Answer:
3.25 h = 3 h 15 m
Step-by-step explanation:
156 mi * 1h/48 mi = 3.25 h = 3 h 15 m
Answer: 195 min or 3 hr and 15 min
Step-by-step explanation:
We can set up a proportion to solve this problem.
[tex]\frac{48mi}{60 min} =\frac{156 mi}{x}[/tex]
[tex]48x=156*60[/tex]
[tex]48x=9360[/tex]
[tex]x=195 min[/tex]
We can also write this in terms of hours and minutes.
[tex]\frac{60 min}{1 hr} =\frac{195 min}{x}[/tex]
[tex]60x=195[/tex]
[tex]x=3.25[/tex]
3 hr and 15 min
If 3^x=10, what is the value of 3^x-3? A. 10/3 B. 10/9 C. 10/27 D. 27/10
Answer:
B.
Step-by-step explanation:
Recall the properties of logs and once you find the value of x, substitute it into the equation.
Option C ([tex]\frac{10}{27}[/tex]) is the correct alternative.
Given expression is:
→ [tex]3^x = 10[/tex]
By taking log both sides, we get
→ [tex]ln \ 3^x = ln \ 10[/tex]
[tex]x \ ln \ 3= ln \ 10[/tex]
[tex]x = \frac{ln \ 10}{ln \ 3}[/tex]
By substituting the value of "x" in "[tex]3^{x-3}[/tex]", we get
→ [tex]3^{x-3}[/tex] = [tex]3^{\frac{ln \ 10}{ln \ 3} -3}[/tex]
= [tex]\frac{10}{27}[/tex]
So the above is the right solution.
Learn more:
https://brainly.com/question/47626
What else would need to be congruent to show that ABC PQR by SSS?
Answer:
A
Step-by-step explanation:
The SSS is the side-side-side way to prove the triangles are congruent, so each side needs to have a match between triangle 1 and triangle 2. Since AC and BC are congruent to sides, that leaves AC. Hope this helps :)
To prove that both triangles are congruent, the additional information needed is: A. AB ≅ PQ.
What is the SSS Congruence Theorem?Two triangles with three pairs of corresponding congruent sides can be proven to be congruent to each other based on the SSS congruence theorem.
We are already given that two pairs of corresponding sides of the two triangles are congruent. To prove that triangles ABC and PQR are congruent by SSS, the additional information needed is: A. AB ≅ PQ.
Learn more about SSS congruence theorem on:
https://brainly.com/question/2102943
#SPJ5
The zeros of a function P(x)=x2-2x-24
Answer:
x = -4 and x = 6
Step-by-step explanation:
Factor using AC method.
1 × -24 = -24
Factors of -24 that add up to -2 are +4 and -6.
P(x) = (x + 4) (x − 6)
Therefore, zeroes of P(x) are x = -4 and 6.
Find the number of real number solutions for the equation.
x2- 6=0
Answer:
x=3
Step-by-step explanation:
2x-6=0
2x=0+6
2x=6
2x/2=6/2
x=3
A football team consists of 18 freshmen and 18 sophomores, 15 juniors, and 12 seniors. Four players are selected at random to serve as captains. Find the probability that at least 1 of the students is a senior.
Answer:
58.05% probability that at least 1 of the students is a senior.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The order in which the captains are chosen is not important. So we use the combinations formula to solve this question.
Combinations formula:
A probability is the number of desired outcomes divided by the number of total outcomes.
In this question:
Either no seniors are captain, or at least one is. The sum of these probabilities is 100%. The easier way to solve this question is finding the probability of no seniors being captain, and subtratcing 100 from this. So
Probability of no seniors being captain:
Desired outcomes:
Four captains, from a set of 18 + 18 + 15 = 51. So
[tex]D = C_{51,4} = \frac{51!}{4!(51-4)!} = 249900[/tex]
Total outcomes:
Four captains from a set of 18 + 18 + 15 + 12 = 63. So
[tex]T = C_{63,4} = \frac{63!}{4!(63-4)!} = 595665[/tex]
Probability:
[tex]p = \frac{D}{T} = \frac{249900}{595665} = 0.4195[/tex]
41.95% probability of no seniors being captains.
Find the probability that at least 1 of the students is a senior.
p + 41.95 = 100
p = 58.05
58.05% probability that at least 1 of the students is a senior.
After Halloween, a variety store had some costumes regularly priced at $20.50 on sale for 20% off the regular price. A few weeks later the costumes that hadn’t sold were reduced an additional 60% off the sale price. What was the final selling price of the remaining costumes? The final selling price of costumes was $
What is the measure of
Answer: C. 84
Step-by-step explanation:
In triangles ABD y BCD:
- AD=CD
- Angle BAD = angle BCD
- BD common side
THEN the triangles are equal because they have two sides and the angle opposite the longest side respectively equal.
CBD = ABD = 42 because the triangles ABD y BCD are equal
ABC = CBD+ABD = 42+42= 84
Answer:
C. 84 degrees
Step-by-step explanation:
Find the slope of the line on the graph.
Write your answer as a fraction or a whole
number, not a mixed number or decimal.
Answer:
-3
Step-by-step explanation:
We can see that the line passes through (0,3) and (1,0)
So using rise over run we have
-3/1
or
-3
Brent is designing a poster that has an area of 1 sq.ft. He is going to paste a photo collage on a section of the poster that is 1⁄3 foot wide and 3⁄5 foot long. What part of a square foot will the photo collage cover?
Answer:
[tex]\dfrac{1}{5} $ square foot.[/tex]
Step-by-step explanation:
Area of the Poster =1 sq.ft.
The collage will cover a part of the poster that is [tex]\dfrac{1}{3} ft.$ wide and \dfrac{3}{5} ft.$ long.[/tex]
Area of the Photo Collage Section
=Length X Width
[tex]=\dfrac{1}{3} X \dfrac{3}{5} \\\\=\dfrac{1}{5} $ square foot.[/tex]
[tex]\text{Therefore, the photo collage will cover }\dfrac{1}{5} $ of a square foot.[/tex]
-3(7+5)
What is the value of
+9(2)?
32
-6
O-2
0 14
O 22
Step-by-step explanation:
-3(7 + 5) + 9(2)
-3(12) + 18
-36 + 18
= - 18
In my opinion this should be the answer
Answer:
14
Step-by-step explanation:
Your welcome!
A new American graduate is contemplating buying a Japanese, German, or an American car. No matter the type of car, he plans to buy a new one at the end of 8 years. Japanese car will cost $30,000 and have a fuel usage of 23 Miles Per gallon (mpg) for the first 2 years, and will decrease by 3% per year thereafter. Repair cost will start at $700 per year, and increase by 3% per year. At the end of year 8, the car can be sold for $5000. Insurance cost will be $700 for the first year, increasing by 2% per year thereafter. A German car will cost $45,000 and have fuel usage of 21mpg for the first 5 years, and decrease by 1% thereafter to year 8. Repair cost will start at $1000 in year 1 and increase by 4% per year. It will have a salvage value of $7000 at the end of year 8. Insurance cost will be $850 the first year, increasing by 2% per year thereafter. The American car will cost $35,000 and have fuel usage of 20mpg for the first 3 years, and will decrease by 3% per year thereafter. Repair cost will be $800 in year 1, increasing by 4% per year thereafter. Being an American, the graduate will price the pride of owning an American car at $0.4 for every 20 miles driven, increasing by 2% per year. Insurance cost will be $800 per year increasing by 2.2% per year. The car can be sold for $5500 at the end of year 8. If the graduate anticipates driving 150000 miles by the end of year 8 and the average interest rate is expected to remain at 5% per year, which car is economically affordable based on present worth analysis? Assume fuel cost will be $3 per gallon in year 1 and increase by an average of 2% per year. Show all your workings.
Answer:
The best option is to buy Japanese Car.
Step-by-step explanation:
Fuel usage per year is 150000/ 8 = 18750 miles per year
Fuel cost (year 1 -8) = $3.0, $3.06, $3.12, $3.18, $3.25, $3.312, $3.38, $3.5
Japanese Car:
Fuel usage 18750 / 23 = 815 * $3 = $2446
Fuel charges (year 1 -8) = $2445, $2494, $2623, $2758. $2900, $3050, $3207, $3372
Repair Cost (year 1 - 8) = $700, $721, $742, $764, $787, $811, $835, $860
Insurance cost (Year 1 - 8) = $700, $714, $728, $742, $757, $772, $788, $804
Present value of cost at 5% = 24674.07
Cost of car is $30,000
Total cost = $54674.07
Amercian Car:
Cost $35,000
Fuel usage 18750/20 = 937.5 * $3 per gallon = $2812.5.
Fuel charges (year 1 -8) = $2812, $2913, $2986, $3011. $3098, $3124, $3176, $3208
Repair Cost (year 1 - 8) = $800, $894, $921, $978, $1109, $1176, $1207, $1301
Insurance cost (Year 1 - 8) = $800, $827, $876, $898, $908, $932, $954, $934
Present value of cost at 5% = 25302.18
Cost of car is $35,000
Total cost = $60302.
German Car:
Cost = $45,000
Fuel usage 18750 / 21 = 892 * $3 = $2678
Fuel charges (year 1 -8) = $2679, $2732, $2786, $2842. $2899, $2987, $3077, $3171
Repair Cost (year 1 - 8) = $1000, $1040, $1081, $1124, $1169, $1216, $1265, $1316
Insurance cost (Year 1 - 8) = $850, $867, $884, $902, $920, $938, $957, $976
Present value of cost at 5% = 27105.73
Cost of car is $45,000
Total cost = $72105.
Kathleen already owns 1 hair band, and additional hair bands are priced at 3 for a dollar.
Write an equation that shows the relationship between the money spent on additional hair
bands d and the total number of hair bands h.
Write your answer as an equation with h first, followed by an equals sign.
Submit
Not feeling ready yet? These can help:
Answer:
h= 1+3d
Step-by-step explanation:
total amount of hairbands equals the 1 hairband she has + 3 times how ever many more she decides to buy
For the results of a certain survey, the estimated probability of being a car owner was 0.77 for males and 0.62 for females. Express each of these as a conditional probability. The probability of being a car owner (CO ) was 0.77 for males (M). Express this as a conditional probability.
Answer:
P(CO|M)=0.77 for male
and P(CO|M^c)=0.62 for female
Step-by-step explanation:
When we talk of conditional probability, we refer to the measure of the probability of an event given that another event has occurred
Here, we are to express the results of the survey as a conditional probability.
For the male part of the question, the conditional probability means that, given that the person is a male, the probability that he is a car owner would be;
P(CO|M)=0.77
For the female part of the question, what we have as a conditional probability is that if a person is a female that she is a car owner is;
P(CO|M^c)=0.62
Please kindly understand that the term M^c refers to M complement in words
Jackie's broker charges her a commission of $8 per ten share of stock she buys or sells. If jackie buys 77 shares of Alesia insurance at 10.34 per share and 120 shares of Ergar Appliances at 8.11 per share, how much will Jackie spend on commission?
Answer:
$157.60
Step-by-step explanation:
77 shares + 120 shares = 197 total shares
197/10 = 19.7
19.7 x 8 = 157.60
Answer:
b on edge hope this helps :)
Step-by-step explanation:
What is the circumference of the circle diameter 33 cm
Answer:
≈103.67cm
Step-by-step explanation:
Shape: Circle
Solved for circumference
Diameter: 33cm
Formula: C=πd
Answer: ≈103.67cm
Hope this helps.
Answer:
C = 33 pi
Step-by-step explanation:
The circumference of a circle is given by
C = pi *d
C = 33 pi
We can approximate pi by 3.14
C = 33*3.14 =103.62
or by using the pi button
C =103.6725576
1. Divide $13 by 42.
O A. 12/13
B. 142
c. 13/12
D. %16
Answer:
To get your answer, you need to fix the typos, but the answer is gonna be the closest thing to 13/42 if that is the real value
Step-by-step explanation:
none of these answers are anywhere near to 13/42
Determine whether the quadrilateral is a parallelogram using the specified method.
D(-8, 1), E(-3, 6), F(7, 4), G(2, -1)
Answer:
It is a parallelogram, I don't know what specified method you want us to use
A population of bacteria is growing at a rate of 20% per hour. If the population starts at 320, what is it an hour later?
Answer:
384.
Step-by-step explanation:
The population is given by the expression;
[tex]Y(t) = Pr^{t}[/tex]
Where;
Y(t) is the number of bacteria at time, t.
P is the starting amount.
r is the growth rate.
Given that, P = 320, r = 20% = 0.2, t = 1
Substituting the parameters into the equation;
[tex]Y(t) = Pr^{t}[/tex]
Y(1) = 320 * 0.2¹
Y(1) = 320 * 0.2
Y(1) = 64.
Therefore, the population of bacteria increases hourly by 64.
An hour later; 320 + 64 = 384.
Find the directional derivative of f at the given point in the direction indicated by the angle θ.
f(x, y) = y cos(xy), (0, 1), θ = π/4
Answer:
Directional derivative = 1/√2
Step-by-step explanation:
We are given f(x, y) = y cos(xy)
Now, we know that;
∇f(x, y) = ycos xy
Thus, applying that to the question, we have;
∇f(x, y) = [-y² sin xy, cos (xy) - xy sin xy]
At coordinates (0,1),we now have;
∇f(0, 1) = [-1²•sin0, (cos 0) - 0]
∇f(0, 1) = [0, 1]
Now, unit vector indicated by the angle θ is given as; u = [cos θ, sin θ]
From the question, since θ = π/4, thus
u = [cos π/4, sin π/4]
Cos π/4 in surd form is; 1/√2
Also, sin π/4 in surd form is; 1/√2
So, u = [1/√2, 1/√2]
Directional derivative = [∇f(0, 1)] • u
= [0, 1] × [1/√2, 1/√2] = 0 + 1/√2 = 1/√2
What is 576 times 36?
The 4th time of me posting this question
A geologist is making repeated measurements (in grams) on the mass of a rock. It is not known whether the measurements are a random sample from an approximately normal population. Below are three sets of replicate measurements, listed in the order they were made. For each set of readings, state whether the assumptions necessary for the validdity of the t-test appear to be met. If the assumptions are not met, explain why?
a. 213.03 212.95 213.04 213.00 212.99 213.01 221.03 213.05
b. 213.05 213.00 212.94 213.09 212.98 213.02 213.06 212.99
c. 212.92 212.95 212.97 213.00 213.01 213.04 213.05 213.06
Answer:
Step-by-step explanation:
The objective of this question is to known whether the assumptions of the known data set follows normal distribution or not and if it is appropriate to use t-test or not.
TO solve this question ; we construct a box plot for the known data set by using a MINITAB software
The process is as follows to guide you:
Enter your data in the worksheet
Then select Graph→ Boxplot→ EDA→ Boxplot
Under One Y , select simple, Then click OK
In Variables , enter measurements. Then click OK
So; for the first data set (a) 213.03 212.95 213.04 213.00 212.99 213.01 221.03 213.05 as shown in the attached file in word document for the boxplot below;
We will observe that there is a outliner present in the data set and the measurement do not meet the normal assumption ; as a result to that ; the t-distribution is not appropriate for this data.
(b) From the document attached below; we will realize and observe that the boxplot doesn't contains an outliner. Also; it is clear and obvious that the shape of the distribution is right skewed. The measurement thus did not meet the assumptions and as a result t- distribution is not appropriate for this data.
(c)
From the document attached below; we will realize and observe that the boxplot doesn't contains an outliner. Also; it is clear and obvious that the shape of the distribution is left skewed. The measurement thus did not meet the assumptions and as a result t- distribution is not appropriate for this data.
(a): The [tex]t-[/tex]distribution is not appropriate for this data.
(b): The [tex]t[/tex]-distribution is not appropriate for this data.
(c): The [tex]t[/tex]- distribution is not appropriate for this data.
[tex]t-[/tex]test:
A t-test is a statistical test that is used to compare the means of two groups.
Part(a):
Check whether the unknown data set follows the normal distribution or not and is it appropriate to use the [tex]t[/tex]-test or not.
Use the following MS excel instructions to draw a box plot.
Step-1: Enter the data into an excel sheet.
Step-2: Click add-ins-Mega stat-Descriptive statistics.
Step-3: Input the data into the input range box.
Step-4: Click Box-Plot -Click OK
It is observed that there is an outlier in the data set.
The obtained output is as follows:
So, the measurements do not satisfy the normality assumption.
Part(b):
Check whether the known data set follows normal distribution or not and is it appropriate to use the [tex]t[/tex]-test or not.
Use the above MS excel instruction to draw a Box plot.
The obtained output is as follows.
It is observed that there is no outlier in the data set and the shape of the distribution is appears to be right-skewed.
So, the measurement do not satisfy the normality assumption.
Part(c):
Check whether the known data set follows normal distribution or not and is it appropriate to use the [tex]t[/tex]- test or not.
Use the above MS Excel instruction to draw a Box-plot.
The obtained output is as follows.
It is observed that there is no outlier in the data set and the shape of the distribution is appears skewed to the left.
So, the measurement do not satisfy the normality assumption.
Learn more about the topic [tex]t-[/tex]test:
https://brainly.com/question/25999758
what is the equation of a quadratic function p with rational coefficients that has a zero of 3-i
Answer:
p = x² − 6x + 10
Step-by-step explanation:
Complex roots come in conjugate pairs. So if 3−i is a root, then 3+i is also a root.
p = (x − (3−i)) (x − (3+i))
p = x² − (3+i)x − (3−i)x + (3−i)(3+i)
p = x² − 3x − ix − 3x + ix + (9 − i²)
p = x² − 6x + 10
You can check your answer using the quadratic formula.
x = [ -b ± √(b² − 4ac) ] / 2a
x = [ 6 ± √(36 − 40) ] / 2
x = (6 ± 2i) / 2
x = 3 ± i
What is the measure of arc BC
Answer:
Option B
Step-by-step explanation:
As property of angle inside circle, arc BC = 2 x angle BDC.
Let's consider triangle ABD.
BAD = 98 deg
ABD = (1/2) x arc AD = (1/2) x 80 = 40 deg
As the property of sum of 3 angles in a triangle,
=> ADB + ABD + BAD = 180
=> ADB = 180 - ABD - BAD = 180 - 40 - 98 = 42 deg
=> BDC = 70 - 42 = 28
=> arc BC = 2 x 28 = 56 deg
=> Option B is correct
The article "Expectation Analysis of the Probability of Failure for Water Supply Pipes"† proposed using the Poisson distribution to model the number of failures in pipelines of various types. Suppose that for cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then X, the number of failures, has a Poisson distribution with μ = 1. (Round your answers to three decimal places.) Obtain P(X ≤ 5) by using the Cumulative Poisson Probabilities What is the probability that X exceeds its mean value by more than one standard deviation?
Answer:
a. P(X ≤ 5) = 0.999
b. P(X > λ+λ) = P(X > 2) = 0.080
Step-by-step explanation:
We model this randome variable with a Poisson distribution, with parameter λ=1.
We have to calculate, using this distribution, P(X ≤ 5).
The probability of k pipeline failures can be calculated with the following equation:
[tex]P(k)=\lambda^{k} \cdot e^{-\lambda}/k!=1^{k} \cdot e^{-1}/k!=e^{-1}/k![/tex]
Then, we can calculate P(X ≤ 5) as:
[tex]P(X\leq5)=P(0)+P(1)+P(2)+P(4)+P(5)\\\\\\P(0)=1^{0} \cdot e^{-1}/0!=1*0.3679/1=0.368\\\\P(1)=1^{1} \cdot e^{-1}/1!=1*0.3679/1=0.368\\\\P(2)=1^{2} \cdot e^{-1}/2!=1*0.3679/2=0.184\\\\P(3)=1^{3} \cdot e^{-1}/3!=1*0.3679/6=0.061\\\\P(4)=1^{4} \cdot e^{-1}/4!=1*0.3679/24=0.015\\\\P(5)=1^{5} \cdot e^{-1}/5!=1*0.3679/120=0.003\\\\\\P(X\leq5)=0.368+0.368+0.184+0.061+0.015+0.003=0.999[/tex]
The standard deviation of the Poisson deistribution is equal to its parameter λ=1, so the probability that X exceeds its mean value by more than one standard deviation (X>1+1=2) can be calculated as:
[tex]P(X>2)=1-(P(0)+P(1)+P(2))\\\\\\P(0)=1^{0} \cdot e^{-1}/0!=1*0.3679/1=0.368\\\\P(1)=1^{1} \cdot e^{-1}/1!=1*0.3679/1=0.368\\\\P(2)=1^{2} \cdot e^{-1}/2!=1*0.3679/2=0.184\\\\\\P(X>2)=1-(0.368+0.368+0.184)=1-0.920=0.080[/tex]
in a survey of employee satisfaction , 60% of the employees are male and 45% of the employees are satisfied. what is the probability of randomly selecting an employee who is male and satisfied
Answer:
27%
Step-by-step explanation:
6/10*45/100=270/1000 which is 27/100 which is 27%
Eric’s average income for the first 4 months of the year is $1,450.25, what must be his
average income for the remaining 8 months so that his average income for the year is
$1,780.75?
Answer:[tex]\$1946[/tex]
Step-by-step explanation:
Given
Eric average income for four months is [tex]S_1=\$1450.25[/tex]
average Annual income [tex]S=\$1780.75[/tex]
Average income [tex]=\dfrac{\text{Total salary}}{\text{Time }}[/tex]
Sum of 4 month salary [tex]=4\times S_1[/tex]
Sum of 4 month salary [tex]=\$5801[/tex]
Average salary of remaining 8 months[tex]=S_2[/tex]
Sum of last 8 months salary[tex]=8\times S_2[/tex]
So average Salary [tex]S=\dfrac{4\times S_1+8\times S_2}{12}[/tex]
[tex]\Rightarrow 1780.75=\dfrac{4\times 1450.25+8\times S_2}{12}[/tex]
[tex]\Rightarrow 21,369=5801+8\times S_2[/tex]
[tex]\Rightarrow 15,568=8\times S_2[/tex]
[tex]\Rightarrow S_2=\dfrac{15,568}{8}=\$1946[/tex]
So, average salary of remaining 8 months is [tex]\$1946[/tex]
99 POINTS AND BRAINLIEST! PLEASE EXPLAIN. Raphael saw a square patio that was 12-feet long on each side. He wants to build a patio that will be 15-feet long on each side. The change in the scale factor is: A) 1/15 B) 3/5 C) 4/5 D) 5/4 The change of scale means that 1 inch represented 4 feet, but now 1 inch represents feet: A) 5 B) 10 C) 12 D) 15
Answer:
5/4 5 ft
Step-by-step explanation:
We are going from 12 to 15
We need to multiply by
15 = k *12
Divide each side by 12
15/12 = k
5/4 = k
The scale factor is 5/4
1 in = 4 ft
Multiply the 4 ft by the scale factor
1 in = 4 * 5/4 = 5 ft
1 in = 5 ft
Answer:
Raphael saw a square patio that was 12-feet long on each side. He wants to build a patio that will be 15-feet long on each side.
The change in the scale factor is
✔ 4/5
.
The change of scale means that 1 inch represented 4 feet, but now 1 inch represents
✔ 5
feet.
Please answer this correctly
Answer:
98.5
Step-by-step explanation:
64.4 x 2= 128.8
325.8-128.8=197.0
197 /2 = 98.5
brainliest plzz
Answer:
Hey mate,
here is your answer. Hope it helps you.
Step-by-step explanation:
x+x+64.4+64.4=325.8
2x+ 128.8 =325.8
2x = 325.8 - 128.8
2x = 197
x= 197/2
=98.5 inches
This problem models pollution effects in the Great Lakes. We assume pollutants are flowing into a lake at a constant rate of I kg/year, and that water is flowing out at a constant rate of F km3/year. We also assume that the pollutants are uniformly distributed throughout the lake. If C(t) denotes the concentration (in kg/km3) of pollutants at time t (in years), then C(t) satisfies the differential equationdC dt = −FVC + IVwhere V is the volume of the lake (in km3). We assume that (pollutant-free) rain and streams flowing into the lake keep the volume of water in the lake constant.A) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time t by solving the differential equation.B) Find lim t→[infinity] C(t) =C) For Lake Erie, V = 458 km3 and F = 175 km3/year. Suppose that one day its pollutant concentration is C0 and that all incoming pollution suddenly stopped (so I = 0). Determine the number of years it would then take for pollution levels to drop to C0/10.D) For Lake Superior, V = 12221 km3 and F = 65.2 km3/year.
SEE BELOW FOR THE CORRECT FORMAT OF THE QUESTION
Answer:
(a) [tex]\mathbf{C_{(t)} =\dfrac{I}{F} [ 1- e \dfrac{-Ft}{v}+ C_oe \dfrac{-Ft}{v} ]}[/tex]
(b) [tex]\mathbf{\lim_{t \to \infty} C_t = \dfrac{I}{F}[1-0+ 0 ] \ = \dfrac{I}{F}}[/tex]
(c) T = 6.02619 years
(d) T = 431.593 years
Step-by-step explanation:
(a)
[tex]\dfrac{dC}{dt} = -\dfrac{F}{v}C + \dfrac{I}{v} \\ \\ \\ \dfrac{dC}{dt} + \dfrac{F}{v}C = \dfrac{I}{v}[/tex]
By integrating the factor of this linear differential equation ; we have :
[tex]= e \int\limits \dfrac{F}{v}t \\ \\ \\ = e \dfrac{Ft}{v}[/tex]
[tex]C* e \frac{Ft}{v}= \int\limits \dfrac{I}{v}*e \dfrac{Ft}{v} dt[/tex]
[tex]C* e \frac{Ft}{v}= \dfrac{I}{v}* \dfrac{e \frac{Ft}{v} }{F/v}+ K[/tex]
[tex]C* e \frac{Ft}{v}= \dfrac{I}{v}* \dfrac{V}{F} e \dfrac{Ft}{v} + K[/tex]
[tex]Ce \dfrac{Ft}{v} = \dfrac{I}{F} \ * \ e \dfrac{Ft}{v} + k \ \ \ \ (at \ t = 0 \ ; C = C_o)[/tex]
[tex]C_o = \dfrac{I}{F} e^o + K[/tex]
[tex]K = C_o - \dfrac{I}{F}[/tex]
[tex]C_{(t)} = [ \dfrac{I}{F}e \dfrac{Ft}{v}+ C_o - \dfrac{I}{F}] e\dfrac{-Ft}{v}[/tex]
[tex]C_{(t)} =\dfrac{I}{F} [ e \dfrac{Ft}{v} * e \dfrac{-Ft}{v}+ C_oe \dfrac{-Ft}{v} - 1* e \dfrac{-Ft}{v}][/tex]
[tex]\mathbf{C_{(t)} =\dfrac{I}{F} [ 1- e \dfrac{-Ft}{v}+ C_oe \dfrac{-Ft}{v} ]}[/tex]
(b)
[tex]\lim_{t \to \infty} C_t = \dfrac{I}{F}[1-e^{- \infty} + C_o e^{- \infty} ][/tex]
since [tex](e^{- \infty} = 0)[/tex]
[tex]\mathbf{\lim_{t \to \infty} C_t = \dfrac{I}{F}[1-0+ 0 ] \ = \dfrac{I}{F}}[/tex]
(c)
V = 458 km³ and F = 175 km³ , I = 0
[tex]\dfrac{dC}{dt} = - \dfrac{-175}{458}C[/tex]
[tex]= \int\limits \ \dfrac{dC}{C} = - \dfrac{175}{458}\int\limits dt[/tex]
[tex]In (C_{(t)}) = - \dfrac{175}{458} t + K[/tex]
[tex]C_{(t)} = e \dfrac{-175}{458}t + K[/tex]
Let at time t = 0 [tex]C_{(t)}} = C_o \to C_o = e^{0+k} = e^K[/tex]
[tex]C_{(t)} = e \dfrac{-175}{458}t[/tex]
Now at time t = T ; [tex]C_{9t)} = \dfrac{C_o}{10}[/tex]
[tex]\dfrac{C_o}{10} = C_o e \dfrac{-175}{458}T \to \dfrac{1}{10} = e \dfrac{-175}{458}T[/tex]
[tex]In ( \dfrac{1}{10}) = \dfrac{-175}{459}T[/tex]
[tex]- In (10) = \dfrac{175}{458}T[/tex]
[tex]T = \dfrac{458}{175} In (10)[/tex]
T = 6.02619 years
(d) V = 12221 km³
F = 65.2 km³/ year
[tex]\mathbf{T = \dfrac{v}{F}In (10)}[/tex]
[tex]T = \dfrac{12221}{65.2}In (10)[/tex]
T = 431.593 years