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Katarina [22]
2 years ago
13

54. Find

ddle" class="latex-formula"> if x^y=y^x\text{.}
Mathematics
1 answer:
Stella [2.4K]2 years ago
7 0

Step-by-step explanation:

Take the natural log of both sides:

ln ({x}^{y} ) = ln ({y}^{x} )

Logarithm rules allow you to bring down the exponents:

yln(x) = xln(y)

Now differentiate. We will have to implicitly differentiate 'y' since it is a function of 'x'. Both sides require the product rule:

\frac{dy}{dx} ln(x) +  \frac{y}{x}  = ln(y) +  \frac{x}{y}  \frac{dy}{dx}

Isolate the terms that have y' since that is what we want:

\frac{dy}{dx} ln(x) -  \frac{x}{y}  \frac{dy}{dx}= ln(y) - \frac{y}{x}

Factor out y' to get:

\frac{dy}{dx}( ln(x) -   \frac{x}{y})= ln(y) - \frac{y}{x}

Therefore:

\frac{dy}{dx} =  \frac{ln(y) -  \frac{y}{x} }{ln(x) -  \frac{x}{y} }

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How do you solve this problem?
jekas [21]
Very nice problem. You should look up the history of this question. It has a very long one having to do with mirrors made between 50 BC and 50 AD.

Step One
Find AB, BC, AC
Before beginning this problem, I'm going to arbitrarily name 
c = AB
a = BC
b = AC It just makes the calculations easier.

AB = radius of the large circle - radius of the medium circle = 20.62 - 8.04 = 12.58
BC = Radii of the two smaller circles added together = 8.04 + 7.35 = 15.39
AC = Radius of the large circle - Radius of the small circle = 20.62 - 7.35 = 13.27

So
a = 15.39
b = 13.27
c = 12.58

Step Two
Find Angle A or <BAC
a^2 = b^2 + c^2 - 2*b*c * Cos(A)
15.39^2 = 12.58^2 + 13.27^2 - 2 * 12.58 * 13.27 * Cos(A)
236.852 = 158.26 + 176.1 -  333.87 * Cos(A)
236.852 = 334.4 - 333.87*Cos(A)
-97.55 = - 333.87 *  Cos(A) 
-97.55 / -333.87 = Cos(A)
0.2922 = cos(A)
A = cos-1(0.2922)
A = 73.01 degrees

Step Three
Just to see if you understand what was done, I'll give you the givens for finding c, and the answer and you can work through the calculations to see if your answer agrees with mine. If it doesn't PM me.
c = 12.58
b = 13.27
a = 15.39

12.58^2 = 13.27^2 + 15.39^2 - 2*13.27*15.39*Cos(C)
C = 51.42

Step Four
Find <B
Every triangle has 180 degrees so
B = 180 - <A - C
B = 180 - 51.42 - 74.01
B = <54.57. 
I have found a moderator who opened the question up so that I can show you why the Cos law is the only way to do this. If the circles are very disproportionate as in this diagram, then no simple assumption can be made. The cos law is all that will work. I would have posted this earlier, but I didn't think anyone would find another method. It's ingenious but not possible for the situation below.

3 0
3 years ago
Plz help with math u will get Brainliest
emmasim [6.3K]

Answer:

Volume = 315 cm³

Step-by-step explanation:

Volume = Length × Width × Height

→ Substitute in the values

Volume = 6 cm × 3.5 cm × 15 cm

→ Simplify

Volume = 315 cm³

4 0
3 years ago
For a normal distribution with μ=500 and σ=100, what is the minimum score necessary to be in the top 60% of the distribution?
STatiana [176]

Answer:

475.

Step-by-step explanation:

We have been given that for a normal distribution with μ=500 and σ=100. We are asked to find the minimum score that is necessary to be in the top 60% of the distribution.

We will use z-score formula and normal distribution table to solve our given problem.

z=\frac{x-\mu}{\sigma}

Top 60% means greater than 40%.

Let us find z-score corresponding to normal score to 40% or 0.40.

Using normal distribution table, we got a z-score of -0.25.

Upon substituting our given values in z-score formula, we will get:

-0.25=\frac{x-500}{100}

-0.25*100=\frac{x-500}{100}*100

-25=x-500

-25+500=x-500+500

x=475

Therefore, the minimum score necessary to be in the top 60% of the distribution is 475.

3 0
2 years ago
Use the Multiplication Property of Equality to complete the following statement: If , 4x divided by 2 = 4
vovangra [49]

Answer:

then x = 2.

Step-by-step explanation:

4 multiplied by 2 = 8

8 divided by 2 equals 4

Putting it together:

4(2)/2=4

4 0
3 years ago
Read 2 more answers
Find x (geometry) plz help
Ilya [14]

Step-by-step explanation:

\frac{ \sin(x) }{12}  =  \frac{ \sin(75) }{22}  \\  \sin(x)  =  \frac{12 \sin(75) }{22}   \\  \sin(x)  = 0.5268 \\ x = 31.79 = 31.8

6 0
2 years ago
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