Answer:
Step-by-step explanation:
concentration = amount of salt/solution
A) Initial concentration= 90/1000 = 0.09
Q = quantity of salt
Q(0) = 90 kg
Inflow rate = 8 l/min
Outflow rate = 8 l/min
Solution = 1000 L at any time t.
Salt inflow = 0.045 * 8 per minute
= 0.36 kg per minute
This is mixed and drains from the tank.
Outflow = [tex]\frac{Q(t)}{1000}[/tex]
Thus rate of change of salt
Q'(t) = inflow - outflow = [tex]0.36-\frac{Q(t)}{1000} \\=\frac{360-Q(t)}{1000}[/tex]
Separate the variables and integrate
[tex]\frac{1000dQ}{360-q(t)} =dt\\-1000 ln |360-Q(t)| = t+C\\ln |360-Q(t)| = -0.001+C'\\360-Q(t) = Ae^{-0.001t} \\Q(t) = 360-Ae^{-0.001t}[/tex]
Use the fact that Q(0) = 90
90 = 360-A
A = 270
So
[tex]Q(t) = 360-270e^{-0.001t}[/tex]
B) Q(t) = 360-270e^-0.004 = 91.07784
C) When t approaches infinity, we get
Q(t) tends to 360
So concentration =360/1000 = 0.36
Final answer:
a) The initial concentration of the solution in the tank is 0.09 kg/L. b) The amount of salt in the tank after 4 hours is 3.6 kg. c) The concentration of salt in the solution in the tank approaches 0.045 kg/L as time approaches infinity.
Explanation:
a) To find the concentration of the solution in the tank initially, we need to calculate the total mass of salt and water in the tank. The concentration is the mass of salt divided by the volume of water. Since 1 liter of water weighs 1 kg, the initial concentration of the solution in the tank is 90 kg of salt divided by 1000 kg of water, which is 0.09 kg/L.
b) To find the amount of salt in the tank after 4 hours, we need to calculate the amount of salt entering the tank and the amount of salt leaving the tank in that time. The amount of salt entering the tank is the concentration of the incoming solution (0.045 kg/L) multiplied by the rate of flow (8 L/min) and the time (4 hours = 240 minutes). This gives us 0.045 kg/L × 8 L/min × 240 min = 86.4 kg. The amount of salt leaving the tank is the concentration of the solution in the tank (0.09 kg/L) multiplied by the rate of flow (8 L/min) and the time (4 hours = 240 minutes). This gives us 0.09 kg/L × 8 L/min × 240 min = 172.8 kg. Therefore, the amount of salt in the tank after 4 hours is the initial amount of salt (90 kg) plus the amount of salt entering the tank (86.4 kg) minus the amount of salt leaving the tank (172.8 kg), which is 3.6 kg.
c) As time approaches infinity, the concentration of salt in the solution in the tank will approach the concentration of the incoming solution, which is 0.045 kg/L.
Peaceful Travel Agency offers vacation packages. Each vacation package includes a city, a month, and an airline. The agency has cities, month, and airlines to choose from. How many different vacation packages do they offer?
Answer:12 packages.
Step-by-step explanation:
The agency has 2 cities,1 month and 6 airlines to choose from.(completion of the question)
Total package of the agency = 2cities×1 month × 6 airlines=12 packages.
The questions on a multiple choice test each have 4 answer choices. Describe a model that you could use to simulate the outcome of guessing tje correct answers to a 50 question test
Answer:
The outcome of guessing the correct answers to a 50 question test is 12.5.
Step-by-step explanation:
Let us first consider one question.
Since it has 4 options so we can say that;
Total number of possible outcomes = 4
Since 1 answer is correct so we can say that;
Favorable outcome = 1
Now we can say that;
Probability for choosing correct answer can be calculated by Favorable outcome divided by Total number of possible outcomes.
framing in equation form we get;
[tex]P(Correct\ Answer) = \frac{1}{4}[/tex]
Now Given:
Total number of question = 50
So we can say that;
Total number of correct answer out of 50 is equal to Probability for choosing correct answer multiplied by Total number of question.
framing in equation form we get;
Guessing number of correct answer out of 50 = [tex]\frac14 \times 50 = 12.5[/tex]
Hence The outcome of guessing the correct answers to a 50 question test is 12.5.
You can use a binomial distribution model to simulate guessing the correct answers on a 50-question test with 4 answer choices per question. The probability of getting at least 30 correct answers.
Explanation:To simulate the outcome of guessing the correct answers to a 50 question test, you can use a binomial distribution model. A binomial distribution represents the probability of a certain number of successful outcomes in a fixed number of independent trials. In this case, the success would be guessing the correct answer.
Each question has 4 answer choices, so the probability of guessing the correct answer for each question is 1/4. The number of successful guesses in a 50-question test can follow a binomial distribution with parameters n = 50 (number of trials) and p = 1/4 (probability of success).
Using this model, you can calculate probabilities of various outcomes, such as the probability of getting exactly 20 correct answers by guessing
Hence, the probability of getting at least 30 correct answers.
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Christy purchased 6.75 pounds of red licorice and 2.37 pounds of black licorice. How much licorice does she need to put in a bag if she divides the total amount into 10 equal-sized bags?
Answer: The amount she puts in a bag = 0.912 pound.
Step-by-step explanation:
Given : Amount of red licorice purchased by Christy = 6.75 pounds
Amount of black licorice purchased by Christy = 2.37 pounds
Total amount of licorice purchased by Christy = 6.75 + 2.37 pounds
= 9.12 pounds
Then, if she divides the total amount into 10 equal-sized bags , the amount she put in a bag = (Total amount of licorice purchased) ÷ 10
= 9.12 ÷ 10 = 0.912 pound
Hence, the amount she puts in a bag = 0.912 pound.
Final answer:
To find how much licorice Christy needs to put into each of 10 bags, add the weights of red and black licorice and then divide by 10. She will need to put 0.912 pounds of licorice into each bag.
Explanation:
Christy purchased 6.75 pounds of red licorice and 2.37 pounds of black licorice. To find the total amount of licorice, we add the weight of the red and black licorice together:
Total weight of licorice = [tex]6.75 pounds + 2.37 pounds = 9.12 pounds.[/tex]
If she divides the total amount into 10 equal-sized bags, we divide the total weight by 10:
Weight per bag = [tex]9.12 pounds / 10 bags = 0.912 pounds per bag.[/tex]
Steve invests 1,800 in an account that earns 3.7% annual interest, compounded continuously. What is the value of the account after 10 years? Round your answer to the nearest dollar.
Answer: the value of the account after 10 years is $2606
Step-by-step explanation:
The formula for continuously compounded interest is
A = P x e (r x t)
Where
A represents the future value of the investment after t years.
P represents the present value or initial amount invested
r represents the interest rate
t represents the time in years for which the investment was made.
e is the mathematical constant approximated as 2.7183.
From the information given,
P = 1800
r = 3.7% = 3.7/100 = 0.037
t = 10 years
Therefore,
A = 1800 x 2.7183^(0.037 x 10)
A = 1800 x 2.7183^(0.37)
A = $2606 to the nearest dollar
To solve such problems we must know about Continuous Compound Interest.
Continuous Compound Interest[tex]A = Pe^{rt}[/tex]
where,
A is the principal amount after t number of years,
r is the rate at which the principal is been compounded, and P is the principal amount.
The value of Steve's account after 10 years will be $2,606.
Given to usSteve invests 1,800 in an account that earns 3.7% annual interest, compounded continuously.Time period money invested for 10 years, SolutionAs it is given in the problem, the account of Steve is compounding continuously.
Statement 1Steve invests 1,800 in an account that earns 3.7% annual interest, compounded continuously.
Principal amount = $1,800 Rate of interest = 3.7% = 0.037Statement 2Time period money invested for 10 years,
time period, t = 10 yearsValue of the accountSubstituting the values in the formula of Continuous Compound Interest,
[tex]A = Pe^{rt}[/tex]
[tex]\begin{aligned}A&=\$1800\times e^{0.037\times10}\\ &=\$2605.9\approx\$2606\\ \end{aligned}[/tex]
Hence, the value of Steve's account after 10 years will be $2,606.
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How many toothpicks do I need to make a acutangle scalene triangle
Answer:
The (minimum) number of toothpicks required to make an acute angle scalene triangle is 3.
Step-by-step explanation:
The (minimum) number of toothpicks required to make an acute angle scalene triangle is 3.
All the angles in a an acute angled scalene triangle are less than 90°.
The above condition can be satisfied with a minimum of three toothpicks.
To create an acute-angled scalene triangle, three toothpicks are required.
To create an acute-angled scalene triangle, you would need 3 toothpicks. You can stick the toothpicks into a marshmallow or a gumdrop and position them in a way that forms a triangle with angles less than 90 degrees and with all sides of different lengths.
Write an expression for the number or floors the building can have for a given building height. Tell what variable in your expression represents. (Each floor will be 12 feet tall)
Final answer:
To calculate the number of floors a building can have for a given height, use the formula 'Number of Floors = h / 12', where 'h' is the total building height in feet and each floor is 12 feet tall.
Explanation:
To write an expression for the number of floors a building can have for a given building height, where each floor is 12 feet tall, we use the following formula:
Number of Floors = Total Building Height (in feet) / Height of One Floor (in feet)
Let 'h' represent the total height of the building in feet. Then the expression becomes:
Number of Floors = h / 12
For example, if a building has a height of 384 feet, the number of floors can be calculated as 384 feet / 12 feet per floor = 32 floors.
If cos(θ) = 6/8 and θ is in the IV quadrant, then fine:
(a) tan(θ)cot(θ)
(b) csc(θ)tan(θ)
(c) sin^2(θ) + cos^2(θ)
Answer:
a) 1
b) [tex]\frac{4}{3}[/tex]
c) = 1
Step-by-step explanation:
We are given the following in the question:
[tex]\cos \theta = \dfrac{6}{8}[/tex]
θ is in the IV quadrant.
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\\\sin \theta = \sqrt{1-\dfrac{36}{64}} = -\dfrac{2\sqrt7}{8}\\\\\tan \theta = \dfrac{\sin \theta}{\cos \theta} = -\dfrac{2\sqrt7}{6}\\\\\csc \theta = \dfrac{1}{\sin \theta} = -\dfrac{8}{2\sqrt7}[/tex]
Evaluate the following:
a)
[tex]\tan \theta\times \cot \theta =\tan \theta\times\dfrac{1}{\tan \theta} = 1[/tex]
b)
[tex]\csc \theta\times \tan \theta\\\\= -\dfrac{8}{2\sqrt7}\times -\dfrac{2\sqrt7}{6} = \dfrac{4}{3}[/tex]
c)
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\text{using the trignometric identity}[/tex]
Mr. Clements painted his barn for 3 3/5 hours in the morning. He painted the barn for 5 3/4 hours in the afternoon. For 15a-15cn select True of False for each statement.
Answer:
True
Step-by-step explanation:
A common denominator of the mixed numbers is 20. select True or false
Mr. Clements painted his barn for 3 3/5 hours in the morning.
[tex]3\frac{3}{5}=\frac{18}{5}[/tex]
He painted the barn for 5 3/4 hours in the afternoon.
[tex]5\frac{3}{4}=\frac{23}{4}[/tex]
Now we consider the denominator of both fractions 18/5 and 23/4
Least common denominator is 5 times 4 = 20
So its true that the common denominator of mixed numbers is 20
The Commerce Department reported receiving the following applications for the Malcolm Baldrige National Quality Award: 23 from large manufacturing firms, 18 from large service firms, and 30 from small businesses.
1. Is type of business a qualitative or quantitative variable?
2. What percentage of the applications came from small businesses (to 1 decimal)?
Answer:
Categorical42.25%Step-by-step explanation:
CATEGORICAL, because the type of business will not be numerical.30 of the 71 (23+18+30) businesses are small businesses.30/71 = 0.4225 = 42.25%
Suppose the wait through immigration at JFK Airport in New York is thought to be bell-shaped and symmetrical with a mean of 22 minutes. It is known that 68 percent of travelers will spend between 16 and 28 minutes waiting to pass through immigration. The standard deviation for the wait time through immigration is_________.
Answer: 6 minutes
Step-by-step explanation:
According to the Empirical rule ,
About 68% of the population lies within one standard deviation from the mean.
Given : Suppose the wait through immigration at JFK Airport in New York is thought to be bell-shaped and symmetrical with a mean of 22 minutes.
It is known that 68 percent of travelers will spend between 16 and 28 minutes waiting to pass through immigration.
i.e. Mean - standard deviation= 16
and Mean + standard deviation =28
Since , Mean=22 , so we have
22+ standard deviation =28
⇒ standard deviation =28 -22 =6
Hence, the standard deviation for the wait time through immigration is 6 minutes .
The standard deviation for the wait time through immigration is 6 minutes.
To find the standard deviation, we can use the properties of the normal distribution, specifically the 68-95-99.7 rule, which states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
Given that 68% of travellers wait between 16 and 28 minutes, we can determine that these times represent one standard deviation from the mean. The mean wait time is 22 minutes. The range from 16 to 28 minutes is 12 minutes wide (28 - 16 = 12). Since this range represents one standard deviation on either side of the mean, we can divide this range by 2 to find the value of one standard deviation:
Standard deviation = (28 - 16) / 2 = 12 / 2 = 6 minutes.
Mary purchased an annuity that pays her $500 per month for the rest of her life. She paid $70,000 for the annuity. Based on IRS annuity tables, Mary's life expectancy is 16 years. How much of the first $500 payment will Mary include in her gross income (round to two decimals)?
Answer: $135.42
Step-by-step explanation:
Based on IRS tables, Mary is expected to receive 192 (16 years x 12 months) annuity payments. Her investment in the annuity is $70,000 and her return of capital for each annuity payment is $70,000/192 = $364.58. The return of capital portion of each annuity payment is not taxable (not included in gross income). Mary must include the excess received ($500.00 – 364.58) of $135.42 in her gross income.
To calculate how much of the first $500 payment Mary will include in her gross income, we need to determine the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary will include $36.45 of the first $500 payment in her gross income.
Explanation:To determine how much of the first $500 payment Mary will include in her gross income, we need to calculate the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary paid $70,000 for the annuity and her life expectancy is 16 years, so the exclusion ratio is $70,000 / ($500/month * 12 months/year * 16 years) = 0.0729. To determine the amount she will include in her gross income, we multiply the monthly payment by the exclusion ratio: $500 * 0.0729 = $36.45. Therefore, Mary will include $36.45 of the first $500 payment in her gross income.
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Suppose a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students. (a) Suppose one of the 20 classes is chosen at random, let Y be the number of students in that class. Write out the p.m.f. for Y and use it to calculate the average class size at this school. (b) Suppose that one of the 1000 students is chosen at random. Let X be the size of the class to which the student belongs. Write out the p.m.f. of X and find E[X]. (c) Fill in the following: On average, a student at this school is in a class with students. On average, a class at this school has students. Comment
Answer:
Step-by-step explanation:
Given that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students.
a) Y = no of students in the class out of 20 selected at random
Y can take values as
Y 25 100 300 Total
freq 16 3 1 20
P(Y) 16/20 3/20 1/20 1
y*p 20 15 15 50
E(Y) = 50
b) X = size of student from which one student is taken at random
X 25 100 300 Total
No of classes 16 3 1 20
Total students
(X*no of classes) 400 300 300 1000
P(x) 0.4 0.3 0.3 1
X*P 160 90 90 240
E(x) =240
c) On an average, a student at this school is in a class with 240 students. On average, a class at this school has 50 students.
i.e. expectation of number of students in a class is 50 while expectation of a student having students in the class is 240
How do you solve this problem? Emma and Anna went to the market to buy 25 fruits. Emma got a few apples that cost $2 each, and Anna bought some oranges which cost $3 each. If their combined total is $60, how many oranges did Anna get?
Anna bought 10 oranges
Explanation:
Let x denote the number of apples.
Let y denote the number of oranges.
The system of equations can be written as
[tex]x+y=25[/tex] and [tex]2x+3y=60[/tex]
Let us solve the equation by substitution method.
Thus, from [tex]x+y=25[/tex], the value of y can be determined such that [tex]y=25-x[/tex]
Using the y value [tex]y=25-x[/tex] in the equation [tex]2x+3y=60[/tex], we get,
[tex]2x+3(25-x)=60[/tex]
Multiplying 3 within the bracket,
[tex]2x+75-3x=60[/tex]
Adding the terms,
[tex]75-x=60[/tex]
Subtracting both sides by 75, we have,
[tex]x=15[/tex]
Thus, substituting [tex]x=15[/tex] in [tex]y=25-x[/tex], we get,
[tex]y=25-15\\y=10[/tex]
Thus, Anna get 10 Oranges.
3. Using techniques from Calculus, show directly that the maximum value of a 1-D Gaussian distribution occurs at the point x = μ.
Answer:
For a scaler variable, the Gaussian distribution has a probability density function of
p(x |µ, σ² ) = N(x; µ, σ² ) = 1 / 2π×[tex]e^{\frac{-(x-u)^{2}}{2s^{2} } }[/tex]
The term will have a maximum value at the top of the slope of the 1-D Gaussian distribution curve that is when exp(0) =1 or when x = µ
Step-by-step explanation:
Gaussian distributions have similar shape, with the mean controlling the location and the variance controls the dispersion
From the graph of the probability distribution function it is seen that the the peak is the point at which the slope = 0, where µ = 0 and σ² = 1 then solution for the peak = exponential function = 0 or x = µ
ALG TWO HELP ASAP?
What is the period to the graph in problem 1?
the amplitude of the function is 2.
Answer:
Therefore the period of the sinusoidal wave = 3.14.
The amplitude of the function = 2.
Step-by-step explanation:
The period of a sinusiodal wave is given by the length of x axis which covers one full cycle of the wave, which one positive half-cycle and one negative half cycle.
From the graph we can see that one of the cycles starts at -3.14 / 4 and ends at 3.14 [tex]\times[/tex] 3/4.
Therefore we can sutract the start point value from the end point value to get the period
Therefore period = (3.14 [tex]\times[/tex] 3/4) - (-3.14 / 4) = (3.14 [tex]\times[/tex] 3/4) + (3.14 / 4) = 3.14
Therefore the period of the sinusoidal wave = 3.14.
The amplitude of the function = 2.
A car with an initial cost of $23,000 is decreasing in value at a rate of 8% each year. Write the exponential decay function described in this situation. Then use your function to determine when the value of the car will be $15,000, to the nearest year.
Answer:
Step-by-step explanation:
We would apply the formula for exponential decay which is expressed as
A = P(1 - r/n)^ nt
Where
A represents the value after t years.
n represents the period for which the decrease in value is calculated
t represents the number of years.
P represents the value population.
r represents rate of decrease.
From the information given,
P = 23000
r = 8% = 8/100 = 0.08
n = 1
Therefore, the exponential decay function described in this situation is
A = 23000(1 - 0.08/n)1)^ 1 × t
A = 23000(0.92)^t
If A = 15000, then
15000 = 23000(0.92)^t
0.92^t = 15000/23000 = 0.6522
Taking log of both sides to base 10
Log 0.92^t = log 0.6522
tlog 0.92 = log 0.6522
- 0.036t = - 0.1856
t = - 0.1856/- 0.036
t = 5 years to the nearest year
The value of the car will be $15,000 after approximately 4 years.
To model the situation with an exponential decay function, we use the formula:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
Given that the initial cost [tex]P[/tex] is $23,000 and the annual depreciation rate r is 8%, or 0.08 in decimal form, we can write the exponential decay function as:
[tex]\[ V(t) = 23000 \times (1 - 0.08)^t \][/tex]
[tex]\[ V(t) = 23000 \times (0.92)^t \][/tex]
To find out when the value of the car will be $15,000, we set [tex]V(t)[/tex] equal to $15,000 and solve for t.
[tex]\[ \frac{15000}{23000} = (0.92)^t \][/tex]
[tex]\[ 0.652173913043478 = (0.92)^t \][/tex]
Using a calculator, we find:
[tex]\[ t \approx \frac{\ln(0.652173913043478)}{\ln(0.92)} \approx 4.037 \][/tex]
Since we are looking for the time in whole years, we round to the nearest year:
[tex]\[ t \approx 4 \][/tex]
Therefore, the value of the car will be $15,000 after approximately 4 years.
A spherical balloon holding 35 lbm of air on earth has a diameter of 10 ft. Determine (a) the specific volume, in unit ft3/lbm, and (b) the weight, in lbf, of the air within the balloon.
Answer:
a) Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
b)The weight of the air within the balloon is 1,126.09 lbf.
Step-by-step explanation:
Mass of air, m = 35 lbm
Volume of the air = V
Diameter of balloon = d = 10 ft
radius of the balloon = r= 0.5 d = 5 ft
Volume of balloon = V
[tex]V=\frac{4}{3}\pr r^3[/tex]
[tex]V=\frac{4}{3}\times 3.14\times (5 ft)^3[/tex]
Specific volume of the air in balloon = S
[tex]S=\frac{V}{m}=\frac{\frac{4}{3}\times 3.14\times (5 ft)^3}{35 lbm}[/tex]
[tex]S=14.96 ft^3/lbm[/tex]
Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
[tex]lbf=32.174 lbm ft/s^2[/tex]
Weight of the air = W
Acceleration due to gravity = [tex]32.174 lbm ft/s^2[/tex]
Weight = m\times g[/tex]
[tex]W=35 lbm\times 32.174 lbm ft/s^2[/tex]
[tex]W=1,126.09 lbf[/tex]
The weight of the air within the balloon is 1,126.09 lbf.
A construction worker leans his ladder against a building making a 60 degree angle with the ground. If the ladder is 20 feet long, how far away is the base or the ladder from the building? Round to the nearest tenth
Answer: the distance from the base of the ladder from the building is 10 feet.
Step-by-step explanation:
The ladder forms a right angle triangle with the wall of the building and the ground.
The length of the ladder represents the hypotenuse of the right angle triangle.
The distance, h of the base of the ladder from the building represents the adjacent side of the triangle.
To determine h, we would apply
the cosine trigonometric ratio.
Cos θ = adjacent side/hypotenuse. Therefore,
Cos 60 = h/20
h = 20Cos60 = 20 × 0.5
h = 10 feet
Final answer:
To solve for the distance from the base of the ladder to the building, the cosine function is used because the scenario forms a right-angled triangle with the ground and the wall. By the equation cos(60°) = Base / Hypotenuse, with the ladder being the hypotenuse, the base is calculated to be 10 feet from the building.
Explanation:
The question involves using trigonometry to determine the distance from the base of the ladder to the building. When a ladder makes a 60° angle with the ground and its length is given (20 feet), this forms a right-angled triangle with the ground and the building's wall. To find the distance from the base of the ladder to the building, we can use the cosine function, which relates the adjacent side to the hypotenuse in a right-angled triangle.
Using the equation:
cos(60°) = Base / Hypotenuse
Here, the hypotenuse is the length of the ladder which is 20 feet. Therefore:
cos(60°) = Base / 20 feet
cos(60°) equals 0.5 when calculating in degrees. Consequently:
0.5 = Base / 20 feet
Multiplying both sides by 20 gives:
Base = 20 feet * 0.5
Base = 10 feet
Thus, the base of the ladder is 10 feet from the building.
Please explain if you can
Answer:
Therefore,
[tex]x=9\\\\y=7[/tex]
Step-by-step explanation:
Given:
D,E,F are the midpoints of side AB, BC,and AC such that
AB = 4x - 18
BC = 2x - 4
DF = x
FE = y
To Find:
x = ?
y = ?
Solution:
Midpoint Theorem:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
E and F are Midpoint of Sides BC and AC, then
[tex]FE=\dfrac{1}{2}AB[/tex] ......By Midpoint Theorem
Substituting the values we get
[tex]x=\dfrac{1}{2}(4x-18)=2x-9\\\\2x-x=9\\\\x=9[/tex]
Similarly,
D and Fare Midpoint of Sides AB and AC, then
[tex]DF=\dfrac{1}{2}BC[/tex] ......By Midpoint Theorem
Substituting the values we get
[tex]y=\dfrac{1}{2}(2x-4)=x-2\\\\Substitute\ x=9\\\\y=9-2=7[/tex]
Therefore,
[tex]x=9\\\\y=7[/tex]
The volume of a cylinder is increasing at a rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
What is the rate of change of the surface area of the cylinder at that instant (in square meters per hour)?
Answer:
13π/2 m²/h
Step-by-step explanation:
Volume of a cylinder is:
V = πr²h
If h is a constant, then taking derivative of V with respect to time:
dV/dt = 2πrh dr/dt
Surface area of a cylinder is:
A = 2πr² + 2πrh
Taking derivative with respect to time:
dA/dt = (4πr + 2πh) dr/dt
Given that dV/dt = 10π, V = 80π, and h = 5, we need to find dA/dt. But first, we need to find r and dr/dt.
V = πr²h
80π = πr² (5)
r = 4
dV/dt = 2πrh dr/dt
10π = 2π (4) (5) dr/dt
dr/dt = 1/4
dA/dt = (4πr + 2πh) dr/dt
dA/dt = (4π (4) + 2π (5)) (1/4)
dA/dt = 13π/2
The surface area of the cylinder is increasing at 13π/2 m²/h.
The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
Given that,
The volume of the cylinder increase with the rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
We have to determine,
What is the rate of change of the surface area of the cylinder at that instant.
According to the question,
Height of cylinder = 5m
Volume of cylinder increase with rate = 10π cubic meters per hour.
At certain instant volume become = 80π cubic meters per hour.
Volume of cylinder is given as,
V = πr²h
Where h is a constant,
[tex]v = \pi r^{2} h\\\\80\pi = \pi r^{2}. (5)\\\\\frac{80\pi }{5\pi } = r^{2} \\\\r^{2} = 16\\\\r = 4[/tex]
Then, taking derivative of V with respect to time:
[tex]\frac{dv}{dt} = 2\pi rh.\frac{dr}{dt} \\\\[/tex]
Where, dv\dt = 10π, V = 80π, and h = 5,
Then,
[tex]10\pi = 2\pi (4)(5). \frac{dr}{dt} \\\\10\pi = 40\pi \frac{dr}{dt} \\\\\frac{10\pi }{40\pi } = \frac{dr}{dt} \\\\\\\frac{dr}{dt} = \frac{1}{4}[/tex]
Then,
Surface area of a cylinder is define as,
[tex]A = 2\pi r^{2} + 2\pi rh\\\\\ \frac{da}{dt} = (4\pi r + 2\pi h)\frac{dr}{dt} \\\\\frac{da}{dt} = (4\pi (4) + 2\pi (5))\times\frac{1}{4} \\\\\frac{da}{dt} = \frac{26\pi }{4} \\\\\frac{da}{dt} = \frac{13\pi }{2}[/tex]
Hence, The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
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You and your friend are going to some candy. You eat 3/4 of a box of candy. Your friend eats 1/2 as much as you do. How much of a box of candy does your friend eat?
Answer:
[tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
Step-by-step explanation:
Given:
You and your friend are going to some candy.
You eat 3/4 of a box of candy.
Your friend eats 1/2 as much as you do.
Now, to find that how much of a box of candy does your friend eat.
Quantity of a box of candy you eat = [tex]\frac{3}{4}[/tex]
Quantity of candy your friend eat = [tex]\frac{1}{2} \ of\ \frac{3}{4}[/tex]
Now, to get how much the candy your friend eat we multiply 1/2 and 3/4:
= [tex]\frac{1}{2} \times \frac{3}{4}[/tex]
= [tex]\frac{3}{8}[/tex] .
Therefore, [tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
A rectangle has width w inches and height h, where the width is twice the height. Both w and h are functions of time,t, measured in seconds. If A represents the area of the rectangle, what is the rate of change of A with respect to t at the instant where the width is 4 inches and the height is increasing at the rate of 2 inches per second?
Answer:
dA/dt = 16 square inches per second
Step-by-step explanation:
Width of rectangle is w
Height of rectangle is h
width = twice height
w = 2h
Area = wh = (2h)*h
A = 2h^2
Differentiating the equation with respect to time
dA/dt = 2+2h dh/dt
dA/dt = 4h dh/dt
According to the given situation when width is 4 inches
h = w/2
h = 2
Rate of change of A is
dA/dt = wh dh/dt
dA/dt = 4(2)(2)
dA/dt = 16 square inches per second
The rate of change of the area of the rectangle with respect to time at the specified instant is 16 square inches per second.
Explanation:The question asks for the rate of change of the area of a rectangle with respect to time, where the rectangle's width is twice its height, both width and height are functions of time and the rate at which height is increasing is given.
The area A of the rectangle is given by the formula A = w*h. Substituting w = 2h into the formula gives A = 2h². The rate of change of the area with respect to time (dA/dt) can be found by differentiating A with respect to time. By the chain rule this gives dA/dt = 2*2h(dh/dt) = 4h*dh/dt.
At the given instant, where the width is 4 inches (so the height is 2 inches) and the rate of change of the height is 2 inches per second, we have dA/dt = 4*2*2 = 16 square inches per second. Therefore, the rate of change of the area of the rectangle with respect to time at that instant is 16 square inches per second.
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Bill says the equation 600 equals 12×50 means 600 is four times as many as 50 Sarah says the equation means 650 more than 12 who do you agree with explain
Answer:
neither
Step-by-step explanation:
The equation means that 600 is twelve times as many as 50 (not four times). Bill needs to rethink the meaning of "four" relative to the meaning of "12".
__
"650 more than 12" is an expression (12+650), not an equation. Sarah seems to be clueless.
__
Neither Bill nor Sarah have described the meaning of the equation.
On average, employees at a particular company historically missed 4 days of work per year. Then there was a management change. Now the average is 7 days. What is the percent of increase in the average number of missed days?
Answer:
75% increase
Step-by-step explanation:
Given:
Number of missed days of work before the management change = 4
Number of missed days of work after the management change = 7
So, increase in the number of missed days of work is given by subtracting the old number from the new number and is equal to:
Increase in missed number of days = 7 - 4 = 3 days
Now, the percent of this increase is calculated by dividing the increase by the old number and then multiplying the result by 100.
Percent of increase is given as:
[tex]Percent\ of\ increase=\frac{Increased\ value}{Old\ value}\times 100\\\\Percent\ of\ increase=\frac{3}{4}\times 100\\\\Percent\ of\ increase=0.75\times 100\\\\Percent\ of\ increase=75\%[/tex]
So, there is a 75% increase in the number of missed days of work after the change in the management.
The admission fee at an amusement park is $1.50 for children and $4.00 for adults. On a certain day, 304 people entered the park, and the admission fees collected totaled $856. Let x be the number of children that were admitted and let y be the number of adults that were admitted.?
Answer:
x = 144 children
y = 160 adults
Step-by-step explanation:
We have parameters already defined. Number of children that entered is x while number of admitted adults is y.
Since the total number of people admitted was 304, this means:
x + y = 304
Total fees for that day was 856
Mathematically:
1.5x + 4y = 856
From the first equation, we can say y = 304 - x
We then substitute this into the second equation:
1.5x + 4(304 - x) = 856
1.5x + 1216 - 4x = 856
1216-856 = 4x-1.5x
2.5x = 360
x = 360/2.5 = 144
Since y = 304 - 144
y = 160
The number of children admitted was 144 and the number of adult is 160
Officials project that between 201020102010 and 205020502050, the population of a certain city will drastically change. The model below gives the projection of the city's population , PPP, in thousands, with respect to time, ttt, in years, where 201020102010 corresponds to t=0t=0t, equals, 0. \quad P = 175 + \dfrac{11}{2}tP=175+ 2 11 tP, equals, 175, plus, start fraction, 11, divided by, 2, end fraction, t What does the 175175175 mean in the equation?
Answer:
175 means the city's population is 175 thousands in 2010.
Step-by-step explanation:
The model below gives the projection of the city's population , P, in thousands, with respect to time, t, in years, where 2010 corresponds to t=0.
The given equation is
[tex]P=175+\dfrac{11}{2}t[/tex]
We need to interpret 175 for the given model.
Substitute t=0 in the above equation.
[tex]P=175+\dfrac{11}{2}(0)[/tex]
[tex]P=175[/tex]
At t=0 the value of P is 175 it means the city's population is 175 thousands in 2010.
In a survey of 545 human resource professionals, each was asked about the importance of the age of a job applicant. The survey subjects were randomly selected by pollsters from a reputable market research firm.
Answer:
The sampling method described in this question appears to be sound
Step-by-step explanation:
It appears to be sound because the data are not biased in any way, since every member and set of members has an equal chance of being included in the sample selection. Also, random samples are usually fairly representative since they don't favor certain members.
The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers.
The volume of the square prism with a side of base decreasing at a rate of 7 km/min and height increasing at 10 km/min is decreasing at a rate of 344 cubic km/min.
Explanation:In this problem, we're dealing with rates of change and the geometric properties of a square prism. A square prism can be characterized by the side length of its base (we'll call this length 's') and its height (h). The volume (V) of a square prism is given by the equation V = s^2 * h. We're told that the side length s is decreasing at a rate of 7 kilometers per minute (ds/dt = -7 km/min) and the height h is increasing at a rate of 10 kilometers per minute (dh/dt = 10 km/min).
To find the rate at which the volume is changing with respect to time, we can take the derivative of the volume equation with respect to time. Thus, dV/dt = 2*s*ds/dt*h + s^2*dh/dt. Substituting the given values, when s = 4 km and h = 9 km, we find: dV/dt = 2*4*(-7)*9 + 4^2*10 = -504 + 160 = -344 cubic kilometers per minute. Therefore, at the given instant, the volume of the square prism is decreasing at a rate of 344 cubic kilometers per minute.
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The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
To solve this problem, let's first start by identifying the formula for the surface area (SA) of a square prism, which is given as [tex]2s^2 + 4sh[/tex] , where s is the side of the base and h is the height.
Now, let's find the rate of change of the surface area with respect to time, dSA/dt. We'll use the chain rule to differentiate the surface area formula:
1. Differentiate the given surface area formula:
[tex]SA = 2s^2 + 4sh[/tex]
[tex]d(SA)/dt = d(2s^2)/dt + d(4sh)/dt[/tex]
2. Apply the chain rule:
d(SA)/dt = 2 * 2s * (ds/dt) + 4 * (s * (dh/dt) + h * (ds/dt))
3. Plug in the given values: s = 4 km, ds/dt = -7 km/min, h = 9 km, dh/dt = 10 km/min:
d(SA)/dt = 4 * 4 * (-7) + 4 * (4 * 10 + 9 * (-7))
d(SA)/dt = -112 + 4 * (40 - 63)
d(SA)/dt = -112 + 4 * (-23)
d(SA)/dt = -112 - 92
d(SA)/dt = -204
The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
Complete Question:- The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers. What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)? Choose 1 answer: A 204 B 148 C -204 D -148 The surface area of a square prism with base side 8 and height h is [tex]2s^2+4sh.[/tex]
The point A (8, 2) is reflected over the line x = 5, and then is reflected over the line x = -1. What are the coordinates of A'?
Answer:
A'(8, -10)
Step-by-step explanation:
Firstly we have to know the distance from the point A to the line, the point is (8, 2) and the line is x = 5 .
the point has two components, at x and y
the component in y will give us the distance we have on the axis of the X thas is 2.
T he line is 5 units from the axis.
Now to know the distance we have to subtract
2 - 5 = -3
Here is the point with respect to the line x = 5
For the reflected over the line we just need to change the sign, now we got +3 . To this we add the value of the line that is 5
3 + 5 = 8.
So the reflected point over the line x = 5 is (8, 8)
We do the same with the line x = -1.
(8, 8) and x = -1
..
8 - (-1) = 9
..
(-9) + (-1) = -10
(8, -10)
Express tan(23°−21°) in terms of tangents of 23∘and 21∘ You do NOT need to type in the degree symbol. Be sure to PREVIEW your answer before submitting!
Answer:
tan(23° -21°) = (tan(23°) -tan(21°))/(1 +tan(23°)tan(21°))
Step-by-step explanation:
The formula for the tangent of the difference of angles is ...
tan(a-b) = (tan(a) -tan(b))/(1 +tan(a)tan(b))
Filling in the values a=23° and b=21°, you get the formula shown above.
The expression for tan(23°−21°) in terms of the tangents of 23° and 21° is: tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°).
To express tan(23°−21°) in terms of the tangents of 23° and 21°, we can use the angle subtraction formula for tangent, which is:
tan(α - β) = (tan α - tan β) / (1 + tan α * tan β)
Applying this to tan(23°−21°), we get:
tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°)
We can't directly calculate the values from the given table since 23° and 21° are not listed. However, we understand the formula required to express tan(23°−21°) in terms of the tangents of these two angles.