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2 years ago

Is 0.15893 rational or irrational?

Mathematics

2 answers:

Pachacha [2.7K]2 years ago

8 0

Irrational to many numbers to the right of the decimal it is going to far

Ganezh [65]2 years ago

5 0

It would be irrational

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Find the point (x,y) of x2+14xy+49y2=100 that is closest to the origin and lies in the first quadrant.

Gala2k [10]

Notice that

$x^2+14xy+49y^2=(x+7y)^2$

so the constraint is a set of two lines,

$(x+7y)^2=100\implies\begin{cases}x+7y=10\\x+7y=10\end{cases}$

and only the first line passes through the first quadrant.

The distance between any point $(x,y)$ in the plane is $\sqrt{x^2+y^2}$ , but we know that $\sqrt{f(x,y)}$ and $f(x,y)$ share the same critical points, so we need only worry about minimizing $x^2+y^2$ . The Lagrangian for this problem is then

$L(x,y,\lambda)=x^2+y^2+\lambda(x+7y-10)$

with partial derivatives (set equal to 0)

$L_x=2x+\lambda=0$

$L_y=2y+7\lambda=0$

$L_\lambda=x+7y-10=0$

We have

$L_y-7L_x=2y-14x=0\implies y=7x$

which tells us that

$x+7y-10=0\iff x+49x=10\implies x=\dfrac15\implies y=\dfrac75$

so that $\left(\dfrac15,\dfrac75\right)$ is a critical point. The Hessian for the target function $x^2+y^2$ is

$H(x,y)=\begin{bmatrix}2&0\\0&2\end{bmatrix}$

which is positive definite for all $x,y$ , so the critical point is the site of a minimum. The minimum distance itself (which we don't seem to care about for this problem, but we might as well state it) is $\sqrt{\left(\dfrac15\right)^2+\left(\dfrac75\right)^2}=2$ .

3 0

2 years ago

Maddie is reading a graph of equivalent ratios. One point on the graph is (4, 5). Maddie says (10, 8) is also on the graph. Is M

dmitriy555 [2]

Yes and no, (4, 5) multiplied by 2 is (10, 8) but reversed. It depends on how you look at it because (10, 8) would be on the other side of the graph than (8, 10) which is the actually equivalent.

6 0

2 years ago

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A stone is dropped into a barrel of water and sinks to the bottom. The ball is completely covered by water. By how much does the

Rom4ik [11]

Answer:

Answer: 5/12 or approx. 0.42

Step-by-step explanation:

The ball will displace as much water as its volume.

Meaning, the volume of the ball and the volume of the increase in water in the barrel will be equal.

Volume of a sphere:

$V_s=\frac{4}{3} \pi r^3$

The radius is half the diameter. The diameter of the sphere is 10, thus:

$r = d/2 = 10/2 = 5$

5^3 = 5*5*5 = 25*5 = 125

$V_s=\frac{4}{3} \pi * 5^3=\frac{4}{3} \pi * 125$

The volume of a cylinder, which will be the shape of the increase in water, is the following:

$V_c=\pi r^2 h$

We know the radius r, as we know the diameter.

$r=d/2=40/2=20$

$V_c=\pi r^2 h=\pi h * 20^2 = \pi h * 400$

We are looking for the height of the cylinder, h, as that will be the height of the water rise.

Since we know these 2 volumes are the same, we'll set them equal to each other and solve the equation:

$V_s = V_c$

$\frac{4}{3} * 125 * \pi = 400 * h * \pi$

We can get rid of pi by dividing both sides by pi:

$\frac{4}{3} * 125 = 400 * h$

And now divide both sides by 400:

$\frac{4}{3} * \frac{125}{400}= h$

Just reversing it and putting h to the left:

$h = \frac{4}{3} * \frac{125}{400} = \frac{4 * 125}{3 * 400}$

I see that I can divide both the numerator and the denominator by 4:

$h =\frac{125}{3 * 100}= \frac{125}{300}$

I'll now divide both the numerator and the denominator by 25 to simplify the expression:

$h=\frac{125}{300} =\frac{125 / 25}{300 / 25} = \frac{5}{12}$

Answer: The water rises by 5/12 (five twelfths).

If I want an approximate decimal answer, I can simply run this expression in a calculator:

$\frac{5}{12}$ ≈ $0.416666667$ ≈ $0.42$

7 0

2 years ago

You and a friend are playing a game by tossing two coins. If both coins land

Novosadov [1.4K]

c. No. You have a probability of winning, while your friend has a
probability of winning.
Or it’s A

3 0

2 years ago

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Mukat has five times as many book as usha.If mukat gave 16 books to usha,they each would have the same number.how many books did

vladimir1956 [14]

Mukat gets 40, Usha gets 8

Step-by-step explanation:

Mukat = 5 × Usha

After Mukat gives Usha 16 books ,Usha gets (16 + initial number of books) and Mukat gets (5 × Usha - 16)

Then Mukat = final number of books for Usha

5 × Usha - 16 = Usha + 16

(5 × Usha ) - Usha = 16+16

4 × Usha = 32

Usha = 8 i.e her number of books.

Mukat = 5×8=40.

I HOPE IT'S OK

6 0

2 years ago