What is the surface area and volume ?

Evaluate the integral Find the volume of material remaining in a hemisphere of radius 2 after a cylindrical hole ofradius 1 is d

algol13

You want to find the volume inside the hemisphere $x^2+y^2+z^2=4$ (i.e. inside the sphere but above the plane $z=0$ ) and outside the cylinder $x^2+y^2=1$ . Call this region $R$ .

In cylindrical coordinates, we have

$\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_1^2\int_0^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$

$\displaystyle=2\pi\int_1^2 r\sqrt{4-r^2}\,\mathrm dr$

$\displaystyle=-\pi\int_3^0\sqrt u\,\mathrm du$

(where $u=4-r^2$ )

$\displaystyle=\pi\int_0^3\sqrt u\,\mathrm du$

$=\dfrac{2\pi}3u^{3/2}\bigg|_0^3=2\sqrt3\,\pi$

7 0

3 years ago

A heated piece of metal cools according to the function c(x) = (.5)x − 11, where x is measured in hours. A device is added that

e-lub [12.9K]

16.5 degrees below the starting temp, whatever that is. If its 0, then -16.5 degrees.

The first equation would be (.5)5-11=-8.5, because the metal has been cooling for 5 hours.

The device that 'aids' in the cooling would be -5-3=-8, because it is a separate variable that cools the metal, so the amount the device cools is independent of the natural cooling amount, and the equation is independent of the natural cooling equation.
You then add -8.5 and -8, because the device has lowered 8.5 degrees and 8 degrees. This equals -16.5 degrees, or a decrease of 16.5 degrees.

7 0

3 years ago

Customers of a phone company can choose between two service plans for long distance calls. The first plan has a $22 monthly fee

vladimir2022 [97]

Company a = 22 + 0.07m
company b = 8 + 0.12m

We are looking for the time when the prices are equal. i.e. the price of company a = the price of company b

company a = company b
22 + 0.07m = 8 + 0.12m
22 - 8 = 0.12m - 0.07m
14 = 0.05m
m = 14/0.05
m = 280

Therefore, the two companies will cost the same if you talk for 280 minutes

5 0

3 years ago

What integer describes a decrease of seven? (A. 7 / B. -7 / C. 0 / D. 1/7)

Pie

Answer:

I thin B -7 cause its negative and negative goes down.

Step-by-step explanation:

3 0

2 years ago

Paul,Linda,and Aaron make homemade cards. Paul can make 1 card for every 3 cards Linda makes. Linda can make 3 cards for every 2

dimaraw [331]

Answer:

Linda makes 72 more cards than Paul

during the week

Step-by-step explanation:

Here, we want to know how many cards Linda makes than Paul during the week.

We proceed as follows;

From the question;

If Paul makes 1 card, then Linda has made three

Now let’s get their relationship with Aaron;

If Aaron makes 2 cards , Linda has made three

Now since Aaron made 72 cards, Let’s say Linda has made x cards

Let’s get the value of x now

To get the value of x, we simply cross multiply;

2 - 3

72 - x

2 * x = 3 * 72

x = (3 * 72)/2 = 108 cards

Now recall , for every 3 cards of Linda, Paul made only 1;

So the number of cards Paul will make if Linda makes 108 will be 108/3 = 36

So the number of more cards that Linda makes than Paul during the week will be ; 108 -36 = 72 cards

3 0

3 years ago