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faltersainse [42]
2 years ago
10

What is the volume of the cube below? А. Зх в. 6x^2 с. 6х^3 D.x^3

Mathematics
1 answer:
OleMash [197]2 years ago
7 0

Answer:

V = x^3

Step-by-step explanation:

Please share the illustration of the cube.

V = x^3 is the only possible correct answer.

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<img src="https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5C0%7D%28x%2F%28tan%E2%81%A1%28x%29%29%5E%28cot%E2%81%A1%28x%29%5E2%20%
Viefleur [7K]

It looks like the limit you want to compute is

\displaystyle L = \lim_{x\to0}\left(\frac x{\tan(x)}\right)^{\cot^2(x)}

Rewrite the limand with an exponential and logarithm:

\left(\dfrac{x}{\tan(x)}\right)^{\cot^2(x)} = \exp\left(\cot^2(x) \ln\left(\dfrac{x}{\tan(x)}\right)\right) = \exp\left(\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right)

Now, since the exponential function is continuous at 0, we can write

\displaystyle L = \lim_{x\to0} \exp\left(\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right) = \exp\left(\lim_{x\to0}\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)}\right)

Let <em>M</em> denote the remaining limit.

We have \dfrac x{\tan(x)}\to1 as x\to0, so \ln\left(\dfrac x{\tan(x)}\right)\to0 and \tan^2(x)\to0. Apply L'Hopital's rule:

\displaystyle M = \lim_{x\to0}\dfrac{\ln\left(\dfrac{x}{\tan(x)}\right)}{\tan^2(x)} \\\\ M = \lim_{x\to0}\dfrac{\dfrac{\tan(x)-x\sec^2(x)}{\tan^2(x)}\times\dfrac{\tan(x)}{x}}{2\tan(x)\sec^2(x)}

Simplify and rewrite this in terms of sin and cos :

\displaystyle M = \lim_{x\to0} \dfrac{\dfrac{\tan(x)-x\sec^2(x)}{\tan^2(x)}\times\dfrac{\tan(x)}{x}}{2\tan(x)\sec^2(x)} \\\\ M= \lim_{x\to0}\dfrac{\sin(x)\cos^3(x) - x\cos^2(x)}{2x\sin^2(x)}

As x\to0, we get another 0/0 indeterminate form. Apply L'Hopital's rule again:

\displaystyle M = \lim_{x\to0} \frac{\sin(x)\cos^3(x) - x\cos^2(x)}{2x\sin^2(x)} \\\\ M = \lim_{x\to0} \frac{\cos^4(x) - 3\sin^2(x)\cos^2(x) - \cos^2(x) + 2x\cos(x)\sin(x)}{2\sin^2(x)+4x\sin(x)\cos(x)}

Recall the double angle identity for sin:

sin(2<em>x</em>) = 2 sin(<em>x</em>) cos(<em>x</em>)

Also, in the numerator we have

cos⁴(<em>x</em>) - cos²(<em>x</em>) = cos²(<em>x</em>) (cos²(<em>x</em>) - 1) = - cos²(<em>x</em>) sin²(<em>x</em>) = -1/4 sin²(2<em>x</em>)

So we can simplify <em>M</em> as

\displaystyle M = \lim_{x\to0} \frac{x\sin(2x) - \sin^2(2x)}{2\sin^2(x)+2x\sin(2x)}

This again yields 0/0. Apply L'Hopital's rule again:

\displaystyle M = \lim_{x\to0} \frac{\sin(2x)+2x\cos(2x)-4\sin(2x)\cos(2x)}{2\sin(2x)+4x\cos(2x)+4\sin(x)\cos(x)} \\\\ M = \lim_{x\to0} \frac{\sin(2x) + 2x\cos(2x) - 2\sin(4x)}{4\sin(2x)+4x\cos(2x)}

Once again, this gives 0/0. Apply L'Hopital's rule one last time:

\displaystyle M = \lim_{x\to0}\frac{2\cos(2x)+2\cos(2x)-4x\sin(2x)-8\cos(4x)}{8\cos(2x)+4\cos(2x)-8x\sin(2x)} \\\\ M = \lim_{x\to0} \frac{4\cos(2x)-4x\sin(2x)-8\cos(4x)}{12\cos(2x)-8x\sin(2x)}

Now as x\to0, the terms containing <em>x</em> and sin(<em>nx</em>) all go to 0, and we're left with

M = \dfrac{4-8}{12} = -\dfrac13

Then the original limit is

L = \exp(M) = e^{-1/3} = \boxed{\dfrac1{\sqrt[3]{e}}}

6 0
1 year ago
Math help please!!!!!!!!
Marina CMI [18]

Answer:

First question: 2.04%

Second question: 2.5

Third question: 339.29

Step-by-step explanation:

First, find the volume of Container A. This can be found by pir^2h. r is 6, and h is 12, so the volume is 1357.17. Next, find the volume of Container B. We will use the same equation. This gives us 1385.44 for the volume of Container B. We will subtract the volume of A from B to see the difference. The difference is 28.27. We will then take the difference and divide is by the volume of B. This is 0.02. This means that 2% of Container B will be empty. Specifically, 2.04%.

For this one we can simply divide 20 by 8. This gives us a scale factor of 2.5.

In order to find the surface area of a cylinder we will use the equation 2pirh + 2pir^2. We will plug in the values of r and h in order to give us the surface area. This gives us the answer 339.29.

8 0
2 years ago
Read 2 more answers
Rae earns $8.40 an hour plus an overtime rate equal to 1(1/2)times her regular pay for each hour worked beyond 40 hours. What ar
Nitella [24]

Answer:

At $8.40 an hour for a 40 hour week the earnings would be $336. With 5 hours overtime the pay would be $12.60 an hour which equals $63 making her earnings a total of $399 for the 45 hour work week. would be Step-by-step explanation:

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Do you know parallelogram WXYZ Parkway 270° around point W what will be the length of the image WZ
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Answer:

5 units

Step-by-step explanation:

Even if there is a transformation, it is asking for the length.

Therefore the length of WZ is 5 units

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1 year ago
I research or chooses a random sample of registered voters in Kingsville he found that three out of every five voters surveyed s
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