Answer with explanation:
For, a 3 × 3, matrix
[tex]r_{1}=(3,-2,0)\\\\r_{2}=(0,1,1)\\\\r_{3}=(2,-1,0)[/tex]
which are entries of First, Second and Third Row Respectively.
So, if written in the form of Matrix (A)
[tex]A=\left[\begin{array}{ccc}3&-2&0\\0&1&1\\2&-1&0\end{array}\right][/tex]
⇒Adjoint A= Transpose of Cofactor of A
[tex]a_{11}=1,a_{12}=2,a_{13}=-2\\\\a_{21}=0,a_{22}=0,a_{23}=-1\\\\a_{31}=-2,a_{32}=- 3,a_{33}=3\\\\Adj.A=\left[\begin{array}{ccc}1&0&-2\\2&0&-3\\-2&-1&3\end{array}\right][/tex]
⇒≡ |Adj.A|=1 ×(0-3) -2×(-2-0)
= -3 +4
=1 --------(1)
⇒For, a Matrix of Order, 3 × 3,
| Adj.A |=| A|²---------(2)
[tex]|2 A^{-2}|=2^3\times |A^{-2}|\\\\=2^3\times |A|^{-2}\\\\=\frac{8}{|A^{2}|}\\\\=\frac{8}{|Adj.A|}\\\\=\frac{8}{1}\\\\=8[/tex]
--------------------------------------------(Using 1 and 2)
[tex]\rightarrow|2 A^{-2}|=8[/tex]
A particle moves along line segments from the origin to the points (2, 0, 0), (2, 4, 1), (0, 4, 1), and back to the origin under the influence of the force field. F(x, y, z) = z^2i + 3xyj + 2y^2k. Find the work done.
The work is equal to the line integral of [tex]\vec F[/tex] over each line segment.
Parameterize the paths
from (0, 0, 0) to (2, 0, 0) by [tex]\vec r_1(t)=t\,\vec\imath[/tex] with [tex]0\le t\le2[/tex],from (2, 0, 0) to (2, 4, 1) by [tex]\vec r_2(t)=2\,\vec\imath+4t\,\vec\jmath+t\,\vec k[/tex] with [tex]0\le t\le1[/tex],from (2, 4, 1) to (0, 4, 1) by [tex]\vec r_3(t)=(2-t)\,\vec\imath+4\,\vec\jmath+\vec k[/tex] with [tex]0\le t\le2[/tex], andfrom (0, 4, 1) to (0, 0, 0) by [tex]\vec r_4(t)=(4-4t)\,\vec\jmath+(1-t)\,\vec k[/tex] with [tex]0\le t\le1[/tex]The work done by [tex]\vec F[/tex] over each segment (call them [tex]C_1,\ldots,C_4[/tex]) is
[tex]\displaystyle\int_{C_1}\vec F\cdot\mathrm d\vec r_1=\int_0^2\vec0\cdot\vec\imath\,\mathrm dt=0[/tex]
[tex]\displaystyle\int_{C_2}\vec F\cdot\mathrm d\vec r_2=\int_0^1(t^2\,\vec\imath+24t\,\vec\jmath+32t^2\,\vec k)\cdot(4\,\vec\jmath+\vec k)\,\mathrm dt=\int_0^1(96t+32t^2)\,\mathrm dt=\frac{176}3[/tex]
[tex]\displaystyle\int_{C_3}\vec F\cdot\mathrm d\vec r_3=\int_0^2(\vec\imath+(24-12t)\,\vec\jmath+32\,\vec k)\cdot(-\vec\imath)\,\mathrm dr=-\int_0^2\mathrm dt=-2[/tex]
[tex]\displaystyle\int_{C_4}\vec F\cdot\mathrm d\vec r_4=\int_0^1((1-t)^2\,\vec\imath+2(4-4t)^2\,\vec k)\cdot(-4\,\vec\jmath-\vec k)\,\mathrm dt=-2\int_0^1(4-4t)^2\,\mathrm dt=-\frac{32}3[/tex]
Then the total work done by [tex]\vec F[/tex] over the particle's path is 46.
The total work done by the force field as the particle follows this path is 4/3 units of work.
Explanation:To find the total work done by the force field as the particle moves along the path, we'll calculate the line integrals for each segment and then sum them up:
For the first segment (from the origin to (2, 0, 0)):
Parameterization: r1(t) = (2t, 0, 0), t from 0 to 1.
Work on this segment: ∫[0,1] F(r1(t)) · r1'(t) dt
F(r1(t)) = (0, 0, 0), as z = 0.
r1'(t) = (2, 0, 0).
∫[0,1] F(r1(t)) · r1'(t) dt = ∫[0,1] (0) dt = 0.
For the second segment (from (2, 0, 0) to (2, 4, 1)):
Parameterization: r2(t) = (2, 4t, t), t from 0 to 1.
Work on this segment: ∫[0,1] F(r2(t)) · r2'(t) dt
F(r2(t)) = (t^2, 0, 8t^2).
r2'(t) = (0, 4, 1).
∫[0,1] F(r2(t)) · r2'(t) dt = ∫[0,1] (0 + 0 + 8t^2) dt = ∫[0,1] 8t^2 dt = [8t^3/3] from 0 to 1 = (8/3).
For the third segment (from (2, 4, 1) to (0, 4, 1)):
Parameterization: r3(t) = (2 - 2t, 4, 1), t from 0 to 1.
Work on this segment: ∫[0,1] F(r3(t)) · r3'(t) dt
F(r3(t)) = (1, 0, 8).
r3'(t) = (-2, 0, 0).
∫[0,1] F(r3(t)) · r3'(t) dt = ∫[0,1] (-2) dt = [-2t] from 0 to 1 = -2.
For the fourth segment (from (0, 4, 1) back to the origin):
Parameterization: r4(t) = (0, 4 - 4t, 1), t from 0 to 1.
Work on this segment: ∫[0,1] F(r4(t)) · r4'(t) dt
F(r4(t)) = (1, 0, 32 - 8t).
r4'(t) = (0, -4, 0).
∫[0,1] F(r4(t)) · r4'(t) dt = ∫[0,1] (0 + 0 + 0) dt = 0.
Now, sum up the work done on each segment:
Total work = 0 + (8/3) - 2 + 0 = 4/3.
So, the total work done by the force field as the particle follows this path is 4/3 units of work.
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(a + 3)(a - 2)
hurry helpppp asapppp
(a+3)(a-2)
Multiply the two brackets together
a^2-2a+3a+3*-2
a^2+a-6
Answer is a^2+a-6
ANSWER
[tex]{a}^{2} + a - 6[/tex]
EXPLANATION
The given expression is
[tex](a + 3)(a - 2)[/tex]
We expand using the distributive property to obtain:
[tex]a(a - 2) + 3(a - 2)[/tex]
We multiply out the parenthesis to get:
[tex] {a}^{2} - 2a + 3a - 6[/tex]
Let us now simplify by combining the middle terms to obtain;
[tex]{a}^{2} + a - 6[/tex]
Which has the greater energy, light of wavelength 519 nm or light with a frequency of 5.42 x 10^8 sec^-1?
Try this solution:
the rule: if f1>f2, then E1>E2, where f1;f2 - frequency, E1;E2 - energy of light.
The formula is L=c/f, where L - the wavelength, c - 3*10⁸ m/s, f - frequency.
Frequency for the wavelength 519 nm. is:
[tex]f=\frac{c}{L}=\frac{3*10^8}{519*10^{-9}}=\frac{3*10^17}{519}=578034682080924=5.78*10^{14}( \frac{1}{sec})[/tex]
Answer: the energy of light of wavelength 519 nm.
What are the solutions of the following system?
Answer:(-6,312), (6,312)
Step by Step explanation:
Solve the first equation for y.10x^2-y=48
y=-48+10x^2
Substitute the given value of y into the equation 2y=16x^2+482(-48+10x^2)=16x^2+48
Solve the equation for x.
x=-6
x=6
Substitute the given value of x into the equation y=-48+10(-6)^y=-48+10(-6)^2
y=-48+10×6^2
Solve the equation for yy=312
y=312
1 -For what value of x is line a parallel to line b
2-For what value of x is line a parallel to line b
Answer:
1) x = 17, line a parallel to line b
2) x = 18, line a parallel to line b
Step-by-step explanation:
If line a parallel to line b then (10x - 40) + 50 = 180
Solve for x
10x - 40 + 50 = 180
Combine like terms
10x + 10 = 180
10x = 170
x = 17
x = 17, line a parallel to line b
-------------------------------------------------
If line a parallel to line b then 5x - 16 = 74
Solve for x
5x - 16 = 74
5x = 90
x = 18
x = 18, line a parallel to line b
The value of x which makes line a parallel to line b can be found by equating the slopes of the two lines and solving for x.
Explanation:In mathematics, two lines a and b are parallel if and only if their slopes are equal. When we are given the equations of the lines and are asked to find the value of x that make the lines parallel, we start by setting the slopes of the two lines equal to each other. Let's assume now that line a is represented by y = mx + b1 and line b by y = nx + b2. In order for line a to be parallel to line b, m must be equal to n. Therefore, you solve for x from the equation m=n.
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Three boxes contain red and green balls. Box 1 has 5 red balls* and 5 green balls*, Box 2 has 7 red balls* and 3 green balls* and Box 3 contains 6 red balls* and 4 green balls*. The respective probabilities of choosing a box are 1/4, 1/2, 1/4. What is the probability that the ball chosen is green?
Final answer:
The probability of choosing a green ball from the three boxes, given their individual selection probabilities and color distributions, is calculated using the law of total probability. The overall probability of selecting a green ball is found to be 29/80, or roughly 36.25%.
Explanation:
The question asks for the probability of choosing a green ball from three different boxes, given their individual probabilities of being chosen and the distribution of red and green balls in each box. To solve this, we employ the law of total probability which combines the probability of each event (selecting a box) with the conditional probability of finding a green ball within that selected box.
Box 1: Probability of green ball = 5 green balls / (5 red + 5 green) = 1/2
Box 2: Probability of green ball = 3 green balls / (7 red + 3 green) = 3/10
Box 3: Probability of green ball = 4 green balls / (6 red + 4 green) = 2/5
The overall probability is calculated as: P(Green) = P(Box 1) * P(Green|Box 1) + P(Box 2) * P(Green|Box 2) + P(Box 3) * P(Green|Box 3) = (1/4) * (1/2) + (1/2) * (3/10) + (1/4) * (2/5) = 1/8 + 3/20 + 1/10 = 29/80.
Therefore, the probability that the ball chosen is green is 29/80 or approximately 36.25%.
Assume that 1400 births are randomly selected and 1378 of the births are girls. Use subjective judgment to describe the number of girls as significantly high, significantly low, or neither significantly low nor significantly high. Choose the correct answer below. A. The number of girls is neither significantly low nor significantly high. B. The number of girls is significantly high. C. The number of girls is significantly low. D. It is impossible to make a judgment with the given information.
Answer: Hence, Option 'B' is correct.
Step-by-step explanation:
Since we have given that
Number of births = 1400
Number of birth of girls = 1378
Number of birth of boys is given by
[tex]1400-1378\\\\=22[/tex]
so, the number of girls is significantly higher than the number of boys.
So, the number of births of girls is significantly high.
Hence, Option 'B' is correct.
B. The number of girls is significantly high.
When evaluating whether the number of girl births in a sample is significantly high or low, we can reference the expected natural ratio of girls to boys, which is typically 100:105. Given that in the scenario 1378 out of 1400 births were girls, this significantly deviates from the expected natural ratio. For comparison, an article in Newsweek states that the natural ratio is 100:105, and in China, it is 100:114, which equals 46.7 percent girls. If we consider a sample where out of 150 births, there are 60 girls (or 40%), this is lower than the expected percentage based on China's statistics but not implausible. However, in the case of the scenario with 1378 girls out of 1400 births, the proportion of girls is approximately 98.43%, which seems very unlikely given the natural ratio, suggesting an unusual or non-random process may be involved.
Therefore, based on subjective judgment and without applying more precise statistical tests, the number of girls being 1378 out of 1400 births is significantly high compared to natural birth rates or the stated birth rate in China. This leads us to select the correct answer: B. The number of girls is significantly high.
What is the solution of the equation 4^(x + 1) = 21? Round your answer to the nearest ten-thousandth.
For this case we must solve the following equation:
[tex]4 ^ {x + 1} = 21[/tex]
We find Neperian logarithm on both sides:
[tex]ln (4 ^ {x + 1}) = ln (21)[/tex]
According to the rules of Neperian logarithm we have:
[tex](x + 1) ln (4) = ln (21)[/tex]
We apply distributive property:
[tex]xln (4) + ln (4) = ln (21)[/tex]
We subtract ln (4) on both sides:
[tex]xln (4) = ln (21) -ln (4)[/tex]
We divide between ln (4) on both sides:
[tex]x = \frac {ln (21)} {ln (4)} - \frac {ln (4)} {ln (4)}\\x = \frac {ln (21)} {ln (4)} - 1\\x = 1,19615871[/tex]
Rounding:
[tex]x = 1.1962[/tex]
Answer:
x = 1.1962
Answer: [tex]x[/tex]≈[tex]1.196[/tex]
Step-by-step explanation:
Given the equation [tex]4^{(x + 1)} = 21[/tex] you need to solve for the variable "x".
Remember that according to the logarithm properties:
[tex]log_b(b)=1[/tex]
[tex]log(a)^n=nlog(a)[/tex]
Then, you can apply [tex]log_4[/tex] on both sides of the equation:
[tex]log_4(4)^{(x + 1)} = log_4(21)\\\\(x + 1)log_4(4) = log_4(21)\\\(x + 1) = log_4(21)[/tex]
Apply the Change of base formula:
[tex]log_b(x) = \frac{log_a( x)}{log_a(b)}[/tex]
Then you get:
[tex]x =\frac{log(21)}{log(4)}-1[/tex]
[tex]x[/tex]≈[tex]1.196[/tex]
Find the given higher-order derivative.
f''(x) = 9- 3/x
f'''(x)=
Answer:
[tex]f'''(x)=\frac{3}{x^{2}}[/tex]
Step-by-step explanation:
We are given with Second-order derivative of function f(x).
[tex]f''(x)=9-\frac{3}{x}[/tex]
We need to find Third-order derivative of the function f(x).
[tex]f''(x)=9-\frac{3}{x}=9-3x^{-1}[/tex]
We know that,
f'''(x) = (f''(x))'
So,
[tex]f'''(x)=\frac{\mathrm{d}\,f''(x)}{\mathrm{d} x}[/tex]
[tex]f'''(x)=\frac{\mathrm{d}\,(9-3x^{-1})}{\mathrm{d} x}[/tex]
[tex]f'''(x)=\frac{\mathrm{d}\,9}{\mathrm{d} x}-\frac{\mathrm{d}\,3x^{-1}}{\mathrm{d} x}[/tex]
[tex]f'''(x)=0-3\frac{\mathrm{d}\,x^{-1}}{\mathrm{d} x}[/tex]
[tex]f'''(x)=-3(-1)x^{-1-1}[/tex]
[tex]f'''(x)=3x^{-2}[/tex]
[tex]f'''(x)=\frac{3}{x^{2}}[/tex]
Therefore, [tex]f'''(x)=\frac{3}{x^{2}}[/tex]
When are two distinct non vertical lines parallel
Answer:
Two lines are parallel when they share the same slope.
Step-by-step explanation:
Two lines are parallel when they share the same slope.
The slope-intercept form of the equation of a line is: y=mx + b, where 'm' is the slope and 'b' the y-intercept.
If two equations have the same value for 'm', then those lines are parallel, for example:
y = 3x + 8 (Red line)
y = 3x + 5 (Blue line)
y = 3x - 10 (Green line)
All the equations stated above are parallel, to show that, I'm attaching the graph of the equations :).
Explain how each of the following rates satisfies the definition of ratio. Given an example of how each is used.
A. 1580 people/square mile
B. 360 kilowatt-hour/4 months
C. 450 people/year
D. 355 calories/6 ounces
Answer:
Step-by-step explanation:
A ratio is defined as a comparison of two amounts by division.It is of the form of [tex]\frac{p}{q}[/tex] where p and q are the quantities.
A. 1580 people/square mile in division form
1580[tex]\frac{people}{\text{square mile}}[/tex]
This satisfies ratio definition as this compares two quantities People and Square mile and is of the form [tex]\frac{p}{q}[/tex]
Where,
p: People, q: Square mile
Example,
[tex]\frac{1580\times people}{1\times squaremile}[/tex]
expresses that per 1 square mile there are 1580 people. Thus rate satisfies the ratio definition.
B. 360 kilowatt-hour/4 months
in division form [tex]\frac{\text{360 kilowatt-hour}}{\text{4months}}[/tex]
This satisfies ratio definition as this compares two quantities kilowatt-hour and months and is of the form [tex]\frac{p}{q}[/tex]
where, p: kilowatt-hour and q: people
Example,
[tex]\frac{\360\times kilowatt-hour}{4\times months}[/tex]
expresses that per 4 months there are 360 kilowatt-hour. Thus, rate satisfies the ratio definition.
C. 450 people/year
In division form 450[tex]\frac{people}{year}[/tex]
This satisfies ratio definition as this compares two quantities people and year and is of the form [tex]\frac{p}{q}[/tex]
where, p: people and q: year
Example,
[tex]\frac{450\times people}{1\times\text{year}}[/tex]
expresses that per year there are 450 people. Thus, rate satisfies the ratio definition.
D. 355 calories/6 ounces
In division form [tex]\frac{355clories}{6ounces}[/tex]
This satisfies ratio definition as this compares two quantities calories and ounces and is of the form [tex]\frac{p}{q}[/tex]
where p: calories and q: ounces
Example,
[tex]\frac{355\times calories}{6\times\text{ounces}}[/tex]
expresses that per 6 ounces there are 355 calories. Thus, rate satisfies the ratio definition.
Find f if f ''(x) = 12x2 + 6x − 4, f(0) = 9, and f(1) = 1.
Answer:
f(x) = x^4 +x^3 -2x^2 -8x +9
Step-by-step explanation:
You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...
f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant
The second antiderivative is ...
f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant
Then the constants can be found from ...
f(0) = q = 9
f(1) = 1 + 1 - 2 +p + 9 = 1
p = -8
The solution is ...
f(x) = x^4 +x^3 -2x^2 -8x +9
_____
The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.
Final answer:
To find f(x) given f''(x) = 12x² + 6x − 4, one must integrate twice and use the initial conditions f(0) = 9 and f(1) = 1 to solve for the constants. The final function is f(x) = x⁴ + x³ - 2x² - 8x + 9.
Explanation:
The question asks to find the antiderivative f(x) given its second derivative f''(x) = 12x² + 6x − 4, and two initial conditions, f(0) = 9, and f(1) = 1. To solve for f(x), we first integrate the second derivative twice to get the original function.
Integrating f''(x), we get:
f'(x) = ∫( 12x² + 6x - 4)dx = 4x³ + 3x² - 4x + C
We then integrate f'(x) to find f(x):
f(x) = ∫(4x³ + 3x² - 4x + C)dx = x⁴ + x³ - 2x² + Cx + D
Using the initial conditions:
For f(0) = 9, we substitute x = 0 and determine D = 9.For f(1) = 1, we substitute x = 1: 1 + 1 - 2 + C + 9 = 1, solving for C gives us C = -8.Therefore, the original function is f(x) = x⁴ + x³ - 2x² - 8x + 9.
What is the value of x? In this figure
A:53
B:43
C:57
D:47
Answer:
should be 53 if im right
Polygon ABCDE and polygon FGHIJ are similar. The area of polygon ABCDE is
40. What is the area of FGHIJ?
Answer: 640
Step-by-step explanation:
Since the two triangles are similar we can simply multiply the lesser triangle's area by a constant to get our answer.
Polygon FGHIJ is ABCDE with a scale change of 4
For the reason that we are dealing with area, we will multiply 40 by 4² in stead of just 4.
40 * 16 = 640
Answer:
B. 640
Step-by-step explanation:
got it right 2021
Suppose a man is 25 years old and would like to retire at age 60. ?Furthermore, he would like to have a retirement fund from which he can draw an income of ?$100,000 per yearlong dash?forever! How can he do? it? Assume a constant APR of 8?%.
He can have a retirement fund from which he can draw ?$100,000 per year by having ?$ ______ in his savings account when he retires.
Answer:
$1314.37
Step-by-step explanation:
We have to calculate final value i.e. balance to earn $100,000 annually from interest.
= [tex]\frac{100,000}{0.08}[/tex] = $1,250,000
Now, N = n × y = 12 × 25 = 300
I = 8% = APR = 0.08
PV = 0 = PMT = 0
FV = 1,250,000 = A
[tex]A=\frac{PMT\times [(1+\frac{apr}{n})^{ny}-1]}{\frac{apr}{n}}[/tex]
[tex]PMT=\frac{A\times (\frac{APR}{n})}{[(1+\frac{APR}{n})^{ny}-1]}[/tex]
[tex]PMT=\frac{1,250,000\times (\frac{0.08}{12})}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]
[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]
[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+0.006667)^{300}-1]}[/tex]
[tex]PMT=\frac{\frac{25000}{3}}{[1.006667^{300}-1]}[/tex]
[tex]PMT=\frac{\frac{25000}{3}}{6.340176}[/tex]
Monthly payment (PMT) = $1314.369409 ≈ $1314.37
$1314.37 is required monthly payment in order to $100,000 interest.
Which complete bipartite graphs Km, are trees? (b) Let T be a full 8-ary tree with 201 vertices. (ii) How many internal vertices does T have? (iii) How many leaves does T have?
Answer:
the answer is a
Step-by-step explanation:
i just know
Piney Woods Conservation is a company that attempts to help offset the effects of deforestation. A local forest contains approximately 500,000 trees. Lumber companies are continuously clearing the forest at a rate of 4.7% per year. Piney Woods Conservation is about to begin planting trees in the region throughout each year at an average rate of 15,000 trees per year. They are curious to know how long it will be before the number of trees they have planted will be equal to the number of trees still remaining in the forest.
Answer:
15.7 years
Step-by-step explanation:
we know that
The deforestation is a exponential function of the form
[tex]y=a(b)^{x}[/tex]
where
y ----> the number of trees still remaining in the forest
x ----> the number of years
a is the initial value (a=500,000 threes)
b is the base
b=100%-4.7%=95.3%=95.3/100=0.953
substitute
[tex]y=500,000(0.953)^{x}[/tex]
The linear equation of planting threes in the region is equal to
[tex]y=15,000x[/tex]
using a graphing tool
Solve the system of equations
The intersection point is (15.7,235,110)
see the attached figure
therefore
For x=15.7 years
The number of trees they have planted will be equal to the number of trees still remaining in the forest
Answer: There is only one solution, and it is viable.
Step-by-step explanation:
Camille Uses a 20 % Off Coupon When Buying a Sweater That Costs $ 47.99 .If, She Also pays 6 % Sales tax on the Purchase , How Many does She Paid For ????
Answer:
take 47.99 x .20 = 9.598
$9.60 off
then take 47.99 - 9.60 = $ 38.39
take 38.39 x .06 = 2.3034
$ 2.30 (tax)
add 38.39 + 2.30 = $40.69 or $40.70 is the final purchase price
(the two amounts depends on your choice answer or how it is rounded)
Step-by-step explanation:
A ball is thrown upward from the top of a building. The function below shows the height of the ball in relation to sea level, f(t), in feet, at different times, t, in seconds: f(t) = −16t2 + 48t + 100 The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is _____feet per second.
Answer:
The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is __-80___feet per second.
Step-by-step explanation:
The average change rate m is calculated using the following formula
[tex]m=\frac{f(t_2)-f(t_1)}{t_2-t_1}[/tex]
In this case [tex]f(t) = -16t^2 + 48t + 100[/tex], [tex]t_2 = 5\ s\ \ , t_1=3\ s[/tex]
Then
[tex]f(t_2) = f(5) =-16(5)^2 + 48(5) + 100[/tex]
[tex]f(t_2) = -60[/tex]
[tex]f(t_1) = f(3) =-16(3)^2 + 48(3) + 100[/tex]
[tex]f(t_1) = 100[/tex]
Finally
[tex]m=\frac{(-60)-100}{5-3}[/tex]
[tex]m=-80[/tex]
What is the possible solution?
[tex]\sin(3x+13)=\cos(4x)\\\sin(3x+13)=\cos(90-4x)\\3x+13=90-4x\\7x=77\\x=11[/tex]
3.17 Scores on stats final. Below are final exam scores of 20 Introductory Statistics students. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 57, 66, 69, 71, 72, 73, 74, 77, 78, 78, 79, 79, 81, 81, 82, 83, 83, 88, 89, 94 (a) The mean score is 77.7 points. with a standard deviation of 8.44 points. Use this information to determine if the scores approximately follow the 68-95-99.7% Rule. (b) Do these data appear to follow a normal distribution? Explain your reasoning using the graphs provided below.
Answer:
Yes they do.
And yes they do follow a normal distribution.
Percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.
What is a normal distribution?It's the probability curve of a continuous distribution that's most likely symmetric around the mean. On the Z curve, at Z=0, the chance is 50-50. A bell-shaped curve is another name for it.
We have a data of final exam scores of 20 Introductory.
a) Range of 1 standard deviation:
(77.7 – 8.44, 77.7 + 8.44) [69.3, 86.1]
Range of 2 standard deviation:
(77.7 – 2(8.44), 77.7 + 2(8.44)) [60.8, 94.6]
Range of 3 standard deviation:
(77.7 – 3(8.44), 77.7 + 3(8.44)) [52.4, 103.0]
Number of data points lie within 1 standard deviation = 14
Percent of data points lie within 1 SD = (14/20)×100 = 70%
Number of data points lie within 2 SD = 19
Percent of data points lie within 1 SD = (19/20)×100 = 95%
Number of data points lie within 3 SD = 20
Percent of data points lie within 1 SD = (20/20)×100 = 100%
Because these percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed.
b)
Because the histogram in the graph is symmetric, and the normal probability plot reveals that the points are very close to a straight line, the data appears to follow a normal distribution.
Thus, percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.
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(a + 8)(b + 3)
ab + 8a + 3b + 24
ab + 3a + 8b + 24
11ab
24ab
Answer:
ab + 3a + 8b + 24
Step-by-step explanation:
(a + 8)(b + 3)
a(b + 3) + 8(b + 3)
ab + 3a + 8b + 24
Suppose that 45% of all adults regularly consume coffee, 40% regularly consume carbonated soda, and 55% regularly consume at least one of these two products. (a) What is the probability that a randomly selected adult regularly consumes both coffee and soda? (b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?
Answer: a) 30% and b) 45%
Step-by-step explanation:
Since we have given that
Probability that adults regularly consume coffee P(C) = 45% = 0.45
Probability that adults regularly consume carbonated soda P(S) = 40% = 0.40
Probability that adults regularly consume atleast one of these two products P(C∪S) = 55% = 0.55
a) What is the probability that a randomly selected adult regularly consumes both coffee and soda?
As we know that
P(C∪S ) = P(C) +P(S)-P(C∩S)
[tex]0.55=0.45+0.40-P(C\cap S)\\\\0.55=0.85-P(C\cap S)\\\\0.55-0.85=-P(C\cap S)\\\\-0.30=-P(C\cap S)\\\\P(C\cap S)=0.30=30\%[/tex]
b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?
P(C∪S)'=n(U)-P(C∪S)
[tex]\\P(C\cup S)'=100-55=45\%[/tex]
Hence, a) 30% and b) 45%
Which is an equation for the nth terms of the sequence 12,15,18,21
[tex]\bf 12~~,~~\stackrel{12+3}{15}~~,~~\stackrel{15+3}{18}~~,~~\stackrel{18+3}{21}~\hspace{10em}\stackrel{\textit{common difference}}{d=3} \\\\[-0.35em] ~\dotfill\\\\ n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=12\\ d=3 \end{cases} \\\\\\ a_n=12+(n-1)3\implies a_n=12+3n-3\implies a_n=3n+9[/tex]
Answer:
tₙ = 3(3 + n)
Step-by-step explanation:
Points to remember
nth term of an AP
tₙ = a + (n - 1)d
Where a - first term of AP
d - Common difference of AP
To find the nth term
The given series is,
12,15,18,21 .....
Here a = 12 and d = 15 - 12 = 3
tₙ = a + (n - 1)d
= 12 + (n - 1)3
=12 + 3n - 3
= 9 + 3n
= 3(3 + n)
Therefore tₙ = 3(3 + n)
Angle measures and segment lengths someone please explain!!!
Intersecting chords theorem:
[tex]7\cdot7=10x[/tex]
i.e. when two chords intersect, the products of the resulting line segments' lengths are equal. Then
[tex]10x=49\implies x=\dfrac{49}{10}=\boxed{4.9}[/tex]
Answer:
4.9
Step-by-step explanation:
got it right
The concept of determining which reactant is limiting and which is in excess is akin to determining the number of sandwiches that can be made from a set number of ingredients. Assuming that a cheese sandwich consists of 2 slices of bread and 3 slices of cheese, determine the number of whole cheese sandwiches that can be prepared from 44 slices of bread and 75 slices of cheese.
Answer: There are 22 whole cheese sandwiches that can be prepared.
Step-by-step explanation:
Since we have given that
Number of slices of bread = 44
Number of slices of cheese = 75
According to question, a cheese sandwich consists of 2 slices of bread and 3 slices of cheese.
So, we need to find the number of whole cheese sandwiches that can be prepared.
Number of sandwich containing only slice of bread is given by
[tex]\dfrac{44}{2}=22[/tex]
Number of sandwich containing only slice of cheese is given by
[tex]\dfrac{75}{3}=25[/tex]
As we know that each sandwich should contain both slice of bread and slice of cheese.
So, Least of (22, 25) = 22
Hence, there are 22 whole cheese sandwiches that can be prepared.
Please help someone
Answer:
1. Y
2. N
3. N
4. N
Step-by-step explanation:
Let's use the second equation, since it seems to be easier to use.
To check if an ordered pair is a solution, plug it in to the equation.
1. [tex]-10+18=8[/tex] --> Y
2. [tex]25-12=13[/tex] --> N
3. [tex]0-9=-9[/tex] --> N
4. [tex]35-27=8[/tex] --> N
Edit : The 4th equation doesn't work for the first equation, whereas the first one still does.
Answer:
(-2,-6)
Step-by-step explanation:
-9x +2y = 6
5x - 3y = 8
1) Make one of the coefficients the same - y.
-9x +2y = 6 * 3
5x - 3y = 8 * 3
-27x +6y = 18
10x - 6y = 16
2) Add the new equations.
(-27x +6y = 18) + (10x - 6y = 16)
(-27x +6y) + (10x - 6y) = 18 + 16
-17x = 34
3) Divide to find the value of x
-17x = 34
x = 34/-17
x = -2
4) Substitute x into either equation to find the value of y.
-9(-2) +2y = 6
18 +2y = 6
2y = -12
y = -12/2
y = -6
5(-2) - 3y = 8
-10 - 3y = 8
-3y = 18
y = 18/-3
y = -6
Your answer is (-2,-6).
Use the Taylor series you just found for sinc(x) to find the Taylor series for f(x) = (integral from 0 to x) of sinc(t)dt based at 0. a.Give your answer using summation notation. b.Give the interval on which the series converges.
In this question (https://brainly.com/question/12792658) I derived the Taylor series for [tex]\mathrm{sinc}\,x[/tex] about [tex]x=0[/tex]:
[tex]\mathrm{sinc}\,x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}[/tex]
Then the Taylor series for
[tex]f(x)=\displaystyle\int_0^x\mathrm{sinc}\,t\,\mathrm dt[/tex]
is obtained by integrating the series above:
[tex]f(x)=\displaystyle\int\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}\,\mathrm dx=C+\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]
We have [tex]f(0)=0[/tex], so [tex]C=0[/tex] and so
[tex]f(x)=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]
which converges by the ratio test if the following limit is less than 1:
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+3}}{(2n+3)^2(2n+2)!}}{\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)^2(2n)!}{(2n+3)^2(2n+2)!}[/tex]
Like in the linked problem, the limit is 0 so the series for [tex]f(x)[/tex] converges everywhere.
The Taylor series for the function f(x) = ∫ sinc(t)dt based at 0 is derived from the Taylor series of sinc(x) by integrating it term by term, given in summation notation as ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞. The series converges for all real numbers (-∞, ∞).
Explanation:In order to find the Taylor series for the function f(x) = ∫ sinc(t)dt based at 0, one can use the Taylor series for sinc(x) and integrate term by term. We know the Taylor series for sinc(x) is x - x³/3! + x⁵/5! - ..., so the Taylor series for f(x) can be written as x²/2 - x⁴/4*3! + x⁶/6*5! - ... . In summation notation, this is ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞.
The Taylor series for any function converges to the function itself within a certain interval called the radius of convergence. For the Taylor series of sinc(x), due to the nature of sine being bounded between -1 and 1, the series will converge for all real numbers (-∞, ∞).
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(1 point) The players on a soccer team wear shirts, with each player having one of the numbers 1, 2, ..., 11 on their backs. The set A contains players with even numbers on their shirts. The set B comprises players wearing an odd number less than 7. The set C contains the defenders, which are those wearing numbers less than 6. Select the correct set that corresponds to each of the following. Part a) A∩(B∪C) A. {1,2,3,4,5} B. ∅ C. {1,3,5} D. {2,4} E. {2} Part b) (A∩Bc)∪(B∩C)c A. {6,7,8,9,11} B. {2,4,6,7,8,9,10,11} C. {2,3,4,5,6,8,10} D. {1,2,3,4,5,6,8,10} E. {6,7,8,10,11}
This question involves operations on sets to identify specific members based on conditions. Part a) resolves to D. {2,4}, while part b) finds the solution to be B. {2,4,6,7,8,9,10,11}, highlighting the application of intersection, union, and complement operations in set theory.
Explanation:To solve these problems, we need to understand the operations on sets such as intersection (A∩B), union (A∪B), and the complement of a set (Bc). For part a), we identify set A as {2,4,6,8,10}, B as {1,3,5}, and C as {1,2,3,4,5}. A∩(B∪C) means we're looking for the intersection of A with the union of B and C. Since B∪C = {1,2,3,4,5}, intersecting this with A gives us D. {2,4} as the answer.
For part b), (A∩Bc)∪(B∩C)c means we're looking at elements in A but not in B, combined with elements not in both B and C. Since Bc = {6,7,8,9,10,11} and (B∩C)c = {6,7,8,9,10,11}, union these two gives us answer B. {2,4,6,7,8,9,10,11}, by including A∩Bc = {2,4,6,8,10} and excluding duplicates when union with (B∩C)c.
The GMAC Insurance company reported that the mean score on the National Drivers Test was 69.9 with a standard deviation of 3.7 points. The test scores are approximately bell-shaped. Approximately 68% of all test scores were between two values A and B. What is the value of A? Write only a number as your answer. Round to one decimal place.
Answer:
66.2
Step-by-step explanation:
We know that the middle 68% of the distribution is between -1 and +1 standard deviations from the mean. -1 standard deviation is a score of ...
69.9 - 3.7 = 66.2
_____
Comment on the question
The way the question is worded, "A" can be any value less than 68.1, down to -∞. The question does does not require the A-B range to be symmetrical.