Answer:
[tex]k=\frac{10}{(100t^2+1)^{\frac{3}{2}}}[/tex]
Or
[tex]k=\frac{10\sqrt{100t^2+1}}{(100t^2+1)^{2}}[/tex]
Step-by-step explanation:
We want to compute the curvature of the parameterized curve, [tex]r(t)=\:<\:6+5t^2,t,0)\:>\:[/tex] using the alternative formula:
[tex]k=\frac{|a\times v|}{|v|^3}[/tex].
We first compute the required ingredients.
The velocity vector is [tex]v=r'(t)=<\:10t,1,0\:>[/tex]
The acceleration vector is given by [tex]a=r''(t)=<\:10,0,0\:>[/tex]
The magnitude of the velocity vector is [tex]|v|=\sqrt{(10t)^2+1^2+0^2}=\sqrt{100t^2+1}[/tex]
The cross product of the velocity vector and the acceleration vector is
[tex]a\times v=\left|\begin{array}{ccc}i&j&k\\10&0&0\\10t&1&0\end{array}\right|=10k[/tex]
We now substitute ingredients into the formula to get:
[tex]k=\frac{|10k|}{(\sqrt{100t^2+1})^3}[/tex].
[tex]k=\frac{10}{(100t^2+1)^{\frac{3}{2}}}[/tex]
Or
[tex]k=\frac{10\sqrt{100t^2+1}}{(100t^2+1)^{2}}[/tex]
Jayanta is raising money for the? homeless, and discovers each church group requires 2 hr of letter writing and 1 hr of? follow-up calls, while each labor union needs 2 hr of letter writing and 3 hr of ?follow-up. She can raise ?$150 from each church group and ?$175 from each union. She has a maximum of 20 hours of letter writing and 14 hours of ?follow-up available each month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.
Answer:
Jayanta needs 2 labor union groups and 8 church groups.
Step-by-step explanation:
Let c denotes churches .
Let l denotes labor unions.
We know that Jayanta can only spend 20 hours letter writing and 14 hour of follow-up.
So, equations becomes:
[tex]2c+2l=20[/tex]
[tex]c+3l=14[/tex]
And total money raised can be shown by = [tex]150c+175l[/tex]
We have to maximize [tex]150c+175l[/tex] keeping in mind that [tex]2c+2l \leq 20[/tex] and [tex]c+3l \leq 14[/tex]
We will solve the two equations: [tex]2c+2l=20[/tex] and [tex]c+3l=14[/tex]
We get l = 2 and c = 8
And total money raised is [tex]150\times8 + 175\times2[/tex] = [tex]1200+350=1550[/tex] dollars.
Hence, Jayanta needs 2 labor union groups and 8 church groups.
Find the slope and the y -intercept of the line.
Write your answers in simplest form.
9x - 3y = -2
Answer:
The slope is: 3
The y-intercept is: [tex]\frac{2}{3}[/tex] or [tex]0.66[/tex]
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope of the line and "b" is the y-intercept.
To write the given equation in this form, we need to solve for "y":
[tex]9x - 3y = -2\\\\- 3y = -9x-2\\\\y=3x+\frac{2}{3}[/tex]
Therefore, you can identify that the slope of this line is:
[tex]m=3[/tex]
And the y-intercept is:
[tex]b=\frac{2}{3}=0.66[/tex]
The function f(x)= x(squared) is similar to: g(x)= -3(x-5)(squared)+4. Describe the transformations. Show Graphs
Answer:
Parent function f(x) is inverted, stretched vertically by 1 : 3, shifted 5 units right and 4 units upwards to form new function g(x).
Step-by-step explanation:
The parent function graphed is f(x) = x²
This graph when inverted (parabola opening down)function becomes
g(x) = -x²
Further stretched vertically by a scale factor of 1:3 then new function becomes as g(x) = -3x²
Then we shift this function by 5 units to the right function will be
g(x) = -3(x - 5)²
At last we shift it 4 units vertically up then function becomes as
g(x) = -3(x - 5)² + 4
If the trapezoid below is reflected across the x-axis, what are the coordinates of B”?
Answer:
B'(3, -8)
Step-by-step explanation:
The image is the mirror image of the trapezoid below the x-axis.
Each x-coordinate remains the same. Each y-coordinate becomes the opposite.
B'(3, -8)
Answer:
(3 , -8)
Step-by-step explanation:
The current coordinates of B. are (3,8). The x-axis is the horizontal line that runs across. This means that if the trapezoid were to be reflected, it would end up upside down. When this happens, only the y value changes its sign.
In short, your Y value would become negative, making 8 change to -8
Conferences and conventions are resources that should be explored as part of a job search? True or False
Answer:
The given statement is true.
Step-by-step explanation:
Conferences and conventions are resources that should be explored as part of a job search.
Yes this statement is true.
One can choose the various conferences being held around them through sources like internet, social media, newspapers etc. These are very helpful for job seekers as one can get a lot of job related advice.
Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y=6x2 and y=x2+2. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?
Answer:[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]
Step-by-step explanation:
Given two upward facing parabolas with equations
[tex]y=6x^2 & y=x^2+2[/tex]
The two intersect at
[tex]6x^2=x^2+2[/tex]
[tex]5x^2=2[/tex]
[tex]x^2[/tex]=[tex]\frac{2}{5}[/tex]
x=[tex]\pm \sqrt{\frac{2}{5}}[/tex]
area enclosed by them is given by
A=[tex]\int_{-\sqrt{\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left [ \left ( x^2+2\right )-\left ( 6x^2\right ) \right ]dx[/tex]
A=[tex]\int_{\sqrt{-\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left ( 2-5x^2\right )dx[/tex]
A=[tex]4\left [ \sqrt{\frac{2}{5}} \right ]-\frac{5}{3}\left [ \left ( \frac{2}{5}\right )^\frac{3}{2}-\left ( -\frac{2}{5}\right )^\frac{3}{2} \right ][/tex]
A=[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]
The area of the enclosed region is -√2/15 square units.
Explanation:To find the area of the enclosed region, we need to find the points of intersection between the two equations y=6x^2 and y=x^2+2. Setting them equal to each other:
6x^2 = x^2 + 2
5x^2 = 2
x^2 = 2/5
x = ±√(2/5)
Substituting these values of x back into one of the equations, we can find the corresponding y values:
For x = √(2/5), y = 6(√(2/5))^2 = 6(2/5) = 12/5
For x = -√(2/5), y = 6(-√(2/5))^2 = 6(2/5) = 12/5
Now we can find the area of the enclosed region by calculating the definite integral of y=6x^2 - (x^2+2) from x = -√(2/5) to x = √(2/5). This can be done using the fundamental theorem of calculus:
∫[√(2/5), -√(2/5)] [6x^2 - (x^2+2)] dx = ∫[√(2/5), -√(2/5)] (5x^2 - 2) dx = [5/3x^3 - 2x] [√(2/5), -√(2/5)] = 2(√(2/5))^3 - 5/3(√(2/5))^3 = 4√2/15 - 5√2/15 = -√2/15
Therefore, the area of the enclosed region is -√2/15 square units.
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The manager of a fashionable restaurant open Wednesday through Saturday says that the restaurant does about 29 percent of its business on Friday night, 31 percent on Saturday night, and 21 percent on Thursday night. What seasonal relatives would describe this situation?(Round your answers to 2 decimal places.)
Wednesday
Thursday
Friday
Saturday
Answer: The seasonal relatives is calculated are as follows:
Step-by-step explanation:
Given that,
restaurant only open from Wednesday to Saturday,
29 percent of its business on Friday31 percent on Saturday night21 percent on Thursday night.∴ The remaining 19% of its business he does on Wednesday
Now, suppose that total production of sales in a given week be 'y'
So, average sales in a week = [tex]\frac{y}{4}[/tex]
If we assume that y = 1
hence, average sales in a week = [tex]\frac{1}{4}[/tex]
= 0.25
Now, we have to calculate the seasonal relatives,
that is,
= [tex]\frac{Sales in a given day}{average sales in a week}[/tex]
Wednesday:
= [tex]\frac{0.19}{0.25}[/tex]
= 0.76
Thursday:
= [tex]\frac{0.21}{0.25}[/tex]
= 0.84
Friday:
= [tex]\frac{0.29}{0.25}[/tex]
= 1.16
Saturday:
= [tex]\frac{0.31}{0.25}[/tex]
= 1.24
- Wednesday:0.76
- Thursday:0.84
- Friday:1.16
- Saturday:1.24
To determine the seasonal relatives for each night, we need to express the business done each night as a percentage of the total business for the week. The given percentages are:
- Friday: 29%
- Saturday: 31%
- Thursday: 21%
First, let's find the total percentage accounted for by Wednesday, Thursday, Friday, and Saturday. Since we're missing Wednesday's percentage, we can sum the given percentages and subtract from 100%.
[tex]\[\text{Total percentage} = 29\% + 31\% + 21\% = 81\%\][/tex]
The remaining percentage for Wednesday is:
[tex]\[\text{Wednesday's percentage} = 100\% - 81\% = 19\%\][/tex]
Now, we'll convert these percentages into seasonal relatives. Seasonal relatives are the ratios of each night's business to the average nightly business across the four nights.
First, compute the average nightly business percentage:
[tex]\[\text{Average nightly business percentage} = \frac{100\%}{4} = 25\%\][/tex]
Next, calculate the seasonal relatives by dividing each night's percentage by the average nightly business percentage:
1. Wednesday:
[tex]\[ \text{Wednesday's seasonal relative} = \frac{19\%}{25\%} = 0.76 \][/tex]
2. Thursday:
[tex]\[ \text{Thursday's seasonal relative} = \frac{21\%}{25\%} = 0.84 \][/tex]
3. Friday:
[tex]\[ \text{Friday's seasonal relative} = \frac{29\%}{25\%} = 1.16 \][/tex]
4. Saturday:
[tex]\[ \text{Saturday's seasonal relative} = \frac{31\%}{25\%} = 1.24 \][/tex]
Rounded to two decimal places, the seasonal relatives are:
- Wednesday:0.76
- Thursday:0.84
- Friday:1.16
- Saturday:1.24
Forty percent of the homes constructed in the Quail Creek area include a security system. Three homes are selected at random: What is the probability all three of the selected homes have a security system
Answer: 0.064
Step-by-step explanation:
Binomial probability formula :-
[tex]P(X)=^nC_x \ p^x\ (1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.
Given : The proportion of the homes constructed in the Quail Creek area include a security system : [tex]p=0.40[/tex]
Now, if three homes are selected at random, then the probability all three of the selected homes have a security system is given by :-
[tex]P(3)=^3C_3 \ (0.40)^3\ (1-0.40)^{3-3}\\\\=(0.40)^3=0.064[/tex]
Hence, the probability all three of the selected homes have a security system = 0.064
The probability that all three homes selected at random in the Quail Creek area have a security system is 6.4%.
The probability that all three of the selected homes in the Quail Creek area have a security system, given that 40% of the homes have a security system, can be calculated by using the rule for independent events in probability.
Since each house is selected at random, we can multiply the probability of each house having a security system together:
P(all three homes have a security system) = P(home 1 has security system) × P(home 2 has security system) × P(home 3 has security system)
As every home has a 40% (or 0.40) chance of having a security system:
P(all three) = 0.40 × 0.40 × 0.40 = 0.064
Therefore, there is a 6.4% chance that all three homes selected will have a security system.
Convert the measurement as indicated. 52 ft. = _____ yd. _____ ft.
options:
17, 1
16, 1
17, 2
18, 2
Answer:
17, 1
Step-by-step explanation:
You can find the yards by multiplying by the conversion factor, then determining what the fraction or remainder means. Since there are 3 ft in 1 yd, 1/3 yd is 1 ft.
52 ft = (52 ft) × (1 yd)/(3 ft) = 52/3 yd = 17 1/3 yd = 17 yd 1 ft
To convert feet to yards, you divide by 3. Therefore, 52 feet is equivalent to 17 yards and 1 foot.
Explanation:In mathematics, particularly in the subject of measurement conversions, it is useful to know that 1 yard (yd) is equal to 3 feet (ft). So, to convert from feet to yards, you divide the number of feet by 3. Applying this principle to the question, you will take the 52 feet and divide it by 3.
52 feet ÷ 3 = 17.3 repeating
However, since the options provided do not include a decimal, we take the whole number part of the answer, which is 17 yards. There is a remainder when you divide 52 by 3, which indicates there are additional feet not making up a full yard. In this case, it is 1 foot (the decimal part times 3), making the final conversion 17 yards 1 foot.
So, 52 ft equals 17 yards and 1 foot.
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If x, y, a and b are greater than zero and x/y lessthanorequalto a/b, prove that x+a/y+b lessthanorequalto a/b
Answer:
Step-by-step explanation:
Given x,y,a&b are greater than zero
also [tex]\frac{x}{y}[/tex][tex]\leq [/tex][tex]\frac{a}{b}[/tex]
since x,y,a&b are greater than zero therefore we can cross multiply them without changing the inequality
therefore
[tex]\frac{x}{a}[/tex][tex]\leq [/tex][tex]\frac{y}{b}[/tex]
adding 1 on both sides we get
[tex]\frac{x}{a}[/tex]+1[tex]\leq [/tex][tex]\frac{y}{b}[/tex]+1
[tex]\frac{x+a}{a}[/tex][tex]\leq [/tex][tex]\frac{y+b}{b}[/tex]
rearranging
[tex]\frac{x+a}{y+b}[/tex][tex]\leq [/tex][tex]\frac{a}{b}[/tex]
Use Newton's method with initial approximation x1 = −2 to find x2, the second approximation to the root of the equation x3 + x + 6 = 0. (Round your answer to four decimal places.)
Answer:
The value of [tex]x_2=-1.6923[/tex].
Step-by-step explanation:
Consider the provided information.
The provided formula is [tex]f(x)=x^3+x+6[/tex]
Substitute [tex]x_1=-2[/tex] in above equation.
[tex]f(x_1)=(-2)^3+(-2)+6[/tex]
[tex]f(x_1)=-8-2+6[/tex]
[tex]f(x_1)=-4[/tex]
Differentiate the provided function and calculate the value of [tex]f'(x_1)[/tex]
[tex]f'(x)=3x^2+1[/tex]
[tex]f'(x)=3(-2)^2+1[/tex]
[tex]f'(x)=13[/tex]
The Newton iteration formula: [tex]x_2=x_1-\frac{f(x_1)}{f'(x_1)}[/tex]
Substitute the respective values in the above formula.
[tex]x_2=-2-\frac{(-4)}{13}[/tex]
[tex]x_2=-2+0.3077[/tex]
[tex]x_2=-1.6923[/tex]
Hence, the value of [tex]x_2=-1.6923[/tex].
To use Newton's method with an initial approximation of -2 on the equation x3 + x + 6 = 0, we identify the function and its derivative. We then substitute into Newton's method's formula, x_2 = x_1 - f(x_1) / f'(x_1). This will provide us with an approximate second root, which can then be refined further.
Explanation:The subject of the question pertains to Newton's method for root finding, which is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. The function in question is x3 + x + 6 = 0 and the initial approximation provided is -2.
First, in order to use Newton's method, we need to identify the function and its derivative. The function (f(x)) is x3 + x + 6. The derivative (f'(x)) would be 3x2 + 1.
Newton's method follows the formula: x_(n+1) = x_n - f(x_n) / f'(x_n).
With x_1 as -2, we substitute into the Newton method's formula to find x_2. Hence, x_2 = -2 - f(-2) / f'(-2), which results in an approximation of the root that can be further refined. Remember to round your answer to four decimal places after calculations.
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Find the general solution of the following nonhomogeneous second order differential equation: y" - 4y = e^2x
Answer:
Solution is [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]
Step-by-step explanation:
the given equation y''-4y[tex]=e^{2x}[/tex] can be written as
[tex]D^{2}y-4y=e^{2x}\\\\(D^{2}-4)y=e^{2x}\\\\[/tex]
The Complementary function thus becomes
y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}
where [tex]m_{1} , m_{2}[/tex] are the roots of the [tex]D^{2}-4[/tex]
The roots of [tex]D^{2}-4[/tex] are +2,-2 Thus the comlementary function becomes
[tex]y=c_{1}e^{2x}+c_{2}e^{-2x}[/tex]
here [tex]c_{1},c_{2}[/tex] are arbitary constants
Now the Particular Integral becomes using standard formula
[tex]y=\frac{e^{ax}}{f(D)}\\\\y=\frac{e^{ax}}{f(a)} (f(a)\neq 0)\\\\y=x\frac{e^{ax}}{f'(a)}(f(a)=0)[/tex]
[tex]y=\frac{e^{2x}}{D^{2}-4}\\\\y=\frac{e^{2x}}{(D+2)(D-2)}\\\\y=\frac{1}{D-2}\times \frac{e^{2x}}{2+2}\\\\y=\frac{1}{4}\times \frac{e^{2x}}{D-2}\\\\y=\frac{1}{4}xe^{2x}[/tex]
Hence the solution is = Complementary function + Particular Integral
Thus Solution becomes [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]
The final general solution is [tex]y(x) = C1e^2x + C2e^-2x + 1/2xe^2x[/tex].
To find the general solution of the given differential equation: y'' - 4y = e2x, we will follow these steps:
1. Solve the Homogeneous Equation
First, solve the homogeneous part: y'' - 4y = 0
The characteristic equation is: r2 - 4 = 0
Solving for r, we get: r = ±2
Thus, the general solution to the homogeneous equation is: yh(x) = C1e2x + C2e-2x
2. Find a Particular Solution
Next, find a particular solution, yp(x), to the non homogeneous equation through the method of undetermined coefficients. Assume a particular solution of the form: yp(x) = Axe2x
Differentiating, we get: yp' = Ae2x + 2Axe2x and yp'' = 4Axe2x + 2Ae2x
Substitute these into the original equation:
4Axe2x + 2Ae2x - 4(Axe2x) = e2x
which simplifies to: 2Ae2x = e2x
Thus, A = 1/2
So, the particular solution is: yp(x) = (1/2)xe2x
3. Form the General Solution
The general solution to the nonhomogeneous differential equation is the sum of the homogeneous solution and the particular solution:
y(x) = yh(x) + yp(x)
Therefore, the general solution is: [tex]y(x) = C1e2x + C2e-2x + (1/2)xe2x[/tex].
A manufacturer produces bearings, but because of variability in the production process, not all of the bearings have the same diameter. The diameters have a normal distribution with a mean of 1.2 centimeters (cm) and a standard deviation of 0.03 cm. The manufacturer has determined that diameters in the range of 1.17 to 1.23 cm are acceptable. What proportion of all bearings falls in the acceptable range? (Round your answer to four decimal places.)
Answer:
68%
Step-by-step explanation:
It is given that the diameters of bearing have a normal distribution.
Mean = u = 1.2 cm
Standard deviation = [tex]\sigma[/tex] = 0.03 cm
We have to find the proportion of values which falls in between 1.17 to 1.23
In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:
[tex]z=\frac{x- \mu}{\sigma}[/tex]
For 1.17:
[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]
For 1.23:
[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]
So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question.
According to the empirical rule:
68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1.
Therefore, the answer to this question is 68%
Professor Jones has to select 6 students out of his English class randomly to participate in a regional contest. There are 36 students in the class. Is this a PERMUTATION or a COMBINATION problem? How many ways can Prof. Jones choose his students?
Answer: This is a combination.
There are 1947792 ways to choose his students.
Step-by-step explanation:
Since we have given that
Number of students in a class = 36
Number of students selected for his English class = 6
We would use "Combination" .
As permutation is used when there is an arrangement.
whereas Combination is used when we have select r from group of n.
So, Number of ways that Prof. Jones can choose his students is given by
[tex]^{36}C_6=1947792[/tex]
Hence, there are 1947792 ways to choose his students.
A ball is thrown vertically upward. After t seconds, its height h (in feet) is given by the function h(t) = 60t - 16t^2 . What is the maximum height that the ball will reach?
Do not round your answer.
Answer:
56.25 feet.
Step-by-step explanation:
h(t) = 60t - 16t^2
Differentiating to find the velocity:
v(t) = 60 -32t
This equals zero when the ball reaches its maximum height, so
60-32t = 0
t = 60/32 = 1.875 seconds
So the maximum height is h(1.875)
= 60* 1.875 - 16(1.875)^2
= 56.25 feet.
Answer: 56.25 feet.
Step-by-step explanation:
For a Quadratic function in the form [tex]f(x)=ax^2+bx+c[/tex], if [tex]a<0[/tex] then the parabola opens downward.
Rewriting the given function as:
[tex]h(t) = - 16t^2+60t[/tex]
You can identify that [tex]a=-16[/tex]
Since [tex]a<0[/tex] then the parabola opens downward.
Therefore, we need to find the vertex.
Find the x-coordinate of the vertex with this formula:
[tex]x=\frac{-b}{2a}[/tex]
Substitute values:
[tex]x=\frac{-60}{2(-16)}=1.875[/tex]
Substitute the value of "t" into the function to find the height in feet that the ball will reach. Then:
[tex]h(1.875)=- 16(1.875)^2+60(1.875)=56.25ft[/tex]
The average assembly time for a Ford Taurus is μ = 38 hrs. An engineer suggests that using a new adhesive to attach moldings will speed up the assembly process. The new adhesive was used for one month. During that month, the average assembly time for 36 cars was = 37.5 hours with a standard deviation s = 1.2 hours. Use α = 0.01. Based on the calculated P-value will you reject or fail to reject the null hypothesis? Select one: a. reject the null hypothesis / data is significant b. fail to reject the null hypothesis c. cannot be determined
Answer:
a) reject null hypothesis since p < 0.01
Step-by-step explanation:
Given that the average assembly time for a Ford Taurus is
[tex]μ = 38 hrs[/tex]
Sample size [tex]n=36[/tex]
[tex]x bar = 37.5\\s=1.2\\SE = 1.2/6 = 0.2[/tex]
Test statistic t = mean diff/se = 0.5/0.2 = 2.5
(Here population std dev not known hence t test is used)
df = 35
p value = 0.008703
a) reject null hypothesis since p < 0.01
Find the m∠p.
54
90°
27°
36°
Let f(x) = (x − 3)−2. Find all values of c in (2, 5) such that f(5) − f(2) = f '(c)(5 − 2). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
The answer does not exist.
Note - The statement has typing mistakes. Correct form is presented below:
Let [tex]f(x) = (x-3)^{-2}[/tex]. Find all values of [tex]c[/tex] in (2, 5) such that [tex]f(5) - f(2) = f'(c) \cdot (5-2)[/tex]. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
In this question we should use the Mean Value Theorem, which states that given a secant line between points A and B, there is at least a point C that belongs to the curve whose derivative exists.
We begin by calculating [tex]f(2)[/tex] and [tex]f(5)[/tex]:
[tex]f(2) = (2-3)^{-2}[/tex]
[tex]f(2) = 1[/tex]
[tex]f(5) = (5-3)^{-2}[/tex]
[tex]f(5) = 1[/tex]
And the slope of the derivative is:
[tex]f'(c) = \frac{f(5) - f(2)}{5-2}[/tex]
[tex]f'(c) = 0[/tex]
Now we find the derivative of the function:
[tex]f'(x) = -2\cdot (x-3)^{-3}[/tex]
[tex]-2\cdot (x-3)^{-3} = 0[/tex]
[tex]-2 = 0[/tex] (ABSURD)
Hence, we conclude that the answer does not exist.
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Write an equation of the line through(2-1) and perpendicular to 2yx-4 Write the equation in the form x The one the Enter your answer in the box and then click Check Answer parts showing i Type here to search
Answer:
[tex]2x+y=3[/tex]
Step-by-step explanation:
Here we aer given a point (2,-1) and a line [tex]2y=x-4[/tex]. We are supposed to find the equation of the line passing through this point and perpendicular to this line.
Let us find the slope of the line perpendicular to [tex]2y=x-4[/tex]
Dividing above equation by 2 we get
[tex]y=\frac{1}{2}x-2[/tex]
Hence we have this equation in slope intercept form and comparing it with
[tex]y=mx+c[/tex] , we get Slope [tex]m = \frac{1}{2}[/tex]
We know that product of slopes of two perpendicular lines in -1
Hence if slope of line perpendicular to [tex]y=\frac{1}{2}x-2[/tex] is m' then
[tex]m\times m' =-1[/tex]
[tex]\frac{1}{2} \times m' =-1[/tex]
[tex]m'=-2[/tex]
Hence the slope of the line we have to find is -2
now we have slope and a point
Hence the equation of the line will be
[tex]\frac{y-(-1)}{x-2}=-2[/tex]
[tex]y+1=-2(x-2)[/tex]
[tex]y+1=-2x+4[/tex]
adding 2x and subtracting on both sides we get
[tex]2x+y=3[/tex]
Which is our equation asked
Calculate the standard deviation for the following set of numbers: 73, 76, 79, 82, 84, 84, 97
Answer:
Standard deviation is 7.16
Step-by-step explanation:
We have given a set of numbers :
73, 76, 79, 82, 84, 84, 97
To calculate the standard deviation of the given data set, first we have to work out the mean.
Mean = [tex]\frac{(73+76+79+82+84+84+97)}{7}[/tex]
= [tex]\frac{575}{7}[/tex] = 82.14
Now for each number subtract the mean and square the result
(73 - 82.14)² = (-9.14)² = 83.54
(76 - 82.14)² = (-6.14)² = 37.70
(79 - 82.14)² = (-3.14)² = 9.86
(82 - 82.14)² = (0.14)² = 0.02
(84 - 82.14)² = (1.86)² = 3.46
(84 - 82.14)² = (1.86)² = 3.46
(97 - 82.14)² = (14.86)²= 220.82
Now we calculate the mean from of those squared differences :
Mean = [tex]\frac{83.54+37.70+9.86+0.02+3.46+3.46+220.82}{7}[/tex]
= [tex]\frac{358.86}{7}[/tex]
= 51.27
Now square root of this mean = standard deviation = √51.27 = 7.16
Therefore, Standard deviation is 7.16
Which of the following is an example of qualitative data? a. Average rainfall on 25 days of the month b. Number of accidents occurred in the month of September c. Weight of 35 students in a class d. Height of 40 students in a class e. Political affiliation of 2,250 randomly selected voters
Answer: Option 'e' is correct.
Step-by-step explanation:
Qualitative data refers to those data which does not include any numerical value.
Average rainfall on 25 days of the month is a quantitative data as it would represented as numerical value.
Number of accidents occurred in the month of September is quantitative data.
Height of 40 students in a class is quantitative data too.
Weight of 35 students in a class is quantitative data too.
But political affiliation of 2250 randomly selected voters is qualitative data as it has not included any numerical value.
Hence, option 'e' is correct.
Option e, 'Political affiliation of 2,250 randomly selected voters,' represents qualitative data because it categorizes voters based on a non-numeric characteristic, their political affiliation.
The question 'Which of the following is an example of qualitative data? e. Political affiliation of 2,250 randomly selected voters' is asking us to identify a type of data among the given options. Qualitative data refers to information that is categorized based on attributes or qualities rather than numerical values. In contrast, quantitative data involve numbers and can either be discrete, which are countable, or continuous, which can take on any value within a range.
To answer the question, option e, 'Political affiliation of 2,250 randomly selected voters,' represents qualitative data because it describes a characteristic (political affiliation) that is non-numeric and is usually expressed in words or categories. Other options such as average rainfall, number of accidents, weight, and height involve numerical measurements and are therefore examples of quantitative data, which can either be discrete or continuous depending on the nature of the measurement.
Option e is unique as the only example of qualitative data in the options provided, because political affiliation does not result from a measurement or count, but rather, it is a category used to describe a voter's preferred political party or stance.
You purchase boxes of cereal until you obtain one with the collector's toy you want. If, on average, you get the toy you want in every 49th cereal box, what is the probability of getting the toy you want in any given cereal box?
Answer:
The probability of getting the toy in any given cereal box is [tex]\frac{1}{49}[/tex].
Step-by-step explanation:
Given,
On average, we get a toy in every 49th cereal box,
That is, in every 49 boxes there is a toy,
So, the total outcomes = 49,
Favourable outcomes ( getting a toy ) = 1
Since, we know that,
[tex]\text{Probability}=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}[/tex]
Hence, the probability of getting the toy in any given cereal box = [tex]\frac{1}{49}[/tex]
Answer:
The probability of getting the toy you want in any given cereal box is of 0.0204 = 2.04%.
Step-by-step explanation:
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
The expected number of trials for r sucesses is:
[tex]E = \frac{r}{p}[/tex]
If, on average, you get the toy you want in every 49th cereal box, what is the probability of getting the toy you want in any given cereal box?
This means that [tex]E = 49, r = 1[/tex]
So
[tex]49 = \frac{1}{p}[/tex]
[tex]49p = 1[/tex]
[tex]p = \frac{1}{49}[/tex]
[tex]p = 0.0204[/tex]
The probability of getting the toy you want in any given cereal box is of 0.0204 = 2.04%.
Verify that y = c_1 + c_2 e^2x is a solution of the ODE y" - 2y' = 0 for all values of c_1 and c_2.
Answer:
For any value of C1 and C2, [tex]y = C1 + C2*e^{2x}[/tex] is a solution.
Step-by-step explanation:
Let's verify the solution, but first, let's find the first and second derivatives of the given solution:
[tex]y = C1 + C2*e^{2x}[/tex]
For the first derivative we have:
[tex]y' = 0 + C2*(2x)'*e^{2x}[/tex]
[tex]y' = C2*(2)*e^{2x}[/tex]
For the second derivative we have:
[tex]y'' = C2*(2)*(2x)'*e^{2x}[/tex]
[tex]y'' = C2*(2)*(2)*e^{2x}[/tex]
[tex]y'' = C2*(4)*e^{2x}[/tex]
Let's solve the ODE by the above equations:
[tex]y'' - 2y' = 0[/tex]
[tex]C2*(4)*e^{2x} - 2*C2*(2)*e^{2x} = 0[/tex]
[tex]C2*(4)*e^{2x} - C2*(4)*e^{2x} = 0[/tex]
From the above equation we can observe that for any value of C2 the equation is solved, and because the ODE only involves first (y') and second (y'') derivatives, C1 can be any value as well, because it does not change the final result.
Find all the solutions for the equation:
2y2 dx - (x+y)2 dy=0
(Introduction to Differential Equations)
[tex]2y^2\,\mathrm dx-(x+y)^2\,\mathrm dy=0[/tex]
Divide both sides by [tex]x^2\,\mathrm dx[/tex] to get
[tex]2\left(\dfrac yx\right)^2-\left(1+\dfrac yx\right)^2\dfrac{\mathrm dy}{\mathrm dx}=0[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2\left(\frac yx\right)^2}{\left(1+\frac yx\right)^2}[/tex]
Substitute [tex]v(x)=\dfrac{y(x)}x[/tex], so that [tex]\dfrac{\mathrm dv(x)}{\mathrm dx}=\dfrac{x\frac{\mathrm dy(x)}{\mathrm dx}-y(x)}{x^2}[/tex]. Then
[tex]x\dfrac{\mathrm dv}{\mathrm dx}+v=\dfrac{2v^2}{(1+v)^2}[/tex]
[tex]x\dfrac{\mathrm dv}{\mathrm dx}=\dfrac{2v^2-v(1+v)^2}{(1+v)^2}[/tex]
[tex]x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{v(1+v^2)}{(1+v)^2}[/tex]
The remaining ODE is separable. Separating the variables gives
[tex]\dfrac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=-\dfrac{\mathrm dx}x[/tex]
Integrate both sides. On the left, split up the integrand into partial fractions.
[tex]\dfrac{(1+v)^2}{v(1+v^2)}=\dfrac{v^2+2v+1}{v(v^2+1)}=\dfrac av+\dfrac{bv+c}{v^2+1}[/tex]
[tex]\implies v^2+2v+1=a(v^2+1)+(bv+c)v[/tex]
[tex]\implies v^2+2v+1=(a+b)v^2+cv+a[/tex]
[tex]\implies a=1,b=0,c=2[/tex]
Then
[tex]\displaystyle\int\frac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=\int\left(\frac1v+\frac2{v^2+1}\right)\,\mathrm dv=\ln|v|+2\tan^{-1}v[/tex]
On the right, we have
[tex]\displaystyle-\int\frac{\mathrm dx}x=-\ln|x|+C[/tex]
Solving for [tex]v(x)[/tex] explicitly is unlikely to succeed, so we leave the solution in implicit form,
[tex]\ln|v(x)|+2\tan^{-1}v(x)=-\ln|x|+C[/tex]
and finally solve in terms of [tex]y(x)[/tex] by replacing [tex]v(x)=\dfrac{y(x)}x[/tex]:
[tex]\ln\left|\frac{y(x)}x\right|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C[/tex]
[tex]\ln|y(x)|-\ln|x|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C[/tex]
[tex]\boxed{\ln|y(x)|+2\tan^{-1}\dfrac{y(x)}x=C}[/tex]
Problem 5.58. Supposef XY and g : Y Z are functions If g of is one-to-one, prove that fmust be one-to-one 2. Find an example where g o f is one-to-one, but g is not one-to-one
f : X → Y and g: Y → Z
Now we have to show:
If gof is one-to-one then f must be one-to-one.
Given:
gof is one-to-one
To prove:
f is one-to-one.
Proof:
Let us assume that f(x) is not one-to-one .
This means that there exist x and y such that x≠y but f(x)=f(y)
On applying both side of the function by the function g we get:
g(f(x))=g(f(y))
i.e. gof(x)=gof(y)
This shows that gof is not one-to-one which is a contradiction to the given statement.
Hence, f(x) must be one-to-one.
Now, example to show that gof is one-to-one but g is not one-to-one.Let A={1,2,3,4} B={1,2,3,4,5} C={1,2,3,4,5,6}
Let f: A → B
be defined by f(x)=x
and g: B → C be defined by:
g(1)=1,g(2)=2,g(3)=3,g(4)=g(5)=4
is not a one-to-one function.
since 4≠5 but g(4)=g(5)
Also, gof : A → C
is a one-to-one function.
A sum of money amounting to P5.15 consists of 10 cents and 25 cents, If there are 32 coins in all, how many 25 cents are there? A. 14 pcs B. 13 pcs C. 15 pcs D. 12 pcs
Answer: Option 'B' is correct.
Step-by-step explanation:
Let the number of 10 cents pcs be 'x'.
Let the number of 25 cents pcs be 'y'.
Since we have given that
Total number of coins = 32
Sum of money = $5.15
As we know that
$1 = 100 cents
$5.15 = 5.15×100 = 515 cents
According to question, we get that
[tex]x+y=32-----------(1)\\\\10x+25y=\$515------------(2)[/tex]
Using the graphing method, we get that
x = 19
y = 13
So, there are 13 pcs of 25 cents.
Hence, Option 'B' is correct.
By creating a system of equations based on the total number of coins (32) and their total value (P5.15), we calculate that there are 13 pieces of 25-cent coins.
Explanation:The student's question involves figuring out the number of 25-cent coins among a total of 32 coins which altogether amount to P5.15. This problem can be solved by setting up a system of equations to account for the total number of coins and the total value in pesos.
Let's denote the number of 10-cent coins as t and the number of 25-cent coins as q. We know from the problem that there are 32 coins in total, so:
(1) t + q = 32
We also know that the total value of the coins is P5.15, or 515 cents. Therefore:
(2) 10t + 25q = 515
By solving this system of equations, we can find the value of q, the number of 25-cent coins. First, we can multiply equation (1) by 10 to eliminate t when we subtract the equations:
10t + 10q = 320
Subtracting this from equation (2) gives us:
15q = 195
Dividing both sides by 15, we find that:
q = 13
So, there are 13 pieces of 25-cent coins, which corresponds to option B.
Determine whether T : R^2 -->R^2,T((x.y)) = (x,y^2) is a linear transformation
Answer: No, the given transformation T is NOT a linear transformation.
Step-by-step explanation: We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :
[tex]T(x,y)=(x,y^2).[/tex]
We know that
a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R
[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]
So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have
[tex]T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)[/tex]
and
[tex]aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).[/tex]
Therefore, we get
[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]
Thus, the given transformation T is NOT a linear transformation.
Solve for the indicated variable.
Answer:
[tex]y=\frac{2x}{9}-2[/tex]
Step-by-step explanation:
The given equation is:
2x-9y=18
To solve for y means we need to isolate y on one side of the equation, carrying all the other variables and terms to the other side so that we get a formula for y. This can be done as shown below:
2x - 9y = 18
Subtracting 2x from both sides, we get:
-9y = 18 - 2x
Dividing both sides by -9, we get:
[tex]\frac{-9y}{-9}=\frac{18}{-9}-\frac{2x}{-9}\\\\ y=-2+\frac{2x}{9}\\\\ y=\frac{2x}{9}-2[/tex]
Answer:
[tex] y = \frac { 2 ( x - 9 ) } { 9 } [/tex]
Step-by-step explanation:
We are given the following expression and we are to solve it for the indicated variable (y):
[tex] 2 x - 9 y = 1 8 [/tex]
Making [tex] y [/tex] the subject of the equation and simplifying it to get:
[tex] 2 x - 1 8 = 9 y \\\\ 9 y = 2 x - 1 8 \\\\ y = \frac { 2 x - 1 8 } { 9 } \\\\ y = \frac { 2 ( x - 9 ) } { 9 } [/tex]
What is the Z-score for the value that is two standard deviations away from the mean?
Answer:
2
Step-by-step explanation:
The z-score is the number of standard deviations the value is above the mean. If that number is 2, then Z=2. If that number is -2, then Z=-2.
___
Both Z=2 and Z=-2 are values that are two standard deviations from the mean.
4. Fraction: Explain what 5/6 means. Write an explanation of the term fraction that should work with 5/6 and %.
Answer:
See below.
Step-by-step explanation:
5/6 is a fraction. The 5 is in the numerator, and the 6 is in the denominator.
The denominator is the number of parts the unit was divided into. In this case, the denominator is 6. That means one unit, 1, was divided into 6 equal parts. Each part is one-sixth.
The numerator is the number of those parts that you use. 5 in the numerator means to use 5 of those parts, each of which is 1/6 of 1.
In other words, 5/6 means divide 1 into 6 equal parts, and take 5 of those parts.
5/6 is the same as 5 divided by 6, so as a decimal it is 0.8333...
As a percent it is 83.333...%