For a randomly selected 40 to 60-year-old man in Anusha's research, the probabilities are as follows: the probability that he will have gray hair is 5/7, the probability that he does not dye his hair given that it is gray is unknown as it is not provided in the information given, and the probability that he does not appear to have gray hair is 2/7.
Anusha's research indicates that out of the men in the 40 to 60-year-old age group, 5 out of 7 have gray hair. Therefore, the probability that a randomly selected man from this group will have gray hair is 5/7.
The probability that a man does not dye his hair given that it is gray is not provided in the information given. To determine this probability, we would need to know the number of men who have gray hair and do not dye their hair. Without this information, we cannot calculate the probability.
On the other hand, the probability that a man does not appear to have gray hair can be determined. Since 5 out of 7 men have gray hair, it means that 2 out of 7 men do not have gray hair. Therefore, the probability that a randomly selected man does not appear to have gray hair is 2/7.
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Find the dot product v•w and the angle between v and w. v=4i - 3j + k, w = i +43 +4k vow=0 (Simplify your answet Type an exact value, using radicals as needed.) The angle between V and wis 0 = 0 (Do not round until the final answer. Then round to the nearest tenth as needed.)
To find the dot product v • w, we multiply the corresponding components of v and w and sum them up.
v = 4i - 3j + k
w = i + 4j + 3k
v • w = (4)(1) + (-3)(4) + (1)(3)
= 4 - 12 + 3
= -5
So, the dot product of v • w is -5.
To find the angle between v and w, we can use the formula:
cosθ = (v • w) / (||v|| ||w||)
where ||v|| and ||w|| are the magnitudes of vectors v and w, respectively.
||v|| = √(4² + (-3)² + 1²) = √(16 + 9 + 1) = √26
||w|| = √(1² + 4² + 3²) = √(1 + 16 + 9) = √26
cosθ = (-5) / (√26 * √26) = -5 / 26
To find the angle, we take the inverse cosine (arccos) of cosθ:
θ = arccos(-5 / 26)
Therefore, the angle between v and w is θ ≈ 98.4 degrees (rounded to the nearest tenth).
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mBEA =
148⁰
212°
74°
180°
B
C 74 A
E
D
Answer:
(b) 212°
Step-by-step explanation:
You want the measure of major arc BEA intercepted by tangent AD and chord AB.
Angle"Inscribed" angle DAB is the supplement of the one given, so is ...
∠DAB = 180° -74° = 106°
ArcThe measure of the arc that angle intercepts is double the measure of the angle:
arc BEA = 2×∠DAB = 2×106°
arc BEA = 212°
__
Additional comment
An inscribed angle is half the measure of the arc it intercepts. If you consider a point D' on arc EA, angle BAD' will have half the measure of arc AED'. This is true for D' located anywhere on arc EA, including at point A.
In the limit, as point D' approaches A, chord AD' approaches tangent AD. It should be no surprise, then, that angle DAB is half the measure of arc AEB.
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A particle moves in the coordinate plane with position vector (x(t). y(t))such that x(t) = R-4t+3 and y(t) = 4 - 5t + 3+ in the time interval Osts 4. Part A: Find the equation of the tangent line to the path of the particle when t= 4.
Part B: At what time (if any) is the particle at rest? Justify your answer.
Part C: Is the speed of the particle increasing or decreasing when t = 42 Justify your answer.
Part D: Write an expression for the total distance traveled by the particle on the interval Osts 4.
Given x(t) = R-4t+3 and y(t) = 4 - 5t + 3. A particle moves in the coordinate plane with position vector (x(t), y(t)).
Part A: Find the equation of the tangent line to the path of the particle when t= 4.Tangent line is defined as the line that touches a point on a curve but is different from the curve at the point.
Therefore, first, we will differentiate the given position vector with respect to time to obtain the derivative of the position vector.r(t) = (x(t), y(t))r′(t) = (x′(t), y′(t))= (-4, -5)
Therefore, the tangent line to the curve at t = 4 is given by:r(t) = r(4) + r′(4)(t - 4)r(t) = (R-13, -13) + (-4, -5)(t - 4)r(t) = (-4t + R - 29, -5t - 17)
Part B: At what time (if any) is the particle at rest? Justify your answer.Using the given velocity function, we can set it to zero to get the time at which the particle is at rest.
To find the velocity function, we differentiate the position function with respect to time.v(t) = (x′(t), y′(t))= (-4, -5)The speed function is given by:s(t) = √x′(t)² + y′(t)²= √16 + 25= √41s(t) is a constant function.
Therefore, the particle never comes to rest.
Part C: Is the speed of the particle increasing or decreasing when t = 42? Justify your answer. The speed of the particle is constant. Therefore, it is neither increasing nor decreasing when t = 42.
Part D: Write an expression for the total distance traveled by the particle on the interval Osts 4.The distance traveled by the particle between t = 0 and t = 4 is given by the integral of speed function on
[0, 4].∫[0,4] s(t) dt= ∫[0,4] √41 d[√41t]₀¹³= 3√41
The distance traveled by the particle between t = 4 and t = t is given by the integral of speed function on [4, t].∫[4,t] s(t) dt= ∫[4,t] √41 dt= [√41t]₄ᵗ= (t - 4) √41
The total distance traveled by the particle on the interval Osts 4 is given by the sum of the distance traveled between t = 0 and t = 4 and the distance traveled between t = 4 and t = t.d = 3√41 + (t - 4) √41= (√41t - √41 + 3√41) units.
The equation of the tangent line to the path of the particle when t = 4 is y = -5x + R - 57.The particle never comes to rest.The speed of the particle is neither increasing nor decreasing when t = 42.The expression for the total distance traveled by the particle on the interval Osts 4 is d = (√41t - √41 + 3√41) units.
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Suppose the Sunglasses Hut Company has a profit function given by P(q) = – 0.03q² +6q – 48, where q is the number of thousands of pairs of sunglasses sold and produced, and P(q) is the total profit, in thousands of dollars, from selling and producing a pairs of sunglasses. A) How many pairs of sunglasses (in thousands) should be sold to maximize profits? (If necessary, round your answer to three decimal places.) Answer: thousand pairs of sunglasses need to be sold. B) What are the actual maximum profits (in thousands) that can be expected? (If necessary, round your answer to three decimal places.) Answer:
$102 thousand is the actual maximum profits that can be expected.
A) The given profit function is,P(q) = – 0.03q² +6q – 48
We need to find out the number of pairs of sunglasses (in thousands) should be sold to maximize the profit.
The profit function is in the quadratic form ax² + bx + c.
Hence, it represents a parabola where "a" is negative.
So, the graph will be downward which means it will have a maximum point.
Therefore, to maximize the profit, we need to find the vertex of the parabola.
To find the vertex of the parabola, we use the following formula,
Vertex = (-b / 2a, P(-b / 2a))From the given function, a = -0.03 and b = 6Substituting the values in the above formula,
Vertex = (-6 / 2(-0.03), P(-6 / 2(-0.03)))= (100, P(100))
Therefore, 100 thousand pairs of sunglasses should be sold to maximize the profits.
B) Now, to find the actual maximum profits that can be expected, we need to substitute q = 100 in the given profit function.
P(q) = – 0.03q² + 6q – 48P(100)
= – 0.03(100)² + 6(100) – 48
= $102.
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According to a recent poll, 80% of adults enjoy watching TV. Choose a group of 3 adults at random. The probability that all of them enjoy watching TV is: O a. 0.267 Ob. 0.512 c. 0.333 d. 2.400
Answer:
B) 0.512
Step-by-step explanation:
Use the binomial probability formula
[tex]P(X=x)=C(n,x)p^xq^{n-x}\\P(X=3)=C(3,3)(0.80)^3(0.20)^0\\P(X=3)=(0.80)^3\\P(X=3)=0.512[/tex]
Therefore, B is the correct option.
Find dy/du, du/dx, and dy/dx. y = u, u³, u = 2x2 - 4 dy / du = du / dx = dy / dx =
So, the derivatives are: dy/du = 1, du/dx = 4x, dy/dx = 4x.
To find dy/du, du/dx, and dy/dx, we can use the chain rule of differentiation.
Given:
y = u
u = u³
u = 2x² - 4
dy/du:
Since y = u, the derivative of y with respect to u is simply 1.
dy/du = 1
du/dx:
To find du/dx, we need to differentiate the equation u = 2x² - 4 with respect to x.
du/dx = d/dx(2x² - 4)
= 4x
dy/dx:
To find dy/dx, we can use the chain rule by multiplying dy/du and du/dx.
dy/dx = (dy/du) * (du/dx)
= (1) * (4x)
= 4x
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If f(x)=3x^(5)+4x^(3)+3, then what is the remainder when f(x) is divided by x-3 ?
The Remainder when dividing f(x) by x - 3 is 840
The remainder when dividing the polynomial f(x) = 3x^5 + 4x^3 + 3 by the binomial x - 3, we can use the polynomial remainder theorem. According to the theorem, if we substitute the divisor, x - 3, into the polynomial f(x) and evaluate it at x = 3, the resulting value will be the remainder.
Let's calculate the remainder using this approach:
Substituting x = 3 into the polynomial f(x), we have:
f(3) = 3(3)^5 + 4(3)^3 + 3
= 3(243) + 4(27) + 3
= 729 + 108 + 3
= 840
Therefore, when f(x) = 3x^5 + 4x^3 + 3 is divided by x - 3, the remainder is 840.
In summary, the remainder when dividing f(x) by x - 3 is 840.
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