Answer:
Step-by-step explanation:
The original equation is [tex]121y''+110y'-24y=0[/tex]. We propose that the solution of this equations is of the form [tex] y = Ae^{rt}[/tex]. Then, by replacing the derivatives we get the following
[tex]121r^2Ae^{rt}+110rAe^{rt}-24Ae^{rt}=0= Ae^{rt}(121r^2+110r-24)[/tex]
Since we want a non trival solution, it must happen that A is different from zero. Also, the exponential function is always positive, then it must happen that
[tex]121r^2+110r-24=0[/tex]
Recall that the roots of a polynomial of the form [tex]ax^2+bx+c[/tex] are given by the formula
[tex] x = \frac{-b \pm \sqrt[]{b^2-4ac}}{2a}[/tex]
In our case a = 121, b = 110 and c = -24. Using the formula we get the solutions
[tex]r_1 = -\frac{12}{11}[/tex]
[tex]r_2 = \frac{2}{11}[/tex]
So, in this case, the general solution is [tex]y = c_1 e^{\frac{-12t}{11}} + c_2 e^{\frac{2t}{11}}[/tex]
a) In the first case, we are given that y(0) = 1 and y'(0) = 0. By differentiating the general solution and replacing t by 0 we get the equations
[tex]c_1 + c_2 = 1[/tex]
[tex]c_1\frac{-12}{11} + c_2\frac{2}{11} = 0[/tex](or equivalently [tex]c_2 = 6c_1[/tex]
By replacing the second equation in the first one, we get [tex]7c_1 = 1 [/tex] which implies that [tex] c_1 = \frac{1}{7}, c_2 = \frac{6}{7}[/tex].
So [tex]y_1 = \frac{1}{7}e^{\frac{-12t}{11}} + \frac{6}{7}e^{\frac{2t}{11}}[/tex]
b) By using y(0) =0 and y'(0)=1 we get the equations
[tex] c_1+c_2 =0[/tex]
[tex]c_1\frac{-12}{11} + c_2\frac{2}{11} = 1[/tex](or equivalently [tex]-12c_1+2c_2 = 11[/tex]
By solving this system, the solution is [tex]c_1 = \frac{-11}{14}, c_2 = \frac{11}{14}[/tex]
Then [tex]y_2 = \frac{-11}{14}e^{\frac{-12t}{11}} + \frac{11}{14} e^{\frac{2t}{11}}[/tex]
c)
The Wronskian of the solutions is calculated as the determinant of the following matrix
[tex]\left| \begin{matrix}y_1 & y_2 \\ y_1' & y_2'\end{matrix}\right|= W(t) = y_1\cdot y_2'-y_1'y_2[/tex]
By plugging the values of [tex]y_1[/tex] and
We can check this by using Abel's theorem. Given a second degree differential equation of the form y''+p(x)y'+q(x)y the wronskian is given by
[tex]e^{\int -p(x) dx}[/tex]
In this case, by dividing the equation by 121 we get that p(x) = 10/11. So the wronskian is
[tex]e^{\int -\frac{10}{11} dx} = e^{\frac{-10x}{11}}[/tex]
Note that this function is always positive, and thus, never zero. So [tex]y_1, y_2[/tex] is a fundamental set of solutions.
The lengths of nails produced in a factory are normally distributed with a mean of 5.02 centimeters and a standard deviation of 0.05 centimeters. Find the two lengths that separate the top 6% and the bottom 6%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.
Answer:
The length that separates the top 6% is 5.1 centimeters.
The length that separates the bottom 6% is 4.94 centimeters.
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question, we have that:
[tex]\mu = 5.02, \sigma = 0.05[/tex]
Find the two lengths that separate the top 6% and the bottom 6%.
Top 6%:
The 100-6 = 94th percentile, which is X when Z has a pvalue of 0.94. So X when Z = 1.555.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.555 = \frac{X - 5.02}{0.05}[/tex]
[tex]X - 5.02 = 1.555*0.05[/tex]
[tex]X = 5.1[/tex]
So the length that separates the top 6% is 5.1 centimeters.
Bottom 6%:
The 6th percentile, which is X when Z has a pvalue of 0.06. So X when Z = -1.555.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.555 = \frac{X - 5.02}{0.05}[/tex]
[tex]X - 5.02 = -1.555*0.05[/tex]
[tex]X = 4.94[/tex]
The length that separates the bottom 6% is 4.94 centimeters.
A powerful computer is purchased for $2000, but loses 20% of its value each year. How much will it be worth 4 years from now?
a. Growth or Decay?
b. What is your multiplier?
c. Is $2000 your zero term or first term? term
d. Write the equation. (do not use spaces in your response; example: f(x)=10.2(1.22)^x )
e. Solve
Answer:
(A)Decay
(b)0.8
(c)First Term
(d)[tex]f(t)=2000(0.8)^t[/tex]
(e)$819.20
Step-by-step explanation:
The exponential function for modelling growth or decay is given as:
[tex]A(t)=A_o(1\pm r)^t[/tex],
Where:
Plus indicates growth and minus indicates decay.
[tex]A_o$ is the Initial Value\\r is the growth/decay rate\\t is the time period[/tex]
For a powerful computer that was purchased for $2000, but loses 20% of its value each year.
(a)Since it loses value, it is a decay.
(b)Multiplier
Its value decays by 20%.
Therefore, our multiplier(1-r) =(1-20&)=1-0.2
Multiplier =0.8
(c)$2000 is our First term (or Initial Value [tex]A_o[/tex])
(d)The function for this problem is therefore:
[tex]f(t)=f_o(1- r)^t\\f(t)=2000(1- 0.2)^t\\\\f(t)=2000(0.8)^t[/tex]
(e)Since we require the worth of the computer after 4 years,
t=4 years
[tex]f(4)=2000(0.8)^4\\f(4)=\$819.20[/tex]
Evaluate the expression 2x-7 for x = -4
Answer:
-15
Step-by-step explanation:
The solution of expression for x = - 4 is,
⇒ - 15
We have to given that,
An expression is,
⇒ 2x - 7
Now, We can simplify the expression for x = - 4 as,
An expression is,
⇒ 2x - 7
Plug x = - 4;
⇒ 2 × - 4 - 7
⇒ - 8 - 7
⇒ - 15
Thus, The solution of expression for x = - 4 is,
⇒ - 15
Learn more about the equation visit:
brainly.com/question/28871326
#SPJ6
Bonita said that the product of 5/6 x 1 2/3 is 7/3.
How can you tell that her answer is wrong.
Answer:
Bonita's product is too large
Step-by-step explanation:
The two factors in the problem are (5/6) and (5/3). The factor 5/6 is less than 1, ensuring that the product will be less than 5/3.
Bonita's result of 7/3 is more than 5/3, so is too large to be the product.
Please answer this correctly
Answer:
28 and 7
35
Step-by-step explanation:
The area of a triangle is base*height/2, no matter the shape.
So the big one is 8*7/2 = 28 in²
And the little one is 2*7/2 = 7 in²
The total trapezoid therefore has an area of 28+7=35 in²
The radius of a circle is 2.6 in. Find the circumference
to the nearest tenth.
Answer:
16.
Step-by-step explanation:
since given the radius and the formula of the circumference of a circle is 2pie*r
Graph the equation below by plotting the
y-intercept and a second point on the
line. When you click Done, your line will
appear
Answer:
Plot the y-intercept at (0, 2). Plot your second point at (-1, -1).
Step-by-step explanation:
I attached an image of what the finished graph should look like when you press done. *rotate the image so the grey is on the bottom*
The volume of a triangular prism is increased by a factor of 8.
By what factor is the surface area of the figure increased?
o 2
o 4
016
o 24
what is 3 43/ 100 as a decimal number.
Answer:
3.43
Step-by-step explanation:
3 is the whole number and 43 out of 100 is a standard fraction that can simply be stated as 0.43. Hope this helps!
Answer:
3.43
Step-by-step explanation:
Used calculator.
A polygon will be dilated on a coordinate grid to create a smaller polygon. The polygon is dilated using the origin as the center of dilation. Which rule could represent this dilation
Answer:
(x, y) → (4/5 x, 4/5 y)
Question:
The answer choices to determine the rule that represent the dilation were not given. Let's consider the following question:
A polygon will be dilated on a coordinate grid to create a smaller polygon. The polygon is dilated using the origin as the center of dilation. Which rule could represent this dilation?
A) (x, y) → (0.5 − x, 0.5 − y)
B) (x, y) → (x − 7, y − 7)
C) (x, y) → ( 5/4 x, 5/4 y)
D) (x, y) → (4/5 x, 4/5 y)
Step-by-step explanation:
To determine the rule that could represent the dilation, we would multiply each coordinate by a dilation factor (a constant) to create a dilation. Since the dilation would be used to create a smaller polygon, the constant multiplied with the coordinates of x and y would be less than 1.
Let's check the options out.
In option (A), the coordinates is subtracted from the constant (0.5).
In option (B), the constant (7) is subtracted from the coordinates.
In option (C), the coordinates are multiplied by constant (5/4).
But 5/4 = 1.25. This is greater than 1.
In option (D), the coordinates are multiplied by constant (4/5).
4/5 = 0.8
The constant multiplied with the coordinates of x and y is less than 1 in option (D) = (x, y) → (4/5 x, 4/5 y)
4/5 = 0.8
0.8 is less than 1
What is the value of x to the nearest tenth? gradpoint
Answer:
5
Step-by-step explanation:
y=2x−4y=−12x+1 Question 1 options: a) (3, 2) b) (0, 2) c) (2, 0) d) (2, 3)
I need help with solving this
Answer:
49
Step-by-step explanation:
Positive 49 not -49
Prove that : sin^4(2x)=3/8-1/2cos4x+1/8cos8x
Someone plzz help!!!
A man spend 3/5 of his money and has$ 90 left . how much did he have initially
Answer:
225 I believe because if he has $90 that is 2/5 so add another 90 whitch is 2/5 as well and divide 90 by 2 and to get 45 which all adds up to 225
What is the answer for this one ?
4x+3y=20 2x+y=7
Answer:
x = 1/2 , y = 6
Explanation:
Step 1 - Align the equations and multiply the second row by 2
4x + 3y = 20
2x + y = 7
4x + 3y = 20
4x + 2y = 14
Step 2 - Subtract them both
4x + 3y = 20
4x + 2y = 14
y = 6
So, y = 6
Step 3 - Substitute y into the first equation
4x + 3y = 20
4x + 3(6) = 20
4x + 18 = 20
Step 4 - Subtract 18 from both sides
4x + 18 = 20
4x + 18 - 18 = 20 - 18
4x = 2
Step 5 - Divide both sides by 4
4x = 2
4x / 4 = 2 / 4
x = 2/4
So, x = 1/2
What is the value of
3/7x0.1/5/21
?
7
А.1/98
B.9/50
С.9/5
D.18/1
Answer:
B
Step-by-step explanation:
[tex]\dfrac{3}{7}\times 0.1 \div \dfrac{5}{21}= \\\\\\\dfrac{3}{7}\times \dfrac{1}{10}\times \dfrac{21}{5}= \\\\\\\dfrac{3\times 1 \times 21}{7 \times 10 \times 5}=\\\\\\\dfrac{63}{350}=\\\\\\\dfrac{9}{50}[/tex]
Therefore, the correct answer is choice B. Hope this helps!
Answer:
The answer to your question is 9/50
What’s the correct answer for this question?
Answer:
The last option is the correct choice 33.5
Step-by-step explanation:
[tex]V=\pi r^2\frac{h}{3} \\=\pi 2^2\frac{8}{3} \\=33.51\\=33.5[/tex]
Answer:
D
Step-by-step explanation:
In the attached file
Find the surface area of this triangular prism. Be sure to include the correct unit in your answer.
15 m
12 m
0
9 m
11 m
Thanks for anyone that answers
4) If the data below contained an outlier, which coordinate would best represent the outlier?
(MGSE8.SP.1)
100
90
80
70
60
50
Weight
(kgs)
40
30
20
10
0
0
200
250
100 150
Height (cms)
A. (150, 60)
B. (50,20)
C. (200, 100)
D. (250, 80)
Answer:
D. (250, 80)
Step-by-step explanation:
a) Outliers are values that "lie outside" the other values in a dataset, because their values are "far away" from the main group of data.
b) In this case, the values of A, B, and C have ratios of their coordinates of about 2.5, but the coordinate ratio of D is more than 3. This makes it to lie far away from the group of data, and therefore an outliner.
c) The Ratios of the Coordinate Values are calculated as follows: A = 2.5 (150/60), B = 2.5 (50/20), C = 2 (200/100), while D = 3.125 (250/80).
What term should be inserted in
p²-
___+36 to make it a perfect
Square
Answer:
12p
Step-by-step explanation:
the perfect square is form
(p - 6)² = p² - 12p + 36
Answer:
12p
Step-by-step explanation:
p² - ? + 36 = p² - ? + (-6)² = p² -2*6*p + (-6)² = p² - 12p + 36 = (p-6)²
A fair die is rolled twice, with outcomes X for the first roll and Y for the second roll. Find the moment generating function MX`Y ptq of X ` Y . Note that your answer should be a function of t and can contain unsimplified finite sums.
Answer:
[tex]\mathbf{\dfrac{e^{2t}}{36} + \dfrac{e^{3t}}{18} + \dfrac{e^{4t}}{12} +\dfrac{e^{5t}}{9} + \dfrac{5e^{6t}}{36} + \dfrac{7e^{7t}}{6} + \dfrac{5e^{8t}}{36} + \dfrac{e^{9t}}{9} + \dfrac{e^{10t}}{12} + \dfrac{e^{11t}}{18} + \dfrac{e^{12t}}{36} }[/tex]
Step-by-step explanation:
The objective is to find the moment generating function of [tex]M_{X+Y}(t) \ of \ X+Y[/tex].
We are being informed that the fair die is rolled twice;
So; X to be the value for the first roll
Y to be the value of the second roll
The outcomes of X are: X = {1,2,3,4,5,6}
Where ;
[tex]P (X=x) = \dfrac{1}{6}[/tex]
The outcomes of Y are: y = {1,2,3,4,5,6}
Where ;
[tex]P (Y=y) = \dfrac{1}{6}[/tex]
The outcome of Z = X+Y
[tex]= \left[\begin{array}{cccccc}(1,1)&(1,2)&(1,3)&(1,4)&(1,5)&(1,6)\\ (2,1)&(2,2)&(2,3)&(2,4)&(2,5)&(2,6)\\ (3,1)&(3,2)&(3,3)&(3,4)&(3,5)&(3,6) \\ (4,1)&(4,2)&(4,3)&(4,4)&(4,5)&(4,6) \\ (5,1)&(5,2)&(5,3)&(5,4)&(5,5)&(5,6) \\ (6,1)&(6,2)&(6,3)&(6,4)&(6,5)&(6,6) \end{array}\right][/tex]
= [2,3,4,5,6,7,8,9,10,11,12]
Here;
[tex]P (Z=z) = \dfrac{1}{36}[/tex]
∴ the moment generating function [tex]M_{X+Y}(t) \ of \ X+Y[/tex]is as follows:
[tex]M_{X+Y}(t) \ of \ X+Y[/tex] = [tex]E(e^{t(X+Y)}) = E(e^{tz})[/tex]
⇒ [tex]\sum \limits^{12}_ {z=2 } et ^z \ P(Z=z)[/tex]
= [tex]\mathbf{\dfrac{e^{2t}}{36} + \dfrac{e^{3t}}{18} + \dfrac{e^{4t}}{12} +\dfrac{e^{5t}}{9} + \dfrac{5e^{6t}}{36} + \dfrac{7e^{7t}}{6} + \dfrac{5e^{8t}}{36} + \dfrac{e^{9t}}{9} + \dfrac{e^{10t}}{12} + \dfrac{e^{11t}}{18} + \dfrac{e^{12t}}{36} }[/tex]
Find the area of the smaller sector.
Round to the nearest tenth.
Help needed fast
Answer:
About 22.2 square feet
Step-by-step explanation:
First, you need to find the area of the full circle. The area of a circle is \pi r^2, which in this case is:
[tex]\pi r^2 = \pi \cdot 7.13^2\approx 159.628[/tex]
Now, since the sector is only 50 out of the total 360 degrees in a circle, you need to multiply this value by 50/360, which yields and area of about 22.2 square feet. Hope this helps!
please help you will get 10 points and brainliest. and explain your answer.
Answer:
Top prism = 262 in.² Bottom prism = 478 in.²
Step-by-step explanation:
top prism:
front + back: 5 x 3 = 15
sides: 19 x 4 x 2 = 152
bottom: 19 x 5 = 95
15 + 152 + 95 = 262
bottom prism:
front + back: 5 x 6 x 2 = 60
sides: 19 x 6 x 2 = 228
top + bottom: 19 x 5 x 2 = 190
60 + 228 + 190 = 478
True or false: as the value of cosx decreases towards 0, the value of secx increases towards infinity
Answer:
True
Step-by-step explanation:
If a random sample of 53 students was asked for the number of semester hours they are taking this semester. The sample standard deviation was found to be s = 4.7 semester hours. How many more students should be included in the sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean for all students at this college?
Answer:
94 more students should be included in the sample.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
How many students we need to sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean?
We need to survey n students.
n is found when M = 1.
We have that [tex]\sigma = 4.7[/tex]
So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]1 = 2.575*\frac{4.7}{\sqrt{n}}[/tex]
[tex]\sqrt{n} = 2.575*4.7[/tex]
[tex](\sqrt{n})^{2} = (2.575*4.7)^{2}[/tex]
[tex]n = 146.47[/tex]
Rounding up
147 students need to be surveyed.
How many more students should be included...?
53 have already been surveyed
147 - 53 = 94
94 more students should be included in the sample.
write this small number in standard form. 0.00078
Answer:
78/100000
Hope it helps you
#2 Jamal is an apprentice on a boat on Long Island Sound. He is helping the captain collect samples of
marine life for an environmental study, and the captain is teaching him about nautical navigation.
When the boat leaves the environmental station, it will return to its home port 9 nautical miles away.
If
the boat maintains a constant speed of 15 knots (nautical miles per hour), how many minutes will the
trip take?
Answer:
The trip will take 36 minutes.
Step-by-step explanation:
This question can be solved using a rule of three.
The boat maintains a constant speed of 15 knots (nautical miles per hour). How many minutes it will take to return to its home port 9 nautical miles away?
So in 60 minutes, 15 nautical miles. How many minutes for 9 nautical miles?
60 minutes - 15 nautical miles
x minutes - 9 nautical miles
[tex]15x = 60*9[/tex]
[tex]x = \frac{60*9}{15}[/tex]
[tex]x = 36[/tex]
The trip will take 36 minutes.
g You run a regression analysis on a bivariate set of data ( n = 14 ). With ¯ x = 27.7 and ¯ y = 26.5 , you obtain the regression equation y = 0.495 x − 14.914 with a correlation coefficient of r = 0.39 . You want to predict what value (on average) for the response variable will be obtained from a value of 110 as the explanatory variable. What is the predicted response value?
Answer:
Predicted response value = 39.536
Note that this predicted response value will most likely be far from the actual response value because the correlation coefficient of the regression equation used (r = 0.39) is too far away from 1.
Step-by-step explanation:
The response variable is the dependent variable (y) whose value is obtained from the expression involving the independent variable (x).
For this question, although the correlation coefficient, r = 0.39, is far from 1, the regression equation is
y = 0.495x - 14.914
The predicted response value will be obtained from the explanatory variable and the regression equation
x = 110
y = 0.495x - 14.914
y = (0.495×110) - 14.914 = 39.536
Note that this predicted response value will most likely be far from the actual response value because the correlation coefficient of the regression equation used (r = 0.39) is too far away from 1.
Hope this Helps!!!
which law would you use to simply the expression 3^10/3^4 quotient power power of a quotient product of powers power of a product
