975 bacteria after 3 hours.
c) The rate of growth after 3 hours is about 3365.3 bacteria per hour.
d) A population of 10,000 will be reached after 4.072 hours.
Step-by-step explanation:
a) The population growth of the bacteria culture is described by this ordinary differential equation:
[tex]\frac{dP}{dt} = k\cdot P[/tex] (1)
Where:
[tex]k[/tex] - Rate of proportionality, in [tex]\frac{1}{h}[/tex].
[tex]P[/tex] - Population of the bacteria culture, no unit.
[tex]t[/tex] - Time, in hours.
The solution of this differential equation is:
[tex]P(t) = P_{o}\cdot e^{k\cdot t}[/tex] (2)
Where:
[tex]P_{o}[/tex] - Initial population, no unit.
[tex]P(t)[/tex] - Current population, no unit.
If we know that [tex]P_{o} = 100[/tex], [tex]t = 1\,h[/tex] and [tex]P(t) = 310[/tex], then the rate of proportionality is:
[tex]P(t) = P_{o}\cdot e^{k\cdot t}[/tex]
[tex]\frac{P(t)}{P_{o}} = e^{k\cdot t}[/tex]
[tex]k\cdot t = \ln \frac{P(t)}{P_{o}}[/tex]
[tex]k = \frac{1}{t}\cdot \ln \frac{P(t)}{P_{o}}[/tex]
[tex]k = \frac{1}{1}\cdot \ln \frac{310}{100}[/tex]
[tex]k\approx 1.131\,\frac{1}{h}[/tex]
Hence, the expression for the number of bacteria is [tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex].
b) If we know that [tex]t = 3\,h[/tex], then the number of bacteria is:
[tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex]
[tex]P(3) = 100\cdot e^{1.131\cdot (3)}[/tex]
[tex]P(3) \approx 2975.508[/tex]
There are 2975 bacteria after 3 hours.
c) The rate of growth of the population is represented by (1):
[tex]\frac{dP}{dt} = k\cdot P[/tex]
If we know that [tex]k\approx 1.131\,\frac{1}{h}[/tex] and [tex]P \approx 2975.508[/tex], then the rate of growth after 3 hours:
[tex]\frac{dP}{dt} = \left(1.131\,\frac{1}{h} \right)\cdot (2975.508)[/tex]
[tex]\frac{dP}{dt} = 3365.3\,\frac{1}{h}[/tex]
The rate of growth after 3 hours is about 3365.3 bacteria per hour.
d) If we know that [tex]P(t) = 10000[/tex], then the time associated with the size of the bacteria culture is:
[tex]P(t) = 100\cdot e^{1.131\cdot t}[/tex]
[tex]10000 = 100\cdot e^{1.131\cdot t}[/tex]
[tex]100 = e^{1.131\cdot t}[/tex]
[tex]\ln 100 = 1.131\cdot t[/tex]
[tex]t = \frac{\ln 100}{1.131}[/tex]
[tex]t \approx 4.072\,h[/tex]
A population of 10,000 will be reached after 4.072 hours.
A bus started from Kathmandu and reached khanikhola,26km far from Kathmandu, in one hour. if the bus had uniform acceleration, calculate the final velocity of the bus and acceleration.
Answer:
a = 0.0040 m/s², v = 14.4 m/s.
Step-by-step explanation:
Given that,
The distance between Kathmandu and Khanikhola, d = 26 km = 26000 m
Time, t = 1 hour = 3600 seconds
Let a is the acceleration of the bus. Using second equation of motion,
[tex]d=ut+\dfrac{1}{2}at^2[/tex]
Where
u is the initial speed of the bus, u = 0
So,
[tex]d=\dfrac{1}{2}at^2\\\\a=\dfrac{2d}{t^2}\\\\a=\dfrac{2\times 26000}{(3600)^2}\\\\a=0.0040\ m/s^2[/tex]
Now using first equation of motion.
Final velocity, v = u +at
So,
v = 0+0.0040(3600)
v = 14.4 m/s
Hence, this is the required solution.
which is the correct answer ?
Answer:
11/12 cups
Step-by-step explanation:
2/3+1/4 = ( 2x4 + 3x1 )/( 3x4 ) = ( 8+3 )/12 = 11/12
A number is divisible by 3 if the sum of the digits of the number is divisible by 3.
Suppose that the probability distribution for birth weights is normal with a mean of 120 ounces and a standard deviation of 20 ounces. The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is [ Select ] 68%. The probability that a randomly selected infant has a birth weight between 110 and 130 is [ Select ] 68%.
Answer:
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is 68%.
The probability that a randomly selected infant has a birth weight between 110 and 130 is 38%.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of 120 ounces and a standard deviation of 20 ounces.
This means that [tex]\mu = 120, \sigma = 20[/tex]
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is
p-value of Z when X = 140 subtracted by the p-value of Z when X = 100.
X = 140
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{140 - 120}{20}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.84
X = 100
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{100 - 120}{20}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a p-value of 0.16
0.84 - 0.16 = 0.68
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is 68%.
The probability that a randomly selected infant has a birth weight between 110 and 130
This is the p-value of Z when X = 130 subtracted by the p-value of Z when X = 110.
X = 130
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{130 - 120}{20}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a p-value of 0.69
X = 110
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{110 - 120}{20}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a p-value of 0.31
0.69 - 0.31 = 0.38 = 38%.
The probability that a randomly selected infant has a birth weight between 110 and 130 is 38%.
Identify the transformation that occurs to create the graph of g(x). g(x)=f(x)-7
Answer:
g(x) is obtained by shift the function f(x) down 7 units by subtracting 7 units from f(x).
Step-by-step explanation:
We are given that
[tex]g(x)=f(x)-7[/tex]
We have to identify the transformation that occurs to create the graph of g(x).
To identify the transformation that occurs to create the graph of g(x)
We will subtract the 7 from f(x).
Let f(x) be any function
[tex]g(x)=f(x)-k[/tex]
It means g(x) obtained by shift the function f(x) down k units by subtracting k units from f(x).
Therefore, g(x) is obtained by shift the function f(x) down 7 units by subtracting 7 units from f(x).
The categories of a categorical variable are given along with the observed counts from a sample. The expected counts from a null hypothesis are given in parentheses. Compute the x-test statistic, and use the x-distribution to find the p-value of the test. Category Observed (Expected) A 25 (20) B 35(40) C 50(60) D 90(80) Round your answer for the chi-square statistic to two decimal places, and your answer for the p-value to four decimal places. chi-square statistic = p-value = i
Answer:
χ² = 4.80
Pvalue = 0.1874
Step-by-step explanation:
Given :
Category Observed (Expected)
A 25 (20)
B 35(40)
C 50(60)
D 90(80)
The Chisquare statistic (χ²) is given by :
χ² = Σ(observed - Expected)² / Expected
χ² = (25-20)²/20 + (35-40)/40 + (50-60)²/60 + (90-80)²/80
χ² = 1.25 + 0.625 + 1.67 + 1.25
χ² = 4.795
χ² = 4.80 (2 decimal places)
Using the Chisquare Pvalue calculator :
df = n - 1 = 4 - 1 = 3
Pvalue = 0.1874
A box is 2,5 dm long and 5 dm high its volume is 62.5 dm3 how wide it is?
Answer:
7.5 dm
Step-by-step explanation:
Plus mo baka tama ako
The speed (S) an object falls varies directly with time. If the speed is 49.0m/s after 5 seconds, then what is the speed after 3 seconds
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Answer:
29.4 m/s
Step-by-step explanation:
Speed is proportional to time, so we have ...
speed / time = s/3 = 49/5
s = 3/5(49) = 29.4
The speed of the object is 29.4 m/s after 3 seconds.
Identify the domain of the function shown in the graph.
PLEASEEEE HELPP MEEEE I NEED HELPPPPPPP PLELASEEEEEE I REALLY DONT GET THIS AT ALL I JUST WANNA PAST THE 6th grade
The domain of the function f(x)=-x3+4
Answer:
Domain= {x:x £|R}
|R=any real number
1 point
Use log10 3-0.4771; log10 5 0.699010810 7 0.8451; log10 11 1.0414 to approximate the value of each expression-
log10 14710910 (147)
Answer:
[tex]\log_{10}(147) = 2.1673[/tex]
Step-by-step explanation:
Given
[tex]\log_{10} 3 = 0.4771[/tex]
[tex]\log_{10} 5 = 0.6990[/tex]
[tex]\log_{10} 7= 0.8451[/tex]
[tex]\log_{10} 11 = 1.0414[/tex]
Required
Evaluate [tex]\log_{10}(147)[/tex]
Expand
[tex]\log_{10}(147) = \log_{10}(49 * 3)[/tex]
Further expand
[tex]\log_{10}(147) = \log_{10}(7 * 7 * 3)[/tex]
Apply product rule of logarithm
[tex]\log_{10}(147) = \log_{10}(7) + \log_{10}(7) + \log_{10}(3)[/tex]
Substitute values for log(7) and log(3)
[tex]\log_{10}(147) = 0.8451 + 0.8451 + 0.4771[/tex]
[tex]\log_{10}(147) = 2.1673[/tex]
Find the equivalent exponential expression.
(543
Answer:
(5) we have multiple the powers
help asap! Might be easy for some of you
Answer:
51
Step-by-step explanation:
(-3)^4-5(5)+6(5)÷(-3)(2)
81-25+30÷-6
81-25-5
81-30
51
(Remember order of operations-PEMDAS)
The sum of 9 and c is less than or
equal to 15.
Answer:
less than or equal to -26
Answer:
9+c < 15
OR
c < 6
Step-by-step explanation:
"the sum of 9 and c" means: 9+c
"is less than or equal to 15" means: < 15
If you need to simplify it, then subtract 9 from both sides, and you get
c < 6
I believe the answer is 7% but it says round to the nearest tenth of a percent so I am not sure if it is a decimal answer or not. Can someone help me out please?
Answer: 6.1% decrease
Note: It appears that your teacher doesn't want you to type in the percent sign, as that's already covered for you.
=========================================================
Explanation:
The salary decreased by 51500-48355 = 3145
Divide this over the initial salary to get 3145/51500 = 0.0611 which is approximate.
This converts to the percentage 6.11% and that rounds to 6.1%
----------------
As an alternative, you can use the formula method below
A = old value = 51500
B = new value = 48355
C = percent change when going from A to B
C = [ (B-A)/A ] * 100%
C = [ (48355-51500)/51500 ] * 100%
C = (-3145/51500)*100%
C = -0.0611*100%
C = -6.11%
C = -6.1%
The negative C value indicates a percent decrease.
A rectangular auditorium seats 1144 people. The number of seats in each row exceeds the number of rows by 18. Find the number of seats in each row.
Answer:
44 seats in each row
Problem:
A rectangular auditorium seats 1144 people. The number of seats in each row exceeds the number of rows by 18. Find the number of seats in each row.
Step-by-step explanation:
Let n be the number of rows.
If the number of seats exceed the number of rows by 18, then the number ot seats can be represented by n+18.
So we have a n by n+18 rectangle whose number of seats in all is 1144.
So we need to solve n(n+18)=1144
Distribute: n^2+18n=1144
Subtract 1144 on both sides" n^2+18n-1144=0
What two numbers multiply to be -1144 but also add to be 18?
Hmmm.. let's break -1144 down a little into smaller factors.
-1144=2(-572)=4(-286)=8(-143)=-8(13)(11)=-26(44)
We found a pair of factors that will work? -26 and 44.
So the factorization of our quadratic equation is (n-26)(n+44)=0.
This implies either n-26=0 or n+44=0 .
n=26 by adding 26 on both sides for first equation.
n=-44 by subtracting 44 on both sides for second equation.
n=26 is the only one that works.
This means there are 26 rows and 26+18 seats in each row.
26 rows
44 seats in each row
That product does equal 1144 seats in all.
help? haha
solve the equation below:)
3x - 5 = 10 + 2x
Step-by-step explanation:
3x-2x=5+10 [taking variables on one side and constant on other]
x=15
soln:
3x-5= 2x+10
3x -5+5=2x+10+5 [ adding 5 on both side]
3x=2x+15
3x-2x=2x+15-2x [subtracting 2x on both side]
x=15
Ans=15
Answer:
[tex]x = 15[/tex]
Step-by-step explanation:
[tex]3x - 5 = 10 + 2x[/tex]
[tex]3x - 2x = 10 + 5[/tex]
[tex]1x = 15[/tex]
[tex]x = 15[/tex]
Hope it is helpful.....Construct the discrete probability distribution for the random variable described. Express the probabilities as simplified fractions. The number of tails in 5 tosses of a coin.
Answer:
[tex]P(X = 0) = 0.03125[/tex]
[tex]P(X = 1) = 0.15625[/tex]
[tex]P(X = 2) = 0.3125[/tex]
[tex]P(X = 3) = 0.3125[/tex]
[tex]P(X = 4) = 0.15625[/tex]
[tex]P(X = 5) = 0.03125[/tex]
Step-by-step explanation:
For each toss, there are only two possible outcomes. Either it is tails, or it is not. The probability of a toss resulting in tails is independent of any other toss, which means that the binomial probability distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Fair coin:
Equally as likely to be heads or tails, so [tex]p = 0.5[/tex]
5 tosses:
This means that [tex]n = 5[/tex]
Probability distribution:
Probability of each outcome, so:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{5,0}.(0.5)^{0}.(0.5)^{5} = 0.03125[/tex]
[tex]P(X = 1) = C_{5,1}.(0.5)^{1}.(0.5)^{4} = 0.15625[/tex]
[tex]P(X = 2) = C_{5,2}.(0.5)^{2}.(0.5)^{3} = 0.3125[/tex]
[tex]P(X = 3) = C_{5,3}.(0.5)^{3}.(0.5)^{2} = 0.3125[/tex]
[tex]P(X = 4) = C_{5,4}.(0.5)^{4}.(0.5)^{1} = 0.15625[/tex]
[tex]P(X = 5) = C_{5,5}.(0.5)^{5}.(0.5)^{0} = 0.03125[/tex]
a rectangle box has length 12 inches, width 15 inches, and a height of 17 inches. Find the angle between the diagonal of the box and the diagonal of its base. The angle should be measured in radiands
Answer:
0.7246 radians
Step-by-step explanation:
According to the Question,
Given that, a rectangle box has length 12 inches, width 15 inches, and a height of 17 inches
The length of the base diagonal (d) can be found using the Pythagorean theorem on length and width:d = √{ (12)² +(15)² } = √(144+225) = √369inches
The tangent of the angle is the ratio of the height of the box to this lengthTan∅ = 17/√369
Taking the [tex]Tan^{-1}[/tex] , we have
∅ = [tex]Tan^{-1}[/tex](17/√369) ≈ 0.7246 radians
What is the factored form of x2 − 4x − 5?
(x + 5)(x − 1)
(x + 5)(x + 1)
(x − 5)(x − 1)
(x − 5)(x + 1)
Answer:
x2 - 4x - 5 factored form is (x - 5)(x + 1)
Answer:
(x − 5)(x + 1)
Step-by-step explanation:
The answer above is correct.
Johnny tripled his baseball card collection. Then he added 6 more cards to the collection. Now he has 24 cards. How many cards did he start with?
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Answer:
6
Step-by-step explanation:
Work backward.
If he has 24 after adding 6, he had 18 before that addition.
If he had 18 after tripling his collection, he had 18/3 = 6 cards to start with.
__
Note that this is the same process you would use if you started with an equation.
3c +6 = 24 . . . . where c is the number of cards Johnny started with
3c = 24 -6 = 18 . . . . . subtract 6 from the final number
c = 18/3 = 6 . . . . . . . . divide the tripled value by 3 to see the original value
Johnny started with 6 cards.
Let Y1 and Y2 denote the proportions of time (out of one workday) during which employees I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of Y1 and Y2 is modeled by the density function.
f (y 1,y2)=y 1+y 2 o<=y 1<=1, 0<=y2<=1(0 elsewhere)
a. Find P (Y1< 1/2,y2>1/4)
b. Find P(Y 1+Y2<=1)
Are Y1 and Y2 independent?
(a) The region Y₁ < 1/2 and Y₂ > 1/4 corresponds to the rectangle,
{(y₁, y₂) : 0 ≤ y₁ < 1/2 and 1/4 < y₂ ≤ 1}
Integrate the joint density over this region:
[tex]P\left(Y_1<\dfrac12,Y_2>\dfrac14\right) = \displaystyle\int_0^{\frac12}\int_{\frac14}^1 (y_1+y_2)\,\mathrm dy_2\,\mathrm dy_1 = \boxed{\dfrac{21}{64}}[/tex]
(b) The line Y₁ + Y₂ = 1 cuts the support in half into a triangular region,
{(y₁, y₂) : 0 ≤ y₁ < 1 and 0 < y₂ ≤ 1 - y₁}
Integrate to get the probability:
[tex]P(Y_1+Y_2\le1) = \displaystyle\int_0^1\int_0^{1-y_1}(y_1+y_2)\,\mathrm dy_2\,\mathrm dy_1 = \boxed{\dfrac13}[/tex]
Y₁ and Y₂ are not independent because
P(Y₁ = y₁, Y₂ = y₂) ≠ P(Y₁ = y₁) P(Y₂ = y₂)
To see this, compute the marginal densities of Y₁ and Y₂.
[tex]P(Y_1=y_1) = \displaystyle\int_0^1 f(y_1,y_2)\,\mathrm dy_2 = \begin{cases}\frac{2y_1+1}2&\text{if }0\le y_1\le1\\0&\text{otherwise}\end{cases}[/tex]
[tex]P(Y_2=y_2) = \displaystyle\int_0^1 f(y_1,y_2)\,\mathrm dy_1 = \begin{cases}\frac{2y_2+1}2&\text{if }0\le y_2\le1\\0&\text{otherwise}\end{cases}[/tex]
[tex]\implies P(Y_1=y_1)P(Y_2=y_2) = \begin{cases}\frac{(2y_1+1)(2y_2_1)}4&\text{if }0\le y_1\le1,0\ley_2\le1\\0&\text{otherwise}\end{cases}[/tex]
but this clearly does not match the joint density.
Mary Katherine has a bag of 3 red apples , 5 yellow apples and 4 green apples , Mary takes a red apples out of the bag and does not replace it. What is the probability that the next apple she takes out is yellow
Answer:
5/11.... you put the 5 which is yellow over the others which is 12 but remember she removed 1 so it would be equal to 11
Answer:
ok so if she takes a red apple out that means
2 red
5 yellow
4 green
11 in total
so 5/11
The answer is D
Hope This Helps!!!
A flashlight is projecting a triangle onto a wall, as shown below.
A picture shows a flashlight projecting a triangle onto a wall. The original triangle and its projection are similar. The original triangle has 2 sides labeled 15 and one side labeled 20. The projected triangle has two sides labeled 30 and one side labeled n. The triangles have congruent angles.
The original triangle and its projection are similar. What is the missing length n on the projection?
Answer:
Hence the correct option is 3rd option. 40
Step-by-step explanation:
If two figures are similar, then the ratio of the corresponding sides is proportional.
[tex]\frac{15}{30} =\frac{20}{n} \\\\n=\frac{30 \times 20}{15} \\\\n= 40.[/tex]
HELP PLSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS
Answer:
12
Step-by-step explanation:
10 - 1/2 x = 12-4/3x
60 - 3x = 72-2x
-12 = - x
Find the inverse of the given function. (pictured below)
Answer:
4
3
0
Step-by-step explanation:
f(x) = y = -1/2 × sqrt(x+3)
2y = -sqrt(x+3)
4y² = x + 3
x = 4y² - 3
now renaming this, so that the normal symbols and names are used for this function definition, so that the input variable is called "x" :
f-1(x) = 4x² - 3
basically, just by itself, this function would be defined for all possible real values of x.
but because it is the inverse of the original function, which generates only values of y<=0, then for the inverse function that same range applies for its input variable x
x<=0
Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. Together they charged a total of $1125. What was the rate charged per hour by each mechanic if the sum of the two rates was $140 per hour?
Answer:
The first mechanic charged $ 85 an hour, and the second mechanic charged $ 55 an hour.
Step-by-step explanation:
Given that two mechanics worked on a car, and the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours, and together they charged a total of $ 1125, to determine what was the rate charged per hour by each mechanic if the sum of the two rates was $ 140 per hour, the following calculation must be performed:
1125/15 = X
75 = X
80 x 10 + 60 x 5 = 800 + 300 = 1100
85 x 10 + 55 x 5 = 850 + 275 = 1125
Therefore, the first mechanic charged $ 85 an hour, and the second mechanic charged $ 55 an hour.
.It is 12:00 and people are lining up for the matinee at the Bijou Cinema Six. In the first five minutes (12:05), 6 people get into line. At the end of the second five minutes (12:10), there are 11 people in line. At the end of the third five minutes (12:15) there are 16 people in line. If the people keep lining up at this rate, what time will it be when there are 81 people in line?
Answer:
65 minutes
Step-by-step explanation:
already 16 people in line
total is 81
81 - 16 equals 65
about 5 people every 5 minutes (basically 1 min per person)
so, answer probably 65
Factorize : 4(x+y)^2 -9(x-y)^2
Answer:
Step-by-step explanation:
[tex]4(x+y)^{2} - 9(x-y)^{2}=4[x^{2}+2xy+y^{2}]-9[x^{2}-2xy+y^{2}]\\\\=4x^{2}+4*2xy + 4y^{2}-9x^{2}-2xy*(-9)+y^{2}*(-9)\\\\= 4x^{2}+8xy+4y^{2}-9x^{2}+18xy-9y^{2}\\\\= 4x^{2}-9x^{2} + 8xy + 18xy +4y^{2} - 9y^{2}\\\\= -5x^{2} + 26xy - 5y^{2}[/tex]
= -5x² + 25xy + xy - 5y²
= 5x(-x + 5y) - y(-x +5y)
= (-x + 5y)(5x - y)