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Nataliya [291]
3 years ago
14

Find the area of the part of the paraboloid z = 9 - x^2 - y^2 that lies above the xy-plane.

Mathematics
1 answer:
svetoff [14.1K]3 years ago
3 0

Parameterize this surface (call it S) by

\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(9-u^2)\,\vec k

with 0\le u\le3 and 0\le v\le2\pi. Take the normal vector to S to be

\vec r_u\times\vec r_v=2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k

Then the area of S is

\displaystyle\iint_S\mathrm dA=\iint_S\|\vec r_u\times\vec r_v\|\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^{2\pi}\int_0^3u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv

=\displaystyle2\pi\int_0^3u\sqrt{1+4u^2}\,\mathrm du=\boxed{\frac{37\sqrt{37}-1}6\pi}

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A snail can move 1/30 miles in 1/3 hour. How fast does a snail move in miles /hours
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Answer:             1/10 miles/hour

To find how fast a snail moves in miles per hour, we can find the unit rate using a proportion.

<h2>Unit Rate</h2>

The unit rate is the amount per one unit. In this problem, the unit is for one hour. One way to find the unit rate is by using a proportion.

<h2>Proportions</h2>

Proportions look like two fractions set equal to each other with a missing variable to solve.

<h3>State our variables</h3>

let <em>x</em> be the number of miles per hour

<h3>Write a proportion</h3>

The first fraction has a number for the numerator (top) and denominator (bottom), because the question gives us the information. The second fraction has a missing value, <em>x</em>, that we are solving for. We can write <em>x</em> in the numerator.

\displaystyle{\frac{\frac{1}{30}\ mi}{\frac{1}{3}\ hr} = \frac{x\ mi}{1\ hr}}

<h2>Solve</h2>

We can solve by finding the scale factor. The scale factor is the number we can multiply the numerator by to find <em>x</em>.

<h3>Find the scale factor</h3>

Start with the proportion.

\displaystyle{\frac{\frac{1}{30}\ mi}{\frac{1}{3}\ hr} = \frac{x\ mi}{1\ hr}}

To find the scale factor,

  1. Take the denominator from the fraction with a variable: 1\ hr
  2. Take the denominator from the fraction that has no variable:  \frac{1}{3}\ hr
  3. Divide step 1 by step 2 and solve:  1\ hr\ \div\ \frac{1}{3}\ hr = 3

Our scale factor is <u>3</u>.

<h3>Solve for x</h3>

We can find <em>x</em> by multiplying the numerator in the first fraction by the scale factor. This is because <em>x</em> is also in the numerator.

\displaystyle{\frac{\frac{1}{30}\ mi}{\frac{1}{3}\ hr} = \frac{(\frac{1}{30}\ mi)*3}{1\ hr}}

\displaystyle{\frac{\frac{1}{30}\ mi}{\frac{1}{3}\ hr} = \frac{\frac{3}{30}\ mi}{1\ hr}}

Simplify 3/30 by dividing the fraction by 3.

\displaystyle{\frac{\frac{1}{30}\ mi}{\frac{1}{3}\ hr} = \frac{\frac{1}{10}\ mi}{1\ hr}}

x = 1/10

∴ a snail moves 1/10 miles per hour.

Learn another way to solve proportions here: brainly.com/question/18437927

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Step-by-step explanation: The actual definition of marginal cost is the cost added by producing one additional unit of a product of service.

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