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.
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Testing: Hop 0.55 H:P +0.55 Your sample consists of 96 subjects, with 55 successes. Calculate the test statistic, rounded to 2 decimal places z Check Answer
The test statistic is approximately 0.45.
To calculate the test statistic for testing the null hypothesis H₀:p = 0.55 against the alternative hypothesis H₁:p ≠ 0.55, you can use the formula for the z-test statistic:
z = (p' - p) / √(p(1-p)/n)
where:
p' = sample proportion
p = hypothesized proportion under the null hypothesis
n = sample size
In this case, the sample proportion is p' = 55/96 = 0.5729 (rounded to 4 decimal places), p = 0.55, and n = 96.
Now let's calculate the test statistic:
z = (0.5729 - 0.55) / √(0.55 × (1-0.55) / 96)
z = (0.0229) / √(0.55 × 0.45 / 96)
z = (0.0229) / √(0.2475 / 96)
z = (0.0229) / √0.002578125
z = (0.0229) / 0.050773383
z ≈ 0.4502 (rounded to 4 decimal places)
Therefore, the test statistic is approximately 0.45.
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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."
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2hours at 30km/h 7 hours at 65 km/h 1/2 hours at 46 km/h 45 minutes at 80 km/h 1 1/2 hours at 55km/h fors numbers 1-5, calculate the distance that you would travel if you drove for;
Answer:
60 km
455 km
23 km
60 km
82.5 km
Step-by-step explanation:
Time = t
Speed = s
Distance is given by
[tex]d=s\times t[/tex]
t = 2 hours
s = 30 km/h
[tex]d=30\times 2=60\ \text{km}[/tex]
Distance driven is 60 km.
t = 7 hours
s = 65 km/h
[tex]d=65\times 7=455\ \text{km}[/tex]
Distance driven is 455 km
t = [tex]\dfrac{1}{2}\ \text{hours}=0.5\ \text{hours}[/tex]
s = 46 km/h
[tex]d=46\times 0.5=23\ \text{km}[/tex]
Distance driven is 23 km
t = 45 minutes = [tex]\dfrac{45}{60}=0.75\ \text{hours}[/tex]
s = 80 km/h
[tex]d=80\times 0.75=60\ \text{km}[/tex]
Distance driven is 60 km
t = [tex]1\dfrac{1}{2}\ \text{hours}=1.5\ \text{hours}[/tex]
s = 55 km/h
[tex]d=55\times 1.5=82.5\ \text{km}[/tex]
The distance driven is 82.5 km
Help me pleaseeeee would be really appreciated
Answer:
Q:12) 9m-1.7m =7.3m
Q:13) 1.5m-0.8m=0.7
Q:14) 60cm-30cm= 30cm
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).
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Based on the Baseball camp example covered in the class, let's assume the segment size is 10000, price per participant is $80, frequency is 1, variable cost per person is $5, TFC = $9,000.
Based on the assumption provided above, what percentage of the segment should participate if the program wants to make $1500 profit?
A) about 3.4%
B) about 0.8%
C) about 2.3%
D) about 1.4%
To determine the percentage of the segment that should participate in the baseball camp in order to make a $1500 profit, we need to calculate the breakeven point and then find the corresponding percentage.
The breakeven point is the point where the revenue equals the total cost, resulting in zero profit. In this case, the breakeven point can be calculated by adding the fixed cost (TFC) to the variable cost per person multiplied by the number of participants.
Given that the price per participant is $80 and the variable cost per person is $5, we can set up the equation:
80x - (5x + 9000) = 0
Simplifying the equation, we have:
75x - 9000 = 0
75x = 9000
x ≈ 120
So, the breakeven point is approximately 120 participants.
To calculate the percentage, we need to divide the breakeven point (120) by the segment size (10000) and multiply by 100:
(120/10000) * 100 ≈ 1.2%
Therefore, the percentage of the segment that should participate in the baseball camp to make a $1500 profit is not exactly 1.2%, but it is closest to option D) about 1.4%.
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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.
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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.
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(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]
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Can someone explain to me what to do and what is the correct answer please? I would really like to know what to do, thank u in advance if u help me!<3
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 $524.19.
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PLEASE HELP:)!!!!!
Which of the following projections is MOST accurate?
A) When the peak temperature is 40°C ,35 units of electricity are used on average
B) When the peak temperature is 40°C, 120 units of electricity are used on average
C) When the peak temperature is 40°C, 48 units of electricity are used on average
D) When the peak temperature is 40°C , 72 units of electricity are used on average
Answer:a
Step-by-step explanation:
HI CAN SOMEONE HELP ME WITH THESE PUNNET SQUARES PLS
Answer:
1. 100% Rr is the genotype. Phenotype would be red rose.
2. Genotype is 100% Rr. Phenotype is Tall bean.
3. Genotypes are Rr or rr. 50/50 chance of getting either one. Phenotype would be red rose if the genotype is Rr, phenotype would be white rose if the genotype is rr.
reply to this answer if you would like instructions for how to fill out the squares.
Step-by-step explanation:
the capital letters have to do with dominant genes. the lower case letters have to do with not dominant genes. if you have Rr, it would be a dominant gene bc the capital takes over. if you have rr it would be not dominant gene bc there are only lower case. if you have RR it would be dominant gene bc there are only capital letters.
TIP; genotype is the formula (RR, Rr, or rr) phenotype is physical characteristic.
Use the binomial formula to find the coefficient of the y^120x² term in the expansion of (y+3x)^22. ?
This coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.
The binomial formula is used to expand binomials of the form (a + b)ⁿ, where a, b, and n are integer.
In general, the formula is given by:
[tex]$(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.
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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 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]
A rectangular sandbox will be constructed for the students at an elementary school. The sandbox will be filled to the top of the sandbox with sand. If each bag of sand contains 2.6 cubic feet of sand, how many bags of sand will need to be purchased to completely fill the sandbox.
Answer:
Approximately 19 bags of sand will be required.
Step-by-step explanation:
The statement misses the drawing of the sandbox which I have attached here. We can find the dimensions mentioned in the picture.
Length of box = 6.5 feet
Width of box = 5 ft
Height of box = 1.5 ft
Calculating the volume = V = length x width x height = 6.5 x 5 x 1.5 = 48.75 cubic ft
As given in the question, one sand bag has 2.6 cubic ft of sand
No of bags required = 48.75 / 2.6 = 18.75
(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
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Use the following returns for X and Y.
Returns
Year X Y
1 22.1 % 27.3 %
2 17.1 4.1
3 10.1 29.3
4 20.2 15.2
5 5.1 33.3
1. Calculate the average returns for X and Y. (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
2. Calculate the variances for X and Y. (Do not round intermediate calculations and round your answers to 6 decimal places, e.g., 32.161616.)
3. Calculate the standard deviations for X and Y. (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
1)The average return for X is 14.9% and for Y is 21.84%.
2)The variance for X is 48.74 and for Y is 149.64.
3)The standard deviation for X is 6.98% and for Y is 12.23%.
What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variability or dispersion in a dataset. It measures how spread out the values in a dataset are around the mean or average value. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.
[tex]\begin{document}\begin{tabular}{ccc}\topruleYear & X (\%) & Y (\%) \\\midrule1 & 22.1 & 27.3 \\2 & 17.1 & 4.1 \\3 & 10.1 & 29.3 \\4 & 20.2 & 15.2 \\5 & 5.1 & 33.3 \\\bottomrule\end{tabular}[/tex]
[tex]\textbf{1. Calculate the average returns for X and Y:}[/tex]
To calculate the average return for X, we sum up all the returns for X and divide by the number of observations:
[tex]\[\text{Average return for X} = \frac{{22.1 + 17.1 + 10.1 + 20.2 + 5.1}}{5} = 14.9\%\][/tex]
To calculate the average return for Y:
[tex]\[\text{Average return for Y} = \frac{{27.3 + 4.1 + 29.3 + 15.2 + 33.3}}{5} = 21.84\%\][/tex]
Therefore, the average return for X is 14.9% and for Y is 21.84%.
[tex]\textbf{2. Calculate the variances for X and Y:}[/tex]
To calculate the variance for X, we need to calculate the squared differences between each return and the average return for X, sum them up, and divide by the number of observations minus one:
[tex]\[\text{Variance for X} = \frac{{(22.1 - 14.9)^2 + (17.1 - 14.9)^2 + (10.1 - 14.9)^2 + (20.2 - 14.9)^2 + (5.1 - 14.9)^2}}{5-1} = 48.74\][/tex]
To calculate the variance for Y:
[tex]\[\text{Variance for Y} = \frac{{(27.3 - 21.84)^2 + (4.1 - 21.84)^2 + (29.3 - 21.84)^2 + (15.2 - 21.84)^2 + (33.3 - 21.84)^2}}{5-1} = 149.64\][/tex]
Therefore, the variance for X is 48.74 and for Y is 149.64.
[tex]\textbf{3. Calculate the standard deviations for X and Y:}[/tex]
To calculate the standard deviation for X, we take the square root of the variance for X:
[tex]\[\text{Standard deviation for X} = \sqrt{48.74} \approx 6.98\%\][/tex]
To calculate the standard deviation for Y:
[tex]\[\text{Standard deviation for Y} = \sqrt{149.64} \approx 12.23\%\][/tex]
Therefore, the standard deviation for X is 6.98% and for Y is 12.23%.
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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/=
what number that you can multiple by 7 that will give you 7/10
Answer:
0.7
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).
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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.
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If 4 is 1/2 , what is the whole?
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
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! :)
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
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#
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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).
Solve for X: x/5 - 7 = 2
Please give the correct answer. Thank You!
Answer:
x = 45
Step-by-step explanation:
[tex]\frac{x}{5} -7=2\\\\\frac{x}{5}=9\\\\x=45[/tex]