Answer:
.
Step-by-step explanation:
in a group of 62 students; 27 are normal, 13 are abnormal, and 32 are normal abnormal. find the probability that a student picked from this group at random is either a normal or abnormal?
In a group of 62 students, 27 are normal, 13 are abnormal, and 32 are normal-abnormal. We want to find the probability that a student picked at random is either normal or abnormal.
To calculate this probability, we need to consider the total number of students who are either normal or abnormal. This includes the students who are solely normal (27), solely abnormal (13), and those who are both normal and abnormal (32). We add these numbers together to get the total count of students who fall into either category, which is 27 + 13 + 32 = 72.
The probability of picking a student who is either normal or abnormal can be calculated by dividing the total count of students who are either normal or abnormal by the total number of students in the group. Therefore, the probability is 72/62 = 1.1613.
To find the probability of picking a student who is either normal or abnormal, we consider the total number of students falling into those categories. Since a student can only be classified as either normal, abnormal, or normal-abnormal, we need to count the students falling into each category and add them together. Dividing this sum by the total number of students gives us the probability. In this case, the probability is greater than 1 because there seems to be an error in the provided data, where the total count of students who are either normal or abnormal (72) exceeds the total number of students in the group (62).
To know more about probability click here: brainly.com/question/31828911
#SPJ11
PLSSS HELP IMMEDIATELY!!!!! i’ll give brainiest, i’m not giving brainiest if u leave a link tho. (pls check whole picture!!)
Answer:
(4,2)
Step-by-step explanation:
Answer:
(4, 2)
Step-by-step explanation:
Determine the number of zeros of the function f(z) = Z^4 – 2z^3 + 9z^2 + z – 1 in the disk D[0,2].
Given the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1. We have to determine the number of zeros of the function in the disk D[0,2].
According to the Fundamental Theorem of Algebra, a polynomial function of degree n has n complex zeros, counting multiplicity. Here, the degree of the given polynomial function is 4. Therefore, it has exactly 4 zeros.Let the zeros of the function f(z) be a, b, c, and d. The function can be written as the product of its factors:$$f(z) = (z-a)(z-b)(z-c)(z-d)$$$$\Rightarrow f(z) = z^4 - (a+b+c+d)z^3 + (ab+ac+ad+bc+bd+cd)z^2 - (abc+abd+acd+bcd)z + abcd$$
According to the Cauchy's Bound, if a polynomial f(z) of degree n is such that the coefficients satisfy a_0, a_1, ..., a_n are real numbers, and M is a real number such that |a_n|≥M>|a_n-1|+...+|a_0|, then any complex zero z of the polynomial satisfies |z|≤1+M/|a_n|.
We can write the polynomial function as $$f(z) = z^4 - 2z^3 + 9z^2 + z - 1 = (z-1)^2(z+1)(z-1+i)(z-1-i)$$The zeros of the function are 1 (multiplicity 2), -1, 1 + i, and 1 - i. We have to count the zeros that are in the disk D[0,2].Zeros in the disk D[0,2] are 1 and -1.Therefore, the number of zeros of the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1 in the disk D[0,2] is 2.
Know more about Fundamental Theorem of Algebra:
https://brainly.com/question/29015928
#SPJ11
What is the y-intercept for the equation y= 11x + 1?
-11
-1
1
11
Answer:
1
Step-by-step explanation:
The y-intercept in the equation is 1 because the equation uses the format y=mx+b. The b in y=mx+b represents the y-intercept So, in this equation the y-intercept is 1 because b=1.
After driving 50 miles, you get caught in a storm and have to slow down by 10 mph. You then drive 75 miles at this slower speed all the way home. Find an equation for the time t of the trip as a function of the speed s of your car before slowing down.
The equation for the time of the trip, t, as a function of the speed, s, is t = (50/s) + (75/(s-10)).
To find the equation for the time of the trip as a function of the speed of the car before slowing down, we need to consider two parts of the journey. The first part is driving 50 miles at the original speed, which takes (50/s) hours, where s is the speed. The second part is driving 75 miles at a slower speed of (s-10) mph, which takes (75/(s-10)) hours.
To calculate the total time, we add the times for both parts: t = (50/s) + (75/(s-10)). This equation allows us to determine the time of the trip for any given speed before slowing down.
To learn more about function
Click here brainly.com/question/17085093
#SPJ11
use calculus to find the volume of the following solid s: the base of s is the triangular region with vertices (0, 0), (3, 0), and (0, 2). cross-sections perpendicular to the y-axis are semicircles.
The volume of the solid S, where the base is a triangular region and cross-sections perpendicular to the y-axis are semicircles, can be found using calculus. The volume of S is (3π/8) cubic units.
In the first part, the volume of the solid S is (3π/8) cubic units.
In the second part, we can find the volume of S by integrating the areas of the cross-sections along the y-axis. Since the cross-sections are semicircles, we need to find the radius of each semicircle at a given y-value.
Let's consider a vertical strip at a distance y from the x-axis. The width of the strip is dy, and the height of the semicircle is the x-coordinate of the triangle at that y-value. From the equation of the line, we have x = (3/2)y.
The radius of the semicircle is half the width of the strip, so it is (1/2)dy. The area of the semicircle is then[tex](1/2)\pi ((1/2)dy)^2 = (\pi /8)dy^2.[/tex]
To find the limits of integration, we note that the base of the triangle extends from y = 0 to y = 2. Therefore, the limits of integration are 0 to 2.
Now, we integrate the area of the semicircles over the interval [0, 2]:
V = ∫[tex](0 to 2) (\pi /8)dy^2 = (\pi /8) [y^3/3][/tex] (evaluated from 0 to 2) = (3π/8).
Learn more about integration here:
https://brainly.com/question/31744185
#SPJ11
Use the binomial formula to find the coefficient of the y^120x² term in the expansion of (y+3x)^22. ?
This coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.
The binomial formula is used to expand binomials of the form (a + b)ⁿ, where a, b, and n are integer.
In general, the formula is given by:
[tex]$(a+b)^n=\sum_{k=0}^{n}{n \choose k}a^{n-k}b^k$[/tex]
The coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² can be found by using the binomial formula.
To find this coefficient, we need to determine the value of k for which the term [tex]y^{22-k} (3x)^k[/tex] has y¹²⁰x² as a product.
Let's write out the first few terms of the expansion of (y + 3x)²²:
[tex]$(y + 3x)^{22} = {22 \choose 0}y^{22}(3x)^0 + {22 \choose 1}y^{21}(3x)^1 + {22 \choose 2}y^{20}(3x)^2 + \cdots$[/tex]
Notice that each term in the expansion has the form {22 choose k}[tex]y^{22-k} (3x)^k[/tex]
Thus, the coefficient of the y¹²⁰ x² term is given by the binomial coefficient {22 choose k}, where k is the value that makes 22 - k equal to the exponent of y in y¹²⁰ (i.e., 120). Therefore, we have:
22 - k = 120k = 22 - 120k = -98
Thus, the coefficient of the y¹²⁰ x² term is given by the binomial coefficient {22 choose -98}.
However, this coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.
To know more about binomials formula, visit:
https://brainly.com/question/30100288
#SPJ11
"
A Bernoulli differential equation is one of the form dy + P(x)y dx Q(x)y"" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n
For values of n other than 0 or 1 in a Bernoulli differential equation, the substitution [tex]u = y^{(1-n)[/tex] is used to transform it into a linear equation.
A Bernoulli differential equation is given by the form:
dy + P(x)y dx = Q(x)[tex]y^n[/tex] (*)
If we consider the case when n = 0 or n = 1, the Bernoulli equation becomes linear. Let's examine each case:
When n = 0:
Substituting[tex]u = y^{(-n) }= y^{(-0)} = 1[/tex], the differential equation becomes:
[tex]dy + P(x)y dx = Q(x)y^0[/tex]
dy + P(x)y dx = Q(x)
This is a linear differential equation of the first order.
When n = 1:
Substituting [tex]u = y^{(-n) }= y^{(-1)},[/tex] we have:
[tex]u = y^{(-1)[/tex]
Taking the derivative of both sides with respect to x:
[tex]du/dx = -y^{(-2)} \times dy/dx[/tex]
Rearranging the equation:
[tex]dy/dx = -y^2\times du/dx[/tex]
Now substituting the expression for dy/dx in the original Bernoulli equation:
[tex]dy + P(x)y dx = Q(x)y^1\\-y^2 \times du/dx + P(x)y dx = Q(x)y\\-y \times du + P(x)y^3 dx = Q(x)y[/tex]
This equation is also a linear differential equation of the first order, but with the variable u instead of y.
In summary, when n is equal to 0 or 1, the Bernoulli equation becomes linear. For other values of n, a substitution u = y^(-n) is typically used to transform the Bernoulli equation into a linear differential equation, allowing for easier analysis and solution.
for such more question on differential equation
https://brainly.com/question/25731911
#SPJ8
Plzzzzz I need help asap thank you
No links plzzzzz
Answer:
-1.4 ; -0.7 ; 0.003 ; 3% ; 0.3 ; 2/3 ; 7/8 ; 100/50
Step-by-step explanation:
0.003 ; -1.4 ; 100/50 = 2 ; 0.85 ; 2/3 = 0.67 ; 3% = 3/100 = 0.03 ; 7/8 = 0.875 ;
0.3 , -0.7
0.003 ; -1.4 ; 2 ; 0.85 ; 0.67 ; 0.03 ; 0.875 ; 0.3 ; -0.7
Least to greatest:
-1.4 ; - 0.7 ; 0.003 ; 0.03 ; 0.3 ; 0.67 ; 0.85 ;2
-1.4 ; -0.7 ; 0.003 ; 3% ; 0.3 ; 2/3 ; 7/8 ; 100/50
Negative numbers have least value.Then in decimal numbers, the number having the least value in tenth is the least
the expression 2 x ( x − 7 ) 2 is equivalent to x 2 b x 49 for all values of x . what is the value of b ?
To determine the value of b in the expression x^2b(x - 7)^2, we can compare it with the given equivalent expression x^2b49. By equating the two expressions, we can solve for b.
In the given expression x^2b(x - 7)^2, we can simplify it by multiplying the exponents:
x^2 * b * (x - 7)^2 = x^2b(x^2 - 14x + 49)
Comparing this with the equivalent expression x^2b49, we can equate the coefficients of the like terms:
x^2b(x^2 - 14x + 49) = x^2b49
From this equation, we can see that the coefficient of the x term is -14b. For it to be equivalent to 49, we have:
-14b = 49
Solving for b, we divide both sides by -14:
b = -49/14 = -7/2
Therefore, the value of b is -7/2.
To know more about algebraic expressions click here: brainly.com/question/28884894
#SPJ11
What is the geometric mean of 4 and 3? Your answer should be a reduced radical, NOT A DECIMAL.
Answer:
[tex] 2 \sqrt{3} [/tex]
Step-by-step explanation:
Geometric mean of 4 and 3
[tex] = \sqrt{4 \times 3} \\ = \sqrt{ {2}^{2} \times 3 } \\ = 2 \sqrt{3} [/tex]
(2/3)^2 without exponents
Answer:
[tex]\frac{4}{9}[/tex]
Step-by-step explanation:
[tex](\frac{2}{3} )^{2} =\frac{2^2}{3^2} =4/9[/tex]
Hope that helps :)
Trigonometry question help,,, NO LINKS
Answer:
87 ft
Step-by-step explanation:
SohCahToa is your best friend here.
You have two values you need to pay attention to:
The length that is adjacent to the 74°C, 25 ft. And the length opposite of the 74°C, the height of how high the rocket traveled.
So adjacent and opposite, O & A. "Toa", find the tangent of 74°C.
tan(74) = [tex]\frac{x}{25}[/tex]
x = (tan(74))(25)
x = 87 ft
The values of certain types of collectibles can often fluctuate greatly over time. Suppose that the value of a limited-edition flamingos riding alligators lawn ornament set is found to be able to be modeled by the function V(t) = 0.06t4 – 1.05t3 + 3.47t? – 8.896 +269.95 for Osts 15 where V(t) is in dollars, t is the number of years after the lawn ornament set was released, and t = 0 corresponds to the year 2006. a) What was the value of the lawn ornament set in the year 2009? b) What is the value of the lawn ornament set in the year 2021? c) What was the instantaneous rate of change of the value of the lawn ornament set in the year 2013? d) What is the instantaneous rate of change of the value of the lawn ornament set in the year 2021? e) Use your answers from parts a-d to ESTIMATE the value of the lawn ornament set in 2022.
The value of the lawn ornament set in the year 2009 was $51.375. The value of the lawn ornament set in the year 2021 was $558.181. The instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986. The instantaneous rate of change of the value of the lawn ornament set in the year 2021 was $351.076. The estimated value of the lawn ornament set in 2022 was $909.257.
a)
To find the value of the lawn ornament set in the year 2009, we have to plug in t = 3, as t = 0 corresponds to the year 2006.
V(3) = 0.06(3)4 – 1.05(3)3 + 3.47(3) – 8.896 + 269.95
V(3) = 51.375
So, the value of the lawn ornament set in the year 2009 was $51.375.
b)
To find the value of the lawn ornament set in the year 2021, we have to plug in t = 15, as t = 0 corresponds to the year 2006.
V(15) = 0.06(15)4 – 1.05(15)3 + 3.47(15) – 8.896 + 269.95
V(15) = $558.181
So, the value of the lawn ornament set in the year 2021 is $558.181.
c)
To find the instantaneous rate of change of the value of the lawn ornament set in the year 2013, we have to find V'(7), where V(t) is the given function.
V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts 15
V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95
V'(7) = 0.24(7)3 – 3.15(7)2 + 10.41(7) + 269.95
V'(7) = $230.986
So, the instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986.
d) To find the instantaneous rate of change of the value of the lawn ornament set in the year 2021, we have to find V'(15), where V(t) is the given function.
V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts
15V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95
V'(15) = 0.24(15)3 – 3.15(15)2 + 10.41(15) + 269.95
V'(15) = $351.076
So, the instantaneous rate of change of the value of the lawn ornament set in the year 2021 is $351.076.
e)
To ESTIMATE the value of the lawn ornament set in 2022, we can use the formula
V(t) ≈ V(a) + V'(a)(t – a),
where a is the year 2021.
V(a) = V(15) = $558.181
V'(a) = V'(15) = $351.076t = 16 (as we need to estimate the value of the lawn ornament set in 2022)
V(t) ≈ V(a) + V'(a)(t – a)
V(t) ≈ 558.181 + 351.076(16 – 15)
V(t) ≈ $909.257
So, the estimated value of the lawn ornament set in 2022 is $909.257.
To learn more about lawn: https://brainly.com/question/30132672
#SPJ11
HI CAN SOMEONE HELP ME WITH THESE PUNNET SQUARES PLS
Answer:
1. 100% Rr is the genotype. Phenotype would be red rose.
2. Genotype is 100% Rr. Phenotype is Tall bean.
3. Genotypes are Rr or rr. 50/50 chance of getting either one. Phenotype would be red rose if the genotype is Rr, phenotype would be white rose if the genotype is rr.
reply to this answer if you would like instructions for how to fill out the squares.
Step-by-step explanation:
the capital letters have to do with dominant genes. the lower case letters have to do with not dominant genes. if you have Rr, it would be a dominant gene bc the capital takes over. if you have rr it would be not dominant gene bc there are only lower case. if you have RR it would be dominant gene bc there are only capital letters.
TIP; genotype is the formula (RR, Rr, or rr) phenotype is physical characteristic.
Use the data set and line plot below. Jerome studied the feather lengths of some adult fox sparrows.
How long are the longest feathers in the data set?
A.
2
2
inches
B.
2
1
4
214
inches
C.
2
1
2
212
inches
D.
2
3
4
234
inches
Answer: 2 1/2
Step-by-step explanation:
the answer is D i took the test here is proof
A random sample of 1,200 households are selected to estimate the mean amount spent on groceries weekly. A 90% confidence interval was determined from the sample results to be ($150, $250). Which of the following is the correct interpretation of this interval? Question 9 options:
There is a 90% chance that the mean amount spent on groceries is between $150 and $250.
90% of the households will have a weekly grocery bill between $150 and $250
We are 90% confident that the mean amount spent on groceries among the 1,200 households is between $150 and $250.
We are 90% confident that the mean amount spent on groceries among all households is between $150 and $250.
The correct interpretation of the given 90% confidence interval ($150, $250) is:
"We are 90% confident that the mean amount spent on groceries among the 1,200 households is between $150 and $250."
Given that a random sample of 1,200 households are selected to estimate the mean amount spent on groceries weekly. A 90% confidence interval was determined from the sample results to be ($150, $250).
This interpretation accurately reflects the concept of a confidence interval. It means that if repeat the sampling process multiple times and construct 90% confidence intervals, approximately 90% of those intervals would contain the true population mean amount spent on groceries. However, it does not imply that there is a 90% chance for any specific household or the mean to fall within this interval.
It is important to note that the interpretation refers specifically to the mean amount spent on groceries among the 1,200 households in the sample. It does not provide information about individual households or the entire population of households.
Therefore, the correct interpretation of the given 90% confidence interval ($150, $250) is:
"We are 90% confident that the mean amount spent on groceries among the 1,200 households is between $150 and $250."
Learn more about confidence intervals, click here:
https://brainly.com/question/29738242
#SPJ4
Consider Z is the subset of R with its usual topology. Find the subspace topology for Z.[r2]
The subspace topology for Z, which is a subset of R with its usual (standard) topology, is the set of open sets in Z.
In other words, the subspace topology on Z is obtained by considering the intersection of Z with open sets in R.
To find the subspace topology for Z, we need to determine which subsets of Z are open. In the usual topology on R, an open set is a set that can be represented as a union of open intervals. Since Z is a subset of R, its open sets will be the intersection of Z with open intervals in R.
For example, let's consider the open interval (a, b) in R. The intersection of (a, b) with Z will be the set of integers between a and b (inclusive) that belong to Z. This intersection is an open set in Z.
By considering all possible open intervals in R and their intersections with Z, we can generate the collection of open sets that form the subspace topology for Z. This collection of open sets will satisfy the axioms of a topology, including the properties of openness, closure under unions, and closure under finite intersections.
To know more about subspace, refer here:
https://brainly.com/question/32552995#
#SPJ11
(q18) The average time to get your order at a restaurant is 15 minutes. What is probability that you will receive your order in the first 10 minutes?
Note:
where µ is the average value.
The correct answer is option (C): 0.487
Given that the average time to receive an order at a restaurant is 15 minutes, we can use the exponential distribution to calculate the probability of receiving the order in the first 10 minutes.
The exponential distribution is defined by the probability density function (PDF): f(x) = (1/µ) * e^(-x/µ), where µ is the average value or mean.
In this case, the mean (µ) is 15 minutes. We want to find P(a ≤ X ≤ b), where a is 0 (the lower bound) and b is 10 (the upper bound).
To calculate this probability, we need to integrate the PDF from a to b:
P(0 ≤ X ≤ 10) = ∫[0 to 10] (1/15) * e^(-x/15) dx
Integrating this expression gives us:
P(0 ≤ X ≤ 10) = [-e^(-x/15)] from 0 to 10
Plugging in the values, we get:
P(0 ≤ X ≤ 10) = [-e^(-10/15)] - [-e^(0/15)]
Simplifying further:
P(0 ≤ X ≤ 10) = -e^(-2/3) + 1
Using a calculator, we can evaluate this expression:
P(0 ≤ X ≤ 10) ≈ 0.487
For more questions on exponential distribution
https://brainly.com/question/13339415
#SPJ8
Factor this expression using the GCF (greatest common factor) and then explain how you can verify your answer:
6ab+8a
Answer:
2ax(3b+4)
Step-by-step explanation:
there you go your answer
What is (-m)⁻³n if m = 2 and n = -24?
Answer:
-3
Step-by-step explanation:
(-2)^-3 x (-24)
(-2)^3 becomes 1/(-2)^3 in order to make the negative exponent a positive one.
then, you do 1/-8 (the -8 is the (-2)^3 simplified) x -24
1/-8 x (24) = 24/-8 = -3.
Hope this helps! :)
1 points For all named stors that have made landfall in the United States since 2000, of interest is to determine the mean sustained wind speed of the storms at the time they made landfall in this scenario, what is the population of interest?
The population of interest in the given scenario is all named storms that have made landfall in the United States since 2000. "All named storms that have made landfall in the United States since 2000".
The given scenario is focusing on determining the mean sustained wind speed of all named storms that have made landfall in the United States since 2000. Therefore, the population of interest in this scenario is all named storms that have made landfall in the United States since 2000. The population of interest is the entire group of individuals, objects, events, or processes that researchers want to investigate to answer their research questions.
The researchers want to determine the mean sustained wind speed of all named storms that have made landfall in the United States since 2000. Hence, they will collect data on the wind speed of all named storms that have made landfall in the United States since 2000, and calculate the mean sustained wind speed for the entire population.
Learn more about research questions: https://brainly.com/question/27824868
#SPJ11
If 4 is 1/2 , what is the whole?
Help meeeeeeeeeeeeee
Answer:
x = 120°
Step-by-step explanation:
this is a 7-sided polygon and the sum of the interior angles is (7-2)×180° = 900°
add all 7 angles together and set equal to 900
x + 150 + x - 20 + 140 + 120 + x + 20 + 130 = 900
combine 'like terms'
3x + 540 = 900
3x = 360
x = 120
A rectangular prism has a base area of 400 square inches. The volume of the prism is 2,400 cubic inches. What is the height of the prism? (7.9A)
Answer:
6
Step-by-step explanation:
2,400 divided by 4 = 6
Convert the following base-ten numerals to a numeral in the indicated bases. a. 861 in base six b. 2157 in base nine C. 131 in base three a. 861 in base six is six
The values of base-ten numerals to the indicated bases are:
a. 861 in base six is 3553.
b. 2157 in base nine is 2856.
c. 131 in base three is 11221.
To convert the base-ten numerals to the indicated bases:
a. 861 in base six:
To convert 861 to base six, we divide the number by six repeatedly and note down the remainder until the quotient becomes zero.
861 ÷ 6 = 143 remainder 3
143 ÷ 6 = 23 remainder 5
23 ÷ 6 = 3 remainder 5
3 ÷ 6 = 0 remainder 3
Reading the remainders in reverse order, the base-six representation of 861 is 3553.
b. 2157 in base nine:
To convert 2157 to base nine, we follow a similar process.
2157 ÷ 9 = 239 remainder 6
239 ÷ 9 = 26 remainder 5
26 ÷ 9 = 2 remainder 8
2 ÷ 9 = 0 remainder 2
Reading the remainders in reverse order, the base-nine representation of 2157 is 2856.
c. 131 in base three:
To convert 131 to base three, we apply the same procedure.
131 ÷ 3 = 43 remainder 2
43 ÷ 3 = 14 remainder 1
14 ÷ 3 = 4 remainder 2
4 ÷ 3 = 1 remainder 1
1 ÷ 3 = 0 remainder 1
Reading the remainders in reverse order, the base-three representation of 131 is 11221.
Therefore:
a. 861 in base six is 3553.
b. 2157 in base nine is 2856.
c. 131 in base three is 11221.
To know more about base-ten numerals refer here:
https://brainly.com/question/24020782
#SPJ11
Enter values to complete the table below.
Please help me.
Answer:
6
0
2
Step-by-step explanation:
I assume we are taking the y-value and dividing it by x, as indicated by the y/x
-6/-1
6
0/1
0
6/3
2
Solve the non-homogeneous IVP: y'(t)=-X(t) (x(0)= 1,7(0) = 0 a. using the matrix exponential method, b. using any other method of your choice. . Find a Fundamental Matrix 0(t) and solve the IVP: x'= 3y 1 y' = 3* (x(0) = 1, y(0)=0 , for x(t) and y(t).
Using the matrix exponential method, the solution to the non-homogeneous IVP y'(t) = -x(t), with initial conditions x(0) = 1 and y(0) = 0, is given by X(t) = [1 - t; -t 1]. Alternatively, solving the system of equations x'(t) = 3y(t) and y'(t) = 3x(t) yields [tex]\[x(t) = \frac{3yt^2}{2} + t\][/tex] and [tex]\[y(t) = \frac{3xt^2}{2}\][/tex] as the solution.
Here is the explanation :
(a) Using the matrix exponential method:
The given system of equations can be written in matrix form as:
X' = A*X + B, where X = [y; x], A = [0 -1; 0 0], and B = [0; -1].
To solve this system using the matrix exponential method, we first need to find the matrix exponential of A*t. The matrix exponential is given by:
[tex]\[e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \dotsb\][/tex]
To find the matrix exponential, we calculate the powers of A:
A² = [0 -1; 0 0] * [0 -1; 0 0] = [0 0; 0 0]
A³ = A² * A = [0 0; 0 0] * [0 -1; 0 0] = [0 0; 0 0]
...
Since A² = A³ = ..., we can see that Aⁿ = 0 for n ≥ 2. Therefore, the matrix exponential becomes:
[tex]\[e^{At} = I + At\][/tex]
Substituting the values of A and t into the matrix exponential, we get:
[tex][e^{At} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & -t \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -t \\ 0 & 1 \end{bmatrix}][/tex]
Now we can find the solution to the non-homogeneous system using the matrix exponential:
[tex]\[X(t) = e^{At} X(0) + \int_0^t e^{A\tau} B d\tau\][/tex]
Substituting the given initial conditions X(0) = [1; 0] and B = [0; -1], we have:
X(t) = [1 -t; 0 1] * [1; 0] + ∫[0, t] [1 -τ; 0 1] * [0; -1] dτ
Simplifying the integral and matrix multiplication, we get:
X(t) = [1 -t; 0 1] * [1; 0] + ∫[0, t] [0; -1] dτ
= [1 -t; 0 1] * [1; 0] + [-t 1]
Finally, we obtain the solution:
X(t) = [1 -t; -t 1]
(b) Using another method:
Given the system of equations:
x' = 3y
y' = 3x
We can solve this system by taking the derivatives of both equations:
x'' = 3y'
y'' = 3x'
Substituting the initial conditions x(0) = 1 and y(0) = 0, we have:
x''(0) = 3y'(0) = 0
y''(0) = 3x'(0) = 3
Integrating the second-order equations, we find:
x'(t) = 3yt + C₁
y'(t) = 3xt + C₂
Applying the initial conditions x'(0) = 0 and y'(0) = 3, we get:
C₁ = 0
C₂ = 3
Integrating once again, we obtain:
[tex]\[\begin{aligned}x(t) &= \frac{3yt^2}{2} + C_1t + C_3 \\y(t) &= \frac{3xt^2}{2} + C_2t + C_4\end{aligned}\][/tex]
Substituting the initial conditions x(0) = 1 and y
(0) = 0, we have:
C₃ = 1
C₄ = 0
Therefore, the solution to the system is:
[tex]\[\begin{aligned}x(t) &= \frac{3yt^2}{2} + t \\y(t) &= \frac{3xt^2}{2}\end{aligned}\][/tex]
Thus, we have obtained the solutions for x(t) and y(t) using an alternative method.
To know more about the matrix exponential method refer here :
https://brainly.com/question/29729397#
#SPJ11
x'(t)= y(t)-1 1. Solve the non-homogeneous IVP: y'(t)=-X(t) (x(0)= 1,7(0) = 0 a. using the matrix exponential method, b. using any other method of your choice. . Find a Fundamental Matrix 0(t) and solve the IVP: x'= 3y 1 y' = 3* (x(0) = 1, y(0)=0 , for x(t) and y(t).
The marked price of a radio is Sh. 12600. If the shopkeeper can allow a discount of 15% on the marked price and still make a profit of 25%.At what price did the shopkeeper buy the radio?
Answer:
13388
Step-by-step explanation:
12600 will be 100% so we want to get at what price its sold when there is a 15%dicount
So will minus 15% from the 100% of the Mp
100%-15%=85%
so if 100%=12600
what about 85%=?
we crossmultiply
85%×12600/100%=10710
so10710 is what the radio will be sold if a 15% dicount is given but we want to get wat price the shopkeeper got in that he made a profit of25%
so if 100%=10710
what about 125%
125%×10710/100%=13387.5 which is 13388/=
The total cost (in dollars) of manufacturing x auto body frames is C(x)=50,000+400x (A) Find the average cost per unit if 400 frames are produced (B) Find the marginal average cost at a production level of 400 units. (C) Use the results from parts (A) and (B) to estimate the average cost per frame if 401 frames are produced
(A) The average cost per unit when 400 frames are produced is $525.
(B) The marginal average cost at a production level of 400 units is approximately $0.999 per frame. (C) The estimated average cost per frame if 401 frames are produced is approximately $524.19.
(A) Average cost per unit = Total cost / Number of frames
= C(x) / x
= (50,000 + 400x) / x
Substituting x = 400:
Average cost per unit = (50,000 + 400 * 400) / 400
= (50,000 + 160,000) / 400
= 210,000 / 400
= 525 dollars
So, the average cost per unit when 400 frames are produced is $525.
To find the marginal average cost at a production level of 400 units, we need to calculate the derivative of the average cost function:
(B) Marginal average cost = d/dx [(50,000 + 400x) / x]
= (400 - 50,000/x^2) / x
Substituting x = 400:
Marginal average cost = (400 - 50,000/400^2) / 400
= (400 - 50,000/160,000) / 400
= (400 - 0.3125) / 400
= 399.6875 / 400
= 0.999
The marginal average cost at a production level of 400 units is approximately 0.999 dollars per frame.
To estimate the average cost per frame if 401 frames are produced, we can use the average cost function:
(C) Average cost per unit = (50,000 + 400x) / x
Substituting x = 401:
Average cost per unit = (50,000 + 400 * 401) / 401
= (50,000 + 160,400) / 401
= 210,400 / 401
≈ 524.19 dollars
Therefore, the estimated average cost per frame when 401 frames are produced is approximately $524.19.
Learn more about average here: https://brainly.com/question/8501033
#SPJ11