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.)

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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$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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Answer:

  (b)  212°

Step-by-step explanation:

You want the measure of major arc BEA intercepted by tangent AD and chord AB.

"Inscribed" angle DAB is the supplement of the one given, so is ...

  ∠DAB = 180° -74° = 106°

The 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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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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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.

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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