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2 years ago

Rewrite the product in exponential form: a.a.a.b.b.b.b

Mathematics

1 answer:

nikitadnepr [17]2 years ago

8 0

It would be a³ b ^4.

Hope This Helps You!
Good Luck Studying :)

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Find the volume of the solid.

dmitriy555 [2]

In Cartesian coordinates, the region (call it $R$ ) is the set

$R = \left\{(x,y,z) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } 2 \le z \le 4-x^2-y^2\right\}$

In the plane $z=2$ , we have

$2 = 4 - x^2 - y^2 \implies x^2 + y^2 = 2 = \left(\sqrt2\right)^2$

which is a circle with radius $\sqrt2$ . Then we can better describe the solid by

$R = \left\{(x,y,z) ~:~ 0 \le x \le \sqrt2 \text{ and } 0 \le y \le \sqrt{2 - x^2} \text{ and } 2 \le z \le 4 - x^2 - y^2 \right\}$

so that the volume is

$\displaystyle \iiint_R dV = \int_0^{\sqrt2} \int_0^{\sqrt{2-x^2}} \int_2^{4-x^2-y^2} dz \, dy \, dx$

While doable, it's easier to compute the volume in cylindrical coordinates.

$\begin{cases} x = r \cos(\theta) \\ y = r\sin(\theta) \\ z = \zeta \end{cases} \implies \begin{cases}x^2 + y^2 = r^2 \\ dV = r\,dr\,d\theta\,d\zeta\end{cases}$

Then we can describe $R$ in cylindrical coordinates by

$R = \left\{(r,\theta,\zeta) ~:~ 0 \le r \le \sqrt2 \text{ and } 0 \le \theta \le\dfrac\pi2 \text{ and } 2 \le \zeta \le 4 - r^2\right\}$

so that the volume is

$\displaystyle \iiint_R dV = \int_0^{\pi/2} \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta \, dr \, d\theta \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta\,dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} r((4 - r^2) - 2) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} (2r-r^3) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \left(\left(\sqrt2\right)^2 - \frac{\left(\sqrt2\right)^4}4\right) = \boxed{\frac\pi2}$

3 0

1 year ago

A shop has a sale that offers 20% off all prices.On the final day reduce all the sale prices by 25%.Linz buys a radio on the fin

Mrac [35]

Answer:

40%

Step-by-step explanation:

Original price = x

Sale = 20% off

20% / 100 = 0.2 to convert from a percent to a decimal

Sale = x - 20% of x = x - 0.2x = 0.8x

Sale price = y = 0.8x

Reduction of sale prices = 25% = 0.25

Reduction = y - 25% of y = y - 0.25y = 0.75y

End price in terms of original price (x):

0.75y = 0.75(0.8)x

= 0.6x

End price = 0.6 or 60% of original price, coming out to a 100-60 = 40% overall percentage reduction

5 0

2 years ago

Which of the following expressions represents the width of the door?

charle [14.2K]

Answer:

Step-by-step explanation:

5 0

2 years ago

1. Find the capacity in litres of a bucket 24cm in

Sav [38]

Answer:

6.36l

Step-by-step explanation:

We compute first the volume:

V=pi×h/3×(R²+r²+rR), where h=20cm is the height, R=24/2=12cm is the top radius, r=16/2=8cm is the bottom radius.

We get: V=pi×20/3(12²+8²+12×8)=

20×pi/3(144+64+96)cm³

V=20×pi/3×304=6363.73cm³

In dm³ we have (divide by 1000) V=6.36dm³

By definition, 1dm³=1l, so the capacity is 6.36l

8 0

3 years ago

Rewrite <br> 16/9<br> as a mixed number.

PilotLPTM [1.2K]

To find the answer, to convert a improper into mixed, first take the denominator (9) and find how many times it can go into the numerator (16) FULLY. The answer is 1 time. 1 is the whole number. Then whatever is leftover is the new numerator, and the old denominator is the same. So your answer is 1 7/9

Hope this helps!

3 0

2 years ago

Read 2 more answers