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ra1l [238]
3 years ago
14

PLEASE HELP ASAP 30 POINTS.

Mathematics
2 answers:
emmasim [6.3K]3 years ago
7 0

When it turns minute 42, they'll make them both at the same time again.

VMariaS [17]3 years ago
5 0
Another 20 mins will pass by in order for them to make the same sandwiches again. 
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More Calculus! (I'm so sorry)
Olenka [21]
Recall that converting from Cartesian to polar coordinates involves the identities

\begin{cases}y(r,\phi)=r\sin\phi\\x(r,\phi)=r\cos\phi\end{cases}

As a function in polar coordinates, r depends on \phi, so you can write r=r(\phi).

Differentiating the identities with respect to \phi gives

\begin{cases}\dfrac{\mathrm dy}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi\\\\\dfrac{\mathrm dx}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi\end{cases}

The slope of the tangent line to r(\phi) is given by

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm d\phi}}{\frac{\mathrm dx}{\mathrm d\phi}}=\dfrac{\frac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi}{\frac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi}

Given r(\phi)=3\cos\phi, you have \dfrac{\mathrm dr}{\mathrm d\phi}=-3\sin\phi. So the tangent line to r(\phi) has a slope of

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{-3\sin^2\phi+3\cos^2\phi}{-3\sin\phi\cos\phi-3\cos\phi\sin\phi}=\dfrac{3\cos2\phi}{-3\sin2\phi}=-\cot2\phi

When \phi=120^\circ=\dfrac{2\pi}3\text{ rad}, the tangent line has slope

\dfrac{\mathrm dy}{\mathrm dx}=-\cot\dfrac{4\pi}3=-\dfrac1{\sqrt3}

This line is tangent to the point (r,\phi)=\left(-\dfrac32,\dfrac{2\pi}3\right) which in Cartesian coordinates is equivalent to (x,y)=\left(\dfrac34,-\dfrac{3\sqrt3}4\right), so the equation of the tangent line is

y+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(x-\dfrac34\right)

In polar coordinates, this line has equation

r\sin\phi+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(r\cos\phi-\dfrac34\right)
\implies r=-\dfrac{3\sqrt3}{2\sqrt3\cos\phi+6\sin\phi}

The tangent line passes through the y-axis when x=0, so the y-intercept is \left(0,-\dfrac{\sqrt3}2\right).

The vector from this point to the point of tangency on r(\phi) is given by the difference of the vector from the origin to the y-intercept (which I'll denote \mathbf a) and the vector from the origin to the point of tangency (denoted by \mathbf b). In the attached graphic, this corresponds to the green arrow.

\mathbf b-\mathbf a=\left(\dfrac34,-\dfrac{3\sqrt3}4\right)-\left(0,-\dfrac{\sqrt3}2\right)=\left(\dfrac34,-\dfrac{\sqrt3}4\right)

The angle between this vector and the vector pointing to the point of tangency is what you're looking for. This is given by

\mathbf b\cdot(\mathbf b-\mathbf a)=\|\mathbf b\|\|\mathbf b-\mathbf a\|\cos\theta
\dfrac98=\dfrac{3\sqrt3}4\cos\theta
\implies\theta=\dfrac\pi6\text{ rad}=30^\circ

The second problem is just a matter of computing the second derivative of \phi with respect to t and plugging in t=2.

\phi(t)=2t^3-6t
\phi'(t)=6t^2-6
\phi''(t)=12t
\implies\phi''(2)=24

6 0
3 years ago
suppose that a biologist is watching a trail known for wildebeest migration. During the first minute, 24 wildebeests migrated pa
Monica [59]

Answer:

=1050 wildebeests

Step-by-step explanation:

We can form an arithmetic series for the wildebeest migration.

Sₙ=n/2(2a+(n-1)d) where  n is the number of terms, d is the common difference and a is the first term.

a=24

n=20

d=3

Sₙ=(20/2)(2(24)+(20-1)3)

Sₙ=10(48+57)

=1050 wildebeests

6 0
3 years ago
Use the diagram of the right triangle above and round your answer to the nearest hundredth. If m <br> Answer choices: <br><br> A
Ivenika [448]
If you use the law of sins, the answer would be B) 6.928 (in the law of sines you round so it would be 6.93)
7 0
3 years ago
Read 2 more answers
How can the Associative, Commutative, and Distributive Properties be applied when performing operations on complex numbers and p
ozzi
By using Associative, Commutative, and Distributive Properties in performing operations on complex numbers and polynomials, you would be able to simplify correctly and simply. To combine complex numbers, you must distribute any coefficients in front of the parentheses. Then identify and<span>combine like terms by combining the real number parts and the imaginary number parts separately.</span>
8 0
3 years ago
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What is the Surface area for a rectangular prism that has the measurements of 2, 12, and 6?
Bond [772]

Answer:

216

Step-by-step explanation:

A=2(wl+hl+hw)=2·(12·2+6·2+6·12)=216

5 0
2 years ago
Read 2 more answers
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