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Scrat [10]
3 years ago
7

The base of a solid in the region bounded by the two parabolas y2 = 8x and x2 = 8y. Cross sections of the solid perpendicular to

the x-axis are semicircles. What is the volume, in cubic units, of the solid?
288 times pi over 35

576 times pi over 35

144 times pi over 35

8π
Mathematics
1 answer:
Usimov [2.4K]3 years ago
6 0

The two curves intersect at two points, (0, 0) and (8, 8):

x^2=8y\implies y=\dfrac{x^2}8

y^2=\dfrac{x^4}{64}=8x\implies\dfrac{x^4}{64}-8x=0\implies\dfrac{x(x-8)(x^2+8x+64)}{64}=0

\implies x=0,x=8\implies y=0,y=8

The area of a semicircle with diameter d is \dfrac{\pi d^2}8. The diameter of each cross-section is determined by the vertical distance between the two curves for any given value of x between 0 and 8. Over this interval, y^2=8x\implies y=\sqrt{8x} and \sqrt{8x}>\dfrac{x^2}8, so the volume of this solid is given by the integral

\displaystyle\frac\pi8\int_0^8\left(\sqrt{8x}-\dfrac{x^2}8\right)^2\,\mathrm dx=\frac{288\pi}{35}

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In the accompanying diagram of AABC, side AB is
guapka [62]

Answer:

x = 10

Step-by-step explanation:

The external angle of a triangle is equal to the sum of the 2 opposite interior angles.

∠CBD is an external angle of the triangle, thus

4x + 30 = 2x + 10 + x + 30, that is

4x + 30 = 3x + 40 ( subtract 3x from both sides )

x + 30 = 40 ( subtract 30 from both sides )

x = 10

3 0
3 years ago
Thomas rents a car for his vaccation the milliage included with the 54 miles forer every mile he drives over 54 miles he needs t
Sauron [17]

Answer:

$27

Step-by-step explanation:

Thomas rents a car for his vacation. The mileage include with the rental is 54 miles. For every mile he drives over 54 miles, he needs to pay $1 4/5. If he drives 69 miles, how much extra does he need to pay?

Total mileage included with the rental = 54 miles

Additional cost per mile after 54 miles = $1 4/5

Total miles Thomas drives = 69 miles

Extra miles Thomas drives = 69 miles - 54 miles

= 15 miles

how much extra does he need to pay?

Extra cost Thomas needs to pay = Additional cost per mile after 54 miles * Extra miles Thomas drives

= $1 4/5 * 15 miles

= 9/5 * 15

= (9 * 15) / 5

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Extra cost Thomas needs to pay = $27

4 0
2 years ago
5 x 1 please?? im teaching my cousin right now
Pachacha [2.7K]
Tell your cousin good luck and that the answer is 5 :)
4 0
2 years ago
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Natasha2012 [34]

Answer: Satisfied for n=1, n=k and n=k+1

Step-by-step explanation:

The induction procedure involves two steps

First is

Basic Step

Here we consider that for the value n=1, there is one car and it will always make the full circle.

Induction Step

Since basic step is satisfied for n=1

Now we do it for n=k+1

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Now ,go through the statement again that the original car gets past the other car and take the gas from it to eliminate it. So now cars remain k instead of k+1 as it's fuel has been taken. Now the car that has taken the fuel can make the full circle. The gas is enough to make a circle now.

So by induction we can find a car that satisfies k+1 induction so for k number of cars, we can also find a car that makes a full circle.

8 0
2 years ago
Name another pair of metric units of length that have the same relationship
Vanyuwa [196]

<span>One pair of metric unit of length would be meters and kilometers. 1 km is equal to 1000 meters. Metric system is the official/standard in measuring length, mass, area, volume, speed, etc. and it is the agreed system of measurement.</span>

6 0
3 years ago
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