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3 years ago

How to simplify fractions

Mathematics

1 answer:

serious [3.7K]3 years ago

5 0

Step-by-step explanation:

$\frac{2}{3} + \frac{2}{9} \\\\\frac{6+2}{9}\\\\\frac{8}{9}$

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(1 point) (a) Find the point Q that is a distance 0.1 from the point P=(6,6) in the direction of v=⟨−1,1⟩. Give five decimal pla

natima [27]

Answer:

following are the solution to the given points:

Step-by-step explanation:

In point a:

$\vec{v} = -\vec{1 i} +\vec{1j}\\\\|\vec{v}| = \sqrt{-1^2+1^2}$

$=\sqrt{1+1}\\\\=\sqrt{2}$

calculating unit vector:

$\frac{\vec{v}}{|\vec{v}|} = \frac{-1i+1j}{\sqrt{2}}$

the point Q is at a distance h from P(6,6) Here, h=0.1

$a=-6+O.1 \times \frac{-1}{\sqrt{2}}\\\\= 5.92928 \\\\b= 6+O.1 \times \frac{-1}{\sqrt{2}} \\\\= 6.07071$

the value of Q= (5.92928 ,6.07071 )

In point b:

Calculating the directional derivative of $f (x, y) = \sqrt{x+3y}$ at P in the direction of $\vec{v}$

$f_{PQ} (P) =\fracx{f(Q)-f(P)}{h}\\\\$

$=\frac{f(5.92928 ,6.07071)-f(6,6)}{0.1}\\\\=\frac{\sqrt{(5.92928+ 3 \times 6.07071)}-\sqrt{(6+ 3\times 6)}}{0.1}\\\\= \frac{0.197651557}{0.1}\\\\= 1.97651557$

$\vec{v} = 1.97651557$

In point C:

Computing the directional derivative using the partial derivatives of f.

$f_x(x,y)= \frac{1}{2 \sqrt{x+3y}}\\\\ f_x (6,6)= \frac{1}{2 \sqrt{22}}\\\\f_x(x,y)= \frac{1}{\sqrt{x+3y}}\\\\ f_x (6,6)= \frac{1}{\sqrt{22}}\\\\f_{(PQ)}(P)= (f_x \vec{i} + f_y \vec{j}) \cdot \frac{\vec{v}}{|\vec{v}|}\\\\= (\frac{1}{2 \sqrt{22}}\vec{i} + \frac{1}{\sqrt{22}} \vec{j}) \cdot \frac{-1}{\sqrt{2}}\vec{i} + \frac{1}{\sqrt{2}} \vec{j}$

4 0

2 years ago

Find a cartesian equation for the curve. r = 9 sin(θ)

Paul [167]

For this case we have the following equation:
r = 9 sin (θ)
In addition, we have the following change of variables:
y = r * sine (θ)
Rewriting the equation we have:
r = 9 sin (θ)
r = 9 (y / r)
r ^ 2 = 9y
On the other hand:
r ^ 2 = x ^ 2 + y ^ 2
Substituting values:
x ^ 2 + y ^ 2 = 9y
Rewriting:
x ^ 2 + y ^ 2 - 9y = 0
Completing squares:
x ^ 2 + y ^ 2 - 9y + (-9/2) ^ 2 = (-9/2) ^ 2
Rewriting:
x ^ 2 + 1/4 (2y-9) ^ 2 = 81/4
4x ^ 2 + (2y-9) ^ 2 = 81
Answer:
The Cartesian equation is:
4x ^ 2 + (2y-9) ^ 2 = 81

6 0

2 years ago

2) Brian purchased pens for 75 cents each and pencils for 40 cents each he bought total of 22 writing utensils for $11.95. How m

d1i1m1o1n [39]

Answer:

13 pencils.

Step-by-step explanation:

Let x be the pens and y be the pencils

Given:

Brian purchased pens for 75 cents each and pencils for 40 cents each he bought total of 22 writing utensils for $11.95.

Total pens and pencils is 22

So, $x+y=22$ ----------------(1)