The solution of the given differential equation approaches zero as t → ∞.
The solution of the given initial value problem and the behavior of the solution as t → ∞ is given below.
Solution:The given initial value problem is
52y" + 17y" + y' = 0, y(0) = −10, y'(0) = 18, y"(0) = 0.
Let's find the solution of the given initial value problem using the following steps.
Step 1: Find the characteristic equation associated with the given differential equation
The characteristic equation associated with the given differential equation is obtained by assuming the solution in the form of y(t) = e^(rt).
Substituting y(t) = e^(rt) into the given differential equation, we get
52r² + 17r + 1 = 0.
The roots of the characteristic equation are
r1,2= [-17 ± √(17²-4(52)(1))]/[2(52)]
= [-17 ± 5√6]/104.
Step 2: Find the general solution of the given differential equation
The general solution of the given differential equation is
y(t) = c1e^(r1t) + c2e^(r2t), where c1 and c2 are constants of integration and r1 and r2 are the roots of the characteristic equation.
Step 3: Apply the initial conditions to find the constants of integration
Differentiating the general solution of the given differential equation with respect to t, we get
y'(t) = c1r1e^(r1t) + c2r2e^(r2t).
Differentiating the y'(t) with respect to t, we get
y"(t) = c1r1²e^(r1t) + c2r2²e^(r2t).
Using the given initial conditions,
y(0) = -10c1 + c2 = -10,
y'(0) = 18c1r1 + c2r2 = 18,
y"(0) = 0c1r1² + c2r2² = 0.
From the first equation, we have
c2 = c1 - 10.Substituting c2 = c1 - 10 into the second equation, we have
c1r1 + (c1 - 10)r2 = 2.
Substituting c2 = c1 - 10 into the third equation, we have
c1r1² + (c1 - 10)r2² = 0.
Solving the above equations, we get
c1 = 20/27 and c2 = -530/27.
Therefore, the solution of the given initial value problem is
y(t) = (20/27)e^((-17 + 5√6)t/104) - (530/27)e^((-17 - 5√6)t/104).
Therefore, the solution of the given differential equation approaches zero as t → ∞.
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HELP PLEASEEEE
The Scooter Company manufactures and sells electric scooters. Each scooter cost $200 to produce, and the company has a fixed cost of $1,500. The Scooter Company earns a total revenue that can be determined by the function R(x) = 400x − 2x2, where x represents each electric scooter sold. Which of the following functions represents the Scooter Company's total profit?
A. −2x2 + 200x − 1,500
B. −2x2 − 200x − 1,500
C. −2x2 + 200x − 1,100
D. −400x3 − 3,000x2 + 80,000x + 600,000
The function that represents the Scooter Company's total profit is option A:
A. [tex]-2x^2[/tex] + 200x - 1,500
To determine the total profit of the Scooter Company, we need to subtract the total cost from the total revenue. The total cost consists of both the variable cost (cost to produce each scooter) and the fixed cost.
Variable cost per scooter = $200
Fixed cost = $1,500
Total cost = (Variable cost per scooter * Number of scooters sold) + Fixed cost
= (200x) + 1,500
Total revenue is given by the function R(x) = 400x - [tex]2x^2.[/tex]
Total profit = Total revenue - Total cost
= (400x -[tex]2x^2[/tex]) - (200x + 1,500)
= -2[tex]x^2[/tex] + 200x - 1,500
Therefore, the function that represents the Scooter Company's total profit is option A:
A. [tex]-2x^2[/tex] + 200x - 1,500
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Suppose that the talk time on the Apple iPhone is approximately normally distributed with mean 9 hours and standard deviation 1.2 hours. (a) What proportion of the time will a fully charged iPhone last at least 6 hours? (b) What is the probability a fully charged iPhone will last less than 5 hours? (c) What talk time would represent the cutoff for the top 5% of all talk times? (d) Would it be unusual for the phone to last more than 11.5 hours? Why?
The mean of a fully charged Apple iPhone's talk time is 9 hours with a standard deviation of 1.2 hours.
(a) To find the proportion of the time that a fully charged iPhone will last at least 6 hours is 0.9938.
(b) The probability that a fully charged iPhone will last less than 5 hours is 0.0004299
(c) The talk time that represents the cutoff for the top 5% of all talk times is approximately 11.97 hours.
(d) The probability that a fully charged iPhone lasts more than 11.5 hours is 0.0475.
(a) To find the proportion of the time that a fully charged iPhone will last at least 6 hours, we can standardize the random variable Z as follows: $Z=\frac{X-\mu}{\sigma}$$Z=\frac{6-9}{1.2}=-2.5$.
To obtain the area to the right of Z = -2.5 from the standard normal distribution table, we will use the complementary property of the normal distribution.$$P(Z>-2.5)=1-P(Z<=-2.5)=1-0.0062=0.9938$$. Therefore, the probability that a fully charged iPhone will last at least 6 hours is 0.9938 or 99.38%.
(b) To obtain the probability that a fully charged iPhone will last less than 5 hours, we will standardize the random variable Z as follows: $Z=\frac{X-\mu}{\sigma}$$Z=\frac{5-9}{1.2}=-3.33$.
To find the area to the left of Z = -3.33 from the standard normal distribution table, we use the table.$$P(Z<-3.33)=0.0004299$$. Therefore, the probability that a fully charged iPhone will last less than 5 hours is 0.0004299 or 0.04%.
(c) To determine the talk time that represents the cutoff for the top 5% of all talk times, we find the Z-value corresponding to 5% at the right tail of the standard normal distribution table.$$P(Z>Z_{0.05})=0.05$$$$Z_{0.05}=1.645$$.
We can solve for the corresponding talk time using the standardized random variable equation: $Z=\frac{X-\mu}{\sigma}$Where:Z = Z-value, X = talk time, μ = mean, and σ = standard deviation$1.645=\frac{X-9}{1.2}$.
Solving for X gives:$X = (1.645*1.2)+9 = 11.97$. Therefore, the talk time that represents the cutoff for the top 5% of all talk times is approximately 11.97 hours.
d) To determine whether it would be unusual for a fully charged iPhone to last more than 11.5 hours, we calculate the Z-score as follows: $Z=\frac{X-\mu}{\sigma}$$Z=\frac{11.5-9}{1.2}=1.67$.
Using the standard normal distribution table, we find the area to the right of Z = 1.67.$$P(Z>1.67)=0.0475$$. Therefore, the probability that a fully charged iPhone lasts more than 11.5 hours is 0.0475 or 4.75%. Since the probability is less than 5%, it would be considered unusual for a fully charged iPhone to last more than 11.5 hours.
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Using the technique of front-end estimation, find an approximate value for each of the following. (a) 573+429 (c) 947-829 (a) 573+429 (Round to the nearest hundred as needed.) (b) 436 +587 (Round to t
These are rough estimates and may not be exact, but they provide a quick approximation for the values using the hundreds place as a reference.
Using the technique of front-end estimation, we can find an approximate value for each of the following calculations:
(a) 573 + 429:
To perform front-end estimation, we look at the hundreds place of each number. In this case, 573 and 429 have the same hundreds place, which is 5. We add the remaining digits together, which gives us 7 + 9 = 16. Since 16 is closer to 20 than 10, we can estimate the sum to be 500 + 20 = 520.
Approximate value: 573 + 429 ≈ 520 (rounded to the nearest hundred).
(c) 947 - 829:
Again, we focus on the hundreds place of each number. The hundreds place of 947 is 9, and the hundreds place of 829 is 8. Since 9 is larger than 8, we subtract the remaining digits, which gives us 4 - 2 = 2. Therefore, we can estimate the difference to be 900 + 2 = 902.
Approximate value: 947 - 829 ≈ 902 (rounded to the nearest hundred).
(b) 436 + 587:
For this calculation, the hundreds place of 436 is 4, and the hundreds place of 587 is 5. We add the remaining digits together, which gives us 3 + 8 = 11. Since 11 is closer to 10 than 20, we can estimate the sum to be 400 + 10 = 410.
Approximate value: 436 + 587 ≈ 410 (rounded to the nearest ten).
Using front-end estimation, we obtained approximate values for the given calculations. Please note that these are rough estimates and may not be exact, but they provide a quick approximation for the values using the hundreds place as a reference.
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Find the modulus for the complex number, -3 +5i. Round to the nearest tenth.
Answer:
Step-by-step explanation:
To find the modulus for the complex number, we need to use the absolute value formula. The modulus or absolute value of a complex number is the distance between the origin and the point representing the complex number in the complex plane.We can use the formula:
|z| = sqrt(x^2 + y^2)
where x and y are the real and imaginary parts of the complex number, respectively.
Given that the complex number is -3+5i, we can substitute in the values of x and y to the formula:
|z| = sqrt((-3)^2 + (5)^2)
|z| = sqrt(9 + 25)
|z| = sqrt(34)
|z| ≈ 5.8
Rounded to the nearest tenth, the modulus of -3+5i is approximately 5.8.
Answer:
the modulus of the complex number -3 + 5i, rounded to the nearest tenth, is approximately 5.8.
Step-by-step explanation:
For the complex number -3 + 5i, the modulus is given by:
| -3 + 5i | = √((-3)^2 + 5^2)
| -3 + 5i | = √(9 + 25)
| -3 + 5i | = √34
Rounding √34 to the nearest tenth, we get:
| -3 + 5i | ≈ 5.8
Therefore, the modulus of the complex number -3 + 5i, rounded to the nearest tenth, is approximately 5.8.
A box-shaped vessel 65 m x 10 m x 6 m is floating
upright on an even keel at 4 m draft in salt water. GM = 0.6 m.
Calculate the dynamical stability to 20 degrees heel.
The dynamical stability of the box-shaped vessel at a 20-degree heel is approximately 5,510,350 Nm.
To calculate the dynamical stability of the box-shaped vessel at a 20-degree heel, we need to consider the changes in the center of buoyancy (B) and the center of gravity (G) due to the heeling angle.
Given:
- Length (L) = 65 m
- Breadth (B) = 10 m
- Depth (D) = 6 m
- Draft (T) = 4 m
- GM = 0.6 m (metacentric height)
To determine the dynamical stability, we need to calculate the righting moment (RM) at a 20-degree heel. The formula for calculating the righting moment is:
RM = (GZ) * (W)
Where:
- GZ is the righting arm, which is the horizontal distance between the center of gravity (G) and the vertical line passing through the center of buoyancy (B)
- W is the weight of the vessel
First, let's calculate the weight of the vessel (W):
W = Density of water * Volume of the immersed portion of the vessel
W = Density of water * Length * Breadth * Draft
Assuming the density of saltwater is approximately 1025 kg/m³, we can calculate the weight as follows:
W = 1025 kg/m³ * 65 m * 10 m * 4 m
W = 26,650,000 kg
Next, we need to calculate the righting arm (GZ) at a 20-degree heel. The formula for calculating GZ is
GZ = GM * sin(heel angle)
GZ = 0.6 m * sin(20°)
GZ ≈ 0.207 m
Finally, we can calculate the dynamical stability (RM) using the formula mentioned earlier:
RM = GZ * W
RM = 0.207 m * 26,650,000 kg
RM ≈ 5,510,350 Nm (Newton-meters)
Therefore, the dynamical stability of the box-shaped vessel at a 20-degree heel is approximately 5,510,350 Nm.
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If money earns 7.20% compounded quarterly, what single payment
in three years would be equivalent to a payment of $2,550 due three
years ago, but not paid, and $500 today?
Round to the nearest cent
If money earns 7.20% compounded quarterly, then what single payment in three years would be equivalent to a payment of $2,550 due three years ago, but not paid, and $500 today Round to the nearest cent.Given information: Principal amount = $2,550Due amount = $500Rate of interest = 7.20% per annum Compounding frequency = Quarterly.
We will use the compound interest formula to find out the required single payment that is equivalent to the given payments. The formula for the future value of a present sum of money is:FV = P × (1 + r/n)^(n*t)where,FV = future value of the amountP = principal amountr = rate of interestn = compounding frequencyt = time in years.
Therefore, the required single payment that is equivalent to the given payments will be the sum of the future values (FV) of the due amount and the present amount, i.e.,$3,162.89 + $619.11= $3,782 (approx)Therefore, the required single payment that is equivalent to the given payments is $3,782 (rounded to the nearest cent).
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Catherine rolls a standard 6-sided die six times. If the product of her rolls is 2700, then how many different sequences of rolls could there have been? (The order of the rolls matters.)
The product of Catherine's rolls is 2700, and she rolls a standard 6-sided die six times. The number of different sequences of rolls there could have been is determined in this solution. So, the number of different sequences of rolls there could have been is 1200.
Break 2700 down into its prime factorization of 2 * 3^3 * 5^2. If we have six rolls of a six-sided die, there are 6! (720) permutations that we can roll. We can split the permutations based on how many times each prime number appears as a roll.
For the prime number 2, there are four permutations: {2,2,2,3,3,5}, {2,2,2,3,5,3}, {2,2,2,5,3,3}, and {2,2,3,2,3,5}.
Similarly, for the prime number 3, there are 20 permutations, and for the prime number 5, there are 15 permutations. Therefore, the number of different sequences of rolls there could have been is 4 * 20 * 15 = 1200. Answer: 1200.
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Answer the question Below.
Answer:
BX = 7 inches
Step-by-step explanation:
Since ABCD is a rectangle, the diagonals are equal and bisect each other
⇒ AC = BD and
AX = CX = BX = DX = AC/2 = BD/2
⇒ BX = A/2
⇒ BX = 14/2
⇒ BX = 7
let us Consider R with cofinite topology . find the closure of A and B where A is finite and B is infinite
The closure of the finite set A is the empty set (Closure(A) = Ø).
The closure of the infinite set B is the set itself (Closure(B) = B).
In the cofinite topology on R (the set of real numbers), a set is open if and only if its complement is finite or empty. Let's consider the sets A and B, where A is finite and B is infinite, and determine their closures.
Set A (finite):
Since A is finite, its complement A' in R is infinite. In the cofinite topology, the closure of a set is the smallest closed set that contains it. Since A' is an open set, its complement (A')' is a closed set. Therefore, the closure of A is given by:
Closure(A) = (A')'.
Since A' is infinite, its complement (A')' is the empty set since the empty set is the only closed set containing an infinite set in the cofinite topology.
Closure(A) = Ø (empty set).
Set B (infinite):
Since B is infinite, its complement B' in R is finite. In the cofinite topology, every finite set is closed. Therefore, the closure of B is given by:
Closure(B) = B.
In the cofinite topology, any set that contains all its limit points is closed. Since B' is finite and B contains all its limit points (which are in B'), B is closed, and hence, its own closure.
Closure(B) = B.
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a catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. the wait time is normally distributed with an average of 2.5 minutes and a standard deviation of 0.5 minutes. what is the probability that a caller will be assisted in less than 1.5 minutes? select one: 0.8413 0.0228 0.9772 0.0505
To find the probability that a caller will be assisted in less than 1.5 minutes, we can use the normal distribution and the given average and standard deviation. Using the Z-score formula, we can calculate the Z-score for 1.5 minutes based on the average and standard deviation provided.
The Z-score is defined as (X - μ) / σ, where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. For this case, X = 1.5 minutes, μ = 2.5 minutes, and σ = 0.5 minutes. Plugging these values into the formula, we get (1.5 - 2.5) / 0.5 = -2. To find the probability corresponding to this Z-score, we can consult a standard normal distribution table or use a calculator. The Z-score of -2 corresponds to a probability of approximately 0.0228. Therefore, the correct answer is "0.0228," representing the probability that a caller will be assisted in less than 1.5 minutes.
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A jar contains 6 red marbles, numbered 1 to 6 , and 12 blue marbles numbered 1 to 12. a) A marble is chosen at random. If youre told the marble is blue, what is the probability that it has the number 5 on it? (Round your answers to four decimal places.) b) The first marble is replaced, and another marble is chosen at random. If you're told the marble has the number 1 on it, what is the probability the marble is blue? (Round your answers to four decimal places.)
a) The probability that a randomly chosen blue marble has the number 5 on it is 0.0769 (rounded to four decimal places).
b) The probability that a randomly chosen marble with the number 1 on it is blue is 0.6667 (rounded to four decimal places).
a) To find the probability that a randomly chosen blue marble has the number 5 on it, we need to determine the number of favorable outcomes (blue marbles with the number 5) and the total number of possible outcomes (all blue marbles).
There are 12 blue marbles in total, and only one of them has the number 5.
Therefore, the probability is 1/12, which is approximately 0.0833.
However, since we are given that the marble is blue, we consider the total number of possible outcomes to be the number of blue marbles (12) instead of the total number of marbles (18).
So, the probability is 1/12, which is approximately 0.0769 after rounding to four decimal places.
b) In this case, we have to find the probability that a randomly chosen marble with the number 1 on it is blue.
Again, we need to determine the number of favorable outcomes (blue marbles with the number 1) and the total number of possible outcomes (marbles with the number 1).
There are 18 marbles with the number 1, out of which 12 are blue.
Therefore, the probability is 12/18, which simplifies to 2/3 or approximately 0.6667 after rounding to four decimal places.
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Use the Product Rule to find the derivative of the given function. b. Find the derivative by expanding the product first. f(x) = (x - 5)(x+4) a. Use the product rule to find the derivative of the function. Select the correct answer below and fill in the answer box(es) to complete your choice. A. The derivative is ()(x - 5). B. The derivative is ()x(1x+4). OC. The derivative is (x - 5)(x+4)). D. The derivative is (x-5) (2)+(x+4) (1) E. The derivative is (x-5)(1x+4)+(). b. Expand the product. (x-5)(x+4)=x²-x-20 (Simplify your answer.) d Using either approach.(x - 5)(x+4)=2x-1
Using the Product Rule to find the derivative of the given function:The derivative of the function f(x) = (x - 5)(x + 4) can be obtained by using the Product Rule of differentiation.
So The product rule is a technique in calculus that is used to find the derivative of a function that is the product of two other functions. It states that the derivative of the product of two functions is equal to the sum of the product of the derivative of the first function and the second function, and the product of the first function and the derivative of the second function.
The formula is: f'(x) = g(x) * d/dx[h(x)] + h(x) * d/dx[g(x)]where,
f'(x) = derivative of the function f(x), g(x) and h(x) are two functions and d/dx is the derivative with respect to x.The given function is f(x) = (x - 5)(x + 4).To find the derivative of the function, we need to apply the product rule, which is;
f'(x) = g(x) * d/dx[h(x)] + h(x) * d/dx[g(x)]where,
g(x) = x - 5 and
h(x) = x + 4The derivative of g(x) with respect to x is;d/dx[g
(x)] = d/dx
[x - 5] = 1The derivative of h(x) with respect to x is;
d/dx[h(x)] = d/dx
[x + 4] = 1So, we have;
f'(x) = (x + 4) * d/dx(x - 5) + (x - 5) * d/dx(x + 4)
f'(x) = (x + 4) * 1 + (x - 5) *
1f'(x) = x + 4 + x - 5f'
(x) = 2x - 1Hence, the correct option is d. The derivative is (x - 5) (2)+(x+4) (1).b. Expanding the Product:To expand the product (x - 5)(x + 4),
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Express the given equation x² + y² - 6y: = 0 in polar coordinates.
The equation x² + y² - 6y = 0 can be expressed in polar coordinates as r² - 6r sin(θ) = 0.
Given that;
The equation is,
x² + y² - 6y = 0
To express the equation x² + y² - 6y = 0 in polar coordinates, substitute x and y with their respective polar coordinate representations:
x = r cos(θ)
y = r sin(θ)
By substituting these values into the equation, we get:
(r cos(θ))² + (r sin(θ))² - 6(r sin(θ)) = 0
Now, let's simplify this expression:
r² cos²(θ) + r² sin²(θ) - 6r sin(θ) = 0
Using the trigonometric identity cos²(θ) + sin²(θ) = 1, we can simplify further:
r² × 1 - 6r sin(θ) = 0
Simplifying again, we have:
r² - 6r sin(θ) = 0
Thus, the equation x² + y² - 6y = 0 can be expressed in polar coordinates as r² - 6r sin(θ) = 0.
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The equation x² + y² - 6y = 0 in polar coordinates is (rcosΘ)² + (rsinΘ - 3)² = 9 after completing the square on the 'y' part and substituting x and y with their polar equivalents.
Explanation:To express the given equation x² + y² - 6y = 0 in polar coordinates, we first complete the square for the 'y' part, which then rewrites to x² + (y - 3)² = 9. This can be recognized as the standard form for the equation of a circle, (x-h)² + (y-k)² = r², which represents a circle of radius 'r' at the center (h, k).
To transform this to polar form, we substitute x and y with their polar equivalents, where x = rcosΘ and y = rsinΘ. Plugging these into our equation gives us (rcosΘ)² + (rsinΘ - 3)² = 9.
Therefore, the given equation in polar form is: (rcosΘ)² + (rsinΘ - 3)² = 9. It is important to note that this equation represents a circle with radius 3 at the origin (0, 3) in rectangular coordinates, or equivalently at (3, π/2) in polar coordinates.
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Examine whether participants who received different lengths of treatment differed significantly in the number of relapses they experienced.
Treatment Length M SD
Short Length (1-4 weeks) 4.95 4.26
Moderate length (5-7 weeks) 5.00 3.88
Long length (8+ weeks) 6.16 3.73
ANOVA
Number of relapses
Sum of Squares df Mean Square F Sig.
Between Groups 22.981 2 11.491 .680 .509
Within Groups 1978.319 117 16.909
Total 2001.300 119
The ANOVA table demonstrates that there was no significant difference between the participants who received treatment for various lengths of time in the number of relapses they experienced. The F-value was 0.680, with a corresponding p-value of 0.509, indicating that the null hypothesis (the three groups are not significantly different from one another) should not be rejected.
In other words, the participants who received different lengths of treatment did not have a significantly different number of relapses. The study's statistical analysis yielded results that were insignificant (F(2,117) = 0.680, p > 0.05), indicating that the number of relapses did not differ significantly based on the length of treatment received by the participants.
The participants' M values were similar across all three treatment duration categories: short length (1-4 weeks) at 4.95, moderate length (5-7 weeks) at 5.00, and long length (8+ weeks) at 6.16, but the difference was insignificant.
The participants' SD values were also similar across all three categories of treatment duration.
The short duration of treatment had an SD of 4.26, the moderate duration of treatment had an SD of 3.88, and the long duration of treatment had an SD of 3.73.
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1.)
b) find the area in square inches of a square with a radius length 8 sqrt 2
2.)
a) find The area in square centimeters of an equiangular triangle with a perimeter of 29.4 cm
b) find the area in square inches of an equiangular triangle with the radius of length 6 inches
1) The area in square inches of a square with a radius length 8 sqrt 2 is:
256 square inches.
2) a) The area in square centimeters of an equiangular triangle with a perimeter of 29.4 cm is: 41.67 square centimeters.
b) The area of the equiangular triangle with a radius length of 6 inches is 108√3 square inches.
Here, we have,
To find the area of a square with a radius length of 8√2, we need to determine the length of one side of the square.
The length of the diagonal of a square is given by d = 2r, where r is the radius of the square. In this case, the diagonal is 2(8√2) = 16√2.
The length of one side of the square can be found using the Pythagorean theorem:
s² + s² = (16√2)²
2s² = 512
s² = 256
s = 16
Therefore, the side length of the square is 16.
The area of a square is given by A = s², where s is the length of one side.
A = 16²
A = 256 square inches
2a)
To find the area of an equiangular triangle with a perimeter of 29.4 cm, we need to determine the side length of the triangle first.
Since it is an equiangular triangle, all three sides are equal in length. Let's denote the side length as s.
The perimeter of the triangle is given by P = 3s, where P is the perimeter.
29.4 = 3s
s = 29.4 / 3
s ≈ 9.8 cm
Now, to find the area of the equiangular triangle, we can use the formula A = (√3/4) * s², where A is the area and s is the side length.
A = (√3/4) * (9.8)²
A ≈ 41.67 square centimeters
2b)
To find the area of an equiangular triangle with a radius length of 6 inches, we need to determine the side length of the triangle.
The radius of the equiangular triangle is equal to the inradius, which is one-third of the height of the equilateral triangle.
Let's denote the side length as s.
The inradius (r) can be found using the formula r = (√3/6) * s, where r is the inradius and s is the side length.
6 = (√3/6) * s
s = 6 * (6/√3)
s = 12√3 inches
Now, to find the area of the equiangular triangle, we can use the formula A = (√3/4) * s², where A is the area and s is the side length.
A = (√3/4) * (12√3)²
A = (√3/4) * (144 * 3)
A = (√3/4) * 432
A = 108√3 square inches
Therefore, the area of the equiangular triangle with a radius length of 6 inches is 108√3 square inches.
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Given that the acceleration vector is a(t)=⟨−9cos(3t),−9sin(3t),3t⟩, the initial velocity is v(0)=<1,0,1>, and the initial position vector is r(0)=<1,1,1>, compute: A. The velocity vector v(t)= i+ i+
The velocity vector v(t) = -3 sin (3t) i + 3 cos (3t) j + 3t k.
The acceleration vector is
a(t)=⟨−9cos(3t),−9sin(3t),3t⟩.
The initial velocity is
v(0)=<1,0,1>,
and the initial position vector is
r(0)=<1,1,1>.
Compute: (A) The velocity vector v(t)
Let v(t) be the velocity vector.
Therefore, the velocity can be computed by integrating the acceleration:
v(t) = ∫a(t) dt
Integrating with respect to x, we get:
vx(t) = ∫−9cos(3t) dt
= -3 sin (3t) + C1
Taking the initial velocity to be
v(0) = <1,0,1>,
we can find the value of C1:
vx(0) = -3 sin (0) + C1 = 1
⇒ C1 = 1
Integrating with respect to y, we get:
vy(t) = ∫−9sin(3t) dt
= 3 cos (3t) + C2
Taking the initial velocity to be
v(0) = <1,0,1>,
we can find the value of C2:
vy(0) = 3 cos (0) + C2
= 0
⇒ C2 = -3
So, the velocity vector is given by:
v(t) = vx(t) i + v y(t) j + vz(t) k
v(t) = -3 sin (3t) i + 3 cos (3t) j + 3t k
The velocity vector
v(t) = -3 sin (3t) i + 3 cos (3t) j + 3t k
Answer: The velocity vector v(t) = -3 sin (3t) i + 3 cos (3t) j + 3t k.
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Find E(x), E(x²), the mean, the variance, and the standard deviation of the random variable whose probability density function is given belo 1 1152*, (0.48) E(x) = (Type an integer or a simplified fraction.) E(x²)=(Type an integer or a simplified fraction.) (Type an integer or a simplified fraction.) ²= (Type an integer or a simplified fraction.) =(Type an exact answer, using radicals as needed.) g= f(x)=
The final answers are `E(x) = 0.02`, `E(x²) = 0.04`, mean = `0.02`, variance = `0.0396` and the standard deviation = `0.199`.
Given that the probability density function of a random variable is `f(x) = (0.48)/1152`, `0 ≤ x ≤ 3`.
To find the `E(x)`, `E(x²)`, the mean, the variance, and the standard deviation of the random variable, use the following formulas; E(x) = ∫x * f(x) dx from `0` to `3`.
E(x²) = ∫x² * f(x) dx from `0` to `3`.
Mean = E(x).Variance
= E(x²) - [E(x)]².
Standard deviation = `√(variance)`.
The calculation of `E(x)` and `E(x²)` is shown below;`
E(x) = ∫x * f(x) dx from 0 to 3
`= `∫x * (0.48)/1152 dx from 0 to 3
`= `(0.48/1152) * ∫x dx from 0 to 3
`= `(0.48/1152) * (x²/2) from 0 to 3
`= `(0.48/1152) * (9/2)` = `0.02`
.Therefore, `E(x) = 0.02`.
Similarly, we can find `E(x²)`;`E(x²)
= ∫x² * f(x) dx from 0 to 3
`= `∫x² * (0.48)/1152 dx from 0 to 3`
= `(0.48/1152) * ∫x² dx from 0 to 3
`= `(0.48/1152) * (x³/3) from 0 to 3
`= `(0.48/1152) * (27/9)` = `0.04`.
Therefore, `E(x²) = 0.04`.
We can find the variance and the standard deviation of the random variable using `E(x)` and `E(x²)` as shown below; Variance = E(x²) - [E(x)]²`
= `0.04 - (0.02)²`
= `0.0396`.
Therefore, the variance of the random variable is `0.0396`.Standard deviation = `√(variance)` = `√(0.0396)` = `0.199`.Hence, the standard deviation of the random variable is `0.199`.
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Suppose that \( f(x, y)=x^{2}-x y+y^{2}-4 x+4 y \) with \( x^{2}+y^{2} \leq 16 \). 1. Absolute minimum of \( f(x, y) \) is 2. Absolute maximum is
According to the question the absolute minimum of [tex]\( f(x, y) \)[/tex] is 2, and the absolute maximum is 16.
To find the absolute minimum and maximum of the function [tex]\( f(x, y) = x^2 - xy + y^2 - 4x + 4y \)[/tex] over the region [tex]\( x^2 + y^2 \leq 16 \),[/tex] we need to consider the critical points and the boundary of the region.
First, let's find the critical points by taking the partial derivatives of [tex]\( f(x, y) \)[/tex] with respect to [tex]\( x \) and \( y \)[/tex] and setting them equal to zero:
[tex]\(\frac{\partial f}{\partial x} = 2x - y - 4 = 0\)[/tex]
[tex]\(\frac{\partial f}{\partial y} = -x + 2y + 4 = 0\)[/tex]
Solving these equations simultaneously, we find that the critical point is [tex]\((x, y) = (2, -2)\).[/tex]
Next, we need to examine the boundary of the region [tex]\( x^2 + y^2 \leq 16 \),[/tex] which is the circle centered at the origin with a radius of 4. We can parameterize the boundary of this circle as follows:
[tex]\(x = 4\cos(t)\)[/tex]
[tex]\(y = 4\sin(t)\)[/tex]
where [tex]\(0 \leq t \leq 2\pi\).[/tex]
Substituting these expressions into [tex]\(f(x, y)\),[/tex] we get:
[tex]\(f(t) = (4\cos(t))^2 - (4\cos(t))(4\sin(t)) + (4\sin(t))^2 - 4(4\cos(t)) + 4(4\sin(t))\)[/tex]
Simplifying further:
[tex]\(f(t) = 16\cos^2(t) - 16\cos(t)\sin(t) + 16\sin^2(t) - 16\cos(t) + 16\sin(t)\)[/tex]
We can now find the maximum and minimum values of [tex]\(f(t)\)[/tex] by evaluating it at the critical point [tex]\((2, -2)\)[/tex] and the endpoints of the parameterization [tex]\(t = 0\) and \(t = 2\pi\).[/tex]
Evaluating [tex]\(f(2, -2)\),[/tex] we get:
[tex]\(f(2, -2) = 2^2 - 2(-2) + (-2)^2 - 4(2) + 4(-2) = 2\)[/tex]
Next, let's evaluate [tex]\(f(t)\) at \(t = 0\):[/tex]
[tex]\(f(0) = 16\cos^2(0) - 16\cos(0)\sin(0) + 16\sin^2(0) - 16\cos(0) + 16\sin(0) = 16\)[/tex]
And finally, let's evaluate [tex]\(f(t)\) at \(t = 2\pi\):[/tex]
[tex]\(f(2\pi) = 16\cos^2(2\pi) - 16\cos(2\pi)\sin(2\pi) + 16\sin^2(2\pi) - 16\cos(2\pi) + 16\sin(2\pi) = 16\)[/tex]
Therefore, the absolute minimum of [tex]\(f(x, y)\)[/tex] is 2, and the absolute maximum is 16.
Hence, the absolute minimum of [tex]\( f(x, y) \)[/tex] is 2, and the absolute maximum is 16.
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Find each product by factoring the tens.
7 × 3, 7 × 30, and 7× 300
The products by factoring the tens are: 7 × 3 = 21, 7 × 30 = 210, and 7 × 300 = 2,100.
To find each product by factoring the tens, we need to separate the given number into its tens and ones place values, and then multiply the tens by the given factor.
7 × 3:
The number 7 has a tens place value of 0, so there are no tens to factor. To find the product, simply multiply 7 by 3:
7 × 3 = 21.
7 × 30:
The number 30 has a tens place value of 3.
To find the product, multiply 7 by 3:
7 × 3 = 21.
Since there is a tens place value of 3, we add a zero to the end:
21 + 0 = 210.
7 × 300:
The number 300 has a tens place value of 30.
To find the product, multiply 7 by 30:
7 × 30 = 210.
Since there is a tens place value of 30, we add two zeros to the end:
210 + 00 = 21,000.
Therefore, the products by factoring the tens are:
7 × 3 = 21,
7 × 30 = 210,
7 × 300 = 21,000.
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When particles with diameters > 50 μm are inhaled, they are more likely to... [2] (a) ...settle in the alveolar ducts due to diffusion and sedimentation. (b) ...settle all the way down to the alveoli due to a high terminal velocity. (c) ...be deposited in the upper airways by inertial impaction. (d) ...be absorbed into the bloodstream compared to small particles due to the greater surface area. (e) None of the above. 1.2. Which of the following explosion hazard control strategies is not a valid approach? [2] (a) Decreasing the oxygen level to below the MOC. (b) Completely inerting a unit using carbon dioxide. (c) Adding moisture to the dust. (d) Providing workers with ori-nasal respirators. (e) None of the above. 1.3. Particles with a density of 1200 kg.m-³ and an average diameter of 10 µm are aerated from below in a fluidized bed setup. Which of the following do you expect to see? [2] (a) Even fluidization without any bubbling. (b) Fluidization with immediate bubble formation. (c) Channel formation. (d) Spouting. 1.4. A slurry consisting of 55 vol% alumina particles suspended in a solution of 0.1 M sodium bicarbonate at a pH of 7 must be transported along a pipeline, but the high viscosity results in excessive pumping requirements. Which of the following strategies would you recommend to decrease the pumping costs? Motivate your answer. [3] (a) Addition of hydrogen chloride. (b) Addition of more sodium bicarbonate. (c) Addition of low molecular weight adsorbing polymers. (d) Addition of more alumina particles. (e) None of the above.
(a) When particles with diameters > 50 μm are inhaled, they are more likely to be deposited in the upper airways by inertial impaction.
(b) Decreasing the oxygen level to below the MOC is not a valid approach for explosion hazard control.
(c) Particles with a density of 1200 kg.m-³ and an average diameter of 10 µm are expected to show fluidization with immediate bubble formation.
(d) To decrease pumping costs for a slurry with high viscosity, the addition of low molecular weight adsorbing polymers would be recommended.
When particles with diameters > 50 μm are inhaled, they are more likely to be deposited in the upper airways by inertial impaction. Inertial impaction occurs when particles with sufficient mass and momentum are unable to follow the airstream and impact the walls of the airways.
Decreasing the oxygen level to below the Minimum Oxygen Concentration (MOC) is not a valid approach for explosion hazard control. The MOC represents the minimum oxygen concentration required for combustion to occur. Depleting oxygen below this level can prevent combustion and reduce the risk of explosions.
Particles with a density of 1200 kg.m-³ and an average diameter of 10 µm are expected to show fluidization with immediate bubble formation. These particles are relatively dense and larger in size, leading to rapid fluidization and the formation of bubbles within the fluidized bed.
To decrease pumping costs for a slurry with high viscosity, the addition of low molecular weight adsorbing polymers would be recommended. These polymers can act as flow aids, reducing the viscosity of the slurry and improving its pump ability. The polymers adsorb onto the surface of the particles, reducing interparticle interactions and increasing fluidity. This helps in reducing the energy required for pumping the slurry through the pipeline.
In summary, particles > 50 μm settle in the upper airways, and decreasing oxygen below the MOC is not a valid explosion hazard control strategy, particles with a density of 1200 kg.m-³ and an average diameter of 10 µm show fluidization with immediate bubble formation, and the addition of low molecular weight adsorbing polymers is recommended to decrease pumping costs for a high-viscosity slurry.
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(5 pts) During a pitot traverse of a duct, the following velocity pressures, in millimeters of water, were measured at the center of equal areas: 13.2,29.1,29.7,20.6,17.8,30.4, 28.4, and 15.2. What was the average of the gas pressure (in mmH 2
O )? What was the standard deviation? What was the confidence interval at 95% level?
Average gas pressure: 22.45 mmH2O
Standard deviation: 6.281 mmH2O
95% confidence interval: (17.175, 27.725) mmH2O
To calculate the average gas pressure, standard deviation, and 95% confidence interval, let's use the given velocity pressure measurements: 13.2, 29.1, 29.7, 20.6, 17.8, 30.4, 28.4, and 15.2 (in mmH2O).
Average gas pressure:
Average = (13.2 + 29.1 + 29.7 + 20.6 + 17.8 + 30.4 + 28.4 + 15.2) / 8
Average = 22.45 mmH2O
Standard deviation:
First, calculate the variance:
Variance = [[tex](13.2 - 22.45)^2[/tex] + [tex](29.1 - 22.45)^2[/tex]+[tex](29.7 - 22.45)^2[/tex]+[tex](20.6 - 22.45)^2[/tex] + [tex](17.8 - 22.45)^2[/tex] + [tex](30.4 - 22.45)^2[/tex] + [tex](28.4 - 22.45)^2[/tex] + [tex](15.2 - 22.45)^2][/tex] / (8 - 1)
Variance = 39.4238 mm[tex]H2O^2[/tex]
Next, calculate the standard deviation by taking the square root of the variance:
Standard Deviation = √(39.4238)
Standard Deviation ≈ 6.281 mmH2O
95% confidence interval:
The critical value for a 95% confidence level with 7 degrees of freedom (8 measurements - 1) is 2.365 (obtained from t-distribution tables).
Margin of Error = (Critical Value) * (Standard Deviation / √n)
Margin of Error = 2.365 * (6.281 / √8)
Margin of Error ≈ 5.275 mmH2O
The confidence interval is given by:
Confidence Interval = (Sample Mean) ± (Margin of Error)
Confidence Interval = 22.45 ± 5.275
Confidence Interval ≈ (17.175, 27.725) mmH2O
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Suppose x is a normally distributed random variable with u = 33 and 6 = 5. Find a value Xo of the random variable x. a. P(x2x)= 5 b. P(XXo) = 10 d. P(x > Xo) = 95
Given that a normally distributed random variable x with mean (μ) = 33 and standard deviation (σ) = 5.
To find the value Xo of the random variable x.
P(x2x)= 5For x2x, it is not clear from where to where we need to find the probability.
Hence, it is not possible to find the value of Xo
P(XXo) = 10Here, we need to find the value of Xo such that P(X ≤ Xo) = 0.
10.Using standard normal distribution formula, z = (X - μ) / σWhere μ = 33, σ = 5, P(X ≤ Xo) = 0.10z = (Xo - 33) / 5From standard normal distribution table, for P(Z ≤ 1.28) = 0.1003 (approximately equal to 0.10).
Therefore, z = 1.28(1.28) = (Xo - 33) / 5Xo - 33 = (1.28)(5)Xo = (1.28)(5) + 33 = 39.4
Hence, the value of Xo is 39.4.(d) P(x > Xo) = 95
Here, we need to find the value of Xo such that P(X > Xo) = 0.95
Using standard normal distribution formula, z = (X - μ) / σWhere μ = 33, σ = 5, P(X > Xo) = 0.95P(Z > z) = 0.95
From the standard normal distribution table, for P(Z > 1.64) = 0.05 (approximately equal to 0.05),P(Z < 1.64) = 1 - 0.05 = 0.95
Therefore, z = 1.64Hence, 1.64 = (Xo - 33) / 5Xo - 33 = 1.64 × 5Xo = 8.2 + 33 = 41.2 ,
Therefore, the value of Xo is 41.2.
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(P (-R (QA -S))), (PR), ((-S v U) T) - T 1. P→(-R (QA-S)) :PRI 2.-(PR) : PR 3. -SVU-T : PR
Given are three propositions: P → (-R (QA-S)), ¬P ∧ R, (-S v U) T - T, and we need to determine whether this sequence is valid or not. We can prove this by assuming the premises are true and then attempting to prove the conclusion with the help of the rules of inference.
Here is the proof:
1. P → (-R (QA-S)) : PRI (Premise)
2. ¬P ∧ R : PR (Premise)
3. -S v U : PR (Premise)
4. ¬P : 2, Simplification
5. -R (QA-S) : 1,4, Modus Tollens
6. R : 2, Simplification
7. -S : 3,4, Disjunctive Syllogism
8. Q : 5, Simplification
9. A : 5, Simplification
10. -S v U : 3, Premise
11. U : 7,10, Disjunctive Syllogism
12. (-S v U) T : 11, Addition
13. T : 12,3, Modus Ponens
Therefore, we can conclude that the sequence is valid because we were able to prove the conclusion from the premises.
The proof uses several rules of inference, including Modus Tollens, Disjunctive Syllogism, Simplification, Addition, and Modus Ponens.
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The dose-response for a specific drug is f(x)=100x2x2+0.31f(x)=100x2x2+0.31, where f(x)f(x) is the percent of relief obtained from a dose of xx grams of a drug, where 0≤x≤1.50≤x≤1.5.
Find f'(0.6)f′(0.6) and select the appropriate units.
f'(0.6)f′(0.6) =
The given function is
f(x) = 100x^2 + 0.31
The required to find the value of f '(0.6) using the above-given function is to differentiate the given function using the power rule of differentiation.
Differentiating with respect to x, we have
f(x) = 100x^2 + 0.31
f'(x) = d/dx(100x^2 + 0.31) = d/
dx(100x^2) + d/dx(0.31)
Note: The derivative of a constant term is zero. f'(x) = 200x.
Now, we can find the value of f'(0.6) as follows;f '(0.6) = 200(0.6) = 120Hence, the value of f'(0.6) is 120. The unit is a percentage per gram.
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Use the given conditions to find the exact value of the expression. cot(α)=− 4/7 ,cos(α)<0,tan(α+ π/6 )
tan (α + π/6) = [(-7/√15) + (√3/3)]/[1 - (-7/√15)*(√3/3)]= - 7(√3) - 5 / 4(√15) - 21
Given the conditions cot(α)=− 4/7 ,
cos(α)<0,
tan(α+ π/6 )
tan(α+ π/6)
Let's find sin α first.
sin α = cos α * cot α= -7/4cos α(From given data cot α= -4/7)
Therefore, sin² α = 1 - cos² α= 1 - (cos α)²
(1)Using (1)sin² α + (cos α)² = 1cos² α + (cos α)² = 1cos² α = 1 - (7/4)²= -15/16
Now, as given cos α < 0
Therefore, cos α = - √15/4=- (√15)/4
Now, tan(α+ π/6)can be written as: tan(α+ π/6) = (tan α + tan (π/6))/[1 - tan α * tan (π/6)]...
(2)tan α = sin α/cos α= - 7/(√15)
Therefore, tan (α + π/6) = [(-7/√15) + (√3/3)]/[1 - (-7/√15)*(√3/3)]= - 7(√3) - 5 / 4(√15) - 21
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Express ∑I=14i2 Without Using Summation Notation. Select The Correct Choice Below And Fill In The Answer Box To Complete Your
The sum of the squares of the numbers from 1 to 4, without using summation notation, is equal to 30
To express the summation ∑(i=1 to 4) i^2 without using summation notation, we can manually calculate the sum by adding the squares of each individual term. Let's proceed with the solution.
The given summation represents the sum of the squares of the numbers from 1 to 4. Therefore, we need to calculate the squares of each number and add them together.
Starting with i = 1, the first term of the summation, we square it: 1^2 = 1.
Moving on to i = 2, the second term, we square it: 2^2 = 4.
Proceeding to i = 3, the third term, we square it: 3^2 = 9.
Finally, for i = 4, the last term, we square it: 4^2 = 16.
Now, we add up these squared terms: 1 + 4 + 9 + 16 = 30.
Hence, the sum of the squares of the numbers from 1 to 4, without using summation notation, is equal to 30.
In summary, by calculating the square of each individual number from 1 to 4 and adding them together, we find that the sum of the squares of these numbers is 30. This method allows us to express the given summation without using summation notation.
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A state has a graduated fine system for speeding, meaning you can pay a base fine and then have more charges added on top. Forexample, the base fine for speeding is $100. But that is just the start. If you are convicted of going more than 10 mph over the speed limit, add $20 for each additional mph you were traveling over the speed limit plus 10 mph. Thus, the amount of the fine y(in dollars) for driving x mph while speeding (when the speed limit is 30 miles per hour) can be represented with the equation below.
y=20(x-40)+100, x>=If someone was fined $220 for speeding, how fast were they going?
The person was driving at a speed of 46 mph when they were fined $220 for speeding.
To determine the speed at which someone was fined $220 for speeding, we need to solve the equation:
y = 20(x - 40) + 100
Given that the fine amount y is $220, we can substitute it into the equation:
220 = 20(x - 40) + 100
Now we can solve for x, the speed at which the person was driving:
220 - 100 = 20(x - 40)
120 = 20(x - 40)
Divide both sides of the equation by 20:
6 = x - 40
Add 40 to both sides of the equation:
46 = x
Therefore, the person was driving at a speed of 46 mph when they were fined $220 for speeding.
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Consider a test of H 0 : μ= 7. For the following case, give the
rejection region for the test in terms of the z-statistic: H a :
μ≠7, α= 0.01
A) z > 2.575
B) z > 2.33
C) |z| > 2.575
D) |
The rejection region for the test, with a null hypothesis (H₀) of μ = 7 and an alternative hypothesis (Hₐ) of μ ≠ 7, and a significance level of α = 0.01, is |z| > 2.575.
To determine the rejection region for the test, we need to consider the significance level and the alternative hypothesis. Since the alternative hypothesis is μ ≠ 7, we are conducting a two-tailed test.
For a significance level of α = 0.01, we divide it equally into the two tails, resulting in α/2 = 0.005 for each tail. We then find the critical z-values corresponding to the tail probabilities.
Using a standard normal distribution table or a z-table calculator, we can find that the critical z-value for a tail probability of 0.005 is approximately 2.575.
Since the rejection region includes values that fall outside the range of -2.575 to 2.575, the rejection region for this test is |z| > 2.575.
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A feed to a continuous fractioning column analyses by weight 28 % benzene and 72 % toluene. The analysis of the distillate shows 52 weight percent benzene and 5 weight percent benzene was found in the bottom product. Calculate the amount of distillate and bottom product per 1000 kg of feed per hour. Also calculate the percent recovery of benzene. (Note: draw the block diagram for distillation of benzene-toluene feed mixture)
The amount of distillate per 1000 kg of feed per hour is 520 kg, and the amount of bottom product per 1000 kg of feed per hour is 50 kg. The percent recovery of benzene is 98%.
To calculate the amount of bottom product per 1000 kg of feed per hour, we subtract the amount of benzene in the distillate from the total amount of benzene in the feed:
Amount of bottom product = Amount of benzene in feed - Amount of benzene in distillate
= 280 kg - 0.52 * 528.85 kg
= 280 kg - 275 kg
= 5 kg
Therefore, the amount of bottom product per 1000 kg of feed per hour is 5 kg.
Finally, to calculate the percent recovery of benzene, we use the formula:
Percent recovery = (Amount of benzene in distillate / Amount of benzene in feed) * 100
Percent recovery = (0.52 * 528.85 kg / 280 kg) * 100
Simplifying the equation:
Percent recovery = (275 kg / 280 kg) * 100
= 98.21%
Rounding it to the nearest whole number, the percent recovery of benzene is 98%.
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Find The Area Of The Triangle Whose Vertices Are (0,4,2),(−1,0,3), And (1,3,4). 3) Three Forces Are Acting On An Object. The First
The area of the triangle whose vertices are (0, 4, 2), (-1, 0, 3), and (1, 3, 4) is `1/2` square units.
The coordinates of three vertices of a triangle are given as: A(0, 4, 2), B(-1, 0, 3), and C(1, 3, 4).
To determine the area of the triangle, we can use the formula that states the area of a triangle in terms of the coordinates of the vertices of the triangle. That formula is given by: `A = 1/2 |(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))|`, where `(x1, y1), (x2, y2),` and `(x3, y3)` are the vertices of the triangle.
According to this formula, we have:`A = 1/2 |(0*(0-3) + (-1)*(3-2) + 1*(4-4))|``A = 1/2 |(-1)| = 1/2`
Therefore, the area of the triangle whose vertices are (0, 4, 2), (-1, 0, 3), and (1, 3, 4) is `1/2` square units.
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