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3 years ago

What is mathematics??what are logarithmic functions??

Mathematics

1 answer:

bulgar [2K]3 years ago

8 0

Answer:

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

Step-by-step explanation:

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A wire of length L is cut at the red point. The segment to the left of the cut is formed into an equilateral triangle, and the s

Elden [556K]

Answer:

$(a) x=0\\(b)\ x=\dfrac{4L}{4+\sqrt{3}}$

Step-by-step explanation:

Imagine that the wire is cut as shown in the picture. We know that the lenght of the segment to the left is $x$ and therefore the lenght of the segment to the right must be $L-x.$

Using the formula for the area of an equilateral triangle of sides of length x, we get that $A_{\bigtriangleup}(x)=\dfrac{\sqrt{3}}{4}\, x^2.$

And using the formula for the area of a square of length L-x, we obtain that $A_{\square}(x)=(L-x)^2.$

Then, the total area of the two shapes is giving by the sum of both areas: $A_T(x)=A_{\bigtriangleup}(x)+A_{\square}(x)=\dfrac{\sqrt{3}}{4}\, x^2 + (L-x)^2.$

Now we have to find the values x where the function $A_T(x)$ attains its maximum and its minimum. For this purpose, we calculate its critical points, which occurs when the derivative vanishes:

$A_T'(x)=A_{\bigtriangleup}'(x)+A_{\square}'(x)=\dfrac{\sqrt{3}}{2}x-2(L-x)=0.$

Solving for x we get that:

$x=\dfrac{4L}{4+\sqrt{3}}.$

This is the only critical point. Using the second derivative test we found that, since $A_T''(x)=\dfrac{1}{2} \left(4 + \sqrt{3}\right) >0,$ then at $x=\dfrac{4L}{4+\sqrt{3}}$ the area of the two shapes is minimized.

Now, the only way we have maximum area is when the red point is on the extreme of the wire, at x = 0. This because in that situation, there is no triangle that can be formed and therefore the area is equals $L^2.$

4 0

4 years ago

Suppose that you are given a bag containing n unbiased coins. You are told that n-1 of these coins are normal, with heads on one

gladu [14]

Answer:

The (conditional) probability that the coin you chose is the fake coin is 2/(1 + n)

Step-by-step explanation:

Given

Total unbiased coin = n

Normal coins =n - 1

Fake = 1

The (conditional) probability that the coin you chose is the fake coin is represented by

P(Fake | Head)

And it's calculated as follows;

P(Fake | Head) = P(Fake, Head) ÷ P(Head) ----- (1)

Where P(Fake, Head) = P(Fake) * P(Head | Fake)

P(Fake) = 1/n --- because only one is fake

P(Head | Fake) = n/n because all coins (including the fake) have head

So, P(Fake, Head) = P(Fake) * P(Head | Fake) becomes

P(Fake, Head) = 1/n * n/n

P(Fake, Head) = 1/n

P(Head) is calculated by

P(Fake) * P(Head | Fake) + P(Normal) * P(Head | Normal)

P(Fake) * P(Head | Fake) = P(Fake, Head) = 1/n (as calculated above)

P(Normal) * P(Head | Normal) = ½ * (n - 1)/n ----- considering that the coin also has a tail with equal probability as that of the head.

Going back to (1)

P(Fake | Head) = P(Fake, Head) ÷ P(Head) becomes

P(Fake | Head) = (1/n) ÷ ((1/n) + (½(n-1)/n))

= (1/n) ÷ ((1/n) + (½(n-1)/n))

= (1/n) ÷ (1/n + (n - 1)/2n)

= (1/n) ÷ (2 + n - 1)/(2n)

= (1/n) ÷ (1 + n)/(2n)

= (1/n) * (2n)/(1 + n)

= 2/(1 + n)

Hence, the (conditional) probability that the coin you chose is the fake coin is 2/(1 + n)

5 0

3 years ago

Simplify4(x−1)−3(2x−5)

Jlenok [28]

Answer:

Step-by-step explanation:4x-4-6x+15

Then -2x-4+16

5 0

4 years ago

Read 2 more answers

Wonka candy bars cost 1.50 each. You have $18 to spend on them write and algebraic equations

guapka [62]

Hello!

The equation: 18 = 1.50x

I hope it helps!

8 0

3 years ago

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Answer the question please

mr_godi [17]

Answer:

117

Step-by-step explanation:

<DAB = <B + <C

8 0

3 years ago

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What is mathematics??what are logarithmic functions??​

What is mathematics??what are logarithmic functions??