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3 years ago

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During strenuous exercise, an athlete can burn 12 calories per minute on a treadmill and 8.5 calories per minute on an elliptica

l machine. If an athlete uses both machines and burns 325 calories In a 30- minute workout, how many minutes does the athlete spend on each machine?

1 answer:

Irina18 [472]3 years ago

3 0

Hope it helps 10 mins on the elliptical and 12 on the treadmil

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David has 36 marbles, 24 of which are black and 12 are blue. What is the ratio of black marbles to blue marbles? Write the ratio

iragen [17]

We know there are a total of 36 marbles. We also know 24 out of 36 are black and 12 out of 36 are blue. A ratio is pretty much a division problem... The problem wants the 24 black marbles over the 12 blue marbles. So, 24/12 = 2/1. Now, what this means is that for every 2 black marbles, there is one 1 blue marble. Good luck!

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3 years ago

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

tino4ka555 [31]

a.

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

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.

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

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

b.

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

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 $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

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3 years ago

What is 1,200/2,000 in a fraction

andreev551 [17]

1200/200 in a simplified form is
3/5

7 0

3 years ago

Parallelogram ABCD is shown.<br>

hichkok12 [17]

Answer:

Step-by-step explanation:

#4 part A= D

#4 part B = 76

Your question only states parrallelogram ABCD is shown, so I assume you only wanted those answers, GL

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2 years ago

In a circle with a radius of 3 ft, an arc is intercepted by a central angle of 2π3 radians. What is the length of the arc?

madreJ [45]

The correct question is
<span>In a circle with a radius of 3 ft, an arc is intercepted by a central angle of 2π/3 radians. What is the length of the arc?

we know that
in a circle
</span>2π radians -----------------> lenght of (2*π*r)
2π/3 radians--------------> X
X=[(2π/3)*(2π*r)]/[2π]=(2π/3)*r

the lenght of the arc=(2π/3)*3=2π ft

the answer is 2π ft

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3 years ago

Read 2 more answers