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3 years ago

What is the volume of this triangular prism?

Mathematics

2 answers:

frez [133]3 years ago

7 0

Answer:

Step-by-step explanation:

Serga [27]3 years ago

3 0

I hope this helps you

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Solve 1/5x+3=10. Show all work and check it

iren2701 [21]

1/5x + 3 = 10

Subtract by 3 on both sides.

1/5x = 7.

Now to isolate the variable, flip the coefficient in front of the variable and multiply both sides by it.

5(1/5x) = 7(5)

x = 7(5)
x = 35

Now to check:

1/5(35) + 3 = 10

35/5 + 3 = 10

35/5 is 7.

7 + 3 = 10

10 = 10

5 0

4 years ago

Read 2 more answers

Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64

tino4ka555 [31]

$x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}$

By Fermat's little theorem, we have

$x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5$

$x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7$

5 and 7 are both prime, so $\varphi(5)=4$ and $\varphi(7)=6$ . By Euler's theorem, we get

$x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5$

$x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7$

Now we can use the Chinese remainder theorem to solve for $x$ . Start with

$x=2\cdot7+5\cdot6$

Taken mod 5, the second term vanishes and $14\equiv4\pmod5$ . Multiply by the inverse of 4 mod 5 (4), then by 2.

$x=2\cdot7\cdot4\cdot2+5\cdot6$

Taken mod 7, the first term vanishes and $30\equiv2\pmod7$ . Multiply by the inverse of 2 mod 7 (4), then by 6.

$x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6$

$\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}$

$x^5\equiv3\pmod{64}$

We have $\varphi(64)=32$ , so by Euler's theorem,

$x^{32}\equiv1\pmod{64}$

Now, raising both sides of the original congruence to the power of 6 gives

$x^{30}\equiv3^6\equiv729\equiv25\pmod{64}$

Then multiplying both sides by $x^2$ gives

$x^{32}\equiv25x^2\equiv1\pmod{64}$

so that $x^2$ is the inverse of 25 mod 64. To find this inverse, solve for $y$ in $25y\equiv1\pmod{64}$ . Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that $(-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}$ .

So we know

$25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}$

Squaring both sides of this gives

$x^4\equiv1681\equiv17\pmod{64}$

and multiplying both sides by $x$ tells us

$x^5\equiv17x\equiv3\pmod{64}$

Use the Euclidean algorithm to solve for $x$ .

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that $(-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}$ , and so $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

5 0

3 years ago

15. A rectangle has a length of 2x - 1 and a width of 3x + 7. Find the perimeter of this rectangle.

andrew11 [14]

Answer:

10x + 12 and 6x² + 11x - 7

Step-by-step explanation:

The perimeter (P) of a rectangle is calculated as

P = 2l + 2w ( l is length and w is width )

= 2(2x - 1) + 2(3x + 7) ← distribute parenthesis

= 4x - 2 + 6x + 14 ← collect like terms

= 10x + 12

--------------------------------------------------

The area (A) of a rectangle is calculated as

A = lw

= (2x - 1)(3x + 7) ← expand using FOIL

= 6x² + 14x - 3x - 7 ← collect like terms

= 6x² + 11x - 7

7 0

3 years ago

Which statement must be true about the diagram?

sertanlavr [38]

Ray KM is an angle bisector of NKL.

8 0

3 years ago

Read 2 more answers

Solve the equation 12 - 27n2 = 0 9, 3 (2/3), (-2/3) (3/2), (-3/2) 2, 3

Norma-Jean [14]

(2/3),=(1/2), 3,4
.
............................

4 0

3 years ago