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Can someone answer this please?

Suppose that the graph of a certain function, y=f(x), has the property that if it is shifted 20 units to the right, then the resulting graph is identical to the original graph of y=f(x).  What is the smallest positive a such that if the graph of y=f(x/5)$ is shifted a units to the right, then we know that the resulting graph is identical to the original graph of $y=f(x/5)?

The answer is 5*20=100

This is when we know for sure that the  translation of a (in the right direction) will make the graph identical.

Here is a simple example

You could use a calculator for this problem, there is a calculator on this website. 

0.1024397636

3* blank = 4

Let's subsitute blank for x

3x = 4

Divide both sides by 3 

x = 4/3

The answer to this problem is 4/3

To find the solution for this, calculate the weighted average:

 (10×55+20×56+70×57) / (10+20+70) = 56.6 amu

.

In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?

\(x^2+y^2=39^2\\x^2+y^2=1521 \)

sqrt(1521-38^2) = 8.7749643873921221

sqrt(1521-37^2) = 12.3288280059379529

sqrt(1521-36^2) = 15

there you go your short sides are  36 and 15

there is probably only one answer but i have not checked that.

The shortest side is 15units

You have the setting out of these off pat Chis :)  I am impressed!

As you know I have given up and just do them on paper and take a photo. 

It says to leave it as a factored expresssion so I think you only need to say

(x-53)(x-22)=0

and leave it as that.

Thanks for the help, Guest! This sure helped a lot!

-Red Glasses 

Expand the following:
(x - 22) (x - 53)

(x - 22) (x - 53) = (x) (x) + (x) (-53) + (-22) (x) + (-22) (-53):
x x - 22 x - 53 x - 22 (-53)

x x = x^2:
x^2 - 22 x - 53 x - 22 (-53)

-22 (-53) = 1166:
x^2 - 22 x - 53 x + 1166

-53 x - 22 x = -75 x:

x^2 - 75x + 1166 =0

In a right-angled triangle, you have to know at least 2 sides, or 1 side and 1 angle to find the other side.

Cool! You're definitely going to pass me up at this rate! 

Next time, make sure not to confuse the x variables for multiplication signs. Here is the difference:

\(\times \ (multiplication \ sign)\\ x \ (variable)\)

Anyways:

\(-4x + 7=3x - 7\)

Add 4x to each side to remove the negative value of x (Rule: always subtract the lesser value when you are solving for a variable [this one was negative, so we added 4 instead of subtracting]):

\(7=7x-7\)

Add 7 to each side (Again, it is negative):

\(7x = 14\)

Divide by the coefficient of the variable you are solving for (x, coefficient is 7 so we divide each side by 7):

\(x=2\)

25 units = a line segment starting at (-24, 0) and ending at (0, n)

Pythagorean theorem: \(625 = 576 - n^2\)

625 - 576 = 49

\(\sqrt{49} = 7\)

n = 7

Now that we have the height and the width, we solve using \(a = \frac{1}{2}bh\)

\(7 \times 48 \div 2=7\times 24 = 168\)

The area of the triangle is 168 square centimeters.

^^^ I honestly don't know how I managed to use that method. ^^^

Find the derivative of the following via implicit differentiation:
d/dy(g(y)) = d/dy(16 (1/y - sqrt(y) + 2 y^3))

The derivative of g(y) is g'(y):
g'(y) = d/dy(16 (1/y - sqrt(y) + 2 y^3))

Factor out constants:
g'(y) = 16 d/dy(1/y - sqrt(y) + 2 y^3)

Differentiate the sum term by term and factor out constants:
g'(y) = 16 d/dy(1/y) - d/dy(sqrt(y)) + 2 d/dy(y^3)

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = -1: d/dy(1/y) = d/dy(y^(-1)) = -y^(-2):
g'(y) = 16 (-(d/dy(sqrt(y))) + 2 (d/dy(y^3)) + (-1)/(y^2))

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = 1/2: d/dy(sqrt(y)) = d/dy(y^(1/2)) = y^(-1/2)/2:
g'(y) = 16 (-1/y^2 + 2 (d/dy(y^3)) - 1/(2 sqrt(y)))

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = 3: d/dy(y^3) = 3 y^2:
g'(y) = 16 (-1/y^2 - 1/(2 sqrt(y)) + 2 3 y^2)

Simplify the expression:
g'(y) = 16 (-1/y^2 - 1/(2 sqrt(y)) + 6 y^2) - Courtesy of "Mathematica 11 Home Edition"

That looks kind of like a Fibonacci Spiral. It is based on the sequence \(t(n)=t(n-1)+t(n-2)\) . Eventually, this will leave you with the constant phi, which equals ​\(1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\dots}}}}=\frac{3-\sqrt{5}}{2}\)

I assumed, based off of the wording, that this was based on displacement. The question is, "How many miles from the starting point was the traveler after completing these 4 legs of the journey"

thx that was correct i was wondering why i got it wrong.

thats theproblem i dont know. That is the whole problem.

I'm assuming this is absolute distance, not distance traveled on foot.

19 - 3 = 16

16 - 4 = 12

\(\sqrt{16^{2} + 12^{2}} = \sqrt{256 + 144} = \sqrt{400} =20\)

He moved an overall of 20 miles from the starting point.

3 facts:

QA is congruent to Q'A (QAQ' is an isosceles triangle)

QB is congruent to Q'B (QBQ' is an isosceles triangle)

QC is congruent to Q'C (\(\leftarrow QC \rightarrow\) is the same as \(\leftarrow Q'C \rightarrow\), C is the point where \(\leftarrow AB \rightarrow\) bisects CC')

Are you looking for distance or displacement?

B; this would give us 6 points on the circle, each equidistant. After connecting them, she would have the regular hexagon inscribed in the circle.

Do get the coordinates of the new image from the previous, divide the points of the original by the corresponding points, after translating them both so that the point of dilation is 0 (found by drawing lines through each corresponding point and seeing where they all intersect). In this case, A'B'C' has a relative scale factor of \(\frac{1}{4}\).