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valentina_108 [34]

3 years ago

10

A regular prism with length l width w and height h has a volume of lwh. What is the volume of a prism which has a base of 5 m by

3m and a height of 4m ?

1 answer:

Sindrei [870]3 years ago

7 0

The answer is 60m
You must multipy base x height x width to find the volume of a rectangular prism

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Rewrite the following integral in spherical coordinates.

lora16 [44]

In cylindrical coordinates, we have $r^2=x^2+y^2$ , so that

$z = \pm \sqrt{2-r^2} = \pm \sqrt{2-x^2-y^2}$

correspond to the upper and lower halves of a sphere with radius $\sqrt2$ . In spherical coordinates, this sphere is $\rho=\sqrt2$ .

$1 \le r \le \sqrt2$ means our region is between two cylinders with radius 1 and $\sqrt2$ . In spherical coordinates, the inner cylinder has equation

$x^2+y^2 = 1 \implies \rho^2\cos^2(\theta) \sin^2(\phi) + \rho^2\sin^2(\theta) \sin^2(\phi) = \rho^2 \sin^2(\phi) = 1 \\\\ \implies \rho^2 = \csc^2(\phi) \\\\ \implies \rho = \csc(\phi)$

This cylinder meets the sphere when

$x^2 + y^2 + z^2 = 1 + z^2 = 2 \implies z^2 = 1 \\\\ \implies \rho^2 \cos^2(\phi) = 1 \\\\ \implies \rho^2 = \sec^2(\phi) \\\\ \implies \rho = \sec(\phi)$

which occurs at

$\csc(\phi) = \sec(\phi) \implies \tan(\phi) = 1 \implies \phi = \dfrac\pi4+n\pi$

where $n\in\Bbb Z$ . Then $\frac\pi4\le\phi\le\frac{3\pi}4$ .

The volume element transforms to

$dx\,dy\,dz = r\,dr\,d\theta\,dz = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi$

Putting everything together, we have

$\displaystyle \int_0^{2\pi} \int_1^{\sqrt2} \int_{-\sqrt{2-r^2}}^{\sqrt{2-r^2}} r \, dz \, dr \, d\theta = \boxed{\int_0^{2\pi} \int_{\pi/4}^{3\pi/4} \int_{\csc(\phi)}^{\sqrt2} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta} = \frac{4\pi}3$

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2 years ago

Evaluate the expression 4(2x + 3) + 2(x + 1)-7

Svetlanka [38]

4 (2x + 3) + 2 (x + 1) - 7

8x + 12 + 2x + 2 - 7

10x + 7

10x = -7
10x/10 = -7/10

x = -7/10

hope this helps

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3 years ago

sergejj [24]

Answer:

D. 4.5 kg

Step-by-step explanation:

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Art [367]

A constant function is characterized by having the same value for f(x) in all it's domain. This means every value of x will have the same value in the axis y. You can see that in the graphic as a horizontal line.

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Elena L [17]

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