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1 year ago

Is 4.55 and 1.23456 a rational number

Mathematics

1 answer:

kaheart [24]1 year ago

7 0

Problem

Is 4.55 and 1.23456 a rational number

Solution

4.55 can be written as:

455/100 so then it could be a rational number

For the second number we have: 1.23456

And we can write this number as:

123456/100000

So then is also ratioanl

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5. How does the length of the long leg compare to the length of the short leg?

Oliga [24]

Answer:

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Step-by-step explanation:

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3 years ago

write the factored form for a quadratic equation with x-intercepts of -6 and -2 and the point (-3,-6) is on the parabola

kenny6666 [7]

F(x)=a(x-x1)(x-x2)
F(x)=a(x- - 6)(x- -2)
F(x)=a(x+6)(x+2)
-6=a(-3+6)(-3+2)
-6=a(3)(-1)
-6=-3a
2=a

Equation: f(x)=2(x+6)(x+2)

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3 years ago

1. Evaluate each expression when x = -2 and y = 3.

borishaifa [10]

a) 3(-2) + 4(3) = -6 + 12 = 6

b) 2(-2) -3(3) +5 = -4 -9 + 5 = -8

c) 4(-2) -(3) = -8 -3 = -11

d) -(-2) -2(3) = 4 -6 = -2

e) (1/2)(-2) +(3) = -1 +3 = 2

f) (2/3)(3) -(1/2)(-2) = 2 + 1 = 3

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3 years ago

How to know if a function is periodic without graphing it ?

zhenek [66]

A function $f(t)$ is periodic if there is some constant $k$ such that $f(t+k)=f(k)$ for all $t$ in the domain of $f(t)$ . Then $k$ is the "period" of $f(t)$ .

Example:

If $f(x)=\sin x$ , then we have $\sin(x+2\pi)=\sin x\cos2\pi+\cos x\sin2\pi=\sin x$ , and so $\sin x$ is periodic with period $2\pi$ .

It gets a bit more complicated for a function like yours. We're looking for $k$ such that

$\pi\sin\left(\dfrac\pi2(t+k)\right)+1.8\cos\left(\dfrac{7\pi}5(t+k)\right)=\pi\sin\dfrac{\pi t}2+1.8\cos\dfrac{7\pi t}5$

Expanding on the left, you have

$\pi\sin\dfrac{\pi t}2\cos\dfrac{k\pi}2+\pi\cos\dfrac{\pi t}2\sin\dfrac{k\pi}2$

and

$1.8\cos\dfrac{7\pi t}5\cos\dfrac{7k\pi}5-1.8\sin\dfrac{7\pi t}5\sin\dfrac{7k\pi}5$

It follows that the following must be satisfied:

$\begin{cases}\cos\dfrac{k\pi}2=1\\\\\sin\dfrac{k\pi}2=0\\\\\cos\dfrac{7k\pi}5=1\\\\\sin\dfrac{7k\pi}5=0\end{cases}$

The first two equations are satisfied whenever $k\in\{0,\pm4,\pm8,\ldots\}$ , or more generally, when $k=4n$ and $n\in\mathbb Z$ (i.e. any multiple of 4).

The second two are satisfied whenever $k\in\left\{0,\pm\dfrac{10}7,\pm\dfrac{20}7,\ldots\right\}$ , and more generally when $k=\dfrac{10n}7$ with $n\in\mathbb Z$ (any multiple of 10/7).

It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when $k$ is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.

Let's verify:

$\sin\left(\dfrac\pi2(t+20)\right)=\sin\dfrac{\pi t}2\underbrace{\cos10\pi}_1+\cos\dfrac{\pi t}2\underbrace{\sin10\pi}_0=\sin\dfrac{\pi t}2$

$\cos\left(\dfrac{7\pi}5(t+20)\right)=\cos\dfrac{7\pi t}5\underbrace{\cos28\pi}_1-\sin\dfrac{7\pi t}5\underbrace{\sin28\pi}_0=\cos\dfrac{7\pi t}5$

More generally, it can be shown that

$f(t)=\displaystyle\sum_{i=1}^n(a_i\sin(b_it)+c_i\cos(d_it))$

is periodic with period $\mbox{lcm}(b_1,\ldots,b_n,d_1,\ldots,d_n)$ .

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